Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity

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The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. Methods This study will examine the combined effects of Weissenberg number, power law index, nanoparticle volume fraction, viscous dissipation, magnetic field, heat generation, nonlinear thermal radiation, temperature ratio, Joule heating, Brownian motion, thermophoresis, porous permeability, variable thermal conductivity, Eckert number, Prandtl number, Schmidt number, chemical reaction, velocity ratio, and Forchheimer number on the electrical conductivity of unsteady flow in tangent hyperbolic ternary hybrid nanofluids. The governing equations are transformed into similarity equations using appropriate transformations and solved numerically with the MATLAB BVP5C package. The results are validated against data from published articles to ensure reproducibility. Results The findings reveal that an increase in the Weissenberg and Forchheimer numbers reduces the velocity profile, while the temperature distribution increases. The variable thermal conductivity parameter (Γ) leads to a higher temperature profile, indicating improved heat transfer. Higher nanoparticle concentrations in the nanofluids and hybrid nanofluids result in enhanced skin friction, Nusselt number, and Sherwood number. Ternary hybrid nanofluids show the most significant improvement in heat transfer and thermal conductivity. Conclusions Ternary hybrid nanofluids significantly enhance heat and mass transfer, showing potential for applications in cooling systems, drug delivery, and polymer processing. The numerical results are consistent with previous research, confirming the reliability and reproducibility of the findings. " } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/14-152/v2", "name": "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in..." } } ] } Home Browse Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Tsegaye Moltot A, Haile Gorfie E, Awgichew Zergaw G and Dessie H. Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.12688/f1000research.158629.2 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Revised Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] Asfaw Tsegaye Moltot https://orcid.org/0000-0001-7610-4058 1 , Eshetu Haile Gorfie https://orcid.org/0000-0002-2428-1842 1 , Gurju Awgichew Zergaw 1 , Hunegnaw Dessie 1 Asfaw Tsegaye Moltot https://orcid.org/0000-0001-7610-4058 1 , Eshetu Haile Gorfie https://orcid.org/0000-0002-2428-1842 1 , Gurju Awgichew Zergaw 1 , Hunegnaw Dessie 1 PUBLISHED 10 Mar 2025 Author details Author details 1 Department of Mathematics, College of Science, Bahir Dar University, P.O. Box 79, Bahir Dar, Ethiopia Asfaw Tsegaye Moltot Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Eshetu Haile Gorfie Roles: Conceptualization, Resources, Supervision, Visualization, Writing – Review & Editing Gurju Awgichew Zergaw Roles: Conceptualization, Resources, Supervision, Visualization, Writing – Review & Editing Hunegnaw Dessie Roles: Conceptualization, Resources, Supervision, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Nanoscience & Nanotechnology gateway. Abstract Background This research investigates the unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet. The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. Methods This study will examine the combined effects of Weissenberg number, power law index, nanoparticle volume fraction, viscous dissipation, magnetic field, heat generation, nonlinear thermal radiation, temperature ratio, Joule heating, Brownian motion, thermophoresis, porous permeability, variable thermal conductivity, Eckert number, Prandtl number, Schmidt number, chemical reaction, velocity ratio, and Forchheimer number on the electrical conductivity of unsteady flow in tangent hyperbolic ternary hybrid nanofluids. The governing equations are transformed into similarity equations using appropriate transformations and solved numerically with the MATLAB BVP5C package. The results are validated against data from published articles to ensure reproducibility. Results The findings reveal that an increase in the Weissenberg and Forchheimer numbers reduces the velocity profile, while the temperature distribution increases. The variable thermal conductivity parameter (Γ) leads to a higher temperature profile, indicating improved heat transfer. Higher nanoparticle concentrations in the nanofluids and hybrid nanofluids result in enhanced skin friction, Nusselt number, and Sherwood number. Ternary hybrid nanofluids show the most significant improvement in heat transfer and thermal conductivity. Conclusions Ternary hybrid nanofluids significantly enhance heat and mass transfer, showing potential for applications in cooling systems, drug delivery, and polymer processing. The numerical results are consistent with previous research, confirming the reliability and reproducibility of the findings. READ ALL READ LESS Keywords Tangent hyperbolic fluid, Ternary Hybrid nanofluid, Viscous Dissipation, non-linear thermal radiation, variable thermal conductivity, permeable stretching sheet Corresponding Author(s) Asfaw Tsegaye Moltot ( [email protected] ) Close Corresponding author: Asfaw Tsegaye Moltot Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2025 Tsegaye Moltot A et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Tsegaye Moltot A, Haile Gorfie E, Awgichew Zergaw G and Dessie H. Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.12688/f1000research.158629.2 ) First published: 04 Feb 2025, 14 :152 ( https://doi.org/10.12688/f1000research.158629.1 ) Latest published: 10 Mar 2025, 14 :152 ( https://doi.org/10.12688/f1000research.158629.2 ) Revised Amendments from Version 1 The revised version of the manuscript incorporates the corrections suggested by the reviewers. We have corrected the capitalization in the title as suggested. The abstract has been revised to explicitly highlight key findings. The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. The introduction was expanded with current bibliographic references, including the suggested ones in the introduction and literature review sections. We have added a detailed discussion in the introduction section outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. We have expanded the discussion on the physical configuration in the formulation method section. We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. In the revised manuscript, we have added that the framework is applicable to thermal management, energy systems, and material processing, with ternary hybrid nanofluids showing potential in drug delivery and enhanced heat transfer in cooling systems. The advantages of the numerical methods used in this study have been elaborated in detail, highlighting their accuracy, efficiency, ability to handle nonlinear equations, flexibility in incorporating boundary conditions, and applicability to complex fluid flow problems. Additionally, the computational stability, convergence, and validation against existing studies have been addressed to ensure the reliability of the results. The results and discussion section has been revised to provide a more in-depth analysis of the figures, ensuring a clearer interpretation of the findings and their validation with previous studies. We have revised the conclusion to clearly emphasize the novel contributions of this study and have included a detailed discussion on future directions. In the revised version, a thorough grammar check and proofreading were conducted to improve clarity, coherence, and formatting. The revised version of the manuscript incorporates the corrections suggested by the reviewers. We have corrected the capitalization in the title as suggested. The abstract has been revised to explicitly highlight key findings. The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. The introduction was expanded with current bibliographic references, including the suggested ones in the introduction and literature review sections. We have added a detailed discussion in the introduction section outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. We have expanded the discussion on the physical configuration in the formulation method section. We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. In the revised manuscript, we have added that the framework is applicable to thermal management, energy systems, and material processing, with ternary hybrid nanofluids showing potential in drug delivery and enhanced heat transfer in cooling systems. The advantages of the numerical methods used in this study have been elaborated in detail, highlighting their accuracy, efficiency, ability to handle nonlinear equations, flexibility in incorporating boundary conditions, and applicability to complex fluid flow problems. Additionally, the computational stability, convergence, and validation against existing studies have been addressed to ensure the reliability of the results. The results and discussion section has been revised to provide a more in-depth analysis of the figures, ensuring a clearer interpretation of the findings and their validation with previous studies. We have revised the conclusion to clearly emphasize the novel contributions of this study and have included a detailed discussion on future directions. In the revised version, a thorough grammar check and proofreading were conducted to improve clarity, coherence, and formatting. See the authors' detailed response to the review by Prasun Choudhary and Dr. Kavita Jat See the authors' detailed response to the review by Sohail Rehman See the authors' detailed response to the review by Asra Anjum READ REVIEWER RESPONSES Introduction Non-Newtonian fluids play a crucial role in many applications due to their diverse behaviors. Non-Newtonian fluids have a viscosity that changes with the shear rate. Their flow behavior can be more complex, exhibiting various properties such as shear-thinning or shear-thickening. A non-Newtonian fluid is a specific type of fluid, like ketchup, toothpaste, and paints, where the relationship between shear and stress is nonlinear. Non-Newtonian fluids have a wide range of applications across various industries due to their unique flow characteristics, like, food industry, cosmetics and personal care, construction, textiles, etc. Tangent hyperbolic fluids are a type of non-Newtonian fluid characterized by their stress-strain relationship involving the hyperbolic tangent function. This model captures complex flow behaviors, making it suitable for describing materials that exhibit both viscous and elastic properties under varying shear rates. Tangent hyperbolic fluids represent a significant category of non-Newtonian fluids that exhibit complex flow behavior, essential for various industrial and scientific applications. Their unique properties enable better modeling and understanding of materials that behave differently than simple fluids, making them valuable in diverse fields such as food science, cosmetics, pharmaceuticals, and biomedical engineering. Hashim et al. 1 examined the heat and mass transfer behavior of non-Newtonian nanofluids in a non-parallel vertical enclosure. Rehman et al. 2 explored the flow behavior of non-Newtonian nanofluids between intersecting planes, including the effects of slip mechanisms at the boundaries. Nadeem et al. 3 examined the unsteady tangent hyperbolic hybrid nanofluid flow across an exponentially stretched sheet. They concluded that analyzing how fuzzy parameters affect the behavior of nanofluids could provide valuable insights into optimal conditions for specific applications. Alqahtani et al. 4 numerically examined how the combination of wedge angle and energy transfer during melting affects the flow of ternary hybrid magnetohydrodynamics (MHD) Nano liquid across a permeable wedge. This exploration reveals the intricate effects of wedge angle, heat sources/sinks, and Lorentz forces on the forced convective flow of tangent hyperbolic nanofluids over a permeable wedge. Amjad et al. 5 investigated the behavior of a tangent hyperbolic MHD nanofluid in a two-dimensional framework over an exponentially stretched sheet. According to their observations, the boundary layer is reduced by both the magnetic parameter and the Weissenberg number. Alkaoud et al. 6 investigated the characteristics of tangent hyperbolic fluid flow across a stretching sheet and discovered that fluid velocity decreases and temperature rises with increasing power-law index, slip velocity parameter, porosity parameter, and magnetic number. Choudhary et al. 7 explored the formation of boundary layers in tangent hyperbolic fluid flow through a diverging permeable channel, analyzing the effects of a porous medium, suction or blowing, and a heat source. Nanofluids are advanced colloidal fluids that consist of a base fluid (such as water, oil, or ethylene glycol) combined with nanoparticles typically ranging in size from 1 to 100 nanometers. These nanoparticles can be made from various materials, including metals (like copper and silver), oxides (such as alumina and silica), and carbon-based materials (like graphene). Nanofluids represent a significant advancement in thermal management technology, offering enhanced heat transfer capabilities and efficiency across various applications. Ongoing research continues to explore their properties, optimize formulations, and expand their applicability in diverse fields. Hybrid nanofluids are advanced fluids that consist of two or more different types of nanoparticles suspended in a base fluid. This combination aims to optimize thermal and physical properties, enhancing heat transfer capabilities beyond what single-component nanofluids can achieve. The presence of multiple types of nanoparticles can lead to a significant increase in thermal conductivity, improving heat transfer efficiency. Hybrid nanofluids offer a promising approach to enhance heat transfer performance in various applications. By leveraging the unique properties of multiple nanoparticles, they provide improved thermal management solutions in diverse fields, from industrial processes to renewable energy systems. Ternary hybrid nanofluids are advanced fluids that incorporate three different types of nanoparticles to enhance thermal and transport properties. By combining multiple nanoparticles, these fluids can achieve superior heat transfer capabilities compared to conventional fluids or even binary hybrid nanofluids. The addition of different types of nanoparticles can optimize properties such as thermal conductivity, viscosity, and stability, making them suitable for a variety of applications, including cooling systems, heat exchangers, and energy storage. Boujelbene et al. 8 explored heat transfer and entropy generation in nanofluid flow within an inclined channel. Alqawasmi et al. 9 investigated the effects of Cattaneo-Christov heat exchange on the flow of ternary hybrid nanofluids over a spinning disc, excluding non-linear radiant heat. Their findings indicated that an increase in the magnitude of the suction and injection parameters leads to a reduction in the velocity distribution profile. Jat et al. 10 investigated the impact of various nanoparticle shapes on the radiating hybrid nanofluid flow, particularly over a nonlinear stretchable porous sheet. Mahboobtosi et al. 11 studied the impact of utilizing a ternary hybrid nanofluid in place of a single nanofluid. They concluded that improvements in the curvature parameter, volume fraction, and shape factor resulted in enhanced velocity and temperature profiles. Farooq 12 examined the mixed convection of stagnation point flow of ternary hybrid nanofluids toward a vertical Riga plate. He concluded that when the mixed convection parameter is introduced to counteract the flow, the temperature profile decreases while the velocity profile increases. Abbas et al. 13 examined the mixed convection of ternary hybrid nanofluids during cilia transport through a curved channel. Their findings indicated that increasing the cilia length parameter enhances the liquid velocity. Nabwey et al. 14 investigated the MHD of two-dimensional incompressible boundary layer flow of non-Newtonian Carreau ternary hybrid nanofluids with heat transfer over an exponentially stretched curved surface. They demonstrated that the heat transfer rate and skin friction increase as one moves from the base fluid to mono, hybrid, and then ternary nanofluids. Jamshed et al. 15 explored in order to find an efficient heat-transmitting fluid to replace standard fluids and revolutionary nanofluids. Noreen 16 utilized heat radiation to study the MHD flow of ternary hybrid nanofluids between double disks. The results indicated that increasing variations in the thermal relaxation parameter lead to a decline in the thermal distribution profile. Al Garalleh 17 investigated heat diffusion and mass transport in the flow of ternary hybrid nanofluids ( TiO 2 − Al 2 O 3 − SiO 2 ) over a stretching/shrinking wedge geometry. The study concluded that the role of ethylene glycol (EG) in heat transport is vital for ensuring the efficiency, safety, and reliability of thermal management systems, making it an essential component in modern engineering solutions for thermal regulation. Priyadharshini et al. 18 explored the optimal design and performance estimation of a ternary hybrid nanofluid using advanced machine learning prediction techniques. Their investigation revealed that the ternary hybrid nanofluid exhibits superior thermal performance compared to hybrid nanofluids. Patil et al. 19 investigated the flow of tangent hyperbolic ternary hybrid nanofluids over a rough-yawed cylinder, induced by impulsive motion in a mixed convection mechanism with periodic magnetohydrodynamics. Flow that varies with time is termed unsteady flow, whereas flow that remains constant over time is referred to as steady flow. Engineers and researchers typically prefer steady flow due to its more manageable and predictable nature, allowing for better control in various applications. However, the influence of unsteadiness on flow behavior is significant and cannot be overlooked. Consequently, researchers must account for the effects of unsteady flow in their analyses and designs to ensure accurate modeling and performance optimization. Unsteady flow is a critical concept in fluid dynamics, impacting various engineering fields. Its study allows for the accurate modeling of dynamic systems, ensuring safety, efficiency, and performance in real-world applications. Understanding unsteady flow phenomena is essential for designing and optimizing systems that experience time-dependent changes. Many practical applications experience unsteady flow, making it crucial for accurate modeling and analysis. Ignoring unsteadiness can lead to incomplete or inaccurate results. Unsteady flow captures important dynamic behaviors such as turbulence, flow separation, and transient heat transfer, which are essential in various engineering disciplines. Kebede et al. 20 investigated the time-dependent flow of Williamson nanofluid and found that both the velocity and temperature gradients decrease with increasing unsteadiness parameters within the boundary layer. Jamal et al. 21 investigated the unsteady flow of MHD incompressible tangent hyperbolic fluid with nanofluid particles along a stretching surface. Their results indicated that the velocity field decreases as the Weissenberg number and power-law index increase, while the thermal and concentration fields continue to rise under the same conditions. Reddy et al. 22 focused on the effects of Lorentz force, Joule heating, and viscous dissipation on the unsteady flow of tangent hyperbolic liquid past a vertical plate. Their findings revealed that an increase in the Weissenberg number and magnetic field weakens the velocity profile. Kumar et al. 23 studied unsteady MHD oscillatory flow over a vertically permeable stretching plate in a viscous, incompressible fluid. Their findings indicated that fluid velocity increases over time. Choudhary et al. 8 analyzed the unsteady laminar flow, heat, and mass transfer of a hybrid nanofluid over a non-linearly stretchable porous sheet, considering the effects of thermal radiation and gyrotactic microorganisms. MHD is the study of the behavior of electrically conducting fluids in the presence of a magnetic field. It combines principles from both fluid dynamics and electromagnetism to analyze the motion of plasmas, liquid metals, and other conductive fluids. MHD plays a crucial role in understanding and predicting the behavior of conductive fluids in many natural and industrial processes. MHD has important applications in various fields, including astrophysics (e.g., solar flares and stellar formation), metallurgy (e.g., casting processes), and engineering (e.g., cooling systems for nuclear reactors), biomedical applications (e.g., magneto-fluid dynamics). Krishna and Chamkha 24 studied the MHD squeezing flow of a water-based nanofluid through a saturated porous medium situated between two parallel disks, while considering the effects of Hall current. In the squeezing flow of a nanofluid, the Lorentz force, which governs the impact of magnetic fields on fluid motion, is predominantly influenced by the external magnetic field. In many practical applications, the magnetic field strength is typically quite weak. When the applied magnetic field is low, the induced magnetic field can become negligible in comparison to the external magnetic field. As a result, the impact of the induced magnetic field can be ignored without substantially influencing the flow behavior. The Hall Effect, which results from the movement of charged particles in a magnetic field, may be minimal if the fluids conductivity is low or if the fluid’s velocity is relatively slow. In water-based nanofluids, conductivity can vary, meaning the Hall Effect might not be significant enough to be considered in the analysis of the squeezing flow. The investigation provides further information on the magnetohydrodynamics tangent hyperbolic nanofluid flow as provided in these references. 25 – 32 A porous medium, or porous material, is a substance that contains pores (voids) capable of being filled with fluids (liquids or gases). These materials are distinguished by their ability to facilitate fluid flow through their structure, influenced by factors such as pore size, shape, distribution, and connectivity. The study of porous media is essential across various fields due to their role in fluid dynamics, resource management, and environmental sustainability. In many practical applications involving porous media, particularly at low flow rates, flow behavior is primarily governed by Darcy’s law, which assumes that flow resistance is proportional to the flow rate. The Forchheimer equation, on the other hand, incorporates nonlinear inertial effects that become significant only at high velocities or in highly permeable media. Chamkha 33 examined non-Darcy fully developed mixed convection flow in a channel embedded in a porous medium (referred to as a porous medium channel), considering the effects of heat generation/absorption and hydromagnetic influences. The Darcy-Forchheimer equation is a model that describes fluid flow through porous media by combining Darcy’s law, which addresses laminar flow, with Forchheimer extension, which accounts for inertial effects occurring in higher velocity flows. The Darcy-Forchheimer equation serves as a crucial tool for understanding and predicting the behavior of fluids in porous media, particularly under varying flow conditions. The Darcy-Forchheimer equation has various applications, including modeling fluid flow in reservoir rocks for oil and gas extraction to enhance extraction techniques. In hydrology, it is used to analyze groundwater movement and contaminant transport. In chemical engineering, it aids in designing reactors and filtration systems that involve flow through porous materials. Jawad et al. 34 studied the effects of heat and mass transfer on the convective Darcy-Forchheimer flow of a Maxwell nanofluid over a linearly stretched porous sheet. Their findings indicated that increased thermal radiation and thermophoresis enhance the temperature distribution. Saeed et al. 35 investigated heat transfer and the effects of electromagnetic forces on the MHD flow of couple-stress hybrid nanofluids over a Darcy-Forchheimer model in a symmetric flow scenario with variable viscosity. Alessa et al. 36 studied the Darcy-Forchheimer flow of water-based Al − Al 2 O 3 /Cu − Al 2 O 3 hybrid nanofluids past a heated stretchable plate, incorporating heat consumption/generation and non-linear radiation effects. Their results indicated that the velocity of the hybrid nanofluid decreased as the magnetic field parameter increased. Viscous dissipation is the process where uneven forces in adjacent fluid layers convert work into heat. Viscous dissipation in tangent hyperbolic fluid flow is a crucial factor that affects thermal behavior, flow dynamics, and overall system efficiency. Understanding its implications helps in optimizing processes involving non-Newtonian fluids, ensuring better performance in various industrial applications. Using convective boundary conditions and viscous dissipation, Hussain et al. 37 concentrated on the thermo-physical characteristics of MHD tangent hyperbolic fluid flow across a non-linear stretched sheet. Nandi and Kumbhakar 38 studied the MHD boundary layer flow of a TH nanofluid past a stretched wedge with velocity slip boundary conditions. Numerous research teams have explored unsteady hybrid nanofluid flow over stretching sheets using various geometries and approaches. The literature includes diverse techniques employed to analyze the heat and mass fluxes associated with this type of flow. Of these researchers, Jamrus et al. 39 performed a numerical investigation to examine the flow characteristics and heat transfer of unsteady flow over a permeable. Stretching sheet, utilizing a ternary hybrid nanofluid ( Al 2 O 3 − Cu − TiO 2 /H 2 O ) under the influence of suction and a magnetic field but the important concepts like viscous dissipation, porous medium, Darcy- Forchheimer flow, non-linear thermal radiation, variable thermal conductance, non-Newtonian like tangent hyperbolic fluid, thermophoresis diffusion, Brownian diffusion, joule heating, chemical reaction, and heat generation/absorption are not considered. Jamal et al. 21 investigated the unsteady flow of an MHD tangent hyperbolic fluid influenced by the presence of nanoparticles over a stretching sheet. On their study, they have not included viscous dissipation, Darcy-Forchheimer flow, variable thermal conductance, joule heating, non-linear thermal radiation, chemical reaction, and heat generation/absorption. Thus, to the authors’ knowledge, the combined effects of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductance, non-Newtonian fluid, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the electrical conductivity of unsteady flow in tangent hyperbolic ternary hybrid nanofluids over a stretching sheet have not been addressed in the existing literature. Therefore, this study will examine the combined effects of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductance, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the electrical conductivity of unsteady flow in tangent hyperbolic ternary hybrid nanofluids. These nanofluids consist of Al 2 O 3 , Cu , and TiO 2 nanoparticles suspended in ethylene glycol, flowing over a stretching sheet. This study is significant as it combines theoretical analysis and practical application, providing valuable insights into the behavior of tangent hyperbolic ternary hybrid nanofluids under complex flow conditions. Such nanofluids have a wide range of applications in industries including cooling systems, heat exchangers, energy systems, biomedical applications, etc. The outcomes will not only advance scientific knowledge but also have meaningful implications in various engineering fields. The governing partial differential equations are transformed into ordinary differential equations using suitable variable transformations. We employed the BVP5C algorithm in MATLAB to solve these transformed equations. The effects of various phenomena on temperature, concentration, velocity profiles, rates of heat and mass transfer, and skin friction coefficients are illustrated through graphical representations and tabular data. Methods Mathematical formulation of the flow problem We examine the ternary hybrid nanofluid is laminar boundary layer, two dimensional, time-dependent, incompressible flow of an electrically conducting Tangent hyperbolic fluid over a permeable stretching sheet, surface with velocity U w ( x , t ) = ax / ( 1 − ct ) , where a is a positive constant, t is time, and c is the time-dependent parameter of this problem. The wall mass suction velocity is assumed to be , V w ( x , t ) . The x -axis is oriented along the direction of the stretching sheet, whereas the y -axis is perpendicular to it. The velocity components are represented as u and v , corresponding to the x - and y -axes, respectively. The well temperature, T w , ambient temperature , T ∞ , well concentration, C w , and ambient concentration, C ∞ are considered constants. Figure 1 graphically illustrate the physical flow model in this study, using Cartesian coordinate. The model for momentum, mass, and heat transfer incorporate thermophoresis, Brownian motion, viscous dissipation, chemical reactions, variable thermal conductivity, non-linear thermal radiation, heat generation, magnetic dissipation, and wall mass suction. When a constant magnetic field is applied perpendicular to the surface in the positive y-direction, the induced magnetic field is considered negligible and ignored. Assuming that the porosity of the porous material remains constant, we also take Darcy-Forchheimer flow into account in our investigation. The energy calculations are made simpler by treating all fluid parameters as constants and assuming that the system enters thermal equilibrium. The energy equation takes into consideration the effects of viscous dissipation, in which viscosity causes kinetic energy to be transformed into thermal energy. Through the dispersion of Al 2 O 3 , Cu , and TiO 2 nanoparticles in C 2 H 6 O 2 medium, the ternary hybrid nanofluid is created. According to Refs. 40 , the heat conductivity of the ternary hybrid nanomaterial are proportional and linearly connected to the distributed temperature. (1) κ thnf ( T ) = [ 1 + Γ T − T ∞ T w − T ∞ ] κ thnf = κ thnf ( 1 + Γ θ ) where Γ is variable thermal conductivity. Figure 1. Flow configuration and coordinate system. Table 1. Thermophysical properties of utilized nanoparticles and base fluid (EG). 