Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet

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These fluids improve thermal conductivity and fluid dynamics, leading to better heat management and energy efficiency. This study investigates the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, and other factors on the three-dimensional flow, heat, and mass transfer of a Williamson hybrid nanofluid. The flow occurs over a stretching porous sheet subjected to an external magnetic field, Joule heating, chemical reactions, and heat generation. Methods Copper (Cu) and aluminum oxide (Al₂O₃) nanoparticles are suspended in ethylene glycol (C₂C₆O₂) to form the hybrid nanofluid. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and solved numerically with MATLAB’s bvp4c solver. The study examines various parameters, including stretching ratio, nanoparticle volume fraction, and relaxation times for concentration and thermal effects. Results are validated against existing literature. Results The findings reveal that a higher stretching ratio reduces velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers, while also lowering skin friction and secondary velocity. Increasing nanoparticle volume fraction decreases velocity and temperature profiles but enhances skin friction, local Nusselt, and Sherwood numbers. Concentration profiles decline with higher concentration relaxation time, while temperature increases with longer thermal relaxation time. Conclusions In conclusion, Cu−Al₂O₃/C₂C₆O₂ hybrid nanofluids demonstrate superior heat and mass transfer capabilities compared to mono-nanofluids. The performance is significantly influenced by parameters such as nanoparticle volume fraction, relaxation times, and the stretching ratio, providing valuable insights for heat and mass transfer applications. " } { "@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/14-210/v1", "name": "Effect of non-linear thermal radiation and Cattaneo-Christov heat..." } } ] } Home Browse Effect of non-linear thermal radiation and Cattaneo-Christov heat... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Tsegaye A, Haile E, Awgichew G and Dessie H. Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.12688/f1000research.160734.1 ) 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 Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] Asfaw Tsegaye https://orcid.org/0000-0001-7610-4058 1 , Eshetu Haile https://orcid.org/0000-0002-2428-1842 1 , Gurju Awgichew 1 , Hunegnaw Dessie 1 Asfaw Tsegaye https://orcid.org/0000-0001-7610-4058 1 , Eshetu Haile https://orcid.org/0000-0002-2428-1842 1 , Gurju Awgichew 1 , Hunegnaw Dessie 1 PUBLISHED 14 Feb 2025 Author details Author details 1 Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Asfaw Tsegaye Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Software, Validation, Writing – Original Draft Preparation Eshetu Haile Roles: Conceptualization, Resources, Supervision, Visualization, Writing – Review & Editing Gurju Awgichew Roles: Resources, Supervision, Visualization, Writing – Review & Editing Hunegnaw Dessie Roles: Resources, Supervision, Visualization, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Nanoscience & Nanotechnology gateway. Abstract Background Hybrid nanofluids, consisting of two distinct nanoparticles dispersed in a base fluid, are widely used in industries requiring enhanced heat and mass transfer, such as cooling systems and heat exchangers. These fluids improve thermal conductivity and fluid dynamics, leading to better heat management and energy efficiency. This study investigates the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, and other factors on the three-dimensional flow, heat, and mass transfer of a Williamson hybrid nanofluid. The flow occurs over a stretching porous sheet subjected to an external magnetic field, Joule heating, chemical reactions, and heat generation. Methods Copper (Cu) and aluminum oxide (Al₂O₃) nanoparticles are suspended in ethylene glycol (C₂C₆O₂) to form the hybrid nanofluid. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and solved numerically with MATLAB’s bvp4c solver. The study examines various parameters, including stretching ratio, nanoparticle volume fraction, and relaxation times for concentration and thermal effects. Results are validated against existing literature. Results The findings reveal that a higher stretching ratio reduces velocity, temperature, concentration profiles, and local Nusselt and Sherwood numbers, while also lowering skin friction and secondary velocity. Increasing nanoparticle volume fraction decreases velocity and temperature profiles but enhances skin friction, local Nusselt, and Sherwood numbers. Concentration profiles decline with higher concentration relaxation time, while temperature increases with longer thermal relaxation time. Conclusions In conclusion, Cu−Al₂O₃/C₂C₆O₂ hybrid nanofluids demonstrate superior heat and mass transfer capabilities compared to mono-nanofluids. The performance is significantly influenced by parameters such as nanoparticle volume fraction, relaxation times, and the stretching ratio, providing valuable insights for heat and mass transfer applications. READ ALL READ LESS Keywords Williamson fluid, Hybrid nanofluid, Cattaneo-Christov heat and mass flux, non-linear thermal radiation, stretching sheet Corresponding Author(s) Asfaw Tsegaye ( [email protected] ) Close Corresponding author: Asfaw Tsegaye 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 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 A, Haile E, Awgichew G and Dessie H. Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.12688/f1000research.160734.1 ) First published: 14 Feb 2025, 14 :210 ( https://doi.org/10.12688/f1000research.160734.1 ) Latest published: 03 Dec 2025, 14 :210 ( https://doi.org/10.12688/f1000research.160734.3 )  There is a newer version of this article available. Suppress this message for one day. 1. Introduction Nowadays, researchers have created novel fluids to suit the requirement for improved heat transmission and thermal conductivity. The heat transfer properties of conventional fluids like water, ethylene glycol, glycerin, and ethanol are limited in applications such as power generation, chemical processes, and heating and cooling systems. Scientists are investigating better heat transmission materials to meet the rising energy demands and address concerns about shortages of resources and environmental impacts. A single kind of nanoparticle can be added to the aforementioned fluids to make up for their deficiency. The term nanofluid refers to this process, which was initially studied by Choi and Eastman. 1 Researchers have considered many combinations of nanoparticles, including semiconductors ( SiO 2 , TiO 2 ), metallic oxides ( Al 2 O 3 , CuO ), and metal nanoparticles ( Al , Cu , and Fe ). Heat transfer through nanofluids has been investigated from various perspectives. Mahian et al. 2 presented the uses of nanofluid in a variety of contexts, such as renewable energy systems. They also covered the advantages of energy systems from an environmental perspective when employing nanofluid. Mansoury et al. 3 investigated the flow of Al 2 O 3 /H 2 O nanofluids through parallel heat exchangers. Unfortunately, to obtain the required thermal performance, a single nanoparticle suspension is insufficient. To attain the required thermal properties, hybrid nanofluid is employed. Several experimental and theoretical models have been published and studied in order to use hybrid nanofluids for more efficient industrial and technological processes. Mahanthesh et al. 4 reviewed the flow behavior of hybrid nanofluids, focusing on the effects of Brownian motion and thermophoresis. Sensitivity analysis revealed that the Brownian motion parameter has the most significant impact on the heat transfer rate. Bilal et al. 5 investigated the Darcy-Forchheimer mixed convection flow of hybrid nanofluids through an inclined, extending cylinder using the homotopy analysis method. Their findings indicated that CNT−Fe 3 O 4 /H 2 O hybrid nanofluids enhance the thermal efficiency of the base fluid more effectively than conventional fluids. The investigation provides further information on the hybrid nanofluid flow as provided in these reference. 6 , 7 Scientists and engineers have been motivated by the non-Newtonian motion of fluids because these ma- terials have multiple applications in science and technology processes. Some examples of non-Newtonian fluid are Paints, mud, Soap, glues, apple sauce, printing ink, shampoos, sugar solutions, tomato paste, etc. Furthermore, non-Newtonian fluids are used in several kinds of fields of study, such as the chemical en- gineering field, biology, and geophysical sciences. The Williamson model is the fluid model that is being studied. In the case of Williamson fluid rheology, the constitutive equation for stress-strain is non-linear. Therefore, viscosity that depends on the shear rate forms the basis of the Williamson fluid model. This model captures both shear-thinning and shear-thickening behavior. Williamson 8 introduced the Williamson fluid model, which defines the relationship between stress and strain in pseudoplastic materials. Kebede et al. 9 studied the heat and mass transport properties of Williamson nanofluid flow. Kumar et al. 10 explored the impact of Brownian motion and thermophoresis on heat and mass transfer in pseudoplastic materials. Heat and mass transport are natural phenomena caused by concentration and temperature differences within or between materials. Many industrial processes, including wire drawing, artificial fiber production, paper manufacturing, chemical waste migration, and distillation, are influenced by this phenomenon. Significant efforts have been made in the past to study heat and mass transport mechanisms using Fourier’s law of heat transfer 11 and Fick’s law of diffusion. 12 Previously,”Fourier’s law of heat conduction” was commonly used to explain heat transfer. However, this law does not fully capture the fundamental nature of heat transfer. To address this issue, Cattaneo 13 introduced thermal relaxation into Fourier’s theory, making heat transfer resemble thermal wave propagation at normal speed. Christov 14 improved the Cattaneo model’s thermal relaxation time by using upper-convected Oldroyd derivatives for frame-invariant formation. Sui et al. 15 extended the Cattaneo-Christov model to mass diffusion problems, applying it to the Maxwell nanofluid mass diffusion across a moving surface. Recent developments on this concept are gathered in. 16 – 20 Magnetohydrodynamics is the study of the interaction between a magnetic field and electrically con- ducting fluids, such as salt water solutions, liquid metals, and plasmas. Magnetohydrodynamic flow studies have significant applications in chemistry, physics, and engineering. Additionally, MHD flow has applica- tions in industrial equipment, pumps, electric transformers, and hydro-magnetic generators, among others. Zhao et al. 21 studied the thermally induced electro-kinetic flow of Al 2 O 3 -water nanofluid through a permeable micro-tube to examine its heat transfer properties, considering the effect of an applied magnetic field. Chamkaa et al. 22 used the finite-difference method to analyze mixed convection in a square cavity filled with Cu-water nanofluid. Recent studies on MHD can be found in references. 23 , 24 Flow over a stretching sheet is a classic fluid mechanics problem with various practical applications in industries such as glass fiber production, glass blowing, wire drawing, and copper wire tinning. The concept of boundary layer flow for solid surface motion at a constant speed was first introduced by Sakiadis. 25 – 27 Several researchers have recently investigated the flow of boundary layers over stretchable surfaces. 28 , 29 In fluid dynamics, a porous medium refers to a solid material that contains a network of interconnected voids or pores, allowing fluids (such as gases or liquids) to flow through it. The structure of the porous medium, including the size, shape, and distribution of its pores, significantly influences the flow behavior and other transport phenomena (e.g., heat and mass transfer) within the medium. Shaw et al. 30 investigated the effects of entropy production in a Casson fluid containing MWCNT/Fe 3 O 4 through a stretched disc and the Darcy-Forchheimer porous medium concept using a numerical approach. There are further investigations on the porous media in. 31 , 32 Nonlinear thermal radiation refers to the heat transfer process where the radiation heat flux does not follow a simple linear relationship with temperature. Unlike linear radiation, where the heat flux is proportional to the fourth power of temperature (as in the Stefan-Boltzmann law), nonlinear thermal radiation accounts for more complex interactions between the radiating bodies and the surrounding medium. This phenomenon arises in high-temperature environments, where the radiation intensity varies non-linearly due to factors such as temperature gradients, optical properties of materials, or the nature of the medium. Alamirew et al. 33 studied the impact of nonlinear thermal radiation, ion slip, and Hall on MHD Williamson nanofluid flow over a stretched sheet. Building on this, other researchers 34 , 35 have explored the effects of thermal radiation in various geometries and methods. Several studies have investigated Williamson nanofluid flow over a stretching sheet using various geometries and methods. Different approaches have been employed in the literature to study the heat and mass fluxes in Williamson nanofluid flow. However, to the best of the authors’ knowledge, no studies have yet considered the combined effects of non-linear thermal radiation, Joule heating, non-Fourier heat flux, non-Fick mass flux, chemical reactions, heat generation/absorption, Brownian motion, and thermophoresis. The present study aims to provide a comprehensive analysis of mass and heat transfer in 3D MHD Williamson hybrid nanofluid flow, which consists of Cu and Al2O3 nanoparticles in ethylene glycol, over a linearly stretching porous sheet. By applying similarity transformations, the governing nonlinear partial differential equations are converted into a system of ordinary differential equations. The reduced mathematical model is then solved using the bvp4c function in MATLAB. The study uses tables and figures to analyze the effects of various parameters, such as velocity, temperature, concentration, skin friction, the Nusselt number, and the Sherwood number. 2. Problem formulation Assume a steady three-dimensional hybrid nanofluid ( Cu − Al 2 O 3 /Ethylene glycol) flow past a stretching porous sheet in the presence of an applied magnetic field. The schematic representation in Cartesian coordinates is displayed in Figure 1 . The flow is incompressible and laminar. The velocities of the stretching sheets are v w = by in y and u w = ax in x , and a constant external magnetic field B is applied in z . The temperature and concentration of the surface are kept constant at T w and C w respectively, which is higher than its ambient temperature, T ∞ and ambient concentration, C ∞ . Furthermore, the double diffusion Cattaneo-Christove theorya generalization of the well-known Fourier and Fick laws was used to study heat and mass transmission. 