3 , 18 , 39 Physical Properties ( C 2 H 6 O 2 )( f ) ( Al 2 O 3 ( ϕ 1 )) ( Cu ( ϕ 2 )) TiO 2 ( ϕ 3 ) ρ 1115 3970 8933 4250 C p 2430 765 385 686.2 κ 0.253 40 400 8.9538 σ 0.107 3.5×10 7 5.96×10 7 1.0×10 −12 Constitutive equation of tangential hyperbolic The tangential hyperbolic model is used to describe the flow behavior of shear-thinning fluids. In this model, the relationship between the shear stress tau and shear rate γ ̇ is given by a hyperbolic tangent function. The following formulas provide the basic equation for the tangential hyperbolic fluid 4 , 7 , 26 , 27 : (2) τ = − PI + E , E = [ μ ∞ + ( μ 0 + μ ∞ ) tanh ( N γ ̇ ) n ] γ ̇ The shear rate γ ˙ is assumed (3) γ ̇ = ( 1 2 ∑ i ∑ j γ ̇ ij γ ̇ ji ) 1 2 = ( 1 2 π ) 1 2 where N is material constant, π is the second invariant strain rate tensor and is specified by π = 1 2 tr [ ∇ V + ( ∇ V ) t ] 2 , and t is transpose. We assuming μ ∞ = 0 , the shear stress tensor E for ( N γ ̇ ) < 1 gives to (4) E = μ 0 [ ( N γ ̇ ) n ] γ ̇ = μ 0 [ ( 1 + N γ ̇ − 1 ) n ] γ ̇ = μ 0 [ 1 + n ( N γ ̇ − 1 ) ] γ ̇ The governing equations for the flow problem are: 7 , 21 , 39 (5) ∂ u ∂ x + ∂ v ∂ y = 0 (6) ∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = μ thnf ρ thnf [ ( 1 − n ) + n Γ 2 ∂ u ∂ y ] ∂ 2 u ∂ y 2 − σ t hnf B 2 ρ t hnf u − μ t hnf ρ t hnf K p u − 1 ρ thnf W c W u 2 (7) ∂ T ∂ t + u ∂ T ∂ x + v ∂ T ∂ y = 1 ( ρ C p ) thnf ∂ ∂ y [ κ thnf ( T ) ∂ T ∂ y ] + 1 ( ρ C p ) thnf 16 σ ∗ 3 κ ∗ ∂ ∂ y ( T 3 ∂ T ∂ y ) + Q 0 ( T − T ∞ ) ( ρCp ) thnf + μ thnf ( ρ C p ) thnf n Γ 2 ∂ u ∂ y ( ∂ u ∂ y ) 2 + μ thnf ( ρ C p ) thnf ( 1 − n ) ( ∂ u ∂ y ) 2 + μ thnf ( ρ C p ) thnf 1 K p u 2 + σ thnf B 0 2 ( ρCp ) thnf u 2 + 1 ( ρ C p ) thnf W c W u 3 + ( ρ C p ) np ( ρ C p ) thnf [ D B ∂ C ∂ z ∂ T ∂ z + D T T ∞ ( ∂ T ∂ z ) 2 ] (8) ∂ C ∂ t + u ∂ C ∂ x + v ∂ C ∂ y = D B ∂ 2 C ∂ y 2 + D T T ∞ ( ∂ 2 T ∂ y 2 ) − C 0 ( C − C ∞ ) Subject to boundary condition given by 39 (9) { v = V w ( x , t ) , u = λ u w ( x , t ) , T = T w , C = C w , at y → 0 u → 0 , T → T ∞ , C → C ∞ , as y → ∞ where ( u , v ) are the velocity components in the axes ( x , y ) , T and C are the temperature and concentration of the THNs, respectively, σ is the electrical conductivity, and a and c represent positive constants, μ thnf , ρ thnf , ( ρ C p ) thnf , σ t hnf , are the effective ternary hybrid nanofluids (THNs) viscosity, density and THNs heat capacitance, electrical conductivity of THNs, respectively. The ternary hybrid nanofluids thermophysical characteristics are described as 14 , 18 , 39 : ρ thnf ρ f = ( 1 − ϕ 1 ) ( 1 − ϕ 2 ) ( 1 − ϕ 3 ) + ϕ 3 ρ s 3 ρ f + ϕ 2 ρ s 2 ρ f + ϕ 1 ρ s 1 ρ f μ thnf μ f = 1 ( 1 − ϕ 1 ) 2.5 ( 1 − ϕ 2 ) 2.5 ( 1 − ϕ 3 ) 2.5 ( ρ C p ) thnf ( ρ C p ) f = ( 1 − ( ϕ 1 + ϕ 2 + ϕ 3 ) ) + ϕ 1 ( ρ C p ) s 1 ( ρ C p ) f + ϕ 2 ( ρ C p ) s 2 ( ρ C p ) f + ϕ 3 ( ρ C p ) s 3 ( ρ C p ) f κ nf κ f = κ s 3 + 2 κ f − 2 ϕ 3 ( κ f − κ s 3 ) κ s 3 + 2 κ f + ϕ 3 ( κ f − κ s 3 ) ; κ hnf κ nf = κ s 2 + 2 κ n f − 2 ϕ 2 ( κ n f − κ s 2 ) κ s 2 + 2 κ n f + ϕ 2 ( κ n f − κ s 2 ) κ thnf κ hnf = κ s 1 + 2 κ h n f − 2 ϕ 1 ( κ h n f − κ s 1 ) κ s 1 + 2 κ h n f + ϕ 1 ( κ h n f − κ s 1 ) σ nf σ f = σ s 3 + 2 σ f − 2 ϕ 3 ( σ f − σ s 3 ) σ s 3 + 2 σ f + ϕ 3 ( σ f − σ s 3 ) ; σ hnf σ nf = σ s 2 + 2 σ n f − 2 ϕ 2 ( σ n f − σ s 2 ) σ s 2 + 2 σ n f + ϕ 2 ( σ n f − σ s 2 ) σ thnf σ hnf = σ s 1 + 2 σ h n f − 2 ϕ 1 ( σ h n f − σ s 1 ) σ s 1 + 2 σ h n f + ϕ 1 ( σ h n f − σ s 1 ) Here, the subscripts f , nf , hnf , and thnf stand for traditional fluid, NF, HNF, and THNs, respectively, and the letters 1,2, and 3 stand for the three selected nanoparticles. Since the governing equations are in partial differential form, they are reduced to a system of ordinary differential equations through the use of similarity transformations. Further, to obtain similarity solutions of Eq. (5) to Eq. (8) , the unsteady magnetic field B 0 is in the form B 0 = B /(1− ct ) 1/2 where B is a constant. The unsteady magnetic field B 0 = B /(1− ct ) 1/2 can induce an electric field according to Maxwell’s equations. Specifically, Faraday’s law of electromagnetic induction states that a time-dependent magnetic field generates a circulating electric field. Since B 0 varies with time due to the presence of (1− ct ) 1/2 its time derivative is nonzero, leading to an induced electric field. 2 This induced electric field can influence charge transport, modify current distributions, and potentially alter the overall nanofluid dynamics, making it an important factor in magnetohydrodynamic (MHD) applications. The similarity transformations are given by 21 , 39 (10) { η = y ( a ν f ( 1 − ct ) ) 1 2 , u = ax 1 − ct f ′ ( η ) , v = − ( a ν f 1 − ct ) 1 2 f ( η ) V w = − a ν f 1 − ct S , θ = T − T ∞ T w − T ∞ , Φ = C − C ∞ C w − C ∞ where S is the constant wall mass transfer parameter. S = 0 and S 0 . Equations (6) to (8) are simplified accordingly: (11) A 1 A 2 [ ( 1 − n ) + nWe f ′ ′ ] f ′ ′ ′ − ( A + A 3 A 2 M + A 1 A 2 K ) f ′ + ( f − Aη 2 ) f ′ ′ − ( 1 + 1 A 2 Fr ) ( f ′ ) 2 = 0 (12) A 4 A 5 Pr ( 1 + Γ θ ) θ ′ ′ + R A 5 Pr [ ( 1 + ( θ w − 1 ) ) 2 × 3 ( θ w − 1 ) θ ′ 2 + ( 1 + ( θ w − 1 ) θ ) 3 θ ′ ′ ] + ( f − Aη 2 ) θ ′ + 1 A 5 Qθ + 1 A 5 [ Nb θ ′ Φ ′ + Nt θ ′ 2 ] + Ec [ A 1 A 5 K + A 3 A 5 M ] ( f ′ ) 2 + A 1 A 5 n 2 WeEc ( f ′ ′ ) 3 + A 1 A 5 ( 1 − n ) Ec ( f ′ ′ ) 2 + 1 A 5 FrEc ( f ′ ) 3 = 0 (13) Φ ′ ′ + Nt Nb θ ′ ′ + Scf Φ ′ − Sc Aη 2 Φ ′ − Sc C r Φ = 0 . with subject to boundary conditions (14) { f ( 0 ) = S , f ′ ( 0 ) = λ , θ ( 0 ) = 1 , Φ ( 0 ) = 1 , at η = 0 f ′ ( η ) → 0 , θ ( η ) → 0 , Φ ( η ) → 0 , as η → ∞ where, the Weissenberg number ( We ), magnetic field parameter ( M ), porosity parameter ( K ), Forchheimer number ( Fr ), Prandtl number ( Pr ), non-linear thermal radiation ( R ), ratio of temperature ( θ w ) , Q is the heat generation ( Q > 0) or absorption parameter ( Q < 0), Brownian motion parameter ( Nb ), Thermophoresis parameter ( Nt ), Eckert number ( Ec ), Schmidt number ( Sc ), Chemical reaction ( C r ), unsteady parameter( A ), and A 1 , A 2 , A 3 , A 4 , and A 5 are dynamic viscosity, density, electrical conductivity, thermal conductivity, and heat capacitance of the ternary hybrid nanofluid, respectively can be described as follows: We = x Γ 1 − ct 2 a 3 ν f ( 1 − ct ) , M = σ f B 2 ρ f a , K = ν f a K p , Fr = W c W x ( 1 − ct ) ρ f , Pr = ν f ( ρCp ) f κ f , R = 16 σ ∗ T ∞ 3 3 κ κ ∗ , θ w = T w T ∞ , Q = Q 0 ( 1 − ct ) a ( ρCp ) f , Nb = D B τ ( C w − C ∞ ) ν f , Nt = D T τ ( T w − T ∞ ) ν f T ∞ , Ec = u w 2 ( T w − T ∞ ) ( Cp ) f ( 1 − ct ) , Sc = ν f D B , C r = C 0 ( 1 − ct ) a , A = c a , A 1 = μ thnf μ f , A 2 = ρ thnf ρ f , A 3 = σ thnf σ f , A 4 = κ t hnf κ f , A 5 = ( ρCp ) thnf ( ρCp ) f In this context, the parameter λ , representing the velocity ratio, assumes various values: λ = 0 corresponds to the static sheet, λ 0 denotes the stretching sheet. The Skin friction coefficient ( C f ) , local Nusselt number ( Nu x ) , and local Sherwood number ( Sh x ) are given by 21 : (15) C fx = τ w ρ f u w 2 , N u x = x q w κ f ( T w − T ∞ ) , S h x = x j m D B ( C w − C ∞ ) , where τ w , q w , and j m represents the wall shear stress, the wall heat flux, and mass flux from the stretching sheets, which is given by: (16) { τ xz = μ thnf [ ( 1 − n ) ∂ u ∂ y + n Γ 2 ( ∂ u ∂ y ) 2 ] y = 0 q w = − [ κ thnf + 16 σ ∗ T ∞ 3 3 κ ∗ ] ( ∂ T ∂ y ) y = 0 , j m = − D B ( ∂ C ∂ y ) y = 0 . The dimensionless skin friction coefficient, Nusselt number, and the Sherwood number are given as follows: (17) { ( R e x ) 1 2 C fx = A 1 [ ( 1 − n ) f ′ ′ ( 0 ) + nWe 2 ( f ′ ′ ( 0 ) ) 2 ] ( R e x ) − 1 2 N u x = − ( A 4 + R ) θ ′ ( 0 ) , ( R e x ) − 1 2 S h x = − Φ ′ ( 0 ) where R e x = x u w ν f is the local Reynolds number. Numerical method Equations (11) to (13) and with the boundary condition (14) are numerically solved by using the MAT LAB software ( https://github.com/asfawmat/Tan_22BVP ) utilizing the bvp5c algorithm. 39 BVP5C is a numerical algorithm in MATLAB designed to solve boundary value problems (BVPs) for ordinary differential equations. It extends the capabilities of previous algorithms, allowing for the efficient handling of complex systems that require multiple boundary conditions. BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. BVP5C offers high accuracy for stiff boundary value problems, adaptive mesh refinement, and robustness in handling nonlinearities. Compared to finite difference methods, it achieves faster convergence with minimal grid dependence. BVP5C is widely used in structural analysis, heat transfer problems, and fluid dynamics to model systems governed by differential equations with boundary conditions. The algorithm is applicable in fields such as quantum mechanics and thermodynamics, where BVPs arise naturally in modeling physical phenomena. In order to apply the BVP5C with the shooting technique, the boundary value problems must be reduced to a system of first-order initial value problems (IVPs), as explained below. We use the following relations to convert the nonlinear higher-order boundary value problems into a system of first-order initial value problems. Now let us defined the new variable by the equation (18) { y 1 = f , y 2 = f ′ , y 3 = f ′ ′ , y 4 = θ y 5 = θ ′ , y 6 = Φ , y 7 = Φ ′ Using Eq.(18) , the three coupled higher order differential equations Eq.(11) to Eq. (13) can be written as (19) { y 1 ′ = y 2 ; y 2 ′ = y 3 ; y 3 ′ = ( A + ( A 3 / A 2 ) M + ( A 1 / A 2 ) K ) y 2 + ( 1 + ( Fr / A 2 ) ) ( y 2 ) 2 − ( y 1 − ( Aη ) / 2 ) y 3 ( 1 − n ) + nWe y 3 y 4 ′ = y 5 ; y 5 ′ = ( − [ ( R / A 5 Pr ) ( 3 ( 1 + ( θ w − 1 ) y 4 ) 2 ( θ w − 1 ) ( y 5 ) 2 ) + ( f − ( Aη ) / 2 ) y 5 + ( 1 / A 5 ) Q y 4 + ( A 1 nWeEc ) / ( 2 A 5 ) ( y 3 ) 3 + ( A 1 / A 5 ) ( 1 − n ) Ec ( y 3 ) 2 + Ec ( ( A 1 / A 5 ) K + ( A 3 / A 5 ) M ) ( y 2 ) 2 + ( A 1 / A 5 ) FrEc ( y 2 ) 3 + ( 1 / A 5 ) ( Nb y 7 y 5 + Nt ( y 5 ) 2 ) ] ) / ( ( A 4 A 5 Pr ) ( 1 + Γ y 4 ) + ( R A 5 Pr ) ( 1 + ( θ w − 1 ) θ ) 3 ) : y 6 ′ = y 7 ; y 7 ′ = ( AηSc / 2 ) y 7 + Sc C r y 6 − Sc y 1 y 7 − Nt Nb y 5 ′ ; with corresponding initial conditions (20) { y 1 ( 0 ) = S , y 2 ( 0 ) = λ , y 3 ( 0 ) = β 1 , y 4 ( 0 ) = 1 , y 5 ( 0 ) = β 2 , y 6 ( 0 ) = 1 , y 7 ( 0 ) = β 3 where β 1 , β 2 , and β 3 are the missing initial conditions and prime denote the differentiation with respect to η . The system is solved in MATLAB using the BVP5C function, with a step size of 0.1 and a mesh size of 101. The shooting method employed provides greater precision, achieving a tolerance of 10 − 7 compared to earlier numerical methods. A comparison of our results with those from previous research is presented in Table 2 , and the findings indicate a generally excellent agreement with earlier studies. Table 2. A comparison between the numerical results of Gorla and Sidawi, 41 Waini et al., 42 Priyadlarshtni et al., 18 and Jamrus et al. 39 and the findings of the current investigation by evaluating the values of θ ′(0) for different values of Pr under the conditions of Nb = 1 × 10 − 25 , when Γ = M = K = A = n = We = Fr = R = θ w = Q = Nt = Ec = Sc = C r = S = H = ϕ 1 = ϕ 2 = ϕ 3 = 0. Pr Gorla and Sidawi 41 Waini et al. 42 Priyadlarshtni et al. 18 Jamrus et al. 39 Present study 2 0.9114 0.911353 0.9113 0.911358 0.911352771 6.13 - 1.759682 - 1.759685 1.759681702 7 1.8954 1.8954 1.8954 1.895403 1.8954 20 3.3539 3.353902 3.3539 3.353904 3.353901836 Results and Discussion In this section, we discuss the numerical solution for the time-dependent MHD hyperbolic tangent ternary hybrid nanofluid flowing over stretched surface. For temperature, velocity, concentration, local skin friction, Nusselt number, and Sherwood number under different physical parameter effects such as Weissenberg number ( We ) , magnetic field parameter ( M ) , spherical-shaped Al 2 O 3 , Cu , and TiO 2 nanoparticle volume fraction ( ϕ 1 , ϕ 2 , ϕ 3 ) , heat generation ( Q ) , Forchheimer number ( Fr ) , Prandtl number ( Pr ) , nonlinear thermal radiation ( R ) , unsteady parameter ( A ) , temperature ratio parameter ( θ w ) , Eckert number ( Ec ) , Schmidt number ( Sc ) , chemical reaction ( C r ) , Porosity parameter ( K ) , Brownian ( Nb ) and thermophoresis ( Nt ) diffusion parameter are displays via graphical and table illustrations. Using Table 1 , to present the results in the form of figures and tables, the following fixed numerical values for the parameters are used in this study: Pr = 7 , Sc = 2 , Nb = 0.5 , Nt = 0.1 , λ = 1 , Γ = 0.3 , We = 0.5 , Q = 0.01 , Fr = 0.3 , K = 0.1 , S = 0.1 , n = 0.1 , M = 0.2 , R = 1.5 , θ w = 1.2 , A = 0.1 , Ec = 0.2 , C r = 0.1 , and ϕ 1 = ϕ 2 = ϕ 3 = 0.05 , unless explicitly stated otherwise in the corresponding graphs. The selection of parametric values in this study is guided by a combination of physical relevance, material properties, flow regime characteristics, heat transfer conditions, and insights from prior empirical research. By aligning these values with specific physical scenarios, the model’s accuracy and applicability are enhanced, enabling more reliable predictions and deeper understanding of the system’s behavior. Velocity characteristics Figures 2 - 10 depict the influence of several factors, like: magnetic field parameter ( M ) , porous permeability parameter ( K ) , Weissenberg number ( We ) , unsteady parameter ( A ) , power law index ( n ) , Forchheimer number ( Fr ) , suction parameter ( S ) , velocity ratio parameter ( λ ) , and nanoparticle volume fraction on scaled velocity ( f ′ ( η ) ) . Figure 2 shows how the scaled velocity profile is affected by the magnetic field parameter ( M ) . Physically, as the magnetic field effect increases, the interplay between the Lorentz force and the fluid’s motion leads to a decline in the velocity distribution, primarily due to increased resistance, damping effects, and changes in the dynamics of the flow. Figure 3 depict the influence of the porous permeability parameter ( K ) , on the velocity profile. When the value of the porous permeability parameter increases, the velocity distribution in a fluid flow can declines. Physically, in porous media, as the permeability increases, the inertial effects may become more pronounced, particularly in non-Darcy flow regimes. This can lead to a situation where the increased inertia counteracts the driving forces, resulting in a decline in the overall velocity. Higher permeability typically allows for easier flow through the porous medium. However, if the permeability increases too much, it can lead to a decrease in the pressure gradient needed to drive the flow. This results in reduced fluid velocity as the flow becomes less confined and more dispersed within the pores. Figure 4 illustrate the impact of power law index ( n ) on the velocity profile. As the power law index ( n ) , increases in a tangent hyperbolic fluid, the resulting increase in viscosity and flow resistance, combined with the fluid’s non-Newtonian characteristics, leads to a decline in the velocity distribution. Figure 5 display the effect of unsteady parameter on ( f ′ ( η ) ) . As the unsteady parameter increases, the balance between inertial and viscous forces shifts, leading to higher velocities due to enhanced acceleration, reduced damping, potential turbulence, and increased momentum transfer. The impact of Forchheimer parameter ( Fr ) on the velocity profile illustrated in Figure 6 . As the Forchheimer parameter increases, the combination of heightened inertial resistance, nonlinear flow behavior, increased energy losses, and complex flow distributions contributes to a diminished velocity profile in porous media. This highlights the interplay between viscous and inertial effects in determining flow behavior. The effect of suction parameter on the velocity distribution demonstrated in Figure 7 . When suction is applied to the stretching sheet, the fluid will be pulled closer to the sheet, resulting in the thinning of the boundary layer. As the suction parameter increases, the combined effects of increased resistance to flow, enhanced viscous effects, flow redistribution, boundary layer dynamics, and reduced kinetic energy contribute to a diminished velocity profile. These factors illustrate how suction influences the overall flow behavior and velocity distribution in the system. The Weissenberg number ( We ) is a dimensionless parameter that characterizes the relative importance of elastic (viscoelastic) effects compared to viscous effects in a fluid flow, particularly for non-Newtonian fluids such as tangent hyperbolic fluids. As the Weissenberg number increases, it indicates that the elastic effects become more significant, as depicted in Figure 8 . As the Weissenberg number increases in a tangent hyperbolic fluid, the combined effects of increased elastic resistance, nonlinear flow behavior, thicker boundary layers, increased energy dissipation, and flow stabilization contribute to a diminished velocity profile. These factors highlight the complex interplay between elasticity and viscosity in determining the flow characteristics of non-Newtonian fluids. Figure 9 show the effect of the velocity ratio parameter ( λ ) on the velocity distribution. When the velocity ratio ( λ ) is high, the interactions between fluid layers (or between phases in a multiphase flow) become more dynamic. This enhances the momentum exchange, leading to higher velocities being transmitted through the flow field, thus increasing the overall velocity profile. As in Figure 10 , as nanoparticle volume fraction ( ϕ 1 ) increases, the interplay of increased viscosity, enhanced drag, altered fluid structure, and boundary effects collectively results in a decrease in the velocity profile of the fluid. Figure 11 depict the comparison of nanofluid ( Al 2 O 3 /C 2 H 6 O 2 ), hybrid nanofluid ( Al 2 O 3 − Cu/C 2 H 6 O 2 ), and ternary hybrid nanofluid ( Al 2 O 3 − Cu − TiO 2 /C 2 H 6 O 2 ) on the velocity distribution. When the concentration of nanoparticles in nanofluids, hybrid nanofluids, and ternary hybrid nanofluids increases, the velocity pro- file typically decreases due to increased viscosity, enhanced drag forces, complex microstructural changes, potential shear-thickening effects, and shifts in flow regime. These factors collectively hinder the smooth flow of the fluid, resulting in lower velocities. Ternary hybrid nanofluids often achieve the best velocity distribution among the three categories. The synergistic effects of multiple nanoparticles can enhance dispersion stability and thermal performance while managing viscosity better. This results in improved flow characteristics and potentially higher velocities compared to the other two types. Figure 2. Velocity with M . Figure 3. Velocity with K . Figure 4. Velocity with n . Figure 5. Velocity with A . Figure 6. Velocity with Fr . Figure 7. Velocity with S . Figure 8. Velocity with We . Figure 9. Velocity with λ . Figure 10. Velocity with ϕ 1 . Thermal characteristics The effects of the nonlinear thermal radiation parameter ( R ) on fluid temperatures are illustrated in Figure 12 . When the values of nonlinear thermal radiation increase, the temperature profile in a fluid or material can also increase. In materials where radiation is a primary mode of heat transfer, an increase in nonlinear thermal radiation means that more energy is available to heat the material. This can lead to higher thermal gradients within the material, resulting in an overall increase in the temperature profile. Figure 13 depicts the thermal distribution against the temperature ratio ( θ w ) . The temperature ratio often refers to the ratio of the temperature of a fluid to that of a boundary or a reference temperature. An increased temperature ratio implies a greater difference between the fluid temperature and a reference or surrounding temperature. This enhanced temperature gradient can lead to more efficient heat transfer from the hotter regions to the cooler ones, resulting in higher temperatures throughout the fluid or material. When the values of the unsteady parameter increase, the temperature profile in a fluid or material can also increase due to several interrelated mechanisms, as shown in Figure 14 . The unsteady parameter often reflects the importance of transient effects relative to steady-state conditions, such as in heat transfer or fluid dynamics. An increase in the unsteady parameter can lead to a higher temperature profile through enhanced energy transport, rapid response to thermal gradients, and increased effective thermal conductivity. These factors illustrate the critical role that unsteady conditions play in influencing the thermal behavior of fluids and materials. Figure 15 present the effect of the thermophoresis diffusion on the temperature profile. As the thermophoresis diffusion parameter increases, particles within the fluid experience a stronger thermophoresis force, which causes them to migrate from regions of lower temperature to regions of higher temperature. This movement can lead to a concentration of particles in hotter areas, effectively increasing the local temperature in those regions. The influence of Brownian motion on the temperature distribution, as shown in Figure 16 . As the Brownian diffusion parameter increases, the microscopic motion of particles (due to Brownian motion) becomes more significant. This enhanced movement leads to improved mixing within the fluid, promoting more uniform temperature distribution and potentially raising the average temperature profile by allowing hotter regions to mix with cooler ones more effectively. As we can see from the temperature distribution graphs in Figure 17 , the temperature functions get better as the Eckert number ( Ec ) increases in value. The Eckert number ( Ec ) is a dimensionless parameter that characterizes the relative importance of kinetic energy to thermal energy in a fluid flow, particularly in situations involving heat transfer. As the Eckert number rises, the kinetic energy associated with the flow increases. This kinetic energy can contribute to the overall thermal energy of the fluid, leading to higher temperatures, especially if the kinetic energy is converted into internal energy during viscous dissipation. The significance of the porous permeability media parameter ( K ) on temperature profiles is depicted in Figure 18 . When the porous permeability parameter ( K ) increases, the temperature profile in a fluid flowing through a porous medium can also increase. In porous media, higher permeability often means a more interconnected pore structure. This can provide a larger surface area for heat exchange between the fluid and the solid matrix of the medium, enhancing heat transfer efficiency and contributing to a higher temperature profile. As Figure 19 illustrate, the temperature increases as the magnetic field parameter ( M) increases. As the magnetic field strength increases, it can exert a damping effect on the fluid flow. This damping can reduce the velocity of the fluid, leading to increased viscous heating. The kinetic energy of the fluid being converted into thermal energy due to viscous dissipation results in a higher temperature profile. Figure 20 shows a graph of the temperature profile as a result of the increasing the power law index ( n ) . An increase in the power law index parameter ( n ) , can lead to a higher temperature profile through increased viscosity and viscous heating, reduced flow velocity resulting in enhanced energy dissipation, altered heat transfer characteristics, greater thermal energy retention, improved mixing and flow stability, and interactions with temperature gradients. These factors collectively illustrate how non-Newtonian fluid behavior influences thermal dynamics within the flow. The effect of the Forchheimer number ( Fr ) on the temperature distribution, demonstrated in Figure 21 . When the Forchheimer number increases, it indicates that inertial forces are becoming more significant compared to viscous forces. As inertial forces become more dominant, the mixing and transport of thermal energy are enhanced. The increased movement of the fluid helps distribute heat more uniformly, resulting in a higher average temperature within the porous medium. As in Figure 22 , an increase in the Prandtl number indicates that the kinematic viscosity is relatively high compared to thermal diffusivity. This means that momentum diffuses more slowly than heat. Higher viscosity can lead to reduced flow velocities, which can decrease the fluid’s ability to carry heat away from hot regions, resulting in a less effective temperature distribution. Figure 23 show the effect of the heat generation on the temperature profile. Increasing the rate of heat generation results in a direct increase in thermal energy, leading to higher temperatures in the fluid or material. This effect is reinforced by heat transfer mechanisms, thermal gradients, and material properties, all contributing to a rising temperature profile. As a result, regions closer to the heat source experience significant temperature increases, influencing the overall thermal behavior of the system. As in Figure 24 when the variable thermal conductivity parameter (Γ) increases, the temperature profile generally increases due to enhanced heat transfer capabilities of the material or fluid. As heat is conducted more effectively, the temperature can rise in regions further away from the heat source, resulting in an increased overall temperature profile. As the Weissenberg number ( We ) increases for a tangent hyperbolic fluid, the fluids viscoelastic properties become more pronounced, leading to increased energy storage and viscous dissipation, as seen Figure 25 . This results in enhanced heat generation, steeper temperature gradients, and improved heat transfer characteristics, collectively contributing to an increase in the temperature profile. The balance between heat generation and dissipation is crucial in determining the extent of this temperature rise. As the nanoparticle volume fraction ( ϕ 1 , ϕ 2 , ϕ 3 ) increases, the overall temperature profile of the nanofluid tends to rise due to enhanced thermal conductivity, increased heat generation through viscous dissipation, reduced specific heat capacity, improved thermal boundary layer effects, and enhanced convection, as seen Figures 26 , 27 , 28 . Collectively, these factors contribute to more effective heat transfer and distribution, resulting in higher temperatures within the nanofluid. Figure 29 show a graph of the comparison of the nanofluid ( Al 2 O 3 /C 2 H 6 O 2 ), hybrid nanofluid ( Al 2 O 3 − Cu/C 2 H 6 O 2 ), and ternary hybrid nanofluid ( Al 2 O 3 − Cu − TiO 2 /C 2 H 6 O 2 ) on the temperature distribution. As the volume fraction of nanoparticles ( ϕ 1 , ϕ 2 , ϕ 3 ) increases in nanofluids, hybrid nanofluids, and ternary hybrid nanofluids, the temperature profile increases due to improved thermal conductivity, enhanced viscous dissipation, and better convective heat transfer. The combination of these factors leads to higher localized and overall average temperatures in the fluid. Generally, while all three types of fluids (nanofluids, hybrid nanofluids, and ternary hybrid nanofluids) show increased temperature profiles and enhanced heat transfer capabilities with higher nanoparticle concentrations, ternary hybrid nanofluids generally provide the best performance in terms of thermal conductivity and heat transfer efficiency. This is due to their ability to leverage the synergistic effects of multiple types of nanoparticles, leading to superior thermal management in various applications. Figure 11. Velocity with ϕ 1 , ϕ 2 , ϕ 3 . Figure 12. Temperature with R . Figure 13. Temperature with θ w . Figure 14. Temperature with A . Figure 15. Temperature with Nt . Figure 16. Temperature with Nb . Figure 17. Temperature with Ec . Figure 18. Temperature with K . Figure 19. Temperature with M . Figure 20. Temperature with n . Figure 21. Temperature with Fr . Figure 22. Variation of temperature with Pr . Figure 23. Temperature with Q . Figure 24. Temperature with Γ. Figure 25. Temperature with We . Figure 26. Temperature with ϕ 1 . Figure 27. Temperature with ϕ 2 . Figure 28. Variation of temperature with ϕ 3 . Figure 29. Temperature with ϕ 1 , ϕ 2 , ϕ 3 . Concentration characteristic The effect of the thermophoresis diffusion parameter ( Nt ) on the concentration profile, as illustrated in Figure 30 . As the thermophoresis parameter increases, the transport of particles within the fluid becomes more effective. This enhanced transport allows for a greater number of particles to move towards and concentrate in regions of higher temperature, further increasing the concentration profile. Figure 31 illustrate the impact of the Brownian diffusion Parameter ( Nb ) on the concentration profile. When the Brownian diffusion parameter ( Nb ) increases, it indicates a stronger influence of Brownian motion on the transport of particles suspended in a fluid. As Brownian diffusion increases, particles undergo more vigorous random motion. This enhanced diffusion can cause particles to spread out more rapidly within the fluid, leading to a decrease in local concentration as particles move away from regions of higher concentration. Figure 32 depict the effect of the Schmidt number ( Sc ) on the concentration profile. The Schmidt number ( Sc ) is a dimensionless parameter that characterizes the relative importance of momentum diffusion (viscous effects) to mass diffusion (concentration effects) in a fluid. When the Schmidt number increases, it typically indicates that momentum diffusion is more significant compared to mass diffusion. When momentum diffusion dominates, it can lead to smaller concentration gradients because the particles are less able to respond to local variations in concentration. This smaller gradient reduces the driving force for diffusion, causing a decrease in the concentration profile over time as the particles do not accumulate effectively. Figure 33 show the effect of the unsteady parameter ( A ) on the concentration profile. When the unsteady parameter ( A ) increases, it typically indicates that transient effects in the system are becoming more significant compared to steady-state effects. The unsteady parameter affects how particles are transported in the fluid. With higher values of the unsteady parameter, the transport mechanisms (like diffusion and convection) can operate differently, potentially leading to increased mixing or accumulation of particles in certain areas, thereby increasing the concentration profile. Figure 34 present that when the chemical reaction parameter increases, the solute’s boundary layer thickness decreases. In reactions where the concentration of a reactant decreases as it transforms into products, an increase in the chemical reaction parameter means that the reactants are being converted to products more rapidly. This results in a lower concentration of the reactant remaining in the system, causing the overall concentration profile to decline. Figure 35 depict the effect of suction parameter ( S ) on the concentration profile. As the values of the suction parameter increase, the concentration profile decreases due to the removal of fluid that carries solutes, leading to a reduction in local concentration. This effect is further amplified by the dynamics of the concentration boundary layer and enhanced mass transfer away from the surface that prioritize the extraction of fluid over the retention of solutes. Consequently, the overall concentration profile in the fluid becomes lower with increased suction. Impacts of parameters on the coefficient of skin friction, local Nusselt and Sherwood number In Table 3 , we present the numerical results for the skin friction ( C fx Re 1 2 ) , the Nusselt number ( Nu x Re − 1 2 ) , and the Sherwood number ( Sh x Re − 1 2 ) . These key parameters are evaluated using various dimensionless parameters for ternary hybrid nanofluids. As the Weissenberg number increases, the skin friction coefficient decreases due to the dominance of elastic effects leading to lower wall shear rates and potentially more stable flow and the Sherwood number decreases as the flow becomes less turbulent and diffusion rates drop, resulting in less effective mass transfer. Also, the Nusselt number increases because the tangent hyperbolic fluid properties can still enhance heat transfer through improved convective mechanisms, allowing for better thermal management despite the reduced mixing and diffusion. The increase in the unsteady parameter leads to a reduction in the effectiveness of both mass and heat transfer, resulting in lower values of the skin friction coefficient, Sherwood number, and Nusselt number. Skin friction coefficient tends to increase with higher magnetic field parameter ( M ), Forchheimer number ( Fr ), and velocity ratio parameter ( λ ), while it decrease with Weissenberg number ( We ), unsteady parameter ( A ), and power law index ( n ). Nusselt numbers decrease as unsteady parameter ( A ), magnetic field parameter ( M ), Forchheimer number ( Fr ), Eckert number ( Ec ), and velocity ratio parameter ( λ ) values increase, whereas they increase with the increase in the values of Weissenberg number ( We ), power law index ( n ), variable thermal conductivity (Γ), and nonlinear thermal radiation ( R ). Increases with enhanced thermal conductivity, allowing for better heat transfer. Also, increases as nonlinear thermal radiation enhances heat transfer, particularly at high temperatures. Sherwood numbers increase with the increased values of chemical reaction ( C r ) and velocity ratio parameter ( λ ), whereas they decrease with the increasing values of Weissenberg number ( We ), unsteady parameter ( A ), and power law index ( n ) . Increases as the chemical reaction can enhance concentration gradients, improving mass transfer. Figure 30. Concentration with Nt . Figure 31. Concentration with Nb . Figure 32. Concentration with Sc . Figure 33. Concentration with A . Figure 34. Concentration with Cr . Figure 35. Concentration with S . Table 3. Mathematical data of skin friction coefficient, Nusselt number and Sherwood number of some values of parameters for A = 0 . 1 , Nb = 0 . 5 , Nt = 0 . 1 , θ w = 1 . 2 , Q = 0 . 01 , K = 0 . 1 , λ = 1 , Γ = 0 . 3 , We = 0 . 5 , Cr = 0 . 1 , R = 1 . 5 , Ec = 0 . 2 , Pr = 7 , M = 0 . 2 , Sc = 2 , n = 0 . 1 , S = 0 . 1 , Fr = 0 . 3, ϕ 1 = ϕ 2 = ϕ 3 = 0.05 . We A M ER n λ R Γ Ec Cr C fx Re 1 2 Nu x Re − 1 2 Sh x Re − 1 2 0.1 1.762244 0.590423 1.062417 1.5 1.688485 0.593556 1.054830 3 1.591519 0.595070 1.043963 0.1 1.742281 0.591422 1.060418 0.3 1.703993 0.288018 0.848307 0.5 1.671153 0.046494 0.662183 0 1.662063 0.691278 1.060752 0.7 1.930985 0.358756 1.060005 1.5 2.203555 0.028154 1.060372 0.1 1.673295 0.652511 1.060683 0.5 1.808894 0.532523 1.060228 1 1.966347 0.393666 1.060034 0.1 1.742281 0.591422 1.060418 0.3 1.443509 0.634711 1.031621 0.5 1.709855 0.634365 0.971353 1 1.742281 0.591422 1.060418 1.1 1.980330 0.469779 1.100296 1.3 2.482471 0.154472 1.182047 0.1 1.742281 0.019870 1.091851 1 1.742281 0.392378 1.067107 2 1.742281 0.788465 1.055787 0.1 1.742281 0.530495 1.064099 1 1.742281 0.758082 1.050365 2 1.742281 0.917032 1.040816 0.1 1.742281 0.957248 1.037816 0.2 1.742281 0.591422 1.060418 0.3 1.742281 0.226086 1.083006 0.1 1.742281 0.591422 1.060418 0.3 1.742281 0.577866 1.237084 0.5 1.742281 0.567410 1.394140 Table 4 explores the effects of ϕ 1 , ϕ 2 , and ϕ 3 on the skin friction coefficient, Nusselt number, and Sherwood number for ternary hybrid nanofluids, hybrid nanofluids, and mono nanofluids. The results indicate that increasing the nanoparticle volume fractions ( ϕ 1 , ϕ 2 , ϕ 3 ) leads to enhancements in the skin friction coefficient, local Nusselt number, and local Sherwood number. The skin friction coefficient is a measure of the frictional resistance experienced by a fluid flowing over a surface. With increased nanoparticle concentration, the skin friction coefficient generally increases, indicating greater resistance due to higher fluid density and viscosity. As nanoparticle concentration rises, both the Nusselt and Sherwood numbers rise, indicating improved heat and mass transfer efficiency, with ternary hybrid nanofluids being the most effective in enhancing heat transfer and thermal conductivity, along with an increase in skin friction coefficient. Table 4. Mathematical data of skin friction coefficient, Nusselt number and Sherwood number of some values of parameters for A = 0 . 1 , Nb = 0 . 5 , Nt = 0 . 1 , θ w = 1 . 2 , Q = 0 . 01 , K = 0 . 1 , λ = 1 , Γ = 0 . 3 , We = 0 . 5 , Cr = 0 . 1 , R = 1 . 5 , Ec = 0 . 2 , Pr = 7 , M = 0 . 2 , Sc = 2 , n = 0 . 1 , S = 0 . 1 , Fr = 0 . 3. ϕ 1 ϕ 2 ϕ 3 C fx Re 1 2 Nu x Re − 1 2 Sh x Re − 1 2 0.01 0 0 1.234185 0.802721 1.034962 0.08 0 0 1.486944 0.862325 1.035621 0.15 0 0 1.821320 0.921726 1.036584 0.01 0.01 0 1.261670 0.814131 1.035321 0.08 0.08 0 1.811703 0.964584 1.037424 0.15 0.15 0 2.715110 1.133760 1.039626 0.01 0.01 0.01 1.291004 0.824010 1.035627 0.08 0.08 0.08 2.213728 1.074752 1.039276 0.15 0.15 0.15 4.041513 1.403616 1.043548 The present study has significant real-world applications in various engineering and industrial fields. The enhanced heat transfer properties of tangent hyperbolic ternary hybrid nanofluids make them highly effective for cooling systems, such as electronic device cooling, heat exchangers, and thermal management in power plants. Additionally, their superior thermal conductivity and stability are beneficial in biomedical applications, including targeted drug delivery and hyperthermia treatments. The findings also have implications in polymer processing industries, where precise thermal control is required for extrusion and casting. Furthermore, these nanofluids can be utilized in aerospace and automotive industries for efficient thermal regulation and improved fuel efficiency. The insights gained from this research contribute to optimizing fluid flow and heat transfer processes in advanced engineering systems. Conclusion This study investigates the unsteady MHD flow, heat transfer, and mass transfer characteristics of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet in a Darcy-Forchheimer porous medium with variable thermal conductivity. The combined effects of viscous dissipation, nonlinear thermal radiation, Joule heating, Brownian motion, thermophoresis, and chemical reactions on the fluid’s thermal and flow behavior are analyzed. The governing equations are transformed into a system of ordinary differential equations using similarity transformations and solved numerically via the BVP5C method in MATLAB. The BVP5C solver provides high accuracy for stiff boundary value problems, leveraging adaptive mesh refinement and efficiently handling nonlinearities. Compared to finite difference methods, it offers faster convergence with minimal dependence on grid size. Graphical analysis is employed to illustrate the effects of these parameters on flow, heat, and mass transfer, providing valuable insights for researchers and engineers working on similar thermal management challenges. The results reveal that increasing the Weissenberg and Forchheimer numbers reduces the velocity profile, while thermal conductivity variation enhances heat transfer. The ternary hybrid nanofluid demonstrates superior heat transfer characteristics compared to conventional and binary hybrid nanofluids, highlighting its potential for applications in cooling systems, biomedical engineering, and industrial thermal management. From the current analysis, we can draw the following conclusions: • As the magnetic field effect intensifies, the interaction between the Lorentz force and the fluid’s motion results in a reduction in the velocity distribution. As the Forchheimer parameter increases, the combination of enhanced inertial resistance, nonlinear flow behavior, increased energy losses, and complex flow distributions results in a reduced velocity profile in porous media. • As the Weissenberg number rises in a tangent hyperbolic fluid, the resulting interplay of greater elastic resistance, nonlinear flow behavior, thicker boundary layers, heightened energy dissipation, and flow stabilization leads to a reduced velocity profile. • When the variable thermal conductivity parameter (Γ) increases, the temperature profile typically rises, reflecting the improved heat transfer capabilities of the material or fluid. • When the non-linear thermal radiation parameter, temperature ratio parameter ( θ w ) , heat generation ( Q 0 > 0), Brownian motion parameter, thermophoresis parameter, magnetic field parameter, Forchheimer number ( Fr ), and Eckert number increases, the temperature profile rises. The temperature profile diminishes, when the Prandtl number increases. • As the volume fraction of nanoparticles ( ϕ 1 , ϕ 2 , ϕ 3 ) increases in nanofluids, hybrid nanofluids, and ternary hybrid nanofluids, the temperature profile increases due to improved thermal conductivity, enhanced viscous dissipation, and better convective heat transfer. • The concentration profile decreases as the Schmidt number, chemical reaction parameter, Brownian diffusion parameter, and suction parameter increase. Conversely, the concentration profile increases with higher values of the unsteady parameter and thermophoresis diffusion. • In general, higher concentrations of nanoparticles in nanofluids, hybrid nanofluids, and ternary hybrid nanofluids result in increased skin friction coefficient, Nusselt number, and Sherwood number. Ternary hybrid nanofluids are often the most effective choice for enhancing heat transfer and thermal conductivity. This study provides valuable theoretical insights but has some limitations. Simplified assumptions, such as treating the nanofluid as a single-phase fluid and neglecting higher-order effects, may not fully capture real-world complexities. The BVP5C numerical method, while effective, has constraints in handling highly nonlinear systems. The findings are based on a specific parameter range, and extrapolation beyond this range requires further validation. Additionally, the assumption of uniform nanoparticle dispersion overlooks possible aggregation or sedimentation effects. To enhance validity and generalizability, future research should incorporate experimental validation and more advanced modeling approaches. CRediT authorship contribution statement Asfaw Tsegaye Moltot contributed to Writing review & editing, conceptualization, methodology, formal analysis, validation, and writing the original draft. Eshetu Haile contributed to Writing the review, editing, supervision conceptualization, and resources. Gurju Awgichew contributed to Writing the review, editing, supervision and resources. Hunegnaw Dessie contributed to Writing the review, editing, supervision and resources. Corresponding author: Asfaw Tsegaye Moltot ( [email protected] ) Institutional email: [email protected] Affiliation: Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Author name: Dr. Eshetu Haile: Email: [email protected] Research gate page: https://scholar.google.com/citations?user=GRVFCVAAAAAJ&hl=en AD Scientific Index ID: 4494034 Affiliation: Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Dr. Gurju Awgichew: Email: [email protected] Research gate page: https://www.researchgate.net/profile/Gurju-Zergaw Affiliation: Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Dr. Hunegnaw Dessie: Email: [email protected] Affiliation: Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Ethics and consent Ethics and consent were not required. Data availability statement Source data A list of detailed material properties used for algorithms and thermophysical properties of nanoparticle model analysis was taken from: https://github.com/asfawmat/BVP-MATLAB-Implementation Underlying data BVP-MATLAB-Implementation and Thermophysical properties: https://github.com/asfawmat/BVP-MATLAB-Implementation This project contains the following data: - Values of physical parameter and Matlab Code License: Data is available under license CC BY 4.0 Software availability All the presented results can be reproduced using the codes available in the GitHub links ( https://github.com/asfawmat/BVP-MATLAB-Implementation ) for Matlab implementation. Source code available from: ( https://github.com/asfawmat/BVP-MATLAB-Implementation ) for Matlab implementation. License: Data is available under license MIT License and CC-by 4.0 Archived source code available from: https://doi.org/10.5281/zenodo.14480542 Data is available under license Creative Commons Attribution 4.0 International References 1. Hashim RS, Mohamed Tag Eldin E, Bafakeeh O, et al. : Coupled energy and mass transport for non-Newtonian nanofluid flow through non-parallel vertical enclosure. Ain Shams Eng. J. 2023; 14 (8): 102023. Publisher Full Text 2. Rehman SH, Alqahtani S, Alshehery S: Modeling a non-Newtonian nanofluid flow between intersecting planes with slip mechanism. Contin. Mech. Thermodyn. 2023; 35 (1): 61–80. Publisher Full Text 3. Nadeem M, Siddique I, Riaz Z, et al. : Numerical study of unsteady tangent hyperbolic fuzzy hybrid nanofluid over an exponentially stretching surface. Sci. Rep. 2023; 13 (1): 15551. PubMed Abstract | Publisher Full Text | Free Full Text 4. Alqahtani AM, Bilal M, Riaz MB, et al. : Forced convective tangent hyperbolic nanofluid flow subject to heat source/sink and Lorentz force over a permeable wedge: Numerical exploration. Nanotechnol. Rev. 2024; 13 (1): 20240014. Publisher Full Text 5. Boujelbene M, Rehman S, Hashim AS, et al. : Optimizing thermal characteristics and entropy degradation with the role of nanofluid flow configuration through an inclined channel. Alex. Eng. J. 2023; 69 : 85–107. Publisher Full Text 6. Ibrahim W: Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition. Results Phys. 2017; 7 : 3723–3731. Publisher Full Text 7. Alkaoud A, Khader MM, Eid A, et al. : Numerical simulation of the flow of a tangent hyperbolic fluid over a stretching sheet within a porous medium, accounting for slip conditions. Heliyon. 2024; 10 (8): e28683. PubMed Abstract | Publisher Full Text | Free Full Text 8. Choudhary P, Choudhary S, Jat K, et al. : Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms. Int. J. Thermofluids. 2024; 23 : 100788. Publisher Full Text 9. Alqawasmi K, Alharbi KAM, Farooq U, et al. : Numerical approach toward ternary hybrid nanofluid flow with nonlinear heat source-sink and fourier heat flux model passing through a disk. Int. J. Thermofluids. 2023; 18 : 100367. Publisher Full Text 10. Jat K, Sharma K, Makinde O: Novel features of radiating hybrid nanofluid flow past a nonlinear stretchable porous sheet with different nanoparticles shape. Numer. Heat Transf. B: Fundam. 2024: 1–25. Publisher Full Text 11. Mahboobtosi M, Hosseinzadeh K, Ganji DD: Investigating the convective flow of ternary hybrid nanofluids and single nanofluids around a stretched cylinder: Parameter analysis and performance enhancement. Int. J. Thermofluids. 2024; 23 : 100752. Publisher Full Text 12. Farooq U, Bibi A, Abbasi JN, et al. : Nonsimilar mixed convection analysis of ternary hybrid nanofluid flow near stagnation point over vertical Riga plate. Multidiscip. Model. Mater. Struct. 2024; 20 (2): 261–278. Publisher Full Text 13. Abbas Z, Arslan MS, Rafiq MY: Enhancement of convective flow in ternary hybrid nanofluid induced by metachronal propulsion. Adv. Mech. Eng. 2023; 15 (10): 16878132231205351. Publisher Full Text 14. Nabwey HA, Rashad AM, Khan WA, et al. : Heat transfer in MHD flow of Carreau ternary-hybrid nanofluid over a curved surface stretched exponentially. Front. Phys. 2023; 11 : 1212715. Publisher Full Text 15. Jamshed W, Prakash M, Devi SSU, et al. : Retracted Article: A brief comparative examination of tangent hyperbolic hybrid nanofluid through a extending surface: numerical KellerBox scheme. Sci. Rep. 2021; 11 (1): 24032. PubMed Abstract | Publisher Full Text | Free Full Text 16. Noreen S, Farooq U, Waqas H, et al. : Comparative study of ternary hybrid nanofluids with role of thermal radiation and Cattaneo-Christov heat flux between double rotating disks. Sci. Rep. 2023; 13 (1): 7795. PubMed Abstract | Publisher Full Text | Free Full Text 17. Garalleh AL, H.: Numerical simulation of heat transport mechanism in chemically influenced ternary hybrid nanofluid flow over a wedge geometry. Discov. Appl. Sci. 2024; 6 (9): 449. Publisher Full Text 18. Priyadharshini P, Archana MV, Shah NA, et al. : Ternary hybrid nanofluid flow emerging on a symmetrically stretching sheet optimization with machine learning prediction scheme. Symmetry. 2023; 15 (6): 1225. Publisher Full Text 19. Patil PM, Goudar B, Sheremet MA: Tangent hyperbolic ternary hybrid nanofluid flow over a rough-yawed cylinder due to impulsive motion. J. Taibah Univ. Sci. 2023; 17 (1): 2199664. Publisher Full Text 20. Kebede T, Haile E, Awgichew G, et al. : Heat and mass transfer in unsteady boundary layer flow of Williamson nanofluids. J. Appl. Math. 2020; 2020 (1): 1–13. Publisher Full Text 21. Jamal MA, Faizan M, Farid A, et al. : Unsteady Flow of an MHD Tangent Hyperbolic Nanouid over a Stretching Sheet. VFAST Trans. Math. 2022; 10 (1): 40–56. Publisher Full Text 22. Reddy GVR, Akgl A, Riaz MB: Unsteady mhd flow of tangent hyperbolic liquid past a vertical porous plate plate. Heliyon. 2023; 9 (8). 23. Kumar VV, Shekar MNR, Bejawada SG: Heat and Mass Transfer Significance on MHD Flow over a Vertical Porous Plate in the Presence of Chemical Reaction and Heat Generation. CFD Lett. 2024; 16 (5): 9–20. Publisher Full Text 24. Krishna MV, Chamkha AJ: Hall effects on MHD squeezing flow of a water-based nanofluid between two parallel disks. J. Porous Media. 2019; 22 (2): 209–223. Publisher Full Text 25. Ibrahim W: Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition. Results Phys. 2017; 7 : 3723–3731. Publisher Full Text 26. Prajapati VJ, Meher R: Analysis of MHD tangent hyperbolic hybrid nanofluid flow with different base fluids over a porous stretched sheet. J. Taibah Univ. Sci. 2024; 18 (1): 2300851. Publisher Full Text 27. Reddy BN, Dodda R, Reddy BS: A Numerical Investigation on Boundary Layer Flow of MHD Tangent Hyperbolic Fluid Flow over a Stretching Sheet with Slip Boundary Conditions. J. Adv. Res. Fluid Mech. Therm. Sci. 2024; 120 (1): 122–139. Publisher Full Text 28. Choudhary S, Choudhary P, Alessa N, et al. : MHD Thermal and Solutal Stratified Stagnation Flow of Tangent Hyperbolic Fluid Induced by Stretching Cylinder with Dual Convection. Mathematics. 2023; 11 (9). Publisher Full Text 29. Choudhary P, Choudhary S, Jat K, et al. : Significances of melting heat transfer and bioconvection phenomena in nanofluid flow over a three different geometries. Int. J. Thermofluids. 2024; 24 : 100855. Publisher Full Text 30. Choudhary P, Choudhary S, Jat K, et al. : Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium. J. Therm. Anal. Calorim. 2025. Publisher Full Text 31. Anjum A, Abdul Gaffar S, Kumar DS, et al. : Bég Computation Of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport From An Inclined Plane With Ion Slip And Hall Current Effects.2024. 32. Lakshmi BN, Dharmaiah G, Anjum A, et al. : Influence of Darcy–Forchheimer Fe 3 O 4-CoFe 2 O 4-H 2 O hybrid nanofluid flow with magnetohydrodynamic and viscous dissipation effects past a permeable stretching sheet: a numerical contribution. Multiscale Multidiscip. Model. Exp. Des. 2025; 8 (1): 115. Publisher Full Text 33. Chamkha AJ: Non-Darcy fully developed mixed convection in a porous medium channel with heat generation/absorption and hydromagnetic effects. Numer. Heat Transf. A Appl. 1997; 32 (6): 653–675. Publisher Full Text 34. Jawad M, Hameed MK, Nisar KS, et al. : Darcy-Forchheimer flow of maxwell nanofluid flow over a porous stretching sheet with Arrhenius activation energy and nield boundary conditions. Case Stud. Therm. Eng. 2023; 44 : 102830. Publisher Full Text 35. Saeed A, Kumam P, Gul T, et al. : Darcy-Forchheimer couple stress hybrid nanofluids flow with variable fluid properties. Sci. Rep. 2021; 11 (1): 19612. PubMed Abstract | Publisher Full Text | Free Full Text 36. Alessa N, Sindhu R, Divya S, et al. : Computational analysis of Darcy-Forchheimer flow of Cu/AlAl2O3 hybrid nanofluid in water over a heated stretchable plate with nonlinear radiation. Micromachines. 2023; 14 (2): 338. PubMed Abstract | Publisher Full Text | Free Full Text 37. Hussain A, Malik MY, Salahuddin T, et al. : Effects of viscous dissipation on MHD tangent hyperbolic fluid over a nonlinear stretching sheet with convective boundary conditions. Results Phys. 2017; 7 : 3502–3509. Publisher Full Text 38. Nandi S, Kumbhakar B: Viscous dissipation and chemical reaction effects on tangent hyperbolic nanofluid flow past a stretching wedge with convective heating and Navier’s slip conditions. Iran. J. Sci. Technol. Trans. Mech. Eng. 2022; 46 (2): 379–397. Publisher Full Text 39. Jamrus FN, Waini I, Ishak A: Time-Depending Flow of Ternary Hybrid Nanofluid past a Stretching Sheet with Suction and Magnetohydrodynamic (MHD) Effects. J. Adv. Res. Fluid Mech. Therm. Sci. 2024; 117 (2): 15–27. Publisher Full Text 40. Mkhatshwa MP: Nonlinear Radiative Mixed Convective Flow of Fourth-Grade Tetra-Hybrid Nanomaterial over a Horizontal Cylindrical Surface. J. Nanofluids. 2024; 13 (4): 1040–1054. Publisher Full Text 41. Reddy Gorla RS, Sidawi I: Free convection on a vertical stretching surface with suction and blowing. Appl. Sci. Res. 1994; 52 : 247–257. Publisher Full Text 42. Waini I, Ishak A, Pop I: Unsteady flow and heat transfer past a stretching/shrinking sheet in a hybrid nanofluid. Int. J. Heat Mass Transf. 