36 , 37 (1) q → + κ ∇ T + λ e [ q → t + V → · ∇ q → − q → · ∇ V → + ( ∇ · V → ) q → ] = 0 (2) J → + D B ∇ C + λ c [ J → t + V → · ∇ J → − J → · ∇ V → + ( ∇ · V → ) J → ] = 0 Figure 1. Coordinate system. In equation (1) and (2) , q → and J → , D B , T , C , κ , V → , λ e and λ c represents the heat and mass fluxes, the Brownian motion, the temperature, the concentration, the thermal conductivity, the velocity field, the energy and concentration relaxation factors, respectively. By letting λ e = 0 = λ c , the famous Fourier’s and Fick’s laws can be obtained. From equations (1) and (2) , which generalize the classic Fourier and Fick’s laws, the steady-state conditions of the nanofluid flow are expressed as follows: (3) q → + κ ∇ T + λ e [ V → · ∇ q → − q → · ∇ V → + ( ∇ · V → ) q → ] = 0 (4) J → + D B ∇ C + λ c [ V → · ∇ J → − J → · ∇ V → + ( ∇ · V → ) J → ] = 0 . The Williamson fluid model is given Refs. 38 , 39 by (5) τ = − P → I + S → (6) S → = [ μ ∞ + μ 0 − μ ∞ 1 − Γ γ ̇ ] A 1 The shear rate ( γ ̇ ) and is defined as (7) γ ̇ = 0.5 trace ( A 1 ) 2 This is taken into consideration for pseudo-plastic fluids μ ∞ = 0 and Γ γ ̇ < 1 . Equation (6) can be written as (8) S → = [ μ 0 / ( 1 − Γ γ ̇ ) ] A 1 Using the binomial expansion to Equation (8) , given as (9) S → ≈ μ 0 [ 1 + Γ γ ̇ ] A 1 . Considering the above assumptions, and after applying the boundary layer approximations, the governing equations are expressed as follows 36 , 40 , 41 (10) ∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 (11) u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z = μ hnf ρ hnf ∂ 2 u ∂ z 2 + 2 Γ μ hnf ρ hnf ∂ u ∂ z ∂ 2 u ∂ z 2 − σ hnf B 2 ρ hnf u − μ hnf ρ hnf K p u (12) u ∂ v ∂ x + v ∂ v ∂ y + w ∂ v ∂ z = μ hnf ρ hnf ∂ 2 v ∂ z 2 + 2 Γ μ hnf ρ hnf ∂ v ∂ z ∂ 2 v ∂ z 2 − σ hnf B 2 ρ hnf v − μ hnf ρ hnf K p v (13) u ∂ T ∂ x + v ∂ T ∂ y + w ∂ T ∂ z = α hnf ∂ 2 T ∂ z 2 − 1 ( ρCp ) hnf ∂ q r ∂ z − λ e [ u 2 ∂ 2 T ∂ x 2 + v 2 ∂ 2 T ∂ y 2 + w 2 ∂ 2 T ∂ z 2 + 2 ( uv ∂ 2 T ∂ x ∂ y + vw ∂ 2 T ∂ y ∂ z + uw ∂ 2 T ∂ x ∂ z ) + ( u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z ) ∂ T ∂ x + ( u ∂ v ∂ x + v ∂ v ∂ y + w ∂ v ∂ z ) ∂ T ∂ y + ( u ∂ w ∂ x + v ∂ w ∂ y + w ∂ w ∂ z ) ∂ T ∂ z ] + Q 0 ( T − T ∞ ) ( ρCp ) hnf + σ hnf B 2 ( u 2 + v 2 ) ( ρCp ) hnf + τ [ D B ∂ C ∂ z ∂ T ∂ z + D T T ∞ ( ∂ T ∂ z ) 2 ] (14) u ∂ C ∂ x + v ∂ C ∂ y + w ∂ C ∂ z + κ c λ c = D B ∂ 2 C ∂ z 2 + D T T ∞ ( ∂ 2 T ∂ z 2 ) − K r ( C − C ∞ ) − λ c [ u 2 ∂ 2 C ∂ x 2 + v 2 ∂ 2 C ∂ y 2 + w 2 ∂ 2 C ∂ z 2 + 2 ( uv ∂ 2 C ∂ x ∂ y + vw ∂ 2 C ∂ y ∂ z + uw ∂ 2 C ∂ x ∂ z ) + ( u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z ) ∂ C ∂ x + ( u ∂ v ∂ x + v ∂ v ∂ y + w ∂ v ∂ z ) ∂ C ∂ y + ( u ∂ w ∂ x + v ∂ w ∂ y + w ∂ w ∂ z ) ∂ C ∂ z ] The boundary conditions for the problem are given by 40 , 42 (15) { w = 0 , v = v w = by , u = u w = ax , T = T w , C = C w , at z → 0 u = v = 0 , T = T ∞ , C = C ∞ , as z → ∞ The hybrid nanofluids thermophysical characteristics are described as 39 , 40 : μ hnf = μ f ( 1 − ϕ 1 ) 2.5 ( 1 − ϕ 2 ) 2.5 , α hnf = κ hnf ( ρCp ) hnf , ρ hnf = ( 1 − ϕ 2 ) [ ( 1 − ϕ 1 ) ρ f + ϕ 1 ρ n 1 ] + ϕ 2 ρ n 2 , ( ρCp ) hnf = ( 1 − ϕ 2 ) [ ( 1 − ϕ 1 ) ( ρCp ) f + ϕ 1 ( ρCp ) n 1 ] + ϕ 2 ( ρCp ) n 2 , κ hnf κ nf = κ n 2 + 2 κ nf − 2 ϕ 2 ( κ nf − κ n 2 ) κ n 2 + 2 κ nf + ϕ 2 ( κ nf − κ n 2 ) , κ nf κ f = κ n 1 + 2 κ f − 2 ϕ 1 ( κ f − κ n 1 ) κ n + 2 κ f + ϕ ( κ f − κ n 1 ) , σ hnf σ nf = σ n 2 + 2 σ nf − 2 ϕ 2 ( σ nf − σ n 2 ) σ n 2 + 2 σ nf + ϕ 2 ( σ nf − σ n 2 ) , σ nf σ f = σ n 1 + 2 σ f − 2 ϕ 1 ( σ f − σ n 1 ) σ n + 2 σ f + ϕ ( σ f − σ n 1 ) , where μ f , ϕ 1 , ϕ 2 , ρ f , κ , ( ρCp ) f , σ hnf , hnf , nf , f , and n the base fluid viscosity, nanoparticle volume fraction of Cu, nanoparticle volume fraction of Al 2 O 3 , density, thermal conductivity, heat capacitance, electrical conductivity of hybrid nanofluid, hybrid nanofluid, nanofluid, base fluid, and nanoparticle. Furthermore, in terms of shape, only spherical nanoparticles are taken into consideration. Using the nonlinear Rosseland diffusion approximation, the radiative heat flux q r can be expressed as: (16) q r = − 4 σ ∗ 3 κ ∗ ∂ T 4 ∂ z (17) q r = − 16 σ ∗ 3 κ ∗ T 3 ∂ T ∂ z and therefore (18) ∂ q r ∂ z = − 16 σ ∗ 3 κ ∗ ∂ ∂ z ( T 3 ∂ T ∂ z ) By means of Equations (16) , (17) and (18) , Equation (13) reduces to (19) u ∂ T ∂ x + v ∂ T ∂ y + w ∂ T ∂ z = α hnf ∂ 2 T ∂ z 2 + 1 ( ρCp ) hnf 16 σ ∗ 3 κ ∗ ∂ ∂ z ( T 3 ∂ T ∂ z ) − λ e [ u 2 ∂ 2 T ∂ x 2 + v 2 ∂ 2 T ∂ y 2 + w 2 ∂ 2 T ∂ z 2 + 2 ( uv ∂ 2 T ∂ x ∂ y + vw ∂ 2 T ∂ y ∂ z + uw ∂ 2 T ∂ x ∂ z ) + ( u ∂ u ∂ x + v ∂ u ∂ y + w ∂ u ∂ z ) ∂ T ∂ x + ( u ∂ v ∂ x + v ∂ v ∂ y + w ∂ v ∂ z ) ∂ T ∂ y + ( u ∂ w ∂ x + v ∂ w ∂ y + w ∂ w ∂ z ) ∂ T ∂ z ] + Q 0 ( T − T ∞ ) ( ρCp ) hnf + σ hnf B 2 ( u 2 + v 2 ) ( ρCp ) hnf + τ [ D B ∂ C ∂ z ∂ T ∂ z + D T T ∞ ( ∂ 2 T ∂ z 2 ) 2 ] The similarity transformations are given by 40 , 42 (20) { w = − a ν f ( f ( η ) + g ( η ) ) , u = ax f ′ ( η ) , v = ay g ′ ( η ) , η = z a ν f T = T ∞ + ( T w − T ∞ ) θ ( η ) , C = C ∞ + ( C w − C ∞ ) Φ ( η ) Equations (10) to (14) are simplified accordingly: (21) A 1 A 2 f ′ ′ ′ [ 1 + We f ′ ′ ] + ( f + g ) f ′ ′ − f ′ 2 − A 3 A 2 M f ′ − A 1 A 2 K f ′ = 0 (22) A 1 A 2 g ′ ′ ′ [ 1 + We g ′ ′ ] + ( f + g ) g ′ ′ − g ′ 2 − A 3 A 2 M g ′ − A 1 A 2 K g ′ = 0 (23) A 4 A 5 Pr θ ′ ′ + ( f + g ) θ ′ − β 1 [ ( f + g ) 2 θ ′ ′ + ( f + g ) ( f ′ + g ′ ) θ ′ ] + 1 A 5 Pr R [ ( 1 + ( θ w − 1 ) ) 2 × 3 ( θ w − 1 ) θ ′ 2 + ( 1 + ( θ w − 1 ) θ ) 3 θ ′ ′ ] + 1 A 5 Qθ + Nb θ ′ Φ ′ + Nt θ 2 + A 3 A 5 M ( Ecx ( f ′ ) 2 + Ecy ( g ′ ) 2 ) = 0 (24) Φ ′ ′ + Nt Nb θ ′ ′ + Sc ( f + g ) Φ ′ − β 2 Sc [ ( f + g ) 2 Φ ′ ′ + ( f + g ) ( f ′ + g ′ ) Φ ′ ] − ScH Φ = 0 with subject to boundary conditions (25) { f ′ ( 0 ) = 1 , f ( 0 ) = 0 , g ( 0 ) = 0 , g ′ ( 0 ) = d , θ ( 0 ) = 1 , Φ ( 0 ) = 1 , at η = 0 f ′ ( η ) = 0 , g ′ ( η ) = 0 , θ ( η ) = 0 , Φ ( η ) = 0 , as η → ∞ where, the Weissenberg number ( We ) , magnetic field parameter ( M ) , porosity parameter ( K ) , 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 ) along the x - and y -axis, thermal relaxation parameter ( β 1 ) , concentration relaxation parameter ( β 2 ) , Schmidt number ( Sc ) , Chemical reaction ( H ) , d is ratio of stretching sheet, 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 hybrid nanofluid, respectively can be described as follows: We = x Γ 2 a 3 ν f , M = σ f B 2 ρ f a , K = ν f a K p , Pr = ν f ( ρCp ) f κ f , R = 16 σ ∗ T ∞ 3 3 κ κ ∗ , θ w = T w T ∞ Q = Q 0 a ( ρCp ) f , Nb = D B τ ( C w − C ∞ ) ν f , Nt = D T τ ( T w − T ∞ ) ν f T ∞ , E c x = u w 2 ( T w − T ∞ ) ( Cp ) f E c y = v w 2 ( T w − T ∞ ) ( C p ) f , β 1 = λ e a , β 2 = λ c a , Sc = ν f D B , H = K r a , d = b a A 1 = μ hnf μ f , A 2 = ρ hnf ρ f , A 3 = σ hnf σ f , A 4 = κ hnf κ f , A 5 = ( ρCp ) hnf ( ρCp ) f The engineering components of the skin friction coefficients C fx and C fy , Nusselt number N u x and Sherwood number S h x are defined as (26) C fx = τ xz ρ f u w 2 , C fy = τ yz ρ f v 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 xz and τ yz are the wall shear stresses in the x- and y-direction respectively, given as (27) τ xz = μ hnf [ ∂ u ∂ z + Γ 2 ( ∂ u ∂ z ) 2 ] z = 0 , τ yz = μ hnf [ ∂ v ∂ z + Γ 2 ( ∂ v ∂ z ) 2 ] z = 0 and the wall heat and mass flux from the sheets, which is given by: (28) q w = − [ κ hnf + 16 σ ∗ T ∞ 3 3 κ ∗ ] ( ∂ T ∂ z ) z = 0 , j m = − D B ( ∂ C ∂ z ) z = 0 By using Equations (26) , (27) and (28) , the dimensionless variables, we obtain: (29) ( R e x ) 1 2 C fx = A 1 [ 1 + We 2 f ′ ′ ( 0 ) ] f ′ ′ ( 0 ) (30) d 1.5 ( R e y ) 1 2 C fy = A 1 [ 1 + We 2 g ′ ′ ( 0 ) ] g ′ ′ ( 0 ) (31) ( R e y ) − 1 2 N u x = − ( A 4 + R ) θ ′ ( 0 ) (32) ( R e y ) − 1 2 S h x = − Φ ′ ( 0 ) where R e y = y v w ν f and R e x = x u w ν f are the local Reynolds numbers. 3. Numerical method Using Table 1 , and the MATLAB software ( https://github.com/asfawmat/BVP-MATLAB-Implementation ) uses the bvp4c technique to numerically solve equations (21) through (24) and the boundary condition (25) . The bvp4c is an effective instrument that provides precision and dependability in addressing boundary value problems for differential equation systems. With a high degree of accuracy specifically, fourth- order accuracy this solver exemplifies a collocation technique that provides a precise, continuous solution. The mesh selection and error control are determined by evaluating the residual of the continuous solution. As discussed later, the boundary value problems must be transformed into a system of first-order initial value problems (IVPs) to apply the bvp4c method using the shooting technique. Now let us defined the new variable by the equation (33) { y 1 = f , y 2 = f ′ , y 3 = f ′ ′ , y 4 = g , y 5 = g ′ , y 6 = g ′ ′ , y 7 = θ , y 8 = θ ′ , y 9 = Φ , y 10 = Φ ′ . y 1 ′ = y 2 y 2 ′ = y 3 y 3 ′ = 1 A 1 A 2 ( 1 + We y 3 ) [ A 3 A 2 M y 2 + A 1 A 2 K y 2 + y 2 2 − ( y 1 + y 4 ) y 3 ] , y 4 ′ = y 5 , y 5 ′ = y 6 , y 6 ′ = 1 A 1 A 2 ( 1 + We y 6 ) [ A 3 A 2 M y 5 + A 1 A 2 K y 5 + y 5 2 − ( y 1 + y 4 ) y 6 ] , y 7 ′ = y 8 , y 8 ′ = − [ 1 ( A 4 A 5 Pr + R A 5 Pr ( 1 + ( θ w − 1 ) y 7 ) 3 − β 1 ( y 1 + y 4 ) 2 ) ] × [ R A 5 Pr ( 1 + ( θ w − 1 ) ) 2 × 3 ( θ w − 1 ) y 8 2 + ( y 1 + y 4 ) y 8 − β 1 ( y 1 + y 4 ) ( y 2 + y 5 ) y 8 + 1 A 5 Q y 7 + Nb y 8 y 10 + Nt y 8 2 + A 3 A 5 M ( E c x ( y 2 ) 2 + E c y ( y 5 ) 2 ) ] , y ′ 9 = y 10 , y 10 ′ = [ 1 ( 1 − β 2 Sc ( y 1 + y 4 ) 2 ) ] × [ β 2 Sc ( ( y 1 + y 4 ) ( y 2 + y 5 ) − Sc ( y 1 + y 4 ) ) y 10 + ScH y 9 − Nt Nb [ − [ 1 ( A 4 A 5 Pr + R A 5 Pr ( 1 + ( θ w − 1 ) θ ) 3 − β 1 ( y 1 + y 4 ) 2 ) ] × [ R A 5 Pr ( ( 1 + ( θ w − 1 ) ) 2 × 3 ( θ w − 1 ) y 8 2 ) + ( y 1 + y 4 ) y 8 − β 1 ( y 1 + y 4 ) ( y 2 + y 5 ) y 8 + 1 A 5 Q y 7 + Nb y 8 y 10 + Nt y 8 2 + A 3 A 5 M ( E c x ( y 2 ) 2 + E c y ( y 5 ) 2 ) ] ] with corresponding initial conditions (34) { y 1 ( 0 ) = 0 , y 2 ( 0 ) = 1 , y 3 ( 0 ) = α 1 , y 4 ( 0 ) = 0 , y 5 ( 0 ) = d y 6 ( 0 ) = α 2 , y 7 ( 0 ) = 1 , y 8 ( 0 ) = α 3 , y 9 ( 0 ) = 1 , y 10 ( 0 ) = α 4 where α 1 , α 2 , α 3 , and α 4 are the missing initial conditions. Table 1. Thermophysical properties. 39 , 43 Physical properties Ethylene glycol( C 2 H 6 O 2 )( f ) Copper ( Cu ( ϕ 1 )) Alumina ( Al 2 O 3 ( ϕ 2 )) ρ 1115 8933 3970 C p 2430 385 765 κ 0.253 400 40 σ 0.107 5.96×10 7 3.5×10 7 Utilizing the bvp4c technique, a system of differential equations of the form y ′ = f ( x , y ) is integrated according to the specified boundary conditions. The bvp4c routine uses finite difference method with an achievable accuracy of about 10 − 7 . 4. Results and Discussion In this section, we present the numerical solutions to the problem, considering various physical effects. The system of equations (21) - (24) is solved using the numerical bvp4c method, which satisfies the boundary conditions given by equation (25) . Additionally, we conducted a comparative analysis of our numerical results with those from previous studies to verify the accuracy of our solution and evaluate its consistency with earlier findings. The comparison of our findings with those from earlier research is presented in Table 2 . Comparing our findings with those of previous studies, the comparison’s results show generally excellent agreement. Using a variety of graphs, the impacts of various physical factors on temperature, concentration, velocity, mass transfer rate, surface drag coefficient, and the hybrid Cu-Al 2 O 3 / ethylene glycol nanofluid phase and nanofluid Al 2 O 3 / ethylene glycol phase are displayed. In each figure, a comparison of mono- and hybrid nanofluids is also provided. The numerical computations are discussed by keeping d = 0.5 , Sc = 1 , Nb = Nt = 0.1 , We = 0.2 , R = θ w = 1.2 , K = 0.1 , Pr = 1 , Ec = 0.2 , β 1 = β 2 = 0.1 , H = 0.2 , Q = 0.1 , M = 1,0.01 ≤ ϕ 1 ≤ 0.05 , 0.01 ≤ ϕ 2 ≤ 0.05 throughout the complete study. Table 2. Comparison of f ″ (0) and g ″ (0). d Wang 44 You & Wang 40 Present f ″ g ″ f ″ g ″ f ″ g ″ 0 -1 0 -1 - -1 0 0.25 -1.04881 -0.19456 -1.04881 - -1.048813 -0.194565 0.5 -1.09310 -0.46521 -1.09310 - -1.093096 -0.465205 0.75 -1.13449 -0.79462 -1.13449 - -1.134486 -0.794619 1 -1.17372 -1.17372 -1.17372 - -1.173721 -1.173721 An analysis of the differences between the numerical results of Wang, 44 You & Wang, 40 and the current study’s conclusions by calculating the values of f " ( 0 ) and g " ( 0 ) under the conditions of Pr = 1, Nb = 1 × 10 − 14, when We = M = K = β 1 = β 2 = R = θ w = Q = Nt = Ec = Sc = H = ϕ 1 = ϕ 2 = 0, as shown in Table 2 below. 4.1 Velocity characteristics Figures 2 and 3 illustrate the impact of the stretching ratio on both primary and secondary velocities. As the stretching ratio increases, the primary velocity profiles decrease, while the secondary velocity profiles of the MHD Williamson hybrid nanofluid flow increase. Because d = b / a it decreases when the stretching rate is applied along the x-axis direction. It is evident that the velocity component (a) in the x-direction and the velocity component (b) in the y-direction have an inverse connection with the stretching ratio parameter. The interaction between primary and secondary velocity with a magnetic field is depicted in Figures 4 and 5 . As the magnetic field strength increases, the resistance force also increases. The Lorentz force, which arises from the magnetic field, represents the resistance to fluid motion and can hinder the flow. As the magnetic field strengthens, both primary and secondary velocity profiles decrease. Figures 6 and 7 illustrate the effect of the Weissenberg number on these velocities. As the Weissenberg number increases, the velocity profiles (both primary and secondary) decrease because higher Weissenberg values reduce the relaxation time of the fluid particles. This increase in viscosity results in greater resistance to the fluid flow. Figures 8 and 9 demonstrate how the primary and secondary velocity profiles fall as the porosity parameter rises. This is because the hybrid nanofluid will pass through the sheet more readily as its permeability increases, influencing the flow. The contribution of volume fraction to the primary and secondary velocities of the mono- and hybrid nanofluids is shown in Figures 10 and 11 . This variation demonstrates how increased volume friction will result in an increase in the fluid’s viscous forces, which are its internal resistive forces. When nanoparticles are added to ethylene glycol, the viscous forces in the fluid increase, and the fluid’s velocity decreases. Figure 2. Primary velocity vs d. Figure 3. Secondary velocity vs d. Figure 4. Primary velocity vs M. Figure 5. Secondary velocity vs M . Figure 6. Primary velocity vs We. Figure 7. Secondary velocity vs We. Figure 8. Primary velocity vs K. Figure 9. Secondary velocity vs K. Figure 10. Primary velocity vs ϕ 1 , ϕ 2 . Figure 11. Secondary velocity vs ϕ 1 , ϕ 2 . 4.2 Thermal characteristics The impacts of the thermal relaxation parameter on fluid temperatures are illustrated in Figure 12 . A greater thermal relaxation parameter occurs with temperature diminish. This trend can be explained by the fact that the thermal relaxation parameter in a non-Fourier heat transfer process quantifies the lag time between the temperature gradient and the flux. The time it takes for a material to return to a particular proportion of its equilibrium temperature following a temperature change is known as its thermal relaxation time. The temperature profile graph’s decline over time suggests that the material is gradually cooling. This is caused by the thermal relaxation time; the temperature will drop more slowly the higher the value. Figure 13 depicts the thermal distribution against R. The temperature of the thermal system rises when R increases because it increases thermal efficiency (conductivity). The temperature distribution and the accompanying thickness of the boundary film both improve with an increase in the R estimate. As a result, the temperature in the boundary layer region rises. In this instance, hybrid nanoparticles outperform mono-nanofluid. The strength of thermal boundaries increases as the source of heat generation grows. This physical process leads to increased heat transfer, which elevates the thermal profiles, as shown in Figure 14 . The increase in thermal profiles is more pronounced for hybrid nanoparticles. The random motion of solid nanoparticles is directly related to higher values of the Brownian motion factor. As a result, the thermal boundary layer strengthens when the internal energy of the solid nanoparticles is converted into kinetic or heat energy. Consequently, the thermal properties rise as Brownian motion increases, as depicted in Figure 15 . Similarly, when the thermophoresis factor increases, additional heat is transferred from a region of higher concentration to a region of lower concentration. Consequently, an increase in the thermophoresis parameter is correlated with a corresponding rise in the thermal properties, as Figure 16 illustrates. The temperature functions improve when the magnetic parameter M grows in value, as seen by the temperature distribution graphs in Figure 17 . The valuable heat produced by the resistivity impact of the Lorentz force accounts for the rising thermal patterns observed for increasing values of magnetic parameters. These figures show that the curves produced by hybrid nanofluids are larger than those produced by mono-nanofluids. Hence, hybrid nanoparticles are noticed as more efficient at enhancing the base fluid temperature. The relationship between profiles of temperature and the Eckert number is seen in Figure 18 , where it is observed that temperature profiles rise as the Eckert number grows. From a physical perspective, a higher Eckert number corresponds to a higher amount of heat energy in the boundary layer. Frictional heating keeps heat energy from being produced. As Figure 19 demonstrates, the temperature falls as the stretching ratio grows. We can observe that temperature reduces as the stretching ratio parameter increases. This could happen as a result of the fluid’s temperature dropping as a result of an increase in the stretching ratio parameter, which increases the flow of the colder fluid at the ambient surface toward the hot surface. Increases in the volumetric fraction of solid nanoparticles provide greater resistance to the flow of fluid because the fluid particles have higher densities. In this process, more heat is transmitted from the hotter zone to the colder zone. A graph of the temperature profile resulting from an increasing volume percentage of nanoparticles is presented in Figure 20 . Physically, increasing the concentration of nanoparticles in the base fluid makes the fluid more viscous, leading to the formation of intermolecular frictional forces within the fluid. This results in a noticeable rise in the temperature curve at higher nanoparticle volume fractions. As a result, the temperature is observed to increase with a higher nanoparticle volume fraction. Figure 12. Temperature vs β 1 . Figure 13. Temperature vs R. Figure 14. Temperature vs Q. Figure 15. Temperature vs Nb. Figure 16. Temperature vs Nt. Figure 17. Temperature vs M. Figure 18. Temperature vs Ec. Figure 19. Temperature vs d. Figure 20. Temperature vs ϕ 1 , ϕ 2 . 