2019; 136 : 288–297. Publisher Full Text Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 04 Feb 2025 ADD YOUR COMMENT Comment Author details Author details 1 Department of Mathematics, College of Science, Bahir Dar University, P.O. Box 79, Bahir Dar, Ethiopia Asfaw Tsegaye Moltot Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Eshetu Haile Gorfie Roles: Conceptualization, Resources, Supervision, Visualization, Writing – Review & Editing Gurju Awgichew Zergaw Roles: Conceptualization, Resources, Supervision, Visualization, Writing – Review & Editing Hunegnaw Dessie Roles: Conceptualization, Resources, Supervision, Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (2) version 2 Revised Published: 10 Mar 2025, 14:152 https://doi.org/10.12688/f1000research.158629.2 version 1 Published: 04 Feb 2025, 14:152 https://doi.org/10.12688/f1000research.158629.1 Copyright © 2025 Tsegaye Moltot A et al . This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article Tsegaye Moltot A, Haile Gorfie E, Awgichew Zergaw G and Dessie H. Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.12688/f1000research.158629.2 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 2 VERSION 2 PUBLISHED 10 Mar 2025 Revised Views 0 Cite How to cite this report: Rehman S. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370297 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370297 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 22 Mar 2025 Sohail Rehman , Qurtuba University of Science and Information Technology, Peshawar, Pakistan Approved VIEWS 0 https://doi.org/10.5256/f1000research.178621.r370297 Accept in ... Continue reading READ ALL Accept in present form Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fluid dynamics, heat and mass transfer, Nanofluid I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Rehman S. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370297 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370297 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Anjum A. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370296 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370296 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 11 Mar 2025 Asra Anjum , University of Technology and Applied Sciences, Salalah, Oman Approved VIEWS 0 https://doi.org/10.5256/f1000research.178621.r370296 Approved ... Continue reading READ ALL Approved for acceptance. Competing Interests: No competing interests were disclosed. Reviewer Expertise: fluid dynamics, nanofluids and hybrid nanofluids I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Anjum A. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370296 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370296 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Choudhary P and Jat DK. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370298 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370298 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 11 Mar 2025 Prasun Choudhary , Chennai Institute of Technology, Chennai, India Dr. Kavita Jat , Centre for Computational Modeling, Chennai Institute of Technology, Chennai, Tamil Nadu, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.178621.r370298 I have reviewed the revised version of the manuscript and I appreciate the authors' efforts in incorporating the suggested comments and changes. The present version is well-structured, clear and significantly improved in terms of readability. As all necessary revisions ... Continue reading READ ALL I have reviewed the revised version of the manuscript and I appreciate the authors' efforts in incorporating the suggested comments and changes. The present version is well-structured, clear and significantly improved in terms of readability. As all necessary revisions have been properly addressed, I find the manuscript suitable for acceptance in its current form. I, therefore, accept and recommend it. Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fluid Flow, Heat Transfer, Mass Transfer, Distinct Geometries, Analytical Methods, Numerical Methods We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Choudhary P and Jat DK. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370298 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370298 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Version 1 VERSION 1 PUBLISHED 04 Feb 2025 Views 0 Cite How to cite this report: Rehman S. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366057 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366057 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 21 Feb 2025 Sohail Rehman , Qurtuba University of Science and Information Technology, Peshawar, Pakistan Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.174254.r366057 The main observations are listed below. The acceptance of the manuscript would depend on the revision. The author needs to provide a point by point response or provide a rebuttal. The manuscript need a through correction ... Continue reading READ ALL The main observations are listed below. The acceptance of the manuscript would depend on the revision. The author needs to provide a point by point response or provide a rebuttal. The manuscript need a through correction of grammatical errors. The title of the manuscript need to be revised. The authors have not explained, what motivation for selecting this fluid model is, and what are the real world applications of this fluid under the conditions studied in this paper. Provide mathematical expression for fluid volume. Check your boundary conditions and similarity variables. As they are dimensionally correct? As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? No Are the conclusions drawn adequately supported by the results? Partly References 1. Hashim, Rehman S, Mohamed Tag Eldin E, Bafakeeh O, et al.: Coupled energy and mass transport for non-Newtonian nanofluid flow through non-parallel vertical enclosure. Ain Shams Engineering Journal . 2023; 14 (8). Publisher Full Text 2. Rehman S, Hashim, Alqahtani S, Alshehery S: Modeling a non-Newtonian nanofluid flow between intersecting planes with slip mechanism. Continuum Mechanics and Thermodynamics . 2023; 35 (1): 61-80 Publisher Full Text 3. Boujelbene M, Rehman S, Hashim, Alqahtani S, et al.: Optimizing thermal characteristics and entropy degradation with the role of nanofluid flow configuration through an inclined channel. Alexandria Engineering Journal . 2023; 69 : 85-107 Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fluid dynamics, heat and mass transfer, Nanofluid I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Rehman S. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366057 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366057 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 10 Mar 2025 Asfaw Tsegaye , Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia 10 Mar 2025 Author Response We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The ... Continue reading We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The manuscript need a thorough correction of grammatical errors. The title of the manuscript need to be revised. Author’s Response: In the revised manuscript, we have carefully reviewed and edited the entire document to correct grammatical and typographical errors. We have revised the title to improve clarity and readability. The corrected title is: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. The authors have not explained, what the motivations for selecting this fluid model is, and what are the real-world applications of this fluid under the conditions studied in this paper. Author’s Response: We have added a detailed discussion in the introduction section and formulation of the method section, outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. This model is chosen due to its ability to accurately describe non-Newtonian fluid behaviors, making it suitable for engineering applications such as: Cooling Systems – Efficient heat dissipation in electronic devices. Biomedical Applications – Targeted drug delivery and hyperthermia treatments. Polymer Processing – Used in extrusion and casting processes. Energy Systems – Utilized in heat exchangers and power plants, etc. These points are now explicitly stated in the revised introduction. 3. Provide a mathematical expression for the fluid volume. Author’s Response: The fluid volume is not considered in our study since the analysis focuses on 2D flow, boundary-layer formulation, or similarity transformation, where volume is not a key variable. 4. Check whether the boundary conditions and similarity variables are dimensionally correct. Author’s Response: We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. The necessary corrections have been made, and a justification for each transformation is provided in the revised manuscript. 5. As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Author’s Response: We have expanded the discussion on the physical configuration in results and discussion section. The justification for key assumptions is also included, emphasizing: -The relevance of the mathematical model in practical scenarios. -The role of each governing parameter in defining the flow behavior. -The impact of approximations made to simplify the problem while maintaining accuracy, etc. 6. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Author’s Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 7. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Author’s Response: We have enriched the results section by comparing our findings with existing literature, including computational studies where available. Additionally, we have included benchmark comparisons in Table 2, demonstrating the reliability of our numerical method. The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 8. Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Author’s Response: We have incorporated the suggested references in the introduction and literature review, explaining their relevance to our work. These studies provide context for our analysis and strengthen the foundation of the research. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The manuscript need a thorough correction of grammatical errors. The title of the manuscript need to be revised. Author’s Response: In the revised manuscript, we have carefully reviewed and edited the entire document to correct grammatical and typographical errors. We have revised the title to improve clarity and readability. The corrected title is: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. The authors have not explained, what the motivations for selecting this fluid model is, and what are the real-world applications of this fluid under the conditions studied in this paper. Author’s Response: We have added a detailed discussion in the introduction section and formulation of the method section, outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. This model is chosen due to its ability to accurately describe non-Newtonian fluid behaviors, making it suitable for engineering applications such as: Cooling Systems – Efficient heat dissipation in electronic devices. Biomedical Applications – Targeted drug delivery and hyperthermia treatments. Polymer Processing – Used in extrusion and casting processes. Energy Systems – Utilized in heat exchangers and power plants, etc. These points are now explicitly stated in the revised introduction. 3. Provide a mathematical expression for the fluid volume. Author’s Response: The fluid volume is not considered in our study since the analysis focuses on 2D flow, boundary-layer formulation, or similarity transformation, where volume is not a key variable. 4. Check whether the boundary conditions and similarity variables are dimensionally correct. Author’s Response: We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. The necessary corrections have been made, and a justification for each transformation is provided in the revised manuscript. 5. As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Author’s Response: We have expanded the discussion on the physical configuration in results and discussion section. The justification for key assumptions is also included, emphasizing: -The relevance of the mathematical model in practical scenarios. -The role of each governing parameter in defining the flow behavior. -The impact of approximations made to simplify the problem while maintaining accuracy, etc. 6. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Author’s Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 7. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Author’s Response: We have enriched the results section by comparing our findings with existing literature, including computational studies where available. Additionally, we have included benchmark comparisons in Table 2, demonstrating the reliability of our numerical method. The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 8. Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Author’s Response: We have incorporated the suggested references in the introduction and literature review, explaining their relevance to our work. These studies provide context for our analysis and strengthen the foundation of the research. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 10 Mar 2025 Asfaw Tsegaye , Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia 10 Mar 2025 Author Response We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The ... Continue reading We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The manuscript need a thorough correction of grammatical errors. The title of the manuscript need to be revised. Author’s Response: In the revised manuscript, we have carefully reviewed and edited the entire document to correct grammatical and typographical errors. We have revised the title to improve clarity and readability. The corrected title is: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. The authors have not explained, what the motivations for selecting this fluid model is, and what are the real-world applications of this fluid under the conditions studied in this paper. Author’s Response: We have added a detailed discussion in the introduction section and formulation of the method section, outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. This model is chosen due to its ability to accurately describe non-Newtonian fluid behaviors, making it suitable for engineering applications such as: Cooling Systems – Efficient heat dissipation in electronic devices. Biomedical Applications – Targeted drug delivery and hyperthermia treatments. Polymer Processing – Used in extrusion and casting processes. Energy Systems – Utilized in heat exchangers and power plants, etc. These points are now explicitly stated in the revised introduction. 3. Provide a mathematical expression for the fluid volume. Author’s Response: The fluid volume is not considered in our study since the analysis focuses on 2D flow, boundary-layer formulation, or similarity transformation, where volume is not a key variable. 4. Check whether the boundary conditions and similarity variables are dimensionally correct. Author’s Response: We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. The necessary corrections have been made, and a justification for each transformation is provided in the revised manuscript. 5. As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Author’s Response: We have expanded the discussion on the physical configuration in results and discussion section. The justification for key assumptions is also included, emphasizing: -The relevance of the mathematical model in practical scenarios. -The role of each governing parameter in defining the flow behavior. -The impact of approximations made to simplify the problem while maintaining accuracy, etc. 6. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Author’s Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 7. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Author’s Response: We have enriched the results section by comparing our findings with existing literature, including computational studies where available. Additionally, we have included benchmark comparisons in Table 2, demonstrating the reliability of our numerical method. The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 8. Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Author’s Response: We have incorporated the suggested references in the introduction and literature review, explaining their relevance to our work. These studies provide context for our analysis and strengthen the foundation of the research. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The manuscript need a thorough correction of grammatical errors. The title of the manuscript need to be revised. Author’s Response: In the revised manuscript, we have carefully reviewed and edited the entire document to correct grammatical and typographical errors. We have revised the title to improve clarity and readability. The corrected title is: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. The authors have not explained, what the motivations for selecting this fluid model is, and what are the real-world applications of this fluid under the conditions studied in this paper. Author’s Response: We have added a detailed discussion in the introduction section and formulation of the method section, outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. This model is chosen due to its ability to accurately describe non-Newtonian fluid behaviors, making it suitable for engineering applications such as: Cooling Systems – Efficient heat dissipation in electronic devices. Biomedical Applications – Targeted drug delivery and hyperthermia treatments. Polymer Processing – Used in extrusion and casting processes. Energy Systems – Utilized in heat exchangers and power plants, etc. These points are now explicitly stated in the revised introduction. 3. Provide a mathematical expression for the fluid volume. Author’s Response: The fluid volume is not considered in our study since the analysis focuses on 2D flow, boundary-layer formulation, or similarity transformation, where volume is not a key variable. 4. Check whether the boundary conditions and similarity variables are dimensionally correct. Author’s Response: We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. The necessary corrections have been made, and a justification for each transformation is provided in the revised manuscript. 5. As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Author’s Response: We have expanded the discussion on the physical configuration in results and discussion section. The justification for key assumptions is also included, emphasizing: -The relevance of the mathematical model in practical scenarios. -The role of each governing parameter in defining the flow behavior. -The impact of approximations made to simplify the problem while maintaining accuracy, etc. 6. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Author’s Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 7. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Author’s Response: We have enriched the results section by comparing our findings with existing literature, including computational studies where available. Additionally, we have included benchmark comparisons in Table 2, demonstrating the reliability of our numerical method. The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 8. Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Author’s Response: We have incorporated the suggested references in the introduction and literature review, explaining their relevance to our work. These studies provide context for our analysis and strengthen the foundation of the research. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Choudhary P and Jat DK. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366062 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366062 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 20 Feb 2025 Prasun Choudhary , Chennai Institute of Technology, Chennai, India Dr. Kavita Jat , Centre for Computational Modeling, Chennai Institute of Technology, Chennai, Tamil Nadu, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.174254.r366062 Title of the Article: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity This paper is interesting. This study examines the combined effects ... Continue reading READ ALL Title of the Article: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity This paper is interesting. This study examines the combined effects of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductance, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the electrical conductivity of unsteady flow in tangent hyperbolic ternary hybrid nanofluids. These nanofluids consist of Al 2 O 3 , Cu and TiO 2 nanoparticles suspended in ethylene glycol, flowing over a stretching sheet. This study is significant as it combines theoretical analysis and practical application, providing valuable insights into the behavior of tangent hyperbolic ternary hybrid nanofluids under complex flow conditions. It has good number of applications in engineering and technology. It may be accepted subject to the following major modifications/corrections: 1. Title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu) and titanium oxide (TiO₂) nanoparticles? 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications? 6. Literature review is also to be enriched with recent articles.Add the below articles in your introduction: MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B 0 =B/(1−ct) 1/2 . Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. 8. Explanation related to boundary condition (9) needs more clarity. 9. Velocity plots with We and ϕ 1 show very less variations. Recheck and justify the reasons for these less variations. 10. What are the benchmarks of your study? 11. How are convergence and stability ensured in the computational analysis? 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? 13. Add novel results in the conclusion section. I have cited few papers because they are closely aligned with the author's manuscript. The cited papers provide a detailed explanation of the tangent hyperbolic fluid model, which is only partially covered in the author's paper. Additionally, the topics of MHD and porous media are extensively discussed in the cited works which offers valuable insights that complement the author's discussion. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Choudhary S, Choudhary P, Alessa N, Loganathan K: MHD Thermal and Solutal Stratified Stagnation Flow of Tangent Hyperbolic Fluid Induced by Stretching Cylinder with Dual Convection. Mathematics . 2023; 11 (9). Publisher Full Text 2. Choudhary P, Choudhary S, Jat K, Loganathan K, et al.: Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms. International Journal of Thermofluids . 2024; 23 . Publisher Full Text 3. Choudhary S, Choudhary P, Pattanaik B: Tangent Hyperbolic Fluid Flow under Condition of Divergent Channel in the Presence of Porous Medium with Suction/Blowing and Heat Source: Emergence of the Boundary Layer. International Journal of Mathematics and Mathematical Sciences . 2023; 2023 (1). Publisher Full Text 4. Choudhary P, Choudhary S, Jat K, Loganathan K, et al.: Significances of melting heat transfer and bioconvection phenomena in nanofluid flow over a three different geometries. International Journal of Thermofluids . 2024; 24 . Publisher Full Text 5. Choudhary P, Choudhary S, Jat K, Loganathan K, et al.: Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium. Journal of Thermal Analysis and Calorimetry . 2025. Publisher Full Text 6. Jat K, Sharma K, Makinde O: Novel features of radiating hybrid nanofluid flow past a nonlinear stretchable porous sheet with different nanoparticles shape. Numerical Heat Transfer, Part B: Fundamentals . 2024. 1-25 Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fluid Flow, Heat Transfer, Mass Transfer, Distinct Geometries, Analytical Methods, Numerical Methods We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Choudhary P and Jat DK. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366062 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366062 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 10 Mar 2025 Asfaw Tsegaye , Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia 10 Mar 2025 Author Response Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive ... Continue reading Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewers’ concerns and outline the corresponding revisions made in the manuscript. 1. The title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. Authors' Response: We have corrected the capitalization in the title as suggested. The revised title now reads: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂) nanoparticles? Authors' Response: The thermal conductivity and viscosity of ethylene glycol-based nanofluids are significantly influenced by the dispersion of nanoparticles such as Al₂O₃, Cu, and TiO₂. In particular, ternary hybrid nanofluids—those containing a combination of three different nanoparticles—exhibit unique thermophysical behaviors. The dispersion of Al₂O₃, Cu, and TiO₂ nanoparticles in ethylene glycol-based ternary hybrid nanofluids leads to enhanced thermal conductivity and increased viscosity. The extent of these changes depends on nanoparticle concentration and operating temperature. Viscosity tends to decrease with rising temperatures. Notably, while thermal conductivity improves with elevated temperatures, viscosity decreases, indicating better flow characteristics at higher temperatures. Balancing these factors is essential for optimizing the performance of such nanofluids in practical applications. While viscosity changes are not explicitly addressed, it is generally understood that nanoparticle addition can increase fluid viscosity, potentially impacting flow behavior. 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? Authors' Response: The governing equations exhibit nonlinearities due to factors such as the tangent hyperbolic fluid model, where the shear stress follows a hyperbolic tangent function, introducing nonlinearity in the momentum equation. Nonlinear thermal radiation further complicates the energy equation with temperature-dependent radiative heat flux, while variable thermal conductivity leads to additional nonlinear terms. Viscous dissipation and Joule heating contribute quadratic velocity and temperature-dependent effects, and the Darcy–Forchheimer drag forces in the porous medium introduce nonlinear velocity terms. These nonlinearities make the transformation of partial differential equations (PDEs) into ordinary differential equations (ODEs) via similarity transformations more complex. The resulting nonlinear ODEs require numerical solutions, such as the BVP5C algorithm in MATLAB, where stability, convergence, and accuracy depend on careful parameter tuning for computational efficiency. 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. Authors' Response: Ternary hybrid nanofluids enhance high-performance cooling systems by combining nanoparticles like Al₂O₃, Cu, and TiO₂, leading to superior thermal conductivity, improved stability, and controlled viscosity for efficient heat transfer. Their adaptability makes them ideal for applications in electronics cooling, automotive radiators, and industrial heat exchangers. By optimizing nanoparticle composition, properties like specific heat and viscosity can be tailored to meet cooling demands. However, these benefits come with challenges such as higher synthesis costs, increased viscosity that may raise pumping power requirements, and potential issues with particle agglomeration, sedimentation, and system compatibility. Additionally, their performance varies with temperature, shear rate, and composition, necessitating extensive testing and optimization. Despite these limitations, ternary hybrid nanofluids remain a promising solution for enhancing cooling efficiency, provided their formulation and application challenges are carefully managed. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications. Authors' Response: Ternary hybrid nanofluids hold significant potential in drug delivery applications due to their enhanced thermal and rheological properties. Their high thermal conductivity allows for precise temperature control, making them suitable for targeted hyperthermia treatments in cancer therapy. The improved stability and dispersion of multiple nanoparticles ensure better drug encapsulation and controlled release, enhancing bioavailability. Their adjustable viscosity enables efficient flow through micro capillaries, improving circulation and targeted delivery to specific tissues. Additionally, the presence of multiple nanoparticles (e.g., Al₂O₃, Cu, TiO₂) can enhance cellular uptake and bioactivity , increasing drug efficacy. However, challenges such as biocompatibility, toxicity, and long-term stability must be addressed to ensure their safe and effective use in biomedical applications. 6. The literature review should be enriched with recent articles. Authors' Response: We have incorporated the suggested references into the introduction and literature review sections, specifically discussing how they relate to our study. The following references have been added: - MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; - Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; -Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; - Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; - Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; - Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B0=B/(1−ct)1/2. Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. Authors' Response: We have addressed this concern in the mathematical formulation section by discussing Maxwell’s equations in the context of our study. This justification has been added to the revised manuscript. 8. Explanation related to boundary condition (9) needs more clarity. Authors' Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 9. Velocity plots with We and ϕ1 show very less variations. Recheck and justify the reasons for these less variations. Authors' Response: The velocity plots with respect to the Weissenberg number ( We ) and nanoparticle volume fraction ( ϕ 1 ​) exhibit minimal variation. This limited influence can be attributed to the dominance of viscous effects over elasticity in the given flow regime, the relatively small effect of nanoparticle volume fraction on velocity, and the constraints imposed by the boundary conditions. If ϕ 1 ​ ​ is varied within a narrow range, the increase in viscosity may not be substantial enough to noticeably affect the velocity profile. 10. What are the benchmarks of your study? Authors' Response: The numerical results have been validated against previously published studies, as shown in Table 2. Our findings show strong agreement with benchmark solutions, confirming the accuracy of our computational approach. 11. How are convergence and stability ensured in the computational analysis? Authors' Response: We have explicitly discussed the convergence criteria of the BVP5C algorithm and verified that the residual errors remain within acceptable limits. The computational analysis employs BVP5C, a MATLAB solver for boundary value problems, ensuring convergence and stability through adaptive mesh refinement, in our study, error control (tolerance 10 -7 ), step size regulation, and a well-posed problem formulation. A grid independence test verifies numerical stability, ensuring accurate and reliable results while minimizing numerical artifacts. 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? Authors' Response: We have elaborated on the role of the magnetic field in modifying velocity and temperature profiles. The Lorentz force acts as a resistive force, reducing velocity and increasing temperature due to Joule heating. 13. Add novel results in the conclusion section. Authors' Response: We have updated the conclusion to explicitly highlight the novel contributions of this study, including: The interplay between ternary hybrid nanoparticles and non-Newtonian fluid properties. The effect of variable thermal conductivity on heat transfer performance. The impact of nonlinear radiation on temperature enhancement, etc. These additions strengthen the conclusion and emphasize the novelty of our findings. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewers’ concerns and outline the corresponding revisions made in the manuscript. 1. The title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. Authors' Response: We have corrected the capitalization in the title as suggested. The revised title now reads: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂) nanoparticles? Authors' Response: The thermal conductivity and viscosity of ethylene glycol-based nanofluids are significantly influenced by the dispersion of nanoparticles such as Al₂O₃, Cu, and TiO₂. In particular, ternary hybrid nanofluids—those containing a combination of three different nanoparticles—exhibit unique thermophysical behaviors. The dispersion of Al₂O₃, Cu, and TiO₂ nanoparticles in ethylene glycol-based ternary hybrid nanofluids leads to enhanced thermal conductivity and increased viscosity. The extent of these changes depends on nanoparticle concentration and operating temperature. Viscosity tends to decrease with rising temperatures. Notably, while thermal conductivity improves with elevated temperatures, viscosity decreases, indicating better flow characteristics at higher temperatures. Balancing these factors is essential for optimizing the performance of such nanofluids in practical applications. While viscosity changes are not explicitly addressed, it is generally understood that nanoparticle addition can increase fluid viscosity, potentially impacting flow behavior. 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? Authors' Response: The governing equations exhibit nonlinearities due to factors such as the tangent hyperbolic fluid model, where the shear stress follows a hyperbolic tangent function, introducing nonlinearity in the momentum equation. Nonlinear thermal radiation further complicates the energy equation with temperature-dependent radiative heat flux, while variable thermal conductivity leads to additional nonlinear terms. Viscous dissipation and Joule heating contribute quadratic velocity and temperature-dependent effects, and the Darcy–Forchheimer drag forces in the porous medium introduce nonlinear velocity terms. These nonlinearities make the transformation of partial differential equations (PDEs) into ordinary differential equations (ODEs) via similarity transformations more complex. The resulting nonlinear ODEs require numerical solutions, such as the BVP5C algorithm in MATLAB, where stability, convergence, and accuracy depend on careful parameter tuning for computational efficiency. 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. Authors' Response: Ternary hybrid nanofluids enhance high-performance cooling systems by combining nanoparticles like Al₂O₃, Cu, and TiO₂, leading to superior thermal conductivity, improved stability, and controlled viscosity for efficient heat transfer. Their adaptability makes them ideal for applications in electronics cooling, automotive radiators, and industrial heat exchangers. By optimizing nanoparticle composition, properties like specific heat and viscosity can be tailored to meet cooling demands. However, these benefits come with challenges such as higher synthesis costs, increased viscosity that may raise pumping power requirements, and potential issues with particle agglomeration, sedimentation, and system compatibility. Additionally, their performance varies with temperature, shear rate, and composition, necessitating extensive testing and optimization. Despite these limitations, ternary hybrid nanofluids remain a promising solution for enhancing cooling efficiency, provided their formulation and application challenges are carefully managed. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications. Authors' Response: Ternary hybrid nanofluids hold significant potential in drug delivery applications due to their enhanced thermal and rheological properties. Their high thermal conductivity allows for precise temperature control, making them suitable for targeted hyperthermia treatments in cancer therapy. The improved stability and dispersion of multiple nanoparticles ensure better drug encapsulation and controlled release, enhancing bioavailability. Their adjustable viscosity enables efficient flow through micro capillaries, improving circulation and targeted delivery to specific tissues. Additionally, the presence of multiple nanoparticles (e.g., Al₂O₃, Cu, TiO₂) can enhance cellular uptake and bioactivity , increasing drug efficacy. However, challenges such as biocompatibility, toxicity, and long-term stability must be addressed to ensure their safe and effective use in biomedical applications. 6. The literature review should be enriched with recent articles. Authors' Response: We have incorporated the suggested references into the introduction and literature review sections, specifically discussing how they relate to our study. The following references have been added: - MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; - Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; -Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; - Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; - Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; - Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B0=B/(1−ct)1/2. Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. Authors' Response: We have addressed this concern in the mathematical formulation section by discussing Maxwell’s equations in the context of our study. This justification has been added to the revised manuscript. 8. Explanation related to boundary condition (9) needs more clarity. Authors' Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 9. Velocity plots with We and ϕ1 show very less variations. Recheck and justify the reasons for these less variations. Authors' Response: The velocity plots with respect to the Weissenberg number ( We ) and nanoparticle volume fraction ( ϕ 1 ​) exhibit minimal variation. This limited influence can be attributed to the dominance of viscous effects over elasticity in the given flow regime, the relatively small effect of nanoparticle volume fraction on velocity, and the constraints imposed by the boundary conditions. If ϕ 1 ​ ​ is varied within a narrow range, the increase in viscosity may not be substantial enough to noticeably affect the velocity profile. 10. What are the benchmarks of your study? Authors' Response: The numerical results have been validated against previously published studies, as shown in Table 2. Our findings show strong agreement with benchmark solutions, confirming the accuracy of our computational approach. 11. How are convergence and stability ensured in the computational analysis? Authors' Response: We have explicitly discussed the convergence criteria of the BVP5C algorithm and verified that the residual errors remain within acceptable limits. The computational analysis employs BVP5C, a MATLAB solver for boundary value problems, ensuring convergence and stability through adaptive mesh refinement, in our study, error control (tolerance 10 -7 ), step size regulation, and a well-posed problem formulation. A grid independence test verifies numerical stability, ensuring accurate and reliable results while minimizing numerical artifacts. 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? Authors' Response: We have elaborated on the role of the magnetic field in modifying velocity and temperature profiles. The Lorentz force acts as a resistive force, reducing velocity and increasing temperature due to Joule heating. 13. Add novel results in the conclusion section. Authors' Response: We have updated the conclusion to explicitly highlight the novel contributions of this study, including: The interplay between ternary hybrid nanoparticles and non-Newtonian fluid properties. The effect of variable thermal conductivity on heat transfer performance. The impact of nonlinear radiation on temperature enhancement, etc. These additions strengthen the conclusion and emphasize the novelty of our findings. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 10 Mar 2025 Asfaw Tsegaye , Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia 10 Mar 2025 Author Response Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive ... Continue reading Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewers’ concerns and outline the corresponding revisions made in the manuscript. 1. The title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. Authors' Response: We have corrected the capitalization in the title as suggested. The revised title now reads: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂) nanoparticles? Authors' Response: The thermal conductivity and viscosity of ethylene glycol-based nanofluids are significantly influenced by the dispersion of nanoparticles such as Al₂O₃, Cu, and TiO₂. In particular, ternary hybrid nanofluids—those containing a combination of three different nanoparticles—exhibit unique thermophysical behaviors. The dispersion of Al₂O₃, Cu, and TiO₂ nanoparticles in ethylene glycol-based ternary hybrid nanofluids leads to enhanced thermal conductivity and increased viscosity. The extent of these changes depends on nanoparticle concentration and operating temperature. Viscosity tends to decrease with rising temperatures. Notably, while thermal conductivity improves with elevated temperatures, viscosity decreases, indicating better flow characteristics at higher temperatures. Balancing these factors is essential for optimizing the performance of such nanofluids in practical applications. While viscosity changes are not explicitly addressed, it is generally understood that nanoparticle addition can increase fluid viscosity, potentially impacting flow behavior. 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? Authors' Response: The governing equations exhibit nonlinearities due to factors such as the tangent hyperbolic fluid model, where the shear stress follows a hyperbolic tangent function, introducing nonlinearity in the momentum equation. Nonlinear thermal radiation further complicates the energy equation with temperature-dependent radiative heat flux, while variable thermal conductivity leads to additional nonlinear terms. Viscous dissipation and Joule heating contribute quadratic velocity and temperature-dependent effects, and the Darcy–Forchheimer drag forces in the porous medium introduce nonlinear velocity terms. These nonlinearities make the transformation of partial differential equations (PDEs) into ordinary differential equations (ODEs) via similarity transformations more complex. The resulting nonlinear ODEs require numerical solutions, such as the BVP5C algorithm in MATLAB, where stability, convergence, and accuracy depend on careful parameter tuning for computational efficiency. 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. Authors' Response: Ternary hybrid nanofluids enhance high-performance cooling systems by combining nanoparticles like Al₂O₃, Cu, and TiO₂, leading to superior thermal conductivity, improved stability, and controlled viscosity for efficient heat transfer. Their adaptability makes them ideal for applications in electronics cooling, automotive radiators, and industrial heat exchangers. By optimizing nanoparticle composition, properties like specific heat and viscosity can be tailored to meet cooling demands. However, these benefits come with challenges such as higher synthesis costs, increased viscosity that may raise pumping power requirements, and potential issues with particle agglomeration, sedimentation, and system compatibility. Additionally, their performance varies with temperature, shear rate, and composition, necessitating extensive testing and optimization. Despite these limitations, ternary hybrid nanofluids remain a promising solution for enhancing cooling efficiency, provided their formulation and application challenges are carefully managed. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications. Authors' Response: Ternary hybrid nanofluids hold significant potential in drug delivery applications due to their enhanced thermal and rheological properties. Their high thermal conductivity allows for precise temperature control, making them suitable for targeted hyperthermia treatments in cancer therapy. The improved stability and dispersion of multiple nanoparticles ensure better drug encapsulation and controlled release, enhancing bioavailability. Their adjustable viscosity enables efficient flow through micro capillaries, improving circulation and targeted delivery to specific tissues. Additionally, the presence of multiple nanoparticles (e.g., Al₂O₃, Cu, TiO₂) can enhance cellular uptake and bioactivity , increasing drug efficacy. However, challenges such as biocompatibility, toxicity, and long-term stability must be addressed to ensure their safe and effective use in biomedical applications. 6. The literature review should be enriched with recent articles. Authors' Response: We have incorporated the suggested references into the introduction and literature review sections, specifically discussing how they relate to our study. The following references have been added: - MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; - Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; -Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; - Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; - Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; - Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B0=B/(1−ct)1/2. Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. Authors' Response: We have addressed this concern in the mathematical formulation section by discussing Maxwell’s equations in the context of our study. This justification has been added to the revised manuscript. 8. Explanation related to boundary condition (9) needs more clarity. Authors' Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 9. Velocity plots with We and ϕ1 show very less variations. Recheck and justify the reasons for these less variations. Authors' Response: The velocity plots with respect to the Weissenberg number ( We ) and nanoparticle volume fraction ( ϕ 1 ​) exhibit minimal variation. This limited influence can be attributed to the dominance of viscous effects over elasticity in the given flow regime, the relatively small effect of nanoparticle volume fraction on velocity, and the constraints imposed by the boundary conditions. If ϕ 1 ​ ​ is varied within a narrow range, the increase in viscosity may not be substantial enough to noticeably affect the velocity profile. 10. What are the benchmarks of your study? Authors' Response: The numerical results have been validated against previously published studies, as shown in Table 2. Our findings show strong agreement with benchmark solutions, confirming the accuracy of our computational approach. 11. How are convergence and stability ensured in the computational analysis? Authors' Response: We have explicitly discussed the convergence criteria of the BVP5C algorithm and verified that the residual errors remain within acceptable limits. The computational analysis employs BVP5C, a MATLAB solver for boundary value problems, ensuring convergence and stability through adaptive mesh refinement, in our study, error control (tolerance 10 -7 ), step size regulation, and a well-posed problem formulation. A grid independence test verifies numerical stability, ensuring accurate and reliable results while minimizing numerical artifacts. 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? Authors' Response: We have elaborated on the role of the magnetic field in modifying velocity and temperature profiles. The Lorentz force acts as a resistive force, reducing velocity and increasing temperature due to Joule heating. 13. Add novel results in the conclusion section. Authors' Response: We have updated the conclusion to explicitly highlight the novel contributions of this study, including: The interplay between ternary hybrid nanoparticles and non-Newtonian fluid properties. The effect of variable thermal conductivity on heat transfer performance. The impact of nonlinear radiation on temperature enhancement, etc. These additions strengthen the conclusion and emphasize the novelty of our findings. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewers’ concerns and outline the corresponding revisions made in the manuscript. 1. The title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. Authors' Response: We have corrected the capitalization in the title as suggested. The revised title now reads: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂) nanoparticles? Authors' Response: The thermal conductivity and viscosity of ethylene glycol-based nanofluids are significantly influenced by the dispersion of nanoparticles such as Al₂O₃, Cu, and TiO₂. In particular, ternary hybrid nanofluids—those containing a combination of three different nanoparticles—exhibit unique thermophysical behaviors. The dispersion of Al₂O₃, Cu, and TiO₂ nanoparticles in ethylene glycol-based ternary hybrid nanofluids leads to enhanced thermal conductivity and increased viscosity. The extent of these changes depends on nanoparticle concentration and operating temperature. Viscosity tends to decrease with rising temperatures. Notably, while thermal conductivity improves with elevated temperatures, viscosity decreases, indicating better flow characteristics at higher temperatures. Balancing these factors is essential for optimizing the performance of such nanofluids in practical applications. While viscosity changes are not explicitly addressed, it is generally understood that nanoparticle addition can increase fluid viscosity, potentially impacting flow behavior. 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? Authors' Response: The governing equations exhibit nonlinearities due to factors such as the tangent hyperbolic fluid model, where the shear stress follows a hyperbolic tangent function, introducing nonlinearity in the momentum equation. Nonlinear thermal radiation further complicates the energy equation with temperature-dependent radiative heat flux, while variable thermal conductivity leads to additional nonlinear terms. Viscous dissipation and Joule heating contribute quadratic velocity and temperature-dependent effects, and the Darcy–Forchheimer drag forces in the porous medium introduce nonlinear velocity terms. These nonlinearities make the transformation of partial differential equations (PDEs) into ordinary differential equations (ODEs) via similarity transformations more complex. The resulting nonlinear ODEs require numerical solutions, such as the BVP5C algorithm in MATLAB, where stability, convergence, and accuracy depend on careful parameter tuning for computational efficiency. 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. Authors' Response: Ternary hybrid nanofluids enhance high-performance cooling systems by combining nanoparticles like Al₂O₃, Cu, and TiO₂, leading to superior thermal conductivity, improved stability, and controlled viscosity for efficient heat transfer. Their adaptability makes them ideal for applications in electronics cooling, automotive radiators, and industrial heat exchangers. By optimizing nanoparticle composition, properties like specific heat and viscosity can be tailored to meet cooling demands. However, these benefits come with challenges such as higher synthesis costs, increased viscosity that may raise pumping power requirements, and potential issues with particle agglomeration, sedimentation, and system compatibility. Additionally, their performance varies with temperature, shear rate, and composition, necessitating extensive testing and optimization. Despite these limitations, ternary hybrid nanofluids remain a promising solution for enhancing cooling efficiency, provided their formulation and application challenges are carefully managed. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications. Authors' Response: Ternary hybrid nanofluids hold significant potential in drug delivery applications due to their enhanced thermal and rheological properties. Their high thermal conductivity allows for precise temperature control, making them suitable for targeted hyperthermia treatments in cancer therapy. The improved stability and dispersion of multiple nanoparticles ensure better drug encapsulation and controlled release, enhancing bioavailability. Their adjustable viscosity enables efficient flow through micro capillaries, improving circulation and targeted delivery to specific tissues. Additionally, the presence of multiple nanoparticles (e.g., Al₂O₃, Cu, TiO₂) can enhance cellular uptake and bioactivity , increasing drug efficacy. However, challenges such as biocompatibility, toxicity, and long-term stability must be addressed to ensure their safe and effective use in biomedical applications. 6. The literature review should be enriched with recent articles. Authors' Response: We have incorporated the suggested references into the introduction and literature review sections, specifically discussing how they relate to our study. The following references have been added: - MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; - Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; -Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; - Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; - Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; - Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B0=B/(1−ct)1/2. Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. Authors' Response: We have addressed this concern in the mathematical formulation section by discussing Maxwell’s equations in the context of our study. This justification has been added to the revised manuscript. 8. Explanation related to boundary condition (9) needs more clarity. Authors' Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 9. Velocity plots with We and ϕ1 show very less variations. Recheck and justify the reasons for these less variations. Authors' Response: The velocity plots with respect to the Weissenberg number ( We ) and nanoparticle volume fraction ( ϕ 1 ​) exhibit minimal variation. This limited influence can be attributed to the dominance of viscous effects over elasticity in the given flow regime, the relatively small effect of nanoparticle volume fraction on velocity, and the constraints imposed by the boundary conditions. If ϕ 1 ​ ​ is varied within a narrow range, the increase in viscosity may not be substantial enough to noticeably affect the velocity profile. 10. What are the benchmarks of your study? Authors' Response: The numerical results have been validated against previously published studies, as shown in Table 2. Our findings show strong agreement with benchmark solutions, confirming the accuracy of our computational approach. 11. How are convergence and stability ensured in the computational analysis? Authors' Response: We have explicitly discussed the convergence criteria of the BVP5C algorithm and verified that the residual errors remain within acceptable limits. The computational analysis employs BVP5C, a MATLAB solver for boundary value problems, ensuring convergence and stability through adaptive mesh refinement, in our study, error control (tolerance 10 -7 ), step size regulation, and a well-posed problem formulation. A grid independence test verifies numerical stability, ensuring accurate and reliable results while minimizing numerical artifacts. 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? Authors' Response: We have elaborated on the role of the magnetic field in modifying velocity and temperature profiles. The Lorentz force acts as a resistive force, reducing velocity and increasing temperature due to Joule heating. 