4.3 Concentration characteristic Higher Schmidt numbers are linked to lower nanoparticle concentrations due to the inverse relationship between molecular diffusivity and the Schmidt number, as shown in Figure 21 . Figure 22 depicts the effect of the concentration relaxation parameter on the concentration field. As the concentration relaxation parameter increases, both the concentration profile and the thickness of the concentration boundary layer decrease. This occurs because the particles need more time to diffuse when the concentration relaxation time parameter increases. Figure 23 shows that both the solute’s boundary layer thickness and the nanoparticle concentration in the hybrid nanofluid decrease as the chemical reaction parameter increases. Changes in the intensity of the chemical reaction affect the fluid’s diffusivity, leading to a reduction in concentration. Finally, Figure 24 illustrates the decrease in the concentration profile as the stretching ratio increases. Figure 21. Concentration vs Sc. Figure 22. Concentration vs β 2 . Figure 23. Concentration vs H. Figure 24. Concentration vs d. 4.4 Influence of factors on Coefficient of skin friction, Nusselt and Sherwood numbers In Table 3 , the numerical results of skin friction drag along x- and y-directions ( − C fx Re 1 2 ) and ( − d 1.5 C fy Re 1 2 ) and the Nusselt number ( Nu x Re − 1 2 ) , the Sherwood number ( Sh x Re − 1 2 ) , the essential emerging parameters, are estimated using different dimensionless parameters for mono-nanofluids and hybrid nanofluids. For mono-nanofluids and hybrid nanofluids, skin friction drags rise in the x- and y-directions as ϕ 1 , ϕ 2 , M , and d grow but diminish as We grows for mono-nanofluids and hybrid nanofluids. For hybrid nanofluids, Nusselt numbers grow with the values of ϕ 1 , ϕ 2 , d and β 1 , but they decrease as We , M , θ w , H , β 2 , and R values increases. According to these findings, thermal radiation, temperature ratio parameters, magnetic forces, relaxation time, and other flow parameters all have a major impact on the rate of heat transfer close to the surface. Sherwood numbers drop with rising values of d , ϕ 1 , ϕ 2 , and β 1 , but increase with values of We , M , R , θ w , and H . These results indicate that the mass transfer rate near the surface is strongly affected by relaxation time, chemical processes, and other flow characteristics. Table 3. Mathematical data of coefficient of skin friction, Nusselt and Sherwood number of some values of parameters. We M ϕ 1 ϕ 2 d β 1 R θ w β 2 H − C fx Re 1 2 − d 3 2 C fy Re 1 2 Nu x Re − 1 2 Sh x Re − 1 2 0.1 1.939382 0.903738 0.488015 0.379954 0.2 1.873674 0.890098 0.478565 0.384044 0.3 1.794227 0.875349 0.467349 0.388856 0 1.479947 0.653997 0.662693 0.284334 1 1.873674 0.890098 0.478565 0.384044 2 2.185542 1.073919 0.347022 0.454532 0 0.01 1.461074 0.698628 0.421414 0.381833 0 0.03 1.545986 0.740268 0.451952 0.375229 0 0.05 1.634068 0.783370 0.484698 0.368544 0.01 0.01 1.506024 0.718540 0.418845 0.385502 0.03 0.03 1.684861 0.801975 0.446383 0.385221 0.05 0.05 1.873674 0.890098 0.478565 0.384044 0.1 1.814054 0.159463 0.342775 0.444855 0.5 1.873674 0.890098 0.478565 0.384044 1 1.940978 0.940978 0.588150 0.334435 0.1 1.873674 0.890098 0.478565 0.384044 0.2 1.873674 0.890098 0.489231 0.381149 0.3 1.873674 0.890098 0.501131 0.377928 0.2 1.873674 0.890098 0.478565 0.384044 0.3 1.873674 0.890098 0.470320 0.403437 0.4 1.873674 0.890098 0.464709 0.418902 1.2 1.873674 0.890098 0.478565 0.384044 1.4 1.873674 0.890098 0.416921 0.417224 1.6 1.873674 0.890098 0.347853 0.452785 0.1 1.873674 0.890098 0.478565 0.384044 0.2 1.873674 0.890098 0.477401 0.409733 0.3 1.873674 0.890098 0.476151 0.439006 0.2 1.873674 0.890098 0.478565 0.384044 1 1.873674 0.890098 0.467503 1.012973 2 1.873674 0.890098 0.464065 1.449359 Mathematical data of coefficient of skin friction, Nusselt and Sherwood number of some values of parameters for d = 0 . 5, Nb = Nt = θ w = Q = K = β 1 = β 2 = 0 . 1 ,We = H = R = Ec = 0 . 2, Pr = M = Sc = 1 , ϕ 1 = ϕ 2 = 0 . 05. 5. Conclusion This section presents the results for a 3D MHD Williamson hybrid nanofluid flow. The study utilizes the Cattaneo-Christov and modified Boungerno’s models to simulate the Cu−Al2O3/Ethylene glycol MHD Williamson hybrid nanofluid flow over a linearly stretching sheet, influenced by factors such as magnetic field, heat generation/absorption, non-linear thermal radiation, Joule heating, Brownian motion, thermophoresis, porosity, Williamson fluid parameter, Eckert number, Prandtl number, Schmidt number, chemical reaction parameter, stretching ratio, thermal relaxation, and concentration relaxation parameters. By applying appropriate similarity variables, the system of partial differential equations is transformed into a non-linear ordinary differential equation formulation. These ordinary differential equations are solved numerically using the bvp4c solver on the MATLAB platform. Similarly, graphical analysis is used to plot the obtained flow parameters so that their effects on flow, heat, and mass can be easily seen. Researchers and engineers working on related issues can benefit greatly from the obtained data. Moreover, our findings demonstrate the effectiveness and utility of hybrid nanofluid performance. The present study leads to the following deductions: • When the stretching ratio parameter rises, the primary velocity profile diminishes, but the secondary velocity profile grows. • The primary and secondary velocities diminishes with increasing porosity, the Williamson fluid, and the magnetic field parameter. • As the volume fractions of mono-nanofluid and hybrid nanofluid increase, both the primary and secondary velocities decrease. The addition of nanoparticles to a base fluid enhances the viscous forces, resulting in a reduction of the fluid’s velocity. However, the primary and secondary velocity profiles of the Williamson hybrid nanofluid model are notably higher compared to those of the mono-nanofluid and conventional base fluid models. • When the non-linear thermal radiation parameter, heat generation ( Q 0 > 0), Brownian motion, thermophoresis, magnetic field parameter, and Eckert number increases, the temperature profile rises. The temperature profile diminishes, when the stretching ratio and thermal relaxation parameter increases. • Temperature curves diminishes with increases estimates of nanofluid volume fraction and hybrid nanofluid volume fraction. When nanoparticles are added to a base fluid, the viscosity increases and the fluid temperature falls. When viscosity increases, temperature decreases. • As the stretching ratio, concentration relaxation, chemical reaction parameter, and Schmidt number rise, the concentration profile decreases. • As the volume fraction of hybrid nanoparticles increases, both the skin friction and Nusselt number of the Williamson hybrid nanofluid rise. However, the local Sherwood number of the MHD Williamson hybrid nanofluid decreases with a higher volume fraction of hybrid nanoparticles. • Finally, based on the current findings, we deduced that hybrid nanofluid performs better in ethylene glycol than mono-nanofluid. 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. Ethics and consent Ethical approval and consent were not required. Data availability statement Underlying data BVP-MATLAB-Implementation and Thermophysical properties: https://data.mendeley.com/datasets/s4447nmmmr/2 . 45 This project contains the following data: - Values of physical parameter and Matlab Code Tan_22: MATLAB Implementation [Data set]. Zenodo. https://doi.org/10.5281/zenodo.14480542 . 46 Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0). Software availability • Source code available from: 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 47 ( https://github.com/asfawmat/BVP-MATLAB-Implementation 47 ) for Matlab implementation. • Archived software available from: https://doi.org/10.5281/zenodo.14480542 46 • License: Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0). References 1. Choi SU, Eastman JA: Enhancing thermal conductivity of fluids with nanoparticles (No. ANL/MSD/CP-84938; CONF-951135-29). Argonne, IL (United States): Argonne National Lab. (ANL); 1995. 2. Mahian O, Bellos E, Markides CN, et al. : Recent advances in using nanofluids in renewable energy systems and the environmental implications of their uptake. Nano Energy. 2021; 86 : 106069. Publisher Full Text 3. 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Comments on this article Comments (0) Version 3 VERSION 3 PUBLISHED 14 Feb 2025 ADD YOUR COMMENT Comment Author details Author details 1 Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia Asfaw Tsegaye Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Software, Validation, Writing – Original Draft Preparation Eshetu Haile Roles: Conceptualization, Resources, Supervision, Visualization, Writing – Review & Editing Gurju Awgichew Roles: Resources, Supervision, Visualization, Writing – Review & Editing Hunegnaw Dessie Roles: Resources, Supervision, Visualization, 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 (3) version 3 Revised Published: 03 Dec 2025, 14:210 https://doi.org/10.12688/f1000research.160734.3 version 2 Revised Published: 31 Mar 2025, 14:210 https://doi.org/10.12688/f1000research.160734.2 version 1 Published: 14 Feb 2025, 14:210 https://doi.org/10.12688/f1000research.160734.1 Copyright © 2025 Tsegaye 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 A, Haile E, Awgichew G and Dessie H. Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.12688/f1000research.160734.1 ) 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 1 VERSION 1 PUBLISHED 14 Feb 2025 Views 0 Cite How to cite this report: K N. Reviewer Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.176670.r366915 ) The direct URL for this report is: https://f1000research.com/articles/14-210/v1#referee-response-366915 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 04 Mar 2025 Nandagopal K , Mohan Babu University, Tirupathi, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.176670.r366915 This study explores the Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet. However it requires some following corrections. The work can be suggested to ... Continue reading READ ALL This study explores the Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet. However it requires some following corrections. The work can be suggested to indexed after the following corrections are fulfilled. 1. The whole article may be checked for typo and other grammatical mistakes. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? 3. What are the advantages of BVP4C over other methods? 4.Justify the selection of transformation to reduce the PDEs into ODEs. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. 6. Add Nomenclature separately 7. What is the significance of studying non-linear thermal radiation? 8. Add some result for stream lines to make their study sound more unique. 9. What is the difference between thermal radiation and non-linear thermal radiation? 10. How current results present directions to the future studies. Summarize in conclusion section. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. The present research work may be suggested to publish in F1000 research after the following corrections are fulfilled. 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? Partly References 1. Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation . 2024. 1-18 Publisher Full Text 2. M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport . 2024. 1-26 Publisher Full Text 3. Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications . 2023. 1-17 Publisher Full Text 4. Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media . 2022. 1-20 Publisher Full Text 5. Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering . 2022; 144 (4). Publisher Full Text 6. Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics . 2025; 13 . Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fluid dynamics , peristalsis, porous medium 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 K N. Reviewer Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.176670.r366915 ) The direct URL for this report is: https://f1000research.com/articles/14-210/v1#referee-response-366915 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 20 May 2025 Asfaw Tsegaye , Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia 20 May 2025 Author Response Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers ... 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. The whole article may be checked for typos and other grammatical mistakes. Author’s reply: We have carefully reviewed the manuscript and corrected all typographical and grammatical errors to ensure clarity and readability. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? Author’s reply: The specific boundary conditions used in this study are justified based on the physical and mathematical framework of the problem: a. Physical Relevance: - The boundary conditions at z=0(w=0, v= v w =by, u= u w =ax, T= T w , C= C w ) ​ represent a stretching sheet, which is common in industrial applications such as polymer extrusion, metal forming, and coating processes. - The prescribed velocity components ( u w =ax, v w =by) mimic the stretching motion of the sheet in two directions, ensuring a controlled boundary layer development. b. Mathematical Consistency: - At the sheet ( z=0) , specifying velocity ensures that the no-slip condition is satisfied, which is fundamental for boundary layer flow. - The temperature and concentration at the sheet ( T w , C w ) ​ are held constant to analyze the effects of heat and mass transfer within the hybrid nanofluid. c. Far-Field Conditions: - As z→∞ , the velocity components (u, v) approach zero, ensuring that the fluid is at rest far from the sheet. - The temperature and concentration also reach ambient conditions ( T ∞ , C ∞ ) , signifying thermal and concentration equilibrium. d. Applicability to Cattaneo-Christov Model: - The use of Cattaneo-Christov heat and mass flux models requires that thermal and concentration relaxation effects be properly accounted for at the boundaries. - The conditions ensure a well-posed problem, allowing numerical solutions via similarity transformations and MATLAB’s bvp4c solver. Overall, these boundary conditions are essential for accurately capturing the physics of the problem while ensuring numerical stability and consistency with previous studies. 3. What are the advantages of BVP4C over other methods? Author’s reply: The BVP4C method in MATLAB offers several advantages over other numerical methods for solving boundary value problems: a. High Accuracy: - The BVP4C solver utilizes a finite difference method with collocation techniques, achieving an accuracy of approximately 10 -7 . This makes it highly precise for solving boundary layer flow problems. b. Built-in Error Control: - BVP4C automatically adjusts the mesh selection and step size based on residual error estimates, ensuring numerical stability and reducing computational errors. c. Efficient Handling of Singularities: - It can handle singular boundary conditions effectively, which is particularly useful in fluid dynamics problems involving stretching sheets , hybrid nanofluids, and thermal radiation effects. d. Adaptive Mesh Refinement: - Unlike shooting methods, which require iterative guessing, BVP4C refines the solution using an adaptive mesh, leading to more efficient computations without divergence issues. e. Well-Suited for Stiff Equations: - Since boundary layer problems often involve stiff differential equations, BVP4C efficiently handles them without requiring excessive computational power. f. Direct Implementation of Boundary Conditions: - Unlike shooting methods, where unknown initial conditions need estimation, BVP4C directly incorporates boundary conditions, making it more robust for complex problems​ 4. Justify the selection of transformation to reduce the PDEs into ODEs. Author’s reply: The selection of similarity transformations to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs) is justified based on several key factors: a. Reduction of Complexity - The governing equations of the hybrid nanofluid flow problem are highly nonlinear PDEs. Using similarity transformations simplifies these equations into a set of ODEs, making them more manageable for numerical or analytical solutions. b. Boundary Layer Theory - The similarity variable η=z a ν f is chosen to convert the spatial dependency of the velocity, temperature, and concentration into a single variable. This transformation is in line with classical boundary layer theory, which seeks to reduce multi-dimensional flow problems into a one- dimensional form for efficient analysis. c. Consistency with Existing Literature - The chosen transformations align with prior studies on fluid flow over stretching sheets, ensuring that the results can be compared with previous findings and validated accordingly. d. Mathematical and Computational Efficiency - The resulting system of ODEs is more suitable for numerical techniques like MATLAB’s bvp4c , which is specifically designed for solving boundary value problems with high accuracy and stability. e. Physical Relevance - The transformed equations retain the fundamental physical characteristics of the original PDEs, capturing key effects like thermal radiation, hybrid nanofluid properties, and the Cattaneo-Christov heat and mass flux model.By using these similarity transformations, the study ensures that the problem formulation is both mathematically rigorous and computationally efficient. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. Author’s reply: The discussion section has been revised to provide deeper insights into the physical mechanisms governing hybrid nanofluid flow. We have elaborated on the effects of dimensionless parameters on velocity, temperature, and concentration fields, supported by quantitative analysis and comparison with existing studies. 6. Add Nomenclature separately Author’s reply: A separate nomenclature section has been included to define symbols and parameters used throughout the study for better readability. 7. What is the significance of studying non-linear thermal radiation? Author’s reply: Studying non-linear thermal radiation is essential for accurately modeling heat transfer in high-temperature environments, as it accounts for complex interactions beyond the Stefan-Boltzmann Law. It plays a crucial role in applications like aerospace, nuclear reactors, and industrial heat exchangers, where extreme temperatures amplify radiative effects. Non-linear models enhance heat flux predictions, particularly in nanofluid-based systems, where radiation significantly influences thermal behavior. In hybrid nanofluids like Cu-Al₂O₃/ethylene glycol, it affects boundary layer characteristics and heat dissipation efficiency. Understanding these effects helps optimize thermal management systems, improving cooling performance in electronics, solar panels, and combustion chambers. Additional discussion on its significance has been included in the introduction section. 8. Add some result for streamlines to make their study sound more unique. Author’s reply: We have outlined future research directions on results for plotting streamlines and will continue exploring this topic in upcoming studies. 9. What is the difference between thermal radiation and non-linear thermal radiation? Author’s reply: Thermal radiation follows the Stefan-Boltzmann law , where heat flux is proportional to the fourth power of temperature and is often linearized under small temperature variations. In contrast, non-linear thermal radiation arises in high-temperature environments where heat flux exhibits a more complex dependence on temperature, influenced by temperature gradients, material optical properties, and medium characteristics . This non-linearity makes it more accurate for modeling extreme heat transfer conditions, such as those found in nanofluids, aerospace applications, and combustion chambers . We have added a clarification in the introduction section. 10. How current results present directions to the future studies. Summarize in conclusion section. Author’s reply: The current study highlights key factors influencing hybrid nanofluid flow, providing directions for future research. Further studies can explore optimized stretching conditions, alternative nanoparticle materials, and advanced thermal models, including non-linear radiation effects. Investigating unsteady flow, non-Newtonian fluids, and three-dimensional effects will enhance the findings' applicability in engineering and biomedical fields. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. - Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation. 2024. 1-18 Publisher Full Text -M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport. 2024. 1-26 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications. 2023. 1-17 Publisher Full Text - - Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media. 2022. 1-20 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering. 2022; 144 (4). Publisher Full Text - - Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics. 2025; 13. Publisher Full Text Author’s reply: 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 manuscript. 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. The whole article may be checked for typos and other grammatical mistakes. Author’s reply: We have carefully reviewed the manuscript and corrected all typographical and grammatical errors to ensure clarity and readability. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? Author’s reply: The specific boundary conditions used in this study are justified based on the physical and mathematical framework of the problem: a. Physical Relevance: - The boundary conditions at z=0(w=0, v= v w =by, u= u w =ax, T= T w , C= C w ) ​ represent a stretching sheet, which is common in industrial applications such as polymer extrusion, metal forming, and coating processes. - The prescribed velocity components ( u w =ax, v w =by) mimic the stretching motion of the sheet in two directions, ensuring a controlled boundary layer development. b. Mathematical Consistency: - At the sheet ( z=0) , specifying velocity ensures that the no-slip condition is satisfied, which is fundamental for boundary layer flow. - The temperature and concentration at the sheet ( T w , C w ) ​ are held constant to analyze the effects of heat and mass transfer within the hybrid nanofluid. c. Far-Field Conditions: - As z→∞ , the velocity components (u, v) approach zero, ensuring that the fluid is at rest far from the sheet. - The temperature and concentration also reach ambient conditions ( T ∞ , C ∞ ) , signifying thermal and concentration equilibrium. d. Applicability to Cattaneo-Christov Model: - The use of Cattaneo-Christov heat and mass flux models requires that thermal and concentration relaxation effects be properly accounted for at the boundaries. - The conditions ensure a well-posed problem, allowing numerical solutions via similarity transformations and MATLAB’s bvp4c solver. Overall, these boundary conditions are essential for accurately capturing the physics of the problem while ensuring numerical stability and consistency with previous studies. 3. What are the advantages of BVP4C over other methods? Author’s reply: The BVP4C method in MATLAB offers several advantages over other numerical methods for solving boundary value problems: a. High Accuracy: - The BVP4C solver utilizes a finite difference method with collocation techniques, achieving an accuracy of approximately 10 -7 . This makes it highly precise for solving boundary layer flow problems. b. Built-in Error Control: - BVP4C automatically adjusts the mesh selection and step size based on residual error estimates, ensuring numerical stability and reducing computational errors. c. Efficient Handling of Singularities: - It can handle singular boundary conditions effectively, which is particularly useful in fluid dynamics problems involving stretching sheets , hybrid nanofluids, and thermal radiation effects. d. Adaptive Mesh Refinement: - Unlike shooting methods, which require iterative guessing, BVP4C refines the solution using an adaptive mesh, leading to more efficient computations without divergence issues. e. Well-Suited for Stiff Equations: - Since boundary layer problems often involve stiff differential equations, BVP4C efficiently handles them without requiring excessive computational power. f. Direct Implementation of Boundary Conditions: - Unlike shooting methods, where unknown initial conditions need estimation, BVP4C directly incorporates boundary conditions, making it more robust for complex problems​ 4. Justify the selection of transformation to reduce the PDEs into ODEs. Author’s reply: The selection of similarity transformations to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs) is justified based on several key factors: a. Reduction of Complexity - The governing equations of the hybrid nanofluid flow problem are highly nonlinear PDEs. Using similarity transformations simplifies these equations into a set of ODEs, making them more manageable for numerical or analytical solutions. b. Boundary Layer Theory - The similarity variable η=z a ν f is chosen to convert the spatial dependency of the velocity, temperature, and concentration into a single variable. This transformation is in line with classical boundary layer theory, which seeks to reduce multi-dimensional flow problems into a one- dimensional form for efficient analysis. c. Consistency with Existing Literature - The chosen transformations align with prior studies on fluid flow over stretching sheets, ensuring that the results can be compared with previous findings and validated accordingly. d. Mathematical and Computational Efficiency - The resulting system of ODEs is more suitable for numerical techniques like MATLAB’s bvp4c , which is specifically designed for solving boundary value problems with high accuracy and stability. e. Physical Relevance - The transformed equations retain the fundamental physical characteristics of the original PDEs, capturing key effects like thermal radiation, hybrid nanofluid properties, and the Cattaneo-Christov heat and mass flux model.By using these similarity transformations, the study ensures that the problem formulation is both mathematically rigorous and computationally efficient. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. Author’s reply: The discussion section has been revised to provide deeper insights into the physical mechanisms governing hybrid nanofluid flow. We have elaborated on the effects of dimensionless parameters on velocity, temperature, and concentration fields, supported by quantitative analysis and comparison with existing studies. 6. Add Nomenclature separately Author’s reply: A separate nomenclature section has been included to define symbols and parameters used throughout the study for better readability. 7. What is the significance of studying non-linear thermal radiation? Author’s reply: Studying non-linear thermal radiation is essential for accurately modeling heat transfer in high-temperature environments, as it accounts for complex interactions beyond the Stefan-Boltzmann Law. It plays a crucial role in applications like aerospace, nuclear reactors, and industrial heat exchangers, where extreme temperatures amplify radiative effects. Non-linear models enhance heat flux predictions, particularly in nanofluid-based systems, where radiation significantly influences thermal behavior. In hybrid nanofluids like Cu-Al₂O₃/ethylene glycol, it affects boundary layer characteristics and heat dissipation efficiency. Understanding these effects helps optimize thermal management systems, improving cooling performance in electronics, solar panels, and combustion chambers. Additional discussion on its significance has been included in the introduction section. 8. Add some result for streamlines to make their study sound more unique. Author’s reply: We have outlined future research directions on results for plotting streamlines and will continue exploring this topic in upcoming studies. 9. What is the difference between thermal radiation and non-linear thermal radiation? Author’s reply: Thermal radiation follows the Stefan-Boltzmann law , where heat flux is proportional to the fourth power of temperature and is often linearized under small temperature variations. In contrast, non-linear thermal radiation arises in high-temperature environments where heat flux exhibits a more complex dependence on temperature, influenced by temperature gradients, material optical properties, and medium characteristics . This non-linearity makes it more accurate for modeling extreme heat transfer conditions, such as those found in nanofluids, aerospace applications, and combustion chambers . We have added a clarification in the introduction section. 10. How current results present directions to the future studies. Summarize in conclusion section. Author’s reply: The current study highlights key factors influencing hybrid nanofluid flow, providing directions for future research. Further studies can explore optimized stretching conditions, alternative nanoparticle materials, and advanced thermal models, including non-linear radiation effects. Investigating unsteady flow, non-Newtonian fluids, and three-dimensional effects will enhance the findings' applicability in engineering and biomedical fields. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. - Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation. 2024. 1-18 Publisher Full Text -M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport. 2024. 1-26 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications. 2023. 1-17 Publisher Full Text - - Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media. 2022. 1-20 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering. 2022; 144 (4). Publisher Full Text - - Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics. 2025; 13. Publisher Full Text Author’s reply: 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 manuscript. 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 20 May 2025 Asfaw Tsegaye , Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia 20 May 2025 Author Response Author’s Response to Reviewers’ Comments We sincerely appreciate the reviewers ... 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. The whole article may be checked for typos and other grammatical mistakes. Author’s reply: We have carefully reviewed the manuscript and corrected all typographical and grammatical errors to ensure clarity and readability. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? Author’s reply: The specific boundary conditions used in this study are justified based on the physical and mathematical framework of the problem: a. Physical Relevance: - The boundary conditions at z=0(w=0, v= v w =by, u= u w =ax, T= T w , C= C w ) ​ represent a stretching sheet, which is common in industrial applications such as polymer extrusion, metal forming, and coating processes. - The prescribed velocity components ( u w =ax, v w =by) mimic the stretching motion of the sheet in two directions, ensuring a controlled boundary layer development. b. Mathematical Consistency: - At the sheet ( z=0) , specifying velocity ensures that the no-slip condition is satisfied, which is fundamental for boundary layer flow. - The temperature and concentration at the sheet ( T w , C w ) ​ are held constant to analyze the effects of heat and mass transfer within the hybrid nanofluid. c. Far-Field Conditions: - As z→∞ , the velocity components (u, v) approach zero, ensuring that the fluid is at rest far from the sheet. - The temperature and concentration also reach ambient conditions ( T ∞ , C ∞ ) , signifying thermal and concentration equilibrium. d. Applicability to Cattaneo-Christov Model: - The use of Cattaneo-Christov heat and mass flux models requires that thermal and concentration relaxation effects be properly accounted for at the boundaries. - The conditions ensure a well-posed problem, allowing numerical solutions via similarity transformations and MATLAB’s bvp4c solver. Overall, these boundary conditions are essential for accurately capturing the physics of the problem while ensuring numerical stability and consistency with previous studies. 3. What are the advantages of BVP4C over other methods? Author’s reply: The BVP4C method in MATLAB offers several advantages over other numerical methods for solving boundary value problems: a. High Accuracy: - The BVP4C solver utilizes a finite difference method with collocation techniques, achieving an accuracy of approximately 10 -7 . This makes it highly precise for solving boundary layer flow problems. b. Built-in Error Control: - BVP4C automatically adjusts the mesh selection and step size based on residual error estimates, ensuring numerical stability and reducing computational errors. c. Efficient Handling of Singularities: - It can handle singular boundary conditions effectively, which is particularly useful in fluid dynamics problems involving stretching sheets , hybrid nanofluids, and thermal radiation effects. d. Adaptive Mesh Refinement: - Unlike shooting methods, which require iterative guessing, BVP4C refines the solution using an adaptive mesh, leading to more efficient computations without divergence issues. e. Well-Suited for Stiff Equations: - Since boundary layer problems often involve stiff differential equations, BVP4C efficiently handles them without requiring excessive computational power. f. Direct Implementation of Boundary Conditions: - Unlike shooting methods, where unknown initial conditions need estimation, BVP4C directly incorporates boundary conditions, making it more robust for complex problems​ 4. Justify the selection of transformation to reduce the PDEs into ODEs. Author’s reply: The selection of similarity transformations to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs) is justified based on several key factors: a. Reduction of Complexity - The governing equations of the hybrid nanofluid flow problem are highly nonlinear PDEs. Using similarity transformations simplifies these equations into a set of ODEs, making them more manageable for numerical or analytical solutions. b. Boundary Layer Theory - The similarity variable η=z a ν f is chosen to convert the spatial dependency of the velocity, temperature, and concentration into a single variable. This transformation is in line with classical boundary layer theory, which seeks to reduce multi-dimensional flow problems into a one- dimensional form for efficient analysis. c. Consistency with Existing Literature - The chosen transformations align with prior studies on fluid flow over stretching sheets, ensuring that the results can be compared with previous findings and validated accordingly. d. Mathematical and Computational Efficiency - The resulting system of ODEs is more suitable for numerical techniques like MATLAB’s bvp4c , which is specifically designed for solving boundary value problems with high accuracy and stability. e. Physical Relevance - The transformed equations retain the fundamental physical characteristics of the original PDEs, capturing key effects like thermal radiation, hybrid nanofluid properties, and the Cattaneo-Christov heat and mass flux model.By using these similarity transformations, the study ensures that the problem formulation is both mathematically rigorous and computationally efficient. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. Author’s reply: The discussion section has been revised to provide deeper insights into the physical mechanisms governing hybrid nanofluid flow. We have elaborated on the effects of dimensionless parameters on velocity, temperature, and concentration fields, supported by quantitative analysis and comparison with existing studies. 6. Add Nomenclature separately Author’s reply: A separate nomenclature section has been included to define symbols and parameters used throughout the study for better readability. 7. What is the significance of studying non-linear thermal radiation? Author’s reply: Studying non-linear thermal radiation is essential for accurately modeling heat transfer in high-temperature environments, as it accounts for complex interactions beyond the Stefan-Boltzmann Law. It plays a crucial role in applications like aerospace, nuclear reactors, and industrial heat exchangers, where extreme temperatures amplify radiative effects. Non-linear models enhance heat flux predictions, particularly in nanofluid-based systems, where radiation significantly influences thermal behavior. In hybrid nanofluids like Cu-Al₂O₃/ethylene glycol, it affects boundary layer characteristics and heat dissipation efficiency. Understanding these effects helps optimize thermal management systems, improving cooling performance in electronics, solar panels, and combustion chambers. Additional discussion on its significance has been included in the introduction section. 8. Add some result for streamlines to make their study sound more unique. Author’s reply: We have outlined future research directions on results for plotting streamlines and will continue exploring this topic in upcoming studies. 9. What is the difference between thermal radiation and non-linear thermal radiation? Author’s reply: Thermal radiation follows the Stefan-Boltzmann law , where heat flux is proportional to the fourth power of temperature and is often linearized under small temperature variations. In contrast, non-linear thermal radiation arises in high-temperature environments where heat flux exhibits a more complex dependence on temperature, influenced by temperature gradients, material optical properties, and medium characteristics . This non-linearity makes it more accurate for modeling extreme heat transfer conditions, such as those found in nanofluids, aerospace applications, and combustion chambers . We have added a clarification in the introduction section. 10. How current results present directions to the future studies. Summarize in conclusion section. Author’s reply: The current study highlights key factors influencing hybrid nanofluid flow, providing directions for future research. Further studies can explore optimized stretching conditions, alternative nanoparticle materials, and advanced thermal models, including non-linear radiation effects. Investigating unsteady flow, non-Newtonian fluids, and three-dimensional effects will enhance the findings' applicability in engineering and biomedical fields. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. - Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation. 2024. 1-18 Publisher Full Text -M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport. 2024. 1-26 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications. 2023. 1-17 Publisher Full Text - - Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media. 2022. 1-20 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering. 2022; 144 (4). Publisher Full Text - - Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics. 2025; 13. Publisher Full Text Author’s reply: 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 manuscript. 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. The whole article may be checked for typos and other grammatical mistakes. Author’s reply: We have carefully reviewed the manuscript and corrected all typographical and grammatical errors to ensure clarity and readability. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? Author’s reply: The specific boundary conditions used in this study are justified based on the physical and mathematical framework of the problem: a. Physical Relevance: - The boundary conditions at z=0(w=0, v= v w =by, u= u w =ax, T= T w , C= C w ) ​ represent a stretching sheet, which is common in industrial applications such as polymer extrusion, metal forming, and coating processes. - The prescribed velocity components ( u w =ax, v w =by) mimic the stretching motion of the sheet in two directions, ensuring a controlled boundary layer development. b. Mathematical Consistency: - At the sheet ( z=0) , specifying velocity ensures that the no-slip condition is satisfied, which is fundamental for boundary layer flow. - The temperature and concentration at the sheet ( T w , C w ) ​ are held constant to analyze the effects of heat and mass transfer within the hybrid nanofluid. c. Far-Field Conditions: - As z→∞ , the velocity components (u, v) approach zero, ensuring that the fluid is at rest far from the sheet. - The temperature and concentration also reach ambient conditions ( T ∞ , C ∞ ) , signifying thermal and concentration equilibrium. d. Applicability to Cattaneo-Christov Model: - The use of Cattaneo-Christov heat and mass flux models requires that thermal and concentration relaxation effects be properly accounted for at the boundaries. - The conditions ensure a well-posed problem, allowing numerical solutions via similarity transformations and MATLAB’s bvp4c solver. Overall, these boundary conditions are essential for accurately capturing the physics of the problem while ensuring numerical stability and consistency with previous studies. 3. What are the advantages of BVP4C over other methods? Author’s reply: The BVP4C method in MATLAB offers several advantages over other numerical methods for solving boundary value problems: a. High Accuracy: - The BVP4C solver utilizes a finite difference method with collocation techniques, achieving an accuracy of approximately 10 -7 . This makes it highly precise for solving boundary layer flow problems. b. Built-in Error Control: - BVP4C automatically adjusts the mesh selection and step size based on residual error estimates, ensuring numerical stability and reducing computational errors. c. Efficient Handling of Singularities: - It can handle singular boundary conditions effectively, which is particularly useful in fluid dynamics problems involving stretching sheets , hybrid nanofluids, and thermal radiation effects. d. Adaptive Mesh Refinement: - Unlike shooting methods, which require iterative guessing, BVP4C refines the solution using an adaptive mesh, leading to more efficient computations without divergence issues. e. Well-Suited for Stiff Equations: - Since boundary layer problems often involve stiff differential equations, BVP4C efficiently handles them without requiring excessive computational power. f. Direct Implementation of Boundary Conditions: - Unlike shooting methods, where unknown initial conditions need estimation, BVP4C directly incorporates boundary conditions, making it more robust for complex problems​ 4. Justify the selection of transformation to reduce the PDEs into ODEs. Author’s reply: The selection of similarity transformations to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs) is justified based on several key factors: a. Reduction of Complexity - The governing equations of the hybrid nanofluid flow problem are highly nonlinear PDEs. Using similarity transformations simplifies these equations into a set of ODEs, making them more manageable for numerical or analytical solutions. b. Boundary Layer Theory - The similarity variable η=z a ν f is chosen to convert the spatial dependency of the velocity, temperature, and concentration into a single variable. This transformation is in line with classical boundary layer theory, which seeks to reduce multi-dimensional flow problems into a one- dimensional form for efficient analysis. c. Consistency with Existing Literature - The chosen transformations align with prior studies on fluid flow over stretching sheets, ensuring that the results can be compared with previous findings and validated accordingly. d. Mathematical and Computational Efficiency - The resulting system of ODEs is more suitable for numerical techniques like MATLAB’s bvp4c , which is specifically designed for solving boundary value problems with high accuracy and stability. e. Physical Relevance - The transformed equations retain the fundamental physical characteristics of the original PDEs, capturing key effects like thermal radiation, hybrid nanofluid properties, and the Cattaneo-Christov heat and mass flux model.By using these similarity transformations, the study ensures that the problem formulation is both mathematically rigorous and computationally efficient. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. Author’s reply: The discussion section has been revised to provide deeper insights into the physical mechanisms governing hybrid nanofluid flow. We have elaborated on the effects of dimensionless parameters on velocity, temperature, and concentration fields, supported by quantitative analysis and comparison with existing studies. 6. Add Nomenclature separately Author’s reply: A separate nomenclature section has been included to define symbols and parameters used throughout the study for better readability. 7. What is the significance of studying non-linear thermal radiation? Author’s reply: Studying non-linear thermal radiation is essential for accurately modeling heat transfer in high-temperature environments, as it accounts for complex interactions beyond the Stefan-Boltzmann Law. It plays a crucial role in applications like aerospace, nuclear reactors, and industrial heat exchangers, where extreme temperatures amplify radiative effects. Non-linear models enhance heat flux predictions, particularly in nanofluid-based systems, where radiation significantly influences thermal behavior. In hybrid nanofluids like Cu-Al₂O₃/ethylene glycol, it affects boundary layer characteristics and heat dissipation efficiency. Understanding these effects helps optimize thermal management systems, improving cooling performance in electronics, solar panels, and combustion chambers. Additional discussion on its significance has been included in the introduction section. 8. Add some result for streamlines to make their study sound more unique. Author’s reply: We have outlined future research directions on results for plotting streamlines and will continue exploring this topic in upcoming studies. 9. What is the difference between thermal radiation and non-linear thermal radiation? Author’s reply: Thermal radiation follows the Stefan-Boltzmann law , where heat flux is proportional to the fourth power of temperature and is often linearized under small temperature variations. In contrast, non-linear thermal radiation arises in high-temperature environments where heat flux exhibits a more complex dependence on temperature, influenced by temperature gradients, material optical properties, and medium characteristics . This non-linearity makes it more accurate for modeling extreme heat transfer conditions, such as those found in nanofluids, aerospace applications, and combustion chambers . We have added a clarification in the introduction section. 10. How current results present directions to the future studies. Summarize in conclusion section. Author’s reply: The current study highlights key factors influencing hybrid nanofluid flow, providing directions for future research. Further studies can explore optimized stretching conditions, alternative nanoparticle materials, and advanced thermal models, including non-linear radiation effects. Investigating unsteady flow, non-Newtonian fluids, and three-dimensional effects will enhance the findings' applicability in engineering and biomedical fields. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. - Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation. 2024. 1-18 Publisher Full Text -M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport. 2024. 1-26 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications. 2023. 1-17 Publisher Full Text - - Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media. 2022. 1-20 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering. 2022; 144 (4). Publisher Full Text - - Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics. 2025; 13. Publisher Full Text Author’s reply: 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 manuscript. 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 Views 0 Cite How to cite this report: Fatunmbi EO. Reviewer Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.176670.r366913 ) The direct URL for this report is: https://f1000research.com/articles/14-210/v1#referee-response-366913 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 28 Feb 2025 Ephesus Olusoji Fatunmbi , Federal Polytechnic, Ilaro, Nigeria Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.176670.r366913 Reviewer comments on the manuscript titled “ Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet” The following are my comments on this manuscript: ... Continue reading READ ALL Reviewer comments on the manuscript titled “ Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet” The following are my comments on this manuscript: 1). The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. 2). The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. 3). Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. 4). Why the choice of the method of solution? Include the importance and limitations of this method in the study. 5). The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. 6). Include the future scope of study after the conclusion section. 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? Partly Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Applied mathematics, Computational fluid dynamics, Nano fluid, Boundary layer flow, etc. 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 Fatunmbi EO. Reviewer Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.176670.r366913 ) The direct URL for this report is: https://f1000research.com/articles/14-210/v1#referee-response-366913 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 20 May 2025 Asfaw Tsegaye , Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia 20 May 2025 Author Response Author’s Response to Reviewers’ Comments We sincerely appreciate the ... 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. The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. Author’s reply: The introduction has been revised to clearly highlight the significance of hybrid nanofluids in industrial applications such as cooling systems, biomedical engineering, and energy systems. The last paragraph of the introduction explicitly identifies the research gap by emphasizing that no prior studies have investigated the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, Joule heating, and chemical reactions on Williamson hybrid nanofluid flow over a stretching porous sheet. The findings of this study have significant implications in various engineering and industrial applications , including aerospace engineering, nuclear power plants, electronic cooling systems, biomedical applications, and polymer and textile industries. 2. The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. Author’s reply: Thank you for your suggestions. A dedicated section has been added to justify the assumptions made in modeling the problem. The inclusion of specific terms in the governing equations (such as Steady-State, Three-Dimensional Flow, Williamson Non-Newtonian Fluid Model, Cattaneo-Christov Heat and Mass Flux Model, Non-Linear Thermal Radiation, Joule Heating, Viscous Dissipation, Porous Medium and Magnetic Field Effects, heat generation, chemical reaction terms, and Boundary Conditions) is now explicitly explained with appropriate references. Additional citations have been included to validate the selection of mathematical models for hybrid nanofluids. 3. Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. Author’s reply: This study utilizes a Cu-Al₂O₃/Ethylene Glycol hybrid nanofluid, chosen for its superior thermal properties, stability, and industrial relevance. Copper (Cu) enhances thermal and electrical conductivity, making it ideal for electronics and automotive cooling, while Aluminum Oxide (Al₂O₃) offers stability, corrosion resistance, and broad industrial applications. Ethylene Glycol serves as the base fluid due to its widespread use in cooling systems, wide temperature stability, and effective nanoparticle dispersion. This hybrid nanofluid is applicable in aerospace, nuclear reactors, biomedical engineering, electronic cooling, and renewable energy systems, making it an optimal choice for advanced heat transfer applications. 4. Why the choice of the method of solution? Include the importance and limitations of this method in the study. Author’s reply: In this study, the MATLAB BVP4C solver is used to solve the transformed ordinary differential equations (ODEs) derived from the governing partial differential equations (PDEs). BVP4C is chosen for its high accuracy, adaptive mesh refinement, and robustness in handling boundary value problems (BVPs), making it well-suited for nonlinear and coupled ODEs arising in Williamson hybrid nanofluid flow. Unlike shooting methods, BVP4C ensures stability and efficiently manages multi-point boundary conditions, making it ideal for fluid flow and heat transfer problems. While alternative methods like Runge-Kutta and finite difference approaches were considered, they were found to be less efficient due to stability and convergence issues. However, BVP4C’s computational intensity requires careful selection of initial guesses. 5. The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. Author’s reply: The revised discussion section now offers a detailed scientific interpretation of velocity, temperature, and concentration trends, supported by physical laws and real-world applications. Each parameter's impact is analyzed with scientific reasoning and quantitative analysis , and comparisons with existing literature are included to validate the findings. 6. Include the future scope of the study after the conclusion section. Author’s reply: A dedicated section on future research directions follows the conclusion, outlining potential extensions of this study. These include examining unsteady flow conditions to capture transient effects, exploring various hybrid nanoparticle combinations to enhance heat transfer performance, and extending the model to incorporate non-Newtonian rheological properties relevant to biomedical and industrial applications. 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] 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. The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. Author’s reply: The introduction has been revised to clearly highlight the significance of hybrid nanofluids in industrial applications such as cooling systems, biomedical engineering, and energy systems. The last paragraph of the introduction explicitly identifies the research gap by emphasizing that no prior studies have investigated the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, Joule heating, and chemical reactions on Williamson hybrid nanofluid flow over a stretching porous sheet. The findings of this study have significant implications in various engineering and industrial applications , including aerospace engineering, nuclear power plants, electronic cooling systems, biomedical applications, and polymer and textile industries. 2. The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. Author’s reply: Thank you for your suggestions. A dedicated section has been added to justify the assumptions made in modeling the problem. The inclusion of specific terms in the governing equations (such as Steady-State, Three-Dimensional Flow, Williamson Non-Newtonian Fluid Model, Cattaneo-Christov Heat and Mass Flux Model, Non-Linear Thermal Radiation, Joule Heating, Viscous Dissipation, Porous Medium and Magnetic Field Effects, heat generation, chemical reaction terms, and Boundary Conditions) is now explicitly explained with appropriate references. Additional citations have been included to validate the selection of mathematical models for hybrid nanofluids. 3. Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. Author’s reply: This study utilizes a Cu-Al₂O₃/Ethylene Glycol hybrid nanofluid, chosen for its superior thermal properties, stability, and industrial relevance. Copper (Cu) enhances thermal and electrical conductivity, making it ideal for electronics and automotive cooling, while Aluminum Oxide (Al₂O₃) offers stability, corrosion resistance, and broad industrial applications. Ethylene Glycol serves as the base fluid due to its widespread use in cooling systems, wide temperature stability, and effective nanoparticle dispersion. This hybrid nanofluid is applicable in aerospace, nuclear reactors, biomedical engineering, electronic cooling, and renewable energy systems, making it an optimal choice for advanced heat transfer applications. 4. Why the choice of the method of solution? Include the importance and limitations of this method in the study. Author’s reply: In this study, the MATLAB BVP4C solver is used to solve the transformed ordinary differential equations (ODEs) derived from the governing partial differential equations (PDEs). BVP4C is chosen for its high accuracy, adaptive mesh refinement, and robustness in handling boundary value problems (BVPs), making it well-suited for nonlinear and coupled ODEs arising in Williamson hybrid nanofluid flow. Unlike shooting methods, BVP4C ensures stability and efficiently manages multi-point boundary conditions, making it ideal for fluid flow and heat transfer problems. While alternative methods like Runge-Kutta and finite difference approaches were considered, they were found to be less efficient due to stability and convergence issues. However, BVP4C’s computational intensity requires careful selection of initial guesses. 5. The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. Author’s reply: The revised discussion section now offers a detailed scientific interpretation of velocity, temperature, and concentration trends, supported by physical laws and real-world applications. Each parameter's impact is analyzed with scientific reasoning and quantitative analysis , and comparisons with existing literature are included to validate the findings. 6. Include the future scope of the study after the conclusion section. Author’s reply: A dedicated section on future research directions follows the conclusion, outlining potential extensions of this study. These include examining unsteady flow conditions to capture transient effects, exploring various hybrid nanoparticle combinations to enhance heat transfer performance, and extending the model to incorporate non-Newtonian rheological properties relevant to biomedical and industrial applications. 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 20 May 2025 Asfaw Tsegaye , Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia 20 May 2025 Author Response Author’s Response to Reviewers’ Comments We sincerely appreciate the ... 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. The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. Author’s reply: The introduction has been revised to clearly highlight the significance of hybrid nanofluids in industrial applications such as cooling systems, biomedical engineering, and energy systems. The last paragraph of the introduction explicitly identifies the research gap by emphasizing that no prior studies have investigated the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, Joule heating, and chemical reactions on Williamson hybrid nanofluid flow over a stretching porous sheet. The findings of this study have significant implications in various engineering and industrial applications , including aerospace engineering, nuclear power plants, electronic cooling systems, biomedical applications, and polymer and textile industries. 2. The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. Author’s reply: Thank you for your suggestions. A dedicated section has been added to justify the assumptions made in modeling the problem. The inclusion of specific terms in the governing equations (such as Steady-State, Three-Dimensional Flow, Williamson Non-Newtonian Fluid Model, Cattaneo-Christov Heat and Mass Flux Model, Non-Linear Thermal Radiation, Joule Heating, Viscous Dissipation, Porous Medium and Magnetic Field Effects, heat generation, chemical reaction terms, and Boundary Conditions) is now explicitly explained with appropriate references. Additional citations have been included to validate the selection of mathematical models for hybrid nanofluids. 3. Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. Author’s reply: This study utilizes a Cu-Al₂O₃/Ethylene Glycol hybrid nanofluid, chosen for its superior thermal properties, stability, and industrial relevance. Copper (Cu) enhances thermal and electrical conductivity, making it ideal for electronics and automotive cooling, while Aluminum Oxide (Al₂O₃) offers stability, corrosion resistance, and broad industrial applications. Ethylene Glycol serves as the base fluid due to its widespread use in cooling systems, wide temperature stability, and effective nanoparticle dispersion. This hybrid nanofluid is applicable in aerospace, nuclear reactors, biomedical engineering, electronic cooling, and renewable energy systems, making it an optimal choice for advanced heat transfer applications. 4. Why the choice of the method of solution? Include the importance and limitations of this method in the study. Author’s reply: In this study, the MATLAB BVP4C solver is used to solve the transformed ordinary differential equations (ODEs) derived from the governing partial differential equations (PDEs). BVP4C is chosen for its high accuracy, adaptive mesh refinement, and robustness in handling boundary value problems (BVPs), making it well-suited for nonlinear and coupled ODEs arising in Williamson hybrid nanofluid flow. Unlike shooting methods, BVP4C ensures stability and efficiently manages multi-point boundary conditions, making it ideal for fluid flow and heat transfer problems. While alternative methods like Runge-Kutta and finite difference approaches were considered, they were found to be less efficient due to stability and convergence issues. However, BVP4C’s computational intensity requires careful selection of initial guesses. 5. The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. Author’s reply: The revised discussion section now offers a detailed scientific interpretation of velocity, temperature, and concentration trends, supported by physical laws and real-world applications. Each parameter's impact is analyzed with scientific reasoning and quantitative analysis , and comparisons with existing literature are included to validate the findings. 6. Include the future scope of the study after the conclusion section. Author’s reply: A dedicated section on future research directions follows the conclusion, outlining potential extensions of this study. These include examining unsteady flow conditions to capture transient effects, exploring various hybrid nanoparticle combinations to enhance heat transfer performance, and extending the model to incorporate non-Newtonian rheological properties relevant to biomedical and industrial applications. 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] 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. The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. Author’s reply: The introduction has been revised to clearly highlight the significance of hybrid nanofluids in industrial applications such as cooling systems, biomedical engineering, and energy systems. The last paragraph of the introduction explicitly identifies the research gap by emphasizing that no prior studies have investigated the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, Joule heating, and chemical reactions on Williamson hybrid nanofluid flow over a stretching porous sheet. The findings of this study have significant implications in various engineering and industrial applications , including aerospace engineering, nuclear power plants, electronic cooling systems, biomedical applications, and polymer and textile industries. 2. The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. Author’s reply: Thank you for your suggestions. A dedicated section has been added to justify the assumptions made in modeling the problem. The inclusion of specific terms in the governing equations (such as Steady-State, Three-Dimensional Flow, Williamson Non-Newtonian Fluid Model, Cattaneo-Christov Heat and Mass Flux Model, Non-Linear Thermal Radiation, Joule Heating, Viscous Dissipation, Porous Medium and Magnetic Field Effects, heat generation, chemical reaction terms, and Boundary Conditions) is now explicitly explained with appropriate references. Additional citations have been included to validate the selection of mathematical models for hybrid nanofluids. 3. Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. Author’s reply: This study utilizes a Cu-Al₂O₃/Ethylene Glycol hybrid nanofluid, chosen for its superior thermal properties, stability, and industrial relevance. Copper (Cu) enhances thermal and electrical conductivity, making it ideal for electronics and automotive cooling, while Aluminum Oxide (Al₂O₃) offers stability, corrosion resistance, and broad industrial applications. Ethylene Glycol serves as the base fluid due to its widespread use in cooling systems, wide temperature stability, and effective nanoparticle dispersion. This hybrid nanofluid is applicable in aerospace, nuclear reactors, biomedical engineering, electronic cooling, and renewable energy systems, making it an optimal choice for advanced heat transfer applications. 4. Why the choice of the method of solution? Include the importance and limitations of this method in the study. Author’s reply: In this study, the MATLAB BVP4C solver is used to solve the transformed ordinary differential equations (ODEs) derived from the governing partial differential equations (PDEs). BVP4C is chosen for its high accuracy, adaptive mesh refinement, and robustness in handling boundary value problems (BVPs), making it well-suited for nonlinear and coupled ODEs arising in Williamson hybrid nanofluid flow. Unlike shooting methods, BVP4C ensures stability and efficiently manages multi-point boundary conditions, making it ideal for fluid flow and heat transfer problems. While alternative methods like Runge-Kutta and finite difference approaches were considered, they were found to be less efficient due to stability and convergence issues. However, BVP4C’s computational intensity requires careful selection of initial guesses. 5. The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. Author’s reply: The revised discussion section now offers a detailed scientific interpretation of velocity, temperature, and concentration trends, supported by physical laws and real-world applications. Each parameter's impact is analyzed with scientific reasoning and quantitative analysis , and comparisons with existing literature are included to validate the findings. 6. Include the future scope of the study after the conclusion section. Author’s reply: A dedicated section on future research directions follows the conclusion, outlining potential extensions of this study. These include examining unsteady flow conditions to capture transient effects, exploring various hybrid nanoparticle combinations to enhance heat transfer performance, and extending the model to incorporate non-Newtonian rheological properties relevant to biomedical and industrial applications. 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 Comments on this article Comments (0) Version 3 VERSION 3 PUBLISHED 14 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 4 Version 3 (revision) 03 Dec 25 read Version 2 (revision) 31 Mar 25 read read read read Version 1 14 Feb 25 read read Ephesus Olusoji Fatunmbi , Federal Polytechnic, Ilaro, Nigeria Nandagopal K , Mohan Babu University, Tirupathi, India Dr. Muhammad Bilal , The University of Chenab, Gujrat, Pakistan Dr. Muhammad Faisal , The University of Azad Jammu and Kashmir, Muzaffarabad, 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 Bilal D. 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. 24 Dec 2025 | for Version 3 Dr. Muhammad Bilal , The University of Chenab, Gujrat, Pakistan 0 Views copyright © 2025 Bilal D. 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 All the comments are addressed, and the article can be published now. Competing Interests No competing interests were disclosed. Reviewer Expertise Computational Fluid Dynamics 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) Bilal DM. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.192060.r438380) 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/14-210/v3#referee-response-438380 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Fatunmbi E. 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. 29 Apr 2025 | for Version 2 Ephesus Olusoji Fatunmbi , Federal Polytechnic, Ilaro, Nigeria 0 Views copyright © 2025 Fatunmbi E. 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 The authors have revised the manuscript as suggested in my review comments. Thus, the article can be accepted for indexing. Competing Interests No competing interests were disclosed. Reviewer Expertise Applied mathematics, Computational fluid dynamics, Nano fluid, Boundary layer flow, etc. 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) Fatunmbi EO. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.179489.r374557) 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/14-210/v2#referee-response-374557 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Faisal D. 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. 28 Apr 2025 | for Version 2 Dr. Muhammad Faisal , The University of Azad Jammu and Kashmir, Muzaffarabad, Pakistan 0 Views copyright © 2025 Faisal D. 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 In this manuscript, copper (Cu) and aluminum oxide (Al₂O₃) nanoparticles are suspended in ethylene glycol (C₂C₆O₂) to form the hybrid nanofluid. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and solved numerically with MATLAB’s bvp4c solver. The study examines various parameters, including stretching ratio, nanoparticle volume fraction, and relaxation times for concentration and thermal effects. Results are validated against existing literature. 1. There are many typos and grammatical errors in the manuscript. Improve the English level. 2. The title of the manuscript is too long. Revise the title. 3. Provide references to those equations that are extracted from literature. 4. What are the properties of the Williamson fluid, and in what category of non-Newtonian fluid does it lie? Explain in the manuscript. 5. Provide the vector forms of governing equations, and also include a reference for boundary conditions. 6. The mathematical expressions for the involved engineering parameters are missing. 7. What are the limitations of the applied numerical technique? 8. The value of the Prandtl number is missing in the investigation. Include its value and its plot? 9. Provide a Nomenclature section. 10. What are the reference velocity and reference temperature in this fluid flow model? 11. Discuss novel findings in the abstract, providing relevant applications. 12. Provide the rate of convergence of the solution obtained from the numerical method. 13. Improve the introduction section by incorporating the latest studies. Some suggested works are given below: On Maxwell slip flow of radiative ternary hybrid nanofluid subject to Smoluchowski-Nield’s constraints using an iterative numerical simulation. Simulation of Casson hybrid nanofluid over bidirectional stretching surface with entropy analysis in stagnated domain. Semi-analytical simulation of chemically reactive Maxwell nanofluid with Cattaneo-Christov heat and mass fluxes. Insight into the bidirectional dynamics of hyperbolic tangent hybrid nanomaterial subject to Nield’s conditions using a numerical approach. Rivlin fluid flow between rotatable stretching disks with Cattaneo Christov heat transport. Stagnation point flow of third-order nanofluid towards a lubricated surface using hybrid homotopy analysis method. Nanoparticles hybridization in bidirectional flowing of Prandtl-Eyring material with temperature-dependent conductivity: A numerical approach. Prescribed thermal effects on the dynamics of radiative and chemically reactive hyperbolic tangent nanofluid. Entropic behavior with activation energy in the dynamics of hyperbolic-tangent mixed convective nanomaterial due to a vertical slendering surface. Natural convective behavior of hybrid nanofluid (Al2O3-Cu/water) in an isosceles triangular cavity with bifurcation analysis. Entropic behavior in bidirectional flow of CeO2-ZnO/water hybrid nanofluid with prescribed surface temperature/heat flux aspects. Darcy-Forchheimer dynamics of hybrid nanofluid due to porous Riga surface capitalizing Cattaneo-Christov theory. Entropy optimization in squeezed nanofluidic dissipative transport of radiative water conveying aluminum alloys. Dynamics of MHD tangent hyperbolic nanofluid with prescribed thermal conditions, random motion, and thermo-migration of nanoparticles. 11. A robust validation is needed. Given the above observations, a minor revision is requested before a final decision. Is the work clearly and accurately presented and does it cite the current literature? Partly 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 Mechanics, Computational Fluid Dynamics, Heat and Mass Transfer 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 03 Dec 2025 Asfaw Tsegaye, Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia Author’s Response to Reviewer’s 4 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. There are many typos and grammatical errors in the manuscript. Improve the English level. Author’s reply: Thank you for this observation. The entire manuscript has been thoroughly revised for grammar, clarity, and academic writing quality. All typographical and linguistic errors have been corrected throughout the text. 2. The title of the manuscript is too long. Revise the title. Author’s reply: We appreciate the suggestion. The title has been shortened and refined to improve clarity and conciseness. 3. Provide references to those equations that are extracted from literature. Author’s reply: Thank you for pointing this out. Appropriate references have now been added to all equations derived or adapted from existing literature. 4. What are the properties of the Williamson fluid, and in what category of non-Newtonian fluid does it lie? Explain in the manuscript. Author’s reply: We have added a detailed explanation in the introduction. The Williamson fluid model is a shear-thinning (pseudoplastic) non-Newtonian fluid , and its rheological characteristics and constitutive equations are now clearly described with supporting citations. 5. Provide the vector forms of governing equations, and also include a reference for boundary conditions. Author’s reply: We have included the full vector forms of the governing momentum, heat, and mass equations, along with references supporting the selected boundary conditions. 6. The mathematical expressions for the involved engineering parameters are missing. Author’s reply: These expressions have now been added, particularly for the skin-friction coefficients, the Nusselt number, and the Sherwood number. 7. What are the limitations of the applied numerical technique? Author’s reply: We have added a paragraph in the numerical method section discussing limitations of the bvp4c approach, including sensitivity to initial guesses, computational cost, and potential difficulty in handling highly stiff systems. 8. The value of the Prandtl number is missing in the investigation. Include its value and its plot? Author’s reply: The Prandtl number has been included in the parameter tables, and an additional figure has been added to show its influence on the temperature field. 9. Provide a Nomenclature section. Author’s reply: A complete Nomenclature section has been added immediately after the abstract. 10. What are the reference velocity and reference temperature in this fluid flow model? Author’s reply: The reference velocity and reference temperature definitions have been added and clearly explained in the problem formulation section. 11. Discuss novel findings in the abstract, providing relevant applications. Author’s reply: The abstract has been revised to highlight the novelty of the study, especially the combined effects of nonlinear radiation, Cattaneo-Christov fluxes, and 3D Williamson hybrid nanofluid flow, and to outline practical applications in cooling systems, polymer manufacturing, and energy technologies. 12. Provide the rate of convergence of the solution obtained from the numerical method. Author’s reply: A convergence table showing the rate of convergence of the bvp4c solver has been added to the numerical method section. 13. Improve the introduction section by incorporating the latest studies. Some suggested works are given below: On Maxwell slip flow of radiative ternary hybrid nanofluid subject to Smoluchowski-Nield’s constraints using an iterative numerical simulation. Simulation of Casson hybrid nanofluid over bidirectional stretching surface with entropy analysis in stagnated domain. Semi- analytical simulation of chemically reactive Maxwell nanofluid with Cattaneo-Christov heat and mass fluxes. Insight into the bidirectional dynamics of hyperbolic tangent hybrid nanomaterial subject to Nield’s conditions using a numerical approach. Rivlin fluid flow between rotatable stretching disks with Cattaneo-Christov heat transport. Stagnation point flow of third-order nanofluid towards a lubricated surface using hybrid Homotopy analysis method. Nanoparticles hybridization in bidirectional flowing of Prandtl-Eyring material with temperature-dependent conductivity: A numerical approach. Prescribed thermal effects on the dynamics of radiative and chemically reactive hyperbolic tangent nanofluid. Entropic behavior with activation energy in the dynamics of hyperbolic-tangent mixed convective nanomaterial due to a vertical slandering surface. Natural convective behavior of hybrid nanofluid (Al2O3-Cu/water) in an isosceles triangular cavity with bifurcation analysis. Entropic behavior in bidirectional flow of CeO2- ZnO/water hybrid nanofluid with prescribed surface temperature/heat flux aspects. Darcy- Forchheimer dynamics of hybrid nanofluid due to porous Riga surface capitalizing Cattaneo- Christov theory. Entropy optimization in squeezed nanofluidic dissipative transport of radiative water conveying aluminum alloys. Dynamics of MHD tangent hyperbolic nanofluid with prescribed thermal conditions, random motion, and thermo-migration of nanoparticles. Author’s reply: Following the reviewer’s recommendations, the introduction has been significantly expanded to include all the suggested recent works on Maxwell slip flow, Casson hybrid nanofluid, third-order nanofluid, tangent hyperbolic fluids, Prandtl-Eyring materials, and entropy analysis–based studies. 14. A robust validation is needed. Author’s reply: A new validation subsection has been added. Table 2 compares our results with existing literature, demonstrating excellent agreement and confirming the accuracy of our numerical scheme. 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 Faisal DM. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.179489.r374718) 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/14-210/v2#referee-response-374718 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 K N. 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. 23 Apr 2025 | for Version 2 Nandagopal K , Mohan Babu University, Tirupathi, India 0 Views copyright © 2025 K N. 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 The author satisfies all the comments. So I recommend that the manuscript is indexed. Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid dynamics , peristalsis, porous medium 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) K N. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.179489.r374558) 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/14-210/v2#referee-response-374558 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Bilal D. 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. 23 Apr 2025 | for Version 2 Dr. Muhammad Bilal , The University of Chenab, Gujrat, Pakistan 0 Views copyright © 2025 Bilal D. 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 This article focuses on the flow of a Williamson hybrid nanofluid over a stretching porous sheet, incorporating the effects of non-linear thermal radiation and Cattaneo–Christov heat and mass fluxes. The article is suitable for indexed after addressing the following minor revisions: There are some grammatical errors in the manuscript, particularly in the Introduction section. The authors are advised to thoroughly proofread the manuscript and correct these errors. Please provide the units of all physical quantities listed in the Nomenclature section. Although the Introduction is generally well written, the authors are encouraged to strengthen it further by incorporating some recent studies . Figure 1 should be redrawn. Kindly remove the background and clearly indicate the key physical aspects of the flow geometry for better visualization. Ensure that all displayed equations are punctuated appropriately—insert commas or full stops at the end as needed. In Equation (23) , please correct the Eckert number symbols along the x- and y-directions to reflect accurate notation. The definition of the Weissenberg number includes the variable xxx, which typically challenges the validity of a similarity solution. The authors should briefly justify how similarity is preserved in this context. In the first-order system of equations , all terms involving “A” must include appropriate subscripts for clarity and precision. 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? Not applicable 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 Computational Fluid Dynamics 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 03 Dec 2025 Asfaw Tsegaye, Bahir Dar University Department of Mathematics, Bahir Dar, Ethiopia Author’s Response to Reviewer’s 3 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. There are some grammatical errors in the manuscript, particularly in the introduction section. The authors are advised to thoroughly proofread the manuscript and correct these errors. Author’s reply: The manuscript requires a comprehensive revision to address numerous grammatical and typographical errors present throughout the text. Careful proofreading is necessary to enhance the overall clarity and readability of the paper. 2. Please provide the units of all physical quantities listed in the Nomenclature section. Author’s reply: The nomenclature section has been updated to include SI units for all parameters. 3. Although the Introduction is generally well-written, the authors are encouraged to strengthen it further by incorporating some recent studies. Author’s reply: The references have been incorporated into the introduction and literature review to contextualize and support our research. For instance, - Novel features of radiating hybrid nanofluid flow past a nonlinear stretchable porous sheet with different nanoparticle shapes. - Computation of Convective Magnetohydrodynamic Buongiorno Nanofluid Transport from an Inclined Plane with Ion Slip And Hall Current Effects. - 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: numerical contribution. 4. Figure 1 should be redrawn. Kindly remove the background and clearly indicate the key physical aspects of the flow geometry for better visualization. Author’s reply: Thank you for the suggestion. Figure 1 has been redrawn to improve clarity. The background has been removed, and key physical aspects of the flow geometry such as the stretching sheet, coordinate axes, and flow direction are now clearly labeled to enhance visualization and understanding. 5. Ensure that all displayed equations are punctuated appropriately—insert commas or full stops at the end as needed. Author’s reply: Corrected 6. In Equation (23), please correct the Eckert number symbols along the x- and y-directions to reflect accurate notation. Author’s reply: Corrected 7. The definition of the Weissenberg number includes the variable xxx, which typically challenges the validity of a similarity solution. The authors should briefly justify how similarity is preserved in this context. Author’s reply: Justification: We acknowledge that the inclusion of the spatial variable x in the definition of the Weissenberg number (We) can, in general, pose a challenge to the existence of the similarity solution. However, in our study, the functional dependence of We on x is such that it varies in a manner consistent with the chosen similarity transformation. Specifically, the definition of We is constructed to preserve the invariance of the transformed equations when expressed in terms of the similarity variable η. This allows the partial differential equations to be reduced to ordinary differential equations, ensuring the validity of the similarity solution. Similar formulations can be found in prior literature dealing with viscoelastic or non-Newtonian flows in boundary layer settings, where such transformations have been successfully applied under analogous assumptions. 8. In the first-order system of equations , all terms involving “A” must include appropriate subscripts for clarity and precision. Author’s reply: It is modified in the revised manuscript. 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 Bilal DM. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.179489.r374716) 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/14-210/v2#referee-response-374716 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 K N. 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. 04 Mar 2025 | for Version 1 Nandagopal K , Mohan Babu University, Tirupathi, India 0 Views copyright © 2025 K N. 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 This study explores the Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet. However it requires some following corrections. The work can be suggested to indexed after the following corrections are fulfilled. 