13. Add novel results in the conclusion section. Authors' Response: We have updated the conclusion to explicitly highlight the novel contributions of this study, including: The interplay between ternary hybrid nanoparticles and non-Newtonian fluid properties. The effect of variable thermal conductivity on heat transfer performance. The impact of nonlinear radiation on temperature enhancement, etc. These additions strengthen the conclusion and emphasize the novelty of our findings. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Anjum A. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366061 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366061 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 18 Feb 2025 Asra Anjum , University of Technology and Applied Sciences, Salalah, Oman Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.174254.r366061 Title: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal c onductivity The paper has made a promising contribution and fresh results. However, the article ... Continue reading READ ALL Title: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal c onductivity The paper has made a promising contribution and fresh results. However, the article needs modifications, and I'd like to ask the authors to revise it due to the following comments, which can be accepted. Review comments; 1. Please briefly state more about salient findings in the abstract. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: 5. The author needs to mention briefly the application of the present work in the real world. 6. Why are such boundary conditions used? What is the physical meaning? 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. 9. Write the advantages of the used numerical methods in detail. 10. The author should give a deep analysis of the results shown in the figures. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? 16 . You should describe the abbreviations while using for first time. 17 . You should add major findings in abstract. 18 . Write the references in correct way. There are some recent references which is important for the manuscript: 19 . Explain the physical significance of boundary conditions. 20. You should add a figure for validation of your code. 21. You should rewrite the first paragraph of Conclusions section. 22 . There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. 23 . What is the impact of this article? What are the future directions? Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: fluid dynamics, nanofluids and hybrid nanofluids I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Anjum A. Reviewer Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366061 ) The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366061 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Author Response 10 Mar 2025 Asfaw Tsegaye , Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia 10 Mar 2025 Author Response ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful ... Continue reading ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewer’s concerns and outline the corresponding revisions made in the manuscript. 1. Please briefly state more about salient findings in the abstract. Authors' reply: The abstract has been revised to explicitly highlight key findings, including: - The velocity profile decreases with increasing Weissenberg and Forchheimer numbers. - Ternary hybrid nanofluids exhibit superior heat transfer enhancement compared to mono/hybrid nanofluids. - Variable thermal conductivity (Γ) and nanoparticle volume fractions significantly improve temperature distribution and skin friction. - Applications in cooling systems, biomedical drug delivery, and polymer processing are emphasized. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. Authors' reply: The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. Authors' reply: A new paragraph was added to the introduction (last paragraph) to clarify the novelty: "To the authors' knowledge, no prior study combines viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, and ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂/EG) under unsteady MHD conditions. This work addresses the gap by analyzing synergistic effects of 12+ parameters, offering insights into optimizing thermal systems." 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: Authors' reply: We have not yet received your suggested references. However, we have added recent studies (2023–2024) on ternary hybrid nanofluids and tangent hyperbolic fluids. Additionally, using your Google Scholar profile, we have incorporated some of your references into the introduction and literature review sections, highlighting their relevance to our study. The following references have been added: - Computation Of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport From An Inclined Plane With Ion Slip And Hall Current Effects - Influence of Darcy-Forchheimer hybrid nanofluid flow with magnetohydrodynamic and viscous dissipation effects past a permeable stretching sheet: a numerical contribution 5. The author needs to mention briefly the application of the present work in the real world. Authors' reply: The study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity" investigates the unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer of ternary hybrid nanofluids over a permeable stretching sheet. The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. 6. Why are such boundary conditions used? What is the physical meaning? Authors' reply: In the study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity," the authors employ specific boundary conditions to accurately model the physical phenomena under investigation. These conditions are essential for representing real-world scenarios and ensuring the validity of the mathematical model. These boundary conditions are crucial for accurately modeling the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. They ensure that the mathematical model reflects the physical realities of the system, including the interaction between the fluid and the surface, the development of boundary layers, and the transition to ambient conditions at a distance from the surface. 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). Authors' reply: In our study , the combined effects of several factors on the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet. Specifically, the research examines the influence of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the velocity profile, temperature distribution, concentration profile, skin friction, Nusselt number, and Sherwood number. In this study, correlation were not suggested. We considering the further direction. 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. Authors' reply: The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 9. Write the advantages of the used numerical methods in detail. Authors' reply: BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. The BVP5C solver provides high accuracy for stiff boundary value problems, leveraging adaptive mesh refinement and efficiently handling nonlinearities. Compared to finite difference methods, it offers faster convergence with minimal dependence on grid size. 10. The author should give a deep analysis of the results shown in the figures. Authors' reply: The detailed analysis of the figures in this study provides a comprehensive understanding of how various parameters influence the flow, heat, and mass transfer characteristics of ternary hybrid nanofluids. These insights are valuable for optimizing the performance of nanofluids in applications such as cooling systems, heat exchangers, and polymer processing. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. Authors' reply: It is modified in the conclusion section in the revised manuscript. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? Authors' reply: This study provides valuable theoretical insights but has some limitations. Simplified assumptions, such as treating the nanofluid as a single-phase fluid and neglecting higher-order effects, may not fully capture real-world complexities. The BVP5C numerical method, while effective, has constraints in handling highly nonlinear systems. The findings are based on a specific parameter range, and extrapolation beyond this range requires further validation. Additionally, the assumption of uniform nanoparticle dispersion overlooks possible aggregation or sedimentation effects. To enhance validity and generalizability, future research should incorporate experimental validation and more advanced modeling approaches. 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? Authors' reply: The study demonstrates that the key physical parameters—especially the nanoparticle concentration, Weissenberg number, Forchheimer number, and variable thermal conductivity—have significant effects on the heat and mass transfer characteristics. These parameters enhance the Nusselt number, Sherwood number, and skin friction, contributing to the improved performance of ternary hybrid nanofluids in heat exchange and mass transfer applications. We have addressed this in the results and conclusion sections. Please refer to page 26. 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? Authors' reply: The mathematical framework developed for unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity provides valuable insights into nanofluid behavior under various conditions. It has potential applications in cooling systems , optimizing thermal management in electronics, power plants, and automotive systems. In biomedical applications , it aids in understanding nanofluid flow for targeted drug delivery and hyperthermia treatments. The framework also supports the design of heat exchangers , enhancing industrial heat transfer efficiency. Additionally, it improves polymer processing by predicting material flow and temperature control in extrusion and casting. In energy systems , it contributes to advanced solar thermal collectors and photovoltaic/thermal (PV/T) systems, promoting sustainable energy solutions. By leveraging this model, researchers can enhance the efficiency of various thermal and industrial applications. 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? Authors' reply: This study introduces several novel aspects in investigating unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. It uniquely integrates multiple effects, including viscous dissipation, Darcy-Forchheimer flow, nonlinear thermal radiation, Joule heating, and thermophoresis, providing a comprehensive understanding of nanofluid behavior. Unlike previous studies on binary hybrid nanofluids, this research explores ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂ in ethylene glycol), offering superior thermal properties. The study’s focus on permeable stretching sheets enhances its relevance for real-world applications such as cooling systems and polymer processing. By incorporating variable thermal conductivity , it improves heat transfer accuracy, while an extensive parametric analysis further clarifies the effects of key influencing factors. These contributions make the study a significant advancement in nanofluid convection and heat transfer research. 16. You should describe the abbreviations while using for first time. Authors' reply: It is modified in the revised manuscript. 17. You should add major findings in abstract. Authors' reply: We incorporate the major finding in the abstract. 18. Write the references in correct way. There are some recent references which is important for the manuscript: Authors' reply: Modified 19. Explain the physical significance of boundary conditions. Authors reply: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→ ∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 20. You should add a figure for validation of your code. Authors' reply: This study provides sufficient validation through comprehensive comparisons with existing literature and numerical benchmarks . 21. You should rewrite the first paragraph of conclusions section. Authors' reply: Modified in the revised manuscript. 22. There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. Authors' reply: Corrected 23. What is the impact of this article? What are the future directions? Authors' reply: This study provides a detailed analysis of unsteady MHD flow in ternary hybrid nanofluids ( Al₂O₃–Cu–TiO₂ in ethylene glycol ) over a permeable stretching sheet with variable thermal conductivity. The findings have significant applications in cooling systems , improving heat transfer efficiency, biomedical applications , aiding in drug delivery and hyperthermia treatments, and polymer processing , optimizing thermal conditions in extrusion and casting. Future research should focus on experimental validation to confirm numerical results, optimization studies to refine nanoparticle concentrations for maximum efficiency, and long-term stability analysis to ensure reliability in industrial applications. These advancements will enhance the practical utility of ternary hybrid nanofluids across various engineering and technological fields. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Email: [email protected] ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewer’s concerns and outline the corresponding revisions made in the manuscript. 1. Please briefly state more about salient findings in the abstract. Authors' reply: The abstract has been revised to explicitly highlight key findings, including: - The velocity profile decreases with increasing Weissenberg and Forchheimer numbers. - Ternary hybrid nanofluids exhibit superior heat transfer enhancement compared to mono/hybrid nanofluids. - Variable thermal conductivity (Γ) and nanoparticle volume fractions significantly improve temperature distribution and skin friction. - Applications in cooling systems, biomedical drug delivery, and polymer processing are emphasized. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. Authors' reply: The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. Authors' reply: A new paragraph was added to the introduction (last paragraph) to clarify the novelty: "To the authors' knowledge, no prior study combines viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, and ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂/EG) under unsteady MHD conditions. This work addresses the gap by analyzing synergistic effects of 12+ parameters, offering insights into optimizing thermal systems." 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: Authors' reply: We have not yet received your suggested references. However, we have added recent studies (2023–2024) on ternary hybrid nanofluids and tangent hyperbolic fluids. Additionally, using your Google Scholar profile, we have incorporated some of your references into the introduction and literature review sections, highlighting their relevance to our study. The following references have been added: - Computation Of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport From An Inclined Plane With Ion Slip And Hall Current Effects - Influence of Darcy-Forchheimer hybrid nanofluid flow with magnetohydrodynamic and viscous dissipation effects past a permeable stretching sheet: a numerical contribution 5. The author needs to mention briefly the application of the present work in the real world. Authors' reply: The study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity" investigates the unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer of ternary hybrid nanofluids over a permeable stretching sheet. The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. 6. Why are such boundary conditions used? What is the physical meaning? Authors' reply: In the study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity," the authors employ specific boundary conditions to accurately model the physical phenomena under investigation. These conditions are essential for representing real-world scenarios and ensuring the validity of the mathematical model. These boundary conditions are crucial for accurately modeling the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. They ensure that the mathematical model reflects the physical realities of the system, including the interaction between the fluid and the surface, the development of boundary layers, and the transition to ambient conditions at a distance from the surface. 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). Authors' reply: In our study , the combined effects of several factors on the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet. Specifically, the research examines the influence of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the velocity profile, temperature distribution, concentration profile, skin friction, Nusselt number, and Sherwood number. In this study, correlation were not suggested. We considering the further direction. 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. Authors' reply: The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 9. Write the advantages of the used numerical methods in detail. Authors' reply: BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. The BVP5C solver provides high accuracy for stiff boundary value problems, leveraging adaptive mesh refinement and efficiently handling nonlinearities. Compared to finite difference methods, it offers faster convergence with minimal dependence on grid size. 10. The author should give a deep analysis of the results shown in the figures. Authors' reply: The detailed analysis of the figures in this study provides a comprehensive understanding of how various parameters influence the flow, heat, and mass transfer characteristics of ternary hybrid nanofluids. These insights are valuable for optimizing the performance of nanofluids in applications such as cooling systems, heat exchangers, and polymer processing. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. Authors' reply: It is modified in the conclusion section in the revised manuscript. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? Authors' reply: This study provides valuable theoretical insights but has some limitations. Simplified assumptions, such as treating the nanofluid as a single-phase fluid and neglecting higher-order effects, may not fully capture real-world complexities. The BVP5C numerical method, while effective, has constraints in handling highly nonlinear systems. The findings are based on a specific parameter range, and extrapolation beyond this range requires further validation. Additionally, the assumption of uniform nanoparticle dispersion overlooks possible aggregation or sedimentation effects. To enhance validity and generalizability, future research should incorporate experimental validation and more advanced modeling approaches. 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? Authors' reply: The study demonstrates that the key physical parameters—especially the nanoparticle concentration, Weissenberg number, Forchheimer number, and variable thermal conductivity—have significant effects on the heat and mass transfer characteristics. These parameters enhance the Nusselt number, Sherwood number, and skin friction, contributing to the improved performance of ternary hybrid nanofluids in heat exchange and mass transfer applications. We have addressed this in the results and conclusion sections. Please refer to page 26. 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? Authors' reply: The mathematical framework developed for unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity provides valuable insights into nanofluid behavior under various conditions. It has potential applications in cooling systems , optimizing thermal management in electronics, power plants, and automotive systems. In biomedical applications , it aids in understanding nanofluid flow for targeted drug delivery and hyperthermia treatments. The framework also supports the design of heat exchangers , enhancing industrial heat transfer efficiency. Additionally, it improves polymer processing by predicting material flow and temperature control in extrusion and casting. In energy systems , it contributes to advanced solar thermal collectors and photovoltaic/thermal (PV/T) systems, promoting sustainable energy solutions. By leveraging this model, researchers can enhance the efficiency of various thermal and industrial applications. 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? Authors' reply: This study introduces several novel aspects in investigating unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. It uniquely integrates multiple effects, including viscous dissipation, Darcy-Forchheimer flow, nonlinear thermal radiation, Joule heating, and thermophoresis, providing a comprehensive understanding of nanofluid behavior. Unlike previous studies on binary hybrid nanofluids, this research explores ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂ in ethylene glycol), offering superior thermal properties. The study’s focus on permeable stretching sheets enhances its relevance for real-world applications such as cooling systems and polymer processing. By incorporating variable thermal conductivity , it improves heat transfer accuracy, while an extensive parametric analysis further clarifies the effects of key influencing factors. These contributions make the study a significant advancement in nanofluid convection and heat transfer research. 16. You should describe the abbreviations while using for first time. Authors' reply: It is modified in the revised manuscript. 17. You should add major findings in abstract. Authors' reply: We incorporate the major finding in the abstract. 18. Write the references in correct way. There are some recent references which is important for the manuscript: Authors' reply: Modified 19. Explain the physical significance of boundary conditions. Authors reply: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→ ∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 20. You should add a figure for validation of your code. Authors' reply: This study provides sufficient validation through comprehensive comparisons with existing literature and numerical benchmarks . 21. You should rewrite the first paragraph of conclusions section. Authors' reply: Modified in the revised manuscript. 22. There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. Authors' reply: Corrected 23. What is the impact of this article? What are the future directions? Authors' reply: This study provides a detailed analysis of unsteady MHD flow in ternary hybrid nanofluids ( Al₂O₃–Cu–TiO₂ in ethylene glycol ) over a permeable stretching sheet with variable thermal conductivity. The findings have significant applications in cooling systems , improving heat transfer efficiency, biomedical applications , aiding in drug delivery and hyperthermia treatments, and polymer processing , optimizing thermal conditions in extrusion and casting. Future research should focus on experimental validation to confirm numerical results, optimization studies to refine nanoparticle concentrations for maximum efficiency, and long-term stability analysis to ensure reliability in industrial applications. These advancements will enhance the practical utility of ternary hybrid nanofluids across various engineering and technological fields. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Email: [email protected] Competing Interests: No competing interests were disclosed. Close Report a concern Respond or Comment COMMENTS ON THIS REPORT Author Response 10 Mar 2025 Asfaw Tsegaye , Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia 10 Mar 2025 Author Response ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful ... Continue reading ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewer’s concerns and outline the corresponding revisions made in the manuscript. 1. Please briefly state more about salient findings in the abstract. Authors' reply: The abstract has been revised to explicitly highlight key findings, including: - The velocity profile decreases with increasing Weissenberg and Forchheimer numbers. - Ternary hybrid nanofluids exhibit superior heat transfer enhancement compared to mono/hybrid nanofluids. - Variable thermal conductivity (Γ) and nanoparticle volume fractions significantly improve temperature distribution and skin friction. - Applications in cooling systems, biomedical drug delivery, and polymer processing are emphasized. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. Authors' reply: The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. Authors' reply: A new paragraph was added to the introduction (last paragraph) to clarify the novelty: "To the authors' knowledge, no prior study combines viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, and ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂/EG) under unsteady MHD conditions. This work addresses the gap by analyzing synergistic effects of 12+ parameters, offering insights into optimizing thermal systems." 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: Authors' reply: We have not yet received your suggested references. However, we have added recent studies (2023–2024) on ternary hybrid nanofluids and tangent hyperbolic fluids. Additionally, using your Google Scholar profile, we have incorporated some of your references into the introduction and literature review sections, highlighting their relevance to our study. The following references have been added: - Computation Of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport From An Inclined Plane With Ion Slip And Hall Current Effects - Influence of Darcy-Forchheimer hybrid nanofluid flow with magnetohydrodynamic and viscous dissipation effects past a permeable stretching sheet: a numerical contribution 5. The author needs to mention briefly the application of the present work in the real world. Authors' reply: The study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity" investigates the unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer of ternary hybrid nanofluids over a permeable stretching sheet. The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. 6. Why are such boundary conditions used? What is the physical meaning? Authors' reply: In the study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity," the authors employ specific boundary conditions to accurately model the physical phenomena under investigation. These conditions are essential for representing real-world scenarios and ensuring the validity of the mathematical model. These boundary conditions are crucial for accurately modeling the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. They ensure that the mathematical model reflects the physical realities of the system, including the interaction between the fluid and the surface, the development of boundary layers, and the transition to ambient conditions at a distance from the surface. 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). Authors' reply: In our study , the combined effects of several factors on the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet. Specifically, the research examines the influence of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the velocity profile, temperature distribution, concentration profile, skin friction, Nusselt number, and Sherwood number. In this study, correlation were not suggested. We considering the further direction. 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. Authors' reply: The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 9. Write the advantages of the used numerical methods in detail. Authors' reply: BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. The BVP5C solver provides high accuracy for stiff boundary value problems, leveraging adaptive mesh refinement and efficiently handling nonlinearities. Compared to finite difference methods, it offers faster convergence with minimal dependence on grid size. 10. The author should give a deep analysis of the results shown in the figures. Authors' reply: The detailed analysis of the figures in this study provides a comprehensive understanding of how various parameters influence the flow, heat, and mass transfer characteristics of ternary hybrid nanofluids. These insights are valuable for optimizing the performance of nanofluids in applications such as cooling systems, heat exchangers, and polymer processing. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. Authors' reply: It is modified in the conclusion section in the revised manuscript. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? Authors' reply: This study provides valuable theoretical insights but has some limitations. Simplified assumptions, such as treating the nanofluid as a single-phase fluid and neglecting higher-order effects, may not fully capture real-world complexities. The BVP5C numerical method, while effective, has constraints in handling highly nonlinear systems. The findings are based on a specific parameter range, and extrapolation beyond this range requires further validation. Additionally, the assumption of uniform nanoparticle dispersion overlooks possible aggregation or sedimentation effects. To enhance validity and generalizability, future research should incorporate experimental validation and more advanced modeling approaches. 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? Authors' reply: The study demonstrates that the key physical parameters—especially the nanoparticle concentration, Weissenberg number, Forchheimer number, and variable thermal conductivity—have significant effects on the heat and mass transfer characteristics. These parameters enhance the Nusselt number, Sherwood number, and skin friction, contributing to the improved performance of ternary hybrid nanofluids in heat exchange and mass transfer applications. We have addressed this in the results and conclusion sections. Please refer to page 26. 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? Authors' reply: The mathematical framework developed for unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity provides valuable insights into nanofluid behavior under various conditions. It has potential applications in cooling systems , optimizing thermal management in electronics, power plants, and automotive systems. In biomedical applications , it aids in understanding nanofluid flow for targeted drug delivery and hyperthermia treatments. The framework also supports the design of heat exchangers , enhancing industrial heat transfer efficiency. Additionally, it improves polymer processing by predicting material flow and temperature control in extrusion and casting. In energy systems , it contributes to advanced solar thermal collectors and photovoltaic/thermal (PV/T) systems, promoting sustainable energy solutions. By leveraging this model, researchers can enhance the efficiency of various thermal and industrial applications. 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? Authors' reply: This study introduces several novel aspects in investigating unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. It uniquely integrates multiple effects, including viscous dissipation, Darcy-Forchheimer flow, nonlinear thermal radiation, Joule heating, and thermophoresis, providing a comprehensive understanding of nanofluid behavior. Unlike previous studies on binary hybrid nanofluids, this research explores ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂ in ethylene glycol), offering superior thermal properties. The study’s focus on permeable stretching sheets enhances its relevance for real-world applications such as cooling systems and polymer processing. By incorporating variable thermal conductivity , it improves heat transfer accuracy, while an extensive parametric analysis further clarifies the effects of key influencing factors. These contributions make the study a significant advancement in nanofluid convection and heat transfer research. 16. You should describe the abbreviations while using for first time. Authors' reply: It is modified in the revised manuscript. 17. You should add major findings in abstract. Authors' reply: We incorporate the major finding in the abstract. 18. Write the references in correct way. There are some recent references which is important for the manuscript: Authors' reply: Modified 19. Explain the physical significance of boundary conditions. Authors reply: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→ ∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 20. You should add a figure for validation of your code. Authors' reply: This study provides sufficient validation through comprehensive comparisons with existing literature and numerical benchmarks . 21. You should rewrite the first paragraph of conclusions section. Authors' reply: Modified in the revised manuscript. 22. There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. Authors' reply: Corrected 23. What is the impact of this article? What are the future directions? Authors' reply: This study provides a detailed analysis of unsteady MHD flow in ternary hybrid nanofluids ( Al₂O₃–Cu–TiO₂ in ethylene glycol ) over a permeable stretching sheet with variable thermal conductivity. The findings have significant applications in cooling systems , improving heat transfer efficiency, biomedical applications , aiding in drug delivery and hyperthermia treatments, and polymer processing , optimizing thermal conditions in extrusion and casting. Future research should focus on experimental validation to confirm numerical results, optimization studies to refine nanoparticle concentrations for maximum efficiency, and long-term stability analysis to ensure reliability in industrial applications. These advancements will enhance the practical utility of ternary hybrid nanofluids across various engineering and technological fields. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Email: [email protected] ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewer’s concerns and outline the corresponding revisions made in the manuscript. 1. Please briefly state more about salient findings in the abstract. Authors' reply: The abstract has been revised to explicitly highlight key findings, including: - The velocity profile decreases with increasing Weissenberg and Forchheimer numbers. - Ternary hybrid nanofluids exhibit superior heat transfer enhancement compared to mono/hybrid nanofluids. - Variable thermal conductivity (Γ) and nanoparticle volume fractions significantly improve temperature distribution and skin friction. - Applications in cooling systems, biomedical drug delivery, and polymer processing are emphasized. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. Authors' reply: The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. Authors' reply: A new paragraph was added to the introduction (last paragraph) to clarify the novelty: "To the authors' knowledge, no prior study combines viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, and ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂/EG) under unsteady MHD conditions. This work addresses the gap by analyzing synergistic effects of 12+ parameters, offering insights into optimizing thermal systems." 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: Authors' reply: We have not yet received your suggested references. However, we have added recent studies (2023–2024) on ternary hybrid nanofluids and tangent hyperbolic fluids. Additionally, using your Google Scholar profile, we have incorporated some of your references into the introduction and literature review sections, highlighting their relevance to our study. The following references have been added: - Computation Of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport From An Inclined Plane With Ion Slip And Hall Current Effects - Influence of Darcy-Forchheimer hybrid nanofluid flow with magnetohydrodynamic and viscous dissipation effects past a permeable stretching sheet: a numerical contribution 5. The author needs to mention briefly the application of the present work in the real world. Authors' reply: The study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity" investigates the unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer of ternary hybrid nanofluids over a permeable stretching sheet. The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. 6. Why are such boundary conditions used? What is the physical meaning? Authors' reply: In the study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity," the authors employ specific boundary conditions to accurately model the physical phenomena under investigation. These conditions are essential for representing real-world scenarios and ensuring the validity of the mathematical model. These boundary conditions are crucial for accurately modeling the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. They ensure that the mathematical model reflects the physical realities of the system, including the interaction between the fluid and the surface, the development of boundary layers, and the transition to ambient conditions at a distance from the surface. 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). Authors' reply: In our study , the combined effects of several factors on the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet. Specifically, the research examines the influence of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the velocity profile, temperature distribution, concentration profile, skin friction, Nusselt number, and Sherwood number. In this study, correlation were not suggested. We considering the further direction. 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. Authors' reply: The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 9. Write the advantages of the used numerical methods in detail. Authors' reply: BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. The BVP5C solver provides high accuracy for stiff boundary value problems, leveraging adaptive mesh refinement and efficiently handling nonlinearities. Compared to finite difference methods, it offers faster convergence with minimal dependence on grid size. 10. The author should give a deep analysis of the results shown in the figures. Authors' reply: The detailed analysis of the figures in this study provides a comprehensive understanding of how various parameters influence the flow, heat, and mass transfer characteristics of ternary hybrid nanofluids. These insights are valuable for optimizing the performance of nanofluids in applications such as cooling systems, heat exchangers, and polymer processing. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. Authors' reply: It is modified in the conclusion section in the revised manuscript. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? Authors' reply: This study provides valuable theoretical insights but has some limitations. Simplified assumptions, such as treating the nanofluid as a single-phase fluid and neglecting higher-order effects, may not fully capture real-world complexities. The BVP5C numerical method, while effective, has constraints in handling highly nonlinear systems. The findings are based on a specific parameter range, and extrapolation beyond this range requires further validation. Additionally, the assumption of uniform nanoparticle dispersion overlooks possible aggregation or sedimentation effects. To enhance validity and generalizability, future research should incorporate experimental validation and more advanced modeling approaches. 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? Authors' reply: The study demonstrates that the key physical parameters—especially the nanoparticle concentration, Weissenberg number, Forchheimer number, and variable thermal conductivity—have significant effects on the heat and mass transfer characteristics. These parameters enhance the Nusselt number, Sherwood number, and skin friction, contributing to the improved performance of ternary hybrid nanofluids in heat exchange and mass transfer applications. We have addressed this in the results and conclusion sections. Please refer to page 26. 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? Authors' reply: The mathematical framework developed for unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity provides valuable insights into nanofluid behavior under various conditions. It has potential applications in cooling systems , optimizing thermal management in electronics, power plants, and automotive systems. In biomedical applications , it aids in understanding nanofluid flow for targeted drug delivery and hyperthermia treatments. The framework also supports the design of heat exchangers , enhancing industrial heat transfer efficiency. Additionally, it improves polymer processing by predicting material flow and temperature control in extrusion and casting. In energy systems , it contributes to advanced solar thermal collectors and photovoltaic/thermal (PV/T) systems, promoting sustainable energy solutions. By leveraging this model, researchers can enhance the efficiency of various thermal and industrial applications. 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? Authors' reply: This study introduces several novel aspects in investigating unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. It uniquely integrates multiple effects, including viscous dissipation, Darcy-Forchheimer flow, nonlinear thermal radiation, Joule heating, and thermophoresis, providing a comprehensive understanding of nanofluid behavior. Unlike previous studies on binary hybrid nanofluids, this research explores ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂ in ethylene glycol), offering superior thermal properties. The study’s focus on permeable stretching sheets enhances its relevance for real-world applications such as cooling systems and polymer processing. By incorporating variable thermal conductivity , it improves heat transfer accuracy, while an extensive parametric analysis further clarifies the effects of key influencing factors. These contributions make the study a significant advancement in nanofluid convection and heat transfer research. 16. You should describe the abbreviations while using for first time. Authors' reply: It is modified in the revised manuscript. 17. You should add major findings in abstract. Authors' reply: We incorporate the major finding in the abstract. 18. Write the references in correct way. There are some recent references which is important for the manuscript: Authors' reply: Modified 19. Explain the physical significance of boundary conditions. Authors reply: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→ ∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 20. You should add a figure for validation of your code. Authors' reply: This study provides sufficient validation through comprehensive comparisons with existing literature and numerical benchmarks . 21. You should rewrite the first paragraph of conclusions section. Authors' reply: Modified in the revised manuscript. 22. There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. Authors' reply: Corrected 23. What is the impact of this article? What are the future directions? Authors' reply: This study provides a detailed analysis of unsteady MHD flow in ternary hybrid nanofluids ( Al₂O₃–Cu–TiO₂ in ethylene glycol ) over a permeable stretching sheet with variable thermal conductivity. The findings have significant applications in cooling systems , improving heat transfer efficiency, biomedical applications , aiding in drug delivery and hyperthermia treatments, and polymer processing , optimizing thermal conditions in extrusion and casting. Future research should focus on experimental validation to confirm numerical results, optimization studies to refine nanoparticle concentrations for maximum efficiency, and long-term stability analysis to ensure reliability in industrial applications. These advancements will enhance the practical utility of ternary hybrid nanofluids across various engineering and technological fields. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Email: [email protected] Competing Interests: No competing interests were disclosed. Close Report a concern COMMENT ON THIS REPORT Comments on this article Comments (0) Version 2 VERSION 2 PUBLISHED 04 Feb 2025 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 3 Version 2 (revision) 10 Mar 25 read read read Version 1 04 Feb 25 read read read Asra Anjum , University of Technology and Applied Sciences, Salalah, Oman Prasun Choudhary , Chennai Institute of Technology, Chennai, India Dr. Kavita Jat , Chennai Institute of Technology, Chennai, India Sohail Rehman , Qurtuba University of Science and Information Technology, Peshawar, Pakistan Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Rehman S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 22 Mar 2025 | for Version 2 Sohail Rehman , Qurtuba University of Science and Information Technology, Peshawar, Pakistan 0 Views copyright © 2025 Rehman S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Accept in present form Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid dynamics, heat and mass transfer, Nanofluid I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Rehman S. Peer Review Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370297) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370297 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Anjum A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 11 Mar 2025 | for Version 2 Asra Anjum , University of Technology and Applied Sciences, Salalah, Oman 0 Views copyright © 2025 Anjum A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Approved for acceptance. Competing Interests No competing interests were disclosed. Reviewer Expertise fluid dynamics, nanofluids and hybrid nanofluids I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Anjum A. Peer Review Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370296) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370296 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Choudhary P et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 11 Mar 2025 | for Version 2 Prasun Choudhary , Chennai Institute of Technology, Chennai, India Dr. Kavita Jat , Centre for Computational Modeling, Chennai Institute of Technology, Chennai, Tamil Nadu, India 0 Views copyright © 2025 Choudhary P et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions I have reviewed the revised version of the manuscript and I appreciate the authors' efforts in incorporating the suggested comments and changes. The present version is well-structured, clear and significantly improved in terms of readability. As all necessary revisions have been properly addressed, I find the manuscript suitable for acceptance in its current form. I, therefore, accept and recommend it. Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid Flow, Heat Transfer, Mass Transfer, Distinct Geometries, Analytical Methods, Numerical Methods We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Choudhary P and Jat DK. Peer Review Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.178621.r370298) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-152/v2#referee-response-370298 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Rehman S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 21 Feb 2025 | for Version 1 Sohail Rehman , Qurtuba University of Science and Information Technology, Peshawar, Pakistan 0 Views copyright © 2025 Rehman S. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The main observations are listed below. The acceptance of the manuscript would depend on the revision. The author needs to provide a point by point response or provide a rebuttal. The manuscript need a through correction of grammatical errors. The title of the manuscript need to be revised. The authors have not explained, what motivation for selecting this fluid model is, and what are the real world applications of this fluid under the conditions studied in this paper. Provide mathematical expression for fluid volume. Check your boundary conditions and similarity variables. As they are dimensionally correct? As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Is the work clearly and accurately presented and does it cite the current literature? No Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Partly Are all the source data underlying the results available to ensure full reproducibility? No Are the conclusions drawn adequately supported by the results? Partly References 1. Hashim, Rehman S, Mohamed Tag Eldin E, Bafakeeh O, et al.: Coupled energy and mass transport for non-Newtonian nanofluid flow through non-parallel vertical enclosure. Ain Shams Engineering Journal . 2023; 14 (8). Publisher Full Text 2. Rehman S, Hashim, Alqahtani S, Alshehery S: Modeling a non-Newtonian nanofluid flow between intersecting planes with slip mechanism. Continuum Mechanics and Thermodynamics . 2023; 35 (1): 61-80 Publisher Full Text 3. Boujelbene M, Rehman S, Hashim, Alqahtani S, et al.: Optimizing thermal characteristics and entropy degradation with the role of nanofluid flow configuration through an inclined channel. Alexandria Engineering Journal . 2023; 69 : 85-107 Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid dynamics, heat and mass transfer, Nanofluid I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (1) Author Response 10 Mar 2025 Asfaw Tsegaye, Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia We sincerely appreciate the valuable and constructive feedback provided by the reviewer. Below, we have provided a point-by-point response, detailing the revisions made to improve the manuscript. 1. The manuscript need a thorough correction of grammatical errors. The title of the manuscript need to be revised. Author’s Response: In the revised manuscript, we have carefully reviewed and edited the entire document to correct grammatical and typographical errors. We have revised the title to improve clarity and readability. The corrected title is: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. The authors have not explained, what the motivations for selecting this fluid model is, and what are the real-world applications of this fluid under the conditions studied in this paper. Author’s Response: We have added a detailed discussion in the introduction section and formulation of the method section, outlining the motivation for selecting the tangent hyperbolic ternary hybrid nanofluid model. This model is chosen due to its ability to accurately describe non-Newtonian fluid behaviors, making it suitable for engineering applications such as: Cooling Systems – Efficient heat dissipation in electronic devices. Biomedical Applications – Targeted drug delivery and hyperthermia treatments. Polymer Processing – Used in extrusion and casting processes. Energy Systems – Utilized in heat exchangers and power plants, etc. These points are now explicitly stated in the revised introduction. 3. Provide a mathematical expression for the fluid volume. Author’s Response: The fluid volume is not considered in our study since the analysis focuses on 2D flow, boundary-layer formulation, or similarity transformation, where volume is not a key variable. 4. Check whether the boundary conditions and similarity variables are dimensionally correct. Author’s Response: We have thoroughly reviewed the boundary conditions and similarity variables to ensure dimensional correctness. The necessary corrections have been made, and a justification for each transformation is provided in the revised manuscript. 5. As the study considers the constitutive modeling of a virtual fluid dynamics phenomena without validating the results from any physical experimental data. While modeling such virtual fluid dynamics phenomena, the authors take many assumptions to formulate the mathematical model, which may not be real and practical. Therefore, further explanation about physical configuration of problem should be included and the results should be discussed with relation to physical parameters involved in the model. The authors should also explain the physical situation is being modeled in the manuscript. Author’s Response: We have expanded the discussion on the physical configuration in results and discussion section. The justification for key assumptions is also included, emphasizing: -The relevance of the mathematical model in practical scenarios. -The role of each governing parameter in defining the flow behavior. -The impact of approximations made to simplify the problem while maintaining accuracy, etc. 6. Explain the physical significance of boundary conditions and the reasoning behind their transformations. Author’s Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 7. Detailed comparison of the current results with results from other literature or commercial (CFD) codes are preferred to show the advantages and disadvantages of the presented model? Author’s Response: We have enriched the results section by comparing our findings with existing literature, including computational studies where available. Additionally, we have included benchmark comparisons in Table 2, demonstrating the reliability of our numerical method. The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 8. Please enrich the introduction section with related studies Hashim, et al., 2023 (Ref 1) Rehman S, Et al., 2022 (Ref 2) Boujelbene M, et al., 2023 (Ref 3) Author’s Response: We have incorporated the suggested references in the introduction and literature review, explaining their relevance to our work. These studies provide context for our analysis and strengthen the foundation of the research. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Rehman S. Peer Review Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366057) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366057 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Choudhary P et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 20 Feb 2025 | for Version 1 Prasun Choudhary , Chennai Institute of Technology, Chennai, India Dr. Kavita Jat , Centre for Computational Modeling, Chennai Institute of Technology, Chennai, Tamil Nadu, India 0 Views copyright © 2025 Choudhary P et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Title of the Article: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity This paper is interesting. This study examines the combined effects of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductance, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the electrical conductivity of unsteady flow in tangent hyperbolic ternary hybrid nanofluids. These nanofluids consist of Al 2 O 3 , Cu and TiO 2 nanoparticles suspended in ethylene glycol, flowing over a stretching sheet. This study is significant as it combines theoretical analysis and practical application, providing valuable insights into the behavior of tangent hyperbolic ternary hybrid nanofluids under complex flow conditions. It has good number of applications in engineering and technology. It may be accepted subject to the following major modifications/corrections: 1. Title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu) and titanium oxide (TiO₂) nanoparticles? 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications? 6. Literature review is also to be enriched with recent articles.Add the below articles in your introduction: MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B 0 =B/(1−ct) 1/2 . Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. 8. Explanation related to boundary condition (9) needs more clarity. 9. Velocity plots with We and ϕ 1 show very less variations. Recheck and justify the reasons for these less variations. 10. What are the benchmarks of your study? 11. How are convergence and stability ensured in the computational analysis? 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? 13. Add novel results in the conclusion section. I have cited few papers because they are closely aligned with the author's manuscript. The cited papers provide a detailed explanation of the tangent hyperbolic fluid model, which is only partially covered in the author's paper. Additionally, the topics of MHD and porous media are extensively discussed in the cited works which offers valuable insights that complement the author's discussion. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Choudhary S, Choudhary P, Alessa N, Loganathan K: MHD Thermal and Solutal Stratified Stagnation Flow of Tangent Hyperbolic Fluid Induced by Stretching Cylinder with Dual Convection. Mathematics . 2023; 11 (9). Publisher Full Text 2. Choudhary P, Choudhary S, Jat K, Loganathan K, et al.: Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms. International Journal of Thermofluids . 2024; 23 . Publisher Full Text 3. Choudhary S, Choudhary P, Pattanaik B: Tangent Hyperbolic Fluid Flow under Condition of Divergent Channel in the Presence of Porous Medium with Suction/Blowing and Heat Source: Emergence of the Boundary Layer. International Journal of Mathematics and Mathematical Sciences . 2023; 2023 (1). Publisher Full Text 4. Choudhary P, Choudhary S, Jat K, Loganathan K, et al.: Significances of melting heat transfer and bioconvection phenomena in nanofluid flow over a three different geometries. International Journal of Thermofluids . 2024; 24 . Publisher Full Text 5. Choudhary P, Choudhary S, Jat K, Loganathan K, et al.: Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium. Journal of Thermal Analysis and Calorimetry . 2025. Publisher Full Text 6. Jat K, Sharma K, Makinde O: Novel features of radiating hybrid nanofluid flow past a nonlinear stretchable porous sheet with different nanoparticles shape. Numerical Heat Transfer, Part B: Fundamentals . 2024. 1-25 Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid Flow, Heat Transfer, Mass Transfer, Distinct Geometries, Analytical Methods, Numerical Methods We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above. reply Respond to this report Responses (1) Author Response 10 Mar 2025 Asfaw Tsegaye, Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia Authors’ Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewers’ concerns and outline the corresponding revisions made in the manuscript. 1. The title is mostly well-structured and conveys the research topic clearly. However, "Darcy-Forchheimer" should be capitalized because it refers to a specific model. Authors' Response: We have corrected the capitalization in the title as suggested. The revised title now reads: "Unsteady MHD Flow of Tangent Hyperbolic Ternary Hybrid Nanofluid in a Darcy-Forchheimer Porous Medium over a Permeable Stretching Sheet with Variable Thermal Conductivity." 2. How do the thermal conductivity and viscosity of ethylene glycol-based nanofluids vary with the dispersion of aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂) nanoparticles? Authors' Response: The thermal conductivity and viscosity of ethylene glycol-based nanofluids are significantly influenced by the dispersion of nanoparticles such as Al₂O₃, Cu, and TiO₂. In particular, ternary hybrid nanofluids—those containing a combination of three different nanoparticles—exhibit unique thermophysical behaviors. The dispersion of Al₂O₃, Cu, and TiO₂ nanoparticles in ethylene glycol-based ternary hybrid nanofluids leads to enhanced thermal conductivity and increased viscosity. The extent of these changes depends on nanoparticle concentration and operating temperature. Viscosity tends to decrease with rising temperatures. Notably, while thermal conductivity improves with elevated temperatures, viscosity decreases, indicating better flow characteristics at higher temperatures. Balancing these factors is essential for optimizing the performance of such nanofluids in practical applications. While viscosity changes are not explicitly addressed, it is generally understood that nanoparticle addition can increase fluid viscosity, potentially impacting flow behavior. 3. What are the nonlinearities in the governing equations and how do they influence the similarity transformations? Authors' Response: The governing equations exhibit nonlinearities due to factors such as the tangent hyperbolic fluid model, where the shear stress follows a hyperbolic tangent function, introducing nonlinearity in the momentum equation. Nonlinear thermal radiation further complicates the energy equation with temperature-dependent radiative heat flux, while variable thermal conductivity leads to additional nonlinear terms. Viscous dissipation and Joule heating contribute quadratic velocity and temperature-dependent effects, and the Darcy–Forchheimer drag forces in the porous medium introduce nonlinear velocity terms. These nonlinearities make the transformation of partial differential equations (PDEs) into ordinary differential equations (ODEs) via similarity transformations more complex. The resulting nonlinear ODEs require numerical solutions, such as the BVP5C algorithm in MATLAB, where stability, convergence, and accuracy depend on careful parameter tuning for computational efficiency. 4. Describe advantages and limits of ternary hybrid nanofluids in high-performance cooling systems. Authors' Response: Ternary hybrid nanofluids enhance high-performance cooling systems by combining nanoparticles like Al₂O₃, Cu, and TiO₂, leading to superior thermal conductivity, improved stability, and controlled viscosity for efficient heat transfer. Their adaptability makes them ideal for applications in electronics cooling, automotive radiators, and industrial heat exchangers. By optimizing nanoparticle composition, properties like specific heat and viscosity can be tailored to meet cooling demands. However, these benefits come with challenges such as higher synthesis costs, increased viscosity that may raise pumping power requirements, and potential issues with particle agglomeration, sedimentation, and system compatibility. Additionally, their performance varies with temperature, shear rate, and composition, necessitating extensive testing and optimization. Despite these limitations, ternary hybrid nanofluids remain a promising solution for enhancing cooling efficiency, provided their formulation and application challenges are carefully managed. 5. Mention the potential biomedical implications of ternary hybrid nanofluids in drug delivery applications. Authors' Response: Ternary hybrid nanofluids hold significant potential in drug delivery applications due to their enhanced thermal and rheological properties. Their high thermal conductivity allows for precise temperature control, making them suitable for targeted hyperthermia treatments in cancer therapy. The improved stability and dispersion of multiple nanoparticles ensure better drug encapsulation and controlled release, enhancing bioavailability. Their adjustable viscosity enables efficient flow through micro capillaries, improving circulation and targeted delivery to specific tissues. Additionally, the presence of multiple nanoparticles (e.g., Al₂O₃, Cu, TiO₂) can enhance cellular uptake and bioactivity , increasing drug efficacy. However, challenges such as biocompatibility, toxicity, and long-term stability must be addressed to ensure their safe and effective use in biomedical applications. 6. The literature review should be enriched with recent articles. Authors' Response: We have incorporated the suggested references into the introduction and literature review sections, specifically discussing how they relate to our study. The following references have been added: - MHD thermal and solutal stratified stagnation flow of tangent hyperbolic fluid induced by stretching cylinder with dual convection; - Impacts of unsteady MHD hybrid nanofluid over a non-linear stretchable porous sheet with thermal radiation and gyrotatic microorganisms; -Tangent hyperbolic fluid flow under condition of divergent channel in the presence of porous medium with suction/blowing and heat source: Emergence of the boundary layer; - Significances of melting heat transfer and bio-convection phenomena in nanofluid flow over a three different geometries; - Computational study of the magnetohydrodynamic copper–water nanofluid flow through concentric cylinders immersed in a porous medium; - Bio-convection of a radiating and reacting nanofluid flow past a nonlinear stretchable permeable sheet in a porous medium. 7. Authors use unsteady magnetic field B0=B/(1−ct)1/2. Can this field induce an electric field according to Maxwell’s equations and what are its implications? Justify and mention it in revised version. Authors' Response: We have addressed this concern in the mathematical formulation section by discussing Maxwell’s equations in the context of our study. This justification has been added to the revised manuscript. 8. Explanation related to boundary condition (9) needs more clarity. Authors' Response: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 9. Velocity plots with We and ϕ1 show very less variations. Recheck and justify the reasons for these less variations. Authors' Response: The velocity plots with respect to the Weissenberg number ( We ) and nanoparticle volume fraction ( ϕ 1 ​) exhibit minimal variation. This limited influence can be attributed to the dominance of viscous effects over elasticity in the given flow regime, the relatively small effect of nanoparticle volume fraction on velocity, and the constraints imposed by the boundary conditions. If ϕ 1 ​ ​ is varied within a narrow range, the increase in viscosity may not be substantial enough to noticeably affect the velocity profile. 10. What are the benchmarks of your study? Authors' Response: The numerical results have been validated against previously published studies, as shown in Table 2. Our findings show strong agreement with benchmark solutions, confirming the accuracy of our computational approach. 11. How are convergence and stability ensured in the computational analysis? Authors' Response: We have explicitly discussed the convergence criteria of the BVP5C algorithm and verified that the residual errors remain within acceptable limits. The computational analysis employs BVP5C, a MATLAB solver for boundary value problems, ensuring convergence and stability through adaptive mesh refinement, in our study, error control (tolerance 10 -7 ), step size regulation, and a well-posed problem formulation. A grid independence test verifies numerical stability, ensuring accurate and reliable results while minimizing numerical artifacts. 12. The flow is in a permeable stretching sheet, what will be the role of the magnetic field? Authors' Response: We have elaborated on the role of the magnetic field in modifying velocity and temperature profiles. The Lorentz force acts as a resistive force, reducing velocity and increasing temperature due to Joule heating. 13. Add novel results in the conclusion section. Authors' Response: We have updated the conclusion to explicitly highlight the novel contributions of this study, including: The interplay between ternary hybrid nanoparticles and non-Newtonian fluid properties. The effect of variable thermal conductivity on heat transfer performance. The impact of nonlinear radiation on temperature enhancement, etc. These additions strengthen the conclusion and emphasize the novelty of our findings. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Choudhary P and Jat DK. Peer Review Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366062) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/14-152/v1#referee-response-366062 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Anjum A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 18 Feb 2025 | for Version 1 Asra Anjum , University of Technology and Applied Sciences, Salalah, Oman 0 Views copyright © 2025 Anjum A. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (1) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Title: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal c onductivity The paper has made a promising contribution and fresh results. However, the article needs modifications, and I'd like to ask the authors to revise it due to the following comments, which can be accepted. Review comments; 1. Please briefly state more about salient findings in the abstract. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: 5. The author needs to mention briefly the application of the present work in the real world. 6. Why are such boundary conditions used? What is the physical meaning? 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. 9. Write the advantages of the used numerical methods in detail. 10. The author should give a deep analysis of the results shown in the figures. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? 16 . You should describe the abbreviations while using for first time. 17 . You should add major findings in abstract. 18 . Write the references in correct way. There are some recent references which is important for the manuscript: 19 . Explain the physical significance of boundary conditions. 20. You should add a figure for validation of your code. 21. You should rewrite the first paragraph of Conclusions section. 22 . There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. 23 . What is the impact of this article? What are the future directions? Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Yes Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise fluid dynamics, nanofluids and hybrid nanofluids I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (1) Author Response 10 Mar 2025 Asfaw Tsegaye, Department of Mathematics, Bahir Dar University, Bahir Dar, Ethiopia ​​​​ Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers for their insightful and constructive comments, which have significantly improved the quality of our manuscript. Below, we provide our responses to each of the reviewer’s concerns and outline the corresponding revisions made in the manuscript. 1. Please briefly state more about salient findings in the abstract. Authors' reply: The abstract has been revised to explicitly highlight key findings, including: - The velocity profile decreases with increasing Weissenberg and Forchheimer numbers. - Ternary hybrid nanofluids exhibit superior heat transfer enhancement compared to mono/hybrid nanofluids. - Variable thermal conductivity (Γ) and nanoparticle volume fractions significantly improve temperature distribution and skin friction. - Applications in cooling systems, biomedical drug delivery, and polymer processing are emphasized. 2. Nomenclature should be mentioned along with SI units. Similarly, subscripts and superscripts can be explained under their subtitles. Authors' reply: The nomenclature section has been updated to include SI units for all parameters. Subscripts and superscripts are now clearly defined under their respective sections. 3. It is recommended the authors try to explain the novelty of the paper as clearly as possible and explain the research gap they are trying to fill. Authors' reply: A new paragraph was added to the introduction (last paragraph) to clarify the novelty: "To the authors' knowledge, no prior study combines viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, and ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂/EG) under unsteady MHD conditions. This work addresses the gap by analyzing synergistic effects of 12+ parameters, offering insights into optimizing thermal systems." 4. The introduction section requires a comprehensive survey of recent literature. The authors should conduct a detailed literature review, clearly outlining the contributions of each relevant study. Additionally, the introduction should explain how the findings from previous works support and inform the current research, making it clear how these studies contribute to the development of the present study's objectives and approach. The following articles will help to enhance the quality of the introduction section: Authors' reply: We have not yet received your suggested references. However, we have added recent studies (2023–2024) on ternary hybrid nanofluids and tangent hyperbolic fluids. Additionally, using your Google Scholar profile, we have incorporated some of your references into the introduction and literature review sections, highlighting their relevance to our study. The following references have been added: - Computation Of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport From An Inclined Plane With Ion Slip And Hall Current Effects - Influence of Darcy-Forchheimer hybrid nanofluid flow with magnetohydrodynamic and viscous dissipation effects past a permeable stretching sheet: a numerical contribution 5. The author needs to mention briefly the application of the present work in the real world. Authors' reply: The study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity" investigates the unsteady magnetohydrodynamic (MHD) flow, heat, and mass transfer of ternary hybrid nanofluids over a permeable stretching sheet. The study considers three types of nanoparticles—aluminum oxide (Al₂O₃), copper (Cu), and titanium oxide (TiO₂)—dispersed in a base fluid of ethylene glycol (C₂H₆O₂). This ternary hybrid nanofluid (Al₂O₃–Cu–TiO₂/C₂H₆O₂) has potential applications in cooling systems, biomedical uses for targeted drug delivery and hyperthermia treatments, heat exchangers, and polymer processing techniques like extrusion and casting. 6. Why are such boundary conditions used? What is the physical meaning? Authors' reply: In the study titled "Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a Darcy-Forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity," the authors employ specific boundary conditions to accurately model the physical phenomena under investigation. These conditions are essential for representing real-world scenarios and ensuring the validity of the mathematical model. These boundary conditions are crucial for accurately modeling the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. They ensure that the mathematical model reflects the physical realities of the system, including the interaction between the fluid and the surface, the development of boundary layers, and the transition to ambient conditions at a distance from the surface. 7. A correlation must be suggested based on the computed results (You can use statistical tools or methods for correlation calculation). Authors' reply: In our study , the combined effects of several factors on the unsteady MHD flow of ternary hybrid nanofluids over a permeable stretching sheet. Specifically, the research examines the influence of viscous dissipation, Darcy-Forchheimer flow, variable thermal conductivity, nonlinear thermal radiation, Joule heating, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis on the velocity profile, temperature distribution, concentration profile, skin friction, Nusselt number, and Sherwood number. In this study, correlation were not suggested. We considering the further direction. 8. What software is used for the simulations? Was the code implemented by the authors or a function already existing in the software was used? If the code for the numerical method was taken from another publication or is part of the software used, please cite the resource. Authors' reply: The MATLAB bvp5c solver was used with a self-developed code (GitHub link provided). The algorithm adapts the shooting method with a tolerance of 10⁻⁷, validated against Gorla & Sidawi [41], Waini et al.[40], Priyadlarshtni et al. [18], and Jamrus et al. [38]. 9. Write the advantages of the used numerical methods in detail. Authors' reply: BVP5C is a powerful tool for solving boundary value problems in various scientific and engineering applications. Its versatility, efficiency, and user-friendly nature make it an essential component in numerical analysis. The BVP5C solver provides high accuracy for stiff boundary value problems, leveraging adaptive mesh refinement and efficiently handling nonlinearities. Compared to finite difference methods, it offers faster convergence with minimal dependence on grid size. 10. The author should give a deep analysis of the results shown in the figures. Authors' reply: The detailed analysis of the figures in this study provides a comprehensive understanding of how various parameters influence the flow, heat, and mass transfer characteristics of ternary hybrid nanofluids. These insights are valuable for optimizing the performance of nanofluids in applications such as cooling systems, heat exchangers, and polymer processing. 11. Revise the article's conclusion to make it more impactful and ensure it effectively reinforces the main points and leaves a lasting impression on the reader. Authors' reply: It is modified in the conclusion section in the revised manuscript. 12. Does the conclusion sufficiently address the study's limitations and explain how these constraints affect the findings' overall validity and generalizability? Authors' reply: This study provides valuable theoretical insights but has some limitations. Simplified assumptions, such as treating the nanofluid as a single-phase fluid and neglecting higher-order effects, may not fully capture real-world complexities. The BVP5C numerical method, while effective, has constraints in handling highly nonlinear systems. The findings are based on a specific parameter range, and extrapolation beyond this range requires further validation. Additionally, the assumption of uniform nanoparticle dispersion overlooks possible aggregation or sedimentation effects. To enhance validity and generalizability, future research should incorporate experimental validation and more advanced modeling approaches. 13. What are the key physical parameters analyzed in the study, and how do they influence the Nusselt number, Sherwood number, skin friction? Authors' reply: The study demonstrates that the key physical parameters—especially the nanoparticle concentration, Weissenberg number, Forchheimer number, and variable thermal conductivity—have significant effects on the heat and mass transfer characteristics. These parameters enhance the Nusselt number, Sherwood number, and skin friction, contributing to the improved performance of ternary hybrid nanofluids in heat exchange and mass transfer applications. We have addressed this in the results and conclusion sections. Please refer to page 26. 14. What potential applications are suggested for the mathematical framework developed in the study, particularly concerning nanofluid behavior? Authors' reply: The mathematical framework developed for unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity provides valuable insights into nanofluid behavior under various conditions. It has potential applications in cooling systems , optimizing thermal management in electronics, power plants, and automotive systems. In biomedical applications , it aids in understanding nanofluid flow for targeted drug delivery and hyperthermia treatments. The framework also supports the design of heat exchangers , enhancing industrial heat transfer efficiency. Additionally, it improves polymer processing by predicting material flow and temperature control in extrusion and casting. In energy systems , it contributes to advanced solar thermal collectors and photovoltaic/thermal (PV/T) systems, promoting sustainable energy solutions. By leveraging this model, researchers can enhance the efficiency of various thermal and industrial applications. 15. What unique aspects of the study contribute to its novelty in investigating nanofluid convection and heat and mass transfer phenomena? Authors' reply: This study introduces several novel aspects in investigating unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluids over a permeable stretching sheet with variable thermal conductivity. It uniquely integrates multiple effects, including viscous dissipation, Darcy-Forchheimer flow, nonlinear thermal radiation, Joule heating, and thermophoresis, providing a comprehensive understanding of nanofluid behavior. Unlike previous studies on binary hybrid nanofluids, this research explores ternary hybrid nanofluids (Al₂O₃–Cu–TiO₂ in ethylene glycol), offering superior thermal properties. The study’s focus on permeable stretching sheets enhances its relevance for real-world applications such as cooling systems and polymer processing. By incorporating variable thermal conductivity , it improves heat transfer accuracy, while an extensive parametric analysis further clarifies the effects of key influencing factors. These contributions make the study a significant advancement in nanofluid convection and heat transfer research. 16. You should describe the abbreviations while using for first time. Authors' reply: It is modified in the revised manuscript. 17. You should add major findings in abstract. Authors' reply: We incorporate the major finding in the abstract. 18. Write the references in correct way. There are some recent references which is important for the manuscript: Authors' reply: Modified 19. Explain the physical significance of boundary conditions. Authors reply: The boundary conditions define the behavior of unsteady MHD flow of a tangent hyperbolic ternary hybrid nanofluid over a permeable stretching sheet , ensuring the model accurately represents real-world fluid dynamics. At the sheet surface ( y→0 ) , the velocity conditions account for fluid suction ( V w > 0 ) , which enhances cooling and stabilizes the flow, and fluid injection ( V w 0 representing stretching (common in polymer processing) and λ<0 representing shrinking (relevant to flow separation studies). The temperature ( T = T w ) and concentration ( C = C w ​ ) conditions define thermal and mass transfer at the surface, impacting heat exchangers, coatings, and drug delivery applications . Far from the sheet ( y→ ∞ ), the fluid reaches zero velocity , thermal equilibrium ( T→ T ∞ ) , and concentration equilibrium ( T→ T ∞ ​ ) , ensuring a physically realistic model. These conditions are essential for accurately predicting nanofluid behavior in industrial, biomedical, and energy systems . 20. You should add a figure for validation of your code. Authors' reply: This study provides sufficient validation through comprehensive comparisons with existing literature and numerical benchmarks . 21. You should rewrite the first paragraph of conclusions section. Authors' reply: Modified in the revised manuscript. 22. There are a lot of errors in typing and word spacing. It must be corrected in the revised manuscript. Authors' reply: Corrected 23. What is the impact of this article? What are the future directions? Authors' reply: This study provides a detailed analysis of unsteady MHD flow in ternary hybrid nanofluids ( Al₂O₃–Cu–TiO₂ in ethylene glycol ) over a permeable stretching sheet with variable thermal conductivity. The findings have significant applications in cooling systems , improving heat transfer efficiency, biomedical applications , aiding in drug delivery and hyperthermia treatments, and polymer processing , optimizing thermal conditions in extrusion and casting. Future research should focus on experimental validation to confirm numerical results, optimization studies to refine nanoparticle concentrations for maximum efficiency, and long-term stability analysis to ensure reliability in industrial applications. These advancements will enhance the practical utility of ternary hybrid nanofluids across various engineering and technological fields. All revisions have been incorporated into the manuscript, enhancing clarity, technical rigor, and alignment with reviewer feedback. We appreciate the opportunity to improve our work. Sincerely, Asfaw Tsegaye Moltot[Corresponding Author] Email: [email protected] View more View less Competing Interests No competing interests were disclosed. reply Respond Report a concern Anjum A. Peer Review Report For: Unsteady MHD flow of tangent hyperbolic ternary hybrid nanofluid in a darcy-forchheimer porous medium over a permeable stretching sheet with variable thermal conductivity [version 2; peer review: 3 approved] . F1000Research 2025, 14 :152 ( https://doi.org/10.5256/f1000research.174254.r366061) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. 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