1. The whole article may be checked for typo and other grammatical mistakes. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? 3. What are the advantages of BVP4C over other methods? 4.Justify the selection of transformation to reduce the PDEs into ODEs. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. 6. Add Nomenclature separately 7. What is the significance of studying non-linear thermal radiation? 8. Add some result for stream lines to make their study sound more unique. 9. What is the difference between thermal radiation and non-linear thermal radiation? 10. How current results present directions to the future studies. Summarize in conclusion section. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. The present research work may be suggested to publish in F1000 research after the following corrections are fulfilled. 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? Partly References 1. Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation . 2024. 1-18 Publisher Full Text 2. M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport . 2024. 1-26 Publisher Full Text 3. Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications . 2023. 1-17 Publisher Full Text 4. Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media . 2022. 1-20 Publisher Full Text 5. Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering . 2022; 144 (4). Publisher Full Text 6. Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics . 2025; 13 . Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid dynamics , peristalsis, porous medium 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 20 May 2025 Asfaw Tsegaye, Bahir Dar University Department of Mathematics, 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. The whole article may be checked for typos and other grammatical mistakes. Author’s reply: We have carefully reviewed the manuscript and corrected all typographical and grammatical errors to ensure clarity and readability. 2. Justification must be provided on the use of a particular set of boundary conditions. Why these specific sets of conditions are suitable for this study? Author’s reply: The specific boundary conditions used in this study are justified based on the physical and mathematical framework of the problem: a. Physical Relevance: - The boundary conditions at z=0(w=0, v= v w =by, u= u w =ax, T= T w , C= C w ) ​ represent a stretching sheet, which is common in industrial applications such as polymer extrusion, metal forming, and coating processes. - The prescribed velocity components ( u w =ax, v w =by) mimic the stretching motion of the sheet in two directions, ensuring a controlled boundary layer development. b. Mathematical Consistency: - At the sheet ( z=0) , specifying velocity ensures that the no-slip condition is satisfied, which is fundamental for boundary layer flow. - The temperature and concentration at the sheet ( T w , C w ) ​ are held constant to analyze the effects of heat and mass transfer within the hybrid nanofluid. c. Far-Field Conditions: - As z→∞ , the velocity components (u, v) approach zero, ensuring that the fluid is at rest far from the sheet. - The temperature and concentration also reach ambient conditions ( T ∞ , C ∞ ) , signifying thermal and concentration equilibrium. d. Applicability to Cattaneo-Christov Model: - The use of Cattaneo-Christov heat and mass flux models requires that thermal and concentration relaxation effects be properly accounted for at the boundaries. - The conditions ensure a well-posed problem, allowing numerical solutions via similarity transformations and MATLAB’s bvp4c solver. Overall, these boundary conditions are essential for accurately capturing the physics of the problem while ensuring numerical stability and consistency with previous studies. 3. What are the advantages of BVP4C over other methods? Author’s reply: The BVP4C method in MATLAB offers several advantages over other numerical methods for solving boundary value problems: a. High Accuracy: - The BVP4C solver utilizes a finite difference method with collocation techniques, achieving an accuracy of approximately 10 -7 . This makes it highly precise for solving boundary layer flow problems. b. Built-in Error Control: - BVP4C automatically adjusts the mesh selection and step size based on residual error estimates, ensuring numerical stability and reducing computational errors. c. Efficient Handling of Singularities: - It can handle singular boundary conditions effectively, which is particularly useful in fluid dynamics problems involving stretching sheets , hybrid nanofluids, and thermal radiation effects. d. Adaptive Mesh Refinement: - Unlike shooting methods, which require iterative guessing, BVP4C refines the solution using an adaptive mesh, leading to more efficient computations without divergence issues. e. Well-Suited for Stiff Equations: - Since boundary layer problems often involve stiff differential equations, BVP4C efficiently handles them without requiring excessive computational power. f. Direct Implementation of Boundary Conditions: - Unlike shooting methods, where unknown initial conditions need estimation, BVP4C directly incorporates boundary conditions, making it more robust for complex problems​ 4. Justify the selection of transformation to reduce the PDEs into ODEs. Author’s reply: The selection of similarity transformations to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs) is justified based on several key factors: a. Reduction of Complexity - The governing equations of the hybrid nanofluid flow problem are highly nonlinear PDEs. Using similarity transformations simplifies these equations into a set of ODEs, making them more manageable for numerical or analytical solutions. b. Boundary Layer Theory - The similarity variable η=z a ν f is chosen to convert the spatial dependency of the velocity, temperature, and concentration into a single variable. This transformation is in line with classical boundary layer theory, which seeks to reduce multi-dimensional flow problems into a one- dimensional form for efficient analysis. c. Consistency with Existing Literature - The chosen transformations align with prior studies on fluid flow over stretching sheets, ensuring that the results can be compared with previous findings and validated accordingly. d. Mathematical and Computational Efficiency - The resulting system of ODEs is more suitable for numerical techniques like MATLAB’s bvp4c , which is specifically designed for solving boundary value problems with high accuracy and stability. e. Physical Relevance - The transformed equations retain the fundamental physical characteristics of the original PDEs, capturing key effects like thermal radiation, hybrid nanofluid properties, and the Cattaneo-Christov heat and mass flux model.By using these similarity transformations, the study ensures that the problem formulation is both mathematically rigorous and computationally efficient. 5. Discussion should be improved, and it should be based on the physical reasoning of the model. Some quantitative analysis must support the results. Author’s reply: The discussion section has been revised to provide deeper insights into the physical mechanisms governing hybrid nanofluid flow. We have elaborated on the effects of dimensionless parameters on velocity, temperature, and concentration fields, supported by quantitative analysis and comparison with existing studies. 6. Add Nomenclature separately Author’s reply: A separate nomenclature section has been included to define symbols and parameters used throughout the study for better readability. 7. What is the significance of studying non-linear thermal radiation? Author’s reply: Studying non-linear thermal radiation is essential for accurately modeling heat transfer in high-temperature environments, as it accounts for complex interactions beyond the Stefan-Boltzmann Law. It plays a crucial role in applications like aerospace, nuclear reactors, and industrial heat exchangers, where extreme temperatures amplify radiative effects. Non-linear models enhance heat flux predictions, particularly in nanofluid-based systems, where radiation significantly influences thermal behavior. In hybrid nanofluids like Cu-Al₂O₃/ethylene glycol, it affects boundary layer characteristics and heat dissipation efficiency. Understanding these effects helps optimize thermal management systems, improving cooling performance in electronics, solar panels, and combustion chambers. Additional discussion on its significance has been included in the introduction section. 8. Add some result for streamlines to make their study sound more unique. Author’s reply: We have outlined future research directions on results for plotting streamlines and will continue exploring this topic in upcoming studies. 9. What is the difference between thermal radiation and non-linear thermal radiation? Author’s reply: Thermal radiation follows the Stefan-Boltzmann law , where heat flux is proportional to the fourth power of temperature and is often linearized under small temperature variations. In contrast, non-linear thermal radiation arises in high-temperature environments where heat flux exhibits a more complex dependence on temperature, influenced by temperature gradients, material optical properties, and medium characteristics . This non-linearity makes it more accurate for modeling extreme heat transfer conditions, such as those found in nanofluids, aerospace applications, and combustion chambers . We have added a clarification in the introduction section. 10. How current results present directions to the future studies. Summarize in conclusion section. Author’s reply: The current study highlights key factors influencing hybrid nanofluid flow, providing directions for future research. Further studies can explore optimized stretching conditions, alternative nanoparticle materials, and advanced thermal models, including non-linear radiation effects. Investigating unsteady flow, non-Newtonian fluids, and three-dimensional effects will enhance the findings' applicability in engineering and biomedical fields. 11. Authors must re-check and update the introduction section. Include some recent literature survey subject to the current title. Moreover, remove old-fashioned studies with latest research. Refer 1 to 6. - Reddy M, Vajravelu K, Ajithkumar M, Sucharitha G, et al.: Numerical treatment of entropy generation in convective MHD Williamson nanofluid flow with Cattaneo–Christov heat flux and suction/injection. International Journal of Modelling and Simulation. 2024. 1-18 Publisher Full Text -M. S. I, Lakshminarayana P, Sucharitha G, Vinodkumar Reddy M, et al.: Analysis of Entropy Optimization in MHD Flow of Non-Newtonian Nanofluids with Chemical Reaction and Thermal Energies. Journal of Computational and Theoretical Transport. 2024. 1-26 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P, Vajravelu K, Sucharitha G: Activation of energy in MHD Casson nanofluid flow through a porous medium in the presence of convective boundary conditions and suction/injection. Numerical Heat Transfer, Part A: Applications. 2023. 1-17 Publisher Full Text - - Vinodkumar Reddy M, Sucharitha G, Vajravelu K, Lakshminarayana P: Convective flow of MHD non-Newtonian nanofluids on a chemically reacting porous sheet with Cattaneo-Christov double diffusion. Waves in Random and Complex Media. 2022. 1-20 Publisher Full Text - Vinodkumar Reddy M, Lakshminarayana P: Higher Order Chemical Reaction and Radiation Effects on Magnetohydrodynamic Flow of a Maxwell Nanofluid With Cattaneo–Christov Heat Flux Model Over a Stretching Sheet in a Porous Medium. Journal of Fluids Engineering. 2022; 144 (4). Publisher Full Text - - Chakradhar K, Nandagopal K, Prashanthi V, Parandhama A, et al.: MHD effect on peristaltic motion of Williamson fluid via porous channel with suction and injection. Partial Differential Equations in Applied Mathematics. 2025; 13. Publisher Full Text Author’s reply: 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 manuscript. 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 K N. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.176670.r366915) 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/14-210/v1#referee-response-366915 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2025 Fatunmbi E. 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. 28 Feb 2025 | for Version 1 Ephesus Olusoji Fatunmbi , Federal Polytechnic, Ilaro, Nigeria 0 Views copyright © 2025 Fatunmbi E. 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 Reviewer comments on the manuscript titled “ Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet” The following are my comments on this manuscript: 1). The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. 2). The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. 3). Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. 4). Why the choice of the method of solution? Include the importance and limitations of this method in the study. 5). The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. 6). Include the future scope of study after the conclusion section. 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? Partly Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Partly Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Applied mathematics, Computational fluid dynamics, Nano fluid, Boundary layer flow, etc. 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 20 May 2025 Asfaw Tsegaye, Bahir Dar University Department of Mathematics, 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. The manuscript introduction lacks basic ingredients to convince the readers about the importance of the study. In the last paragraph of the introduction, the authors should clearly identify and state areas of application for the study. More so, the existing gap which the authors intend to fill should be clearly stated. Author’s reply: The introduction has been revised to clearly highlight the significance of hybrid nanofluids in industrial applications such as cooling systems, biomedical engineering, and energy systems. The last paragraph of the introduction explicitly identifies the research gap by emphasizing that no prior studies have investigated the combined effects of non-linear thermal radiation, Cattaneo-Christov heat and mass fluxes, Joule heating, and chemical reactions on Williamson hybrid nanofluid flow over a stretching porous sheet. The findings of this study have significant implications in various engineering and industrial applications , including aerospace engineering, nuclear power plants, electronic cooling systems, biomedical applications, and polymer and textile industries. 2. The assumptions for modeling the problems are not adequately stated in the manuscript. Authors should justify the inclusion of various terms in the governing equations and give details of their derivation with appropriate references. Author’s reply: Thank you for your suggestions. A dedicated section has been added to justify the assumptions made in modeling the problem. The inclusion of specific terms in the governing equations (such as Steady-State, Three-Dimensional Flow, Williamson Non-Newtonian Fluid Model, Cattaneo-Christov Heat and Mass Flux Model, Non-Linear Thermal Radiation, Joule Heating, Viscous Dissipation, Porous Medium and Magnetic Field Effects, heat generation, chemical reaction terms, and Boundary Conditions) is now explicitly explained with appropriate references. Additional citations have been included to validate the selection of mathematical models for hybrid nanofluids. 3. Why the choice of the nanoparticles and base fluid used in the study? It is important to state the significance and applications of the nanoparticles used. Author’s reply: This study utilizes a Cu-Al₂O₃/Ethylene Glycol hybrid nanofluid, chosen for its superior thermal properties, stability, and industrial relevance. Copper (Cu) enhances thermal and electrical conductivity, making it ideal for electronics and automotive cooling, while Aluminum Oxide (Al₂O₃) offers stability, corrosion resistance, and broad industrial applications. Ethylene Glycol serves as the base fluid due to its widespread use in cooling systems, wide temperature stability, and effective nanoparticle dispersion. This hybrid nanofluid is applicable in aerospace, nuclear reactors, biomedical engineering, electronic cooling, and renewable energy systems, making it an optimal choice for advanced heat transfer applications. 4. Why the choice of the method of solution? Include the importance and limitations of this method in the study. Author’s reply: In this study, the MATLAB BVP4C solver is used to solve the transformed ordinary differential equations (ODEs) derived from the governing partial differential equations (PDEs). BVP4C is chosen for its high accuracy, adaptive mesh refinement, and robustness in handling boundary value problems (BVPs), making it well-suited for nonlinear and coupled ODEs arising in Williamson hybrid nanofluid flow. Unlike shooting methods, BVP4C ensures stability and efficiently manages multi-point boundary conditions, making it ideal for fluid flow and heat transfer problems. While alternative methods like Runge-Kutta and finite difference approaches were considered, they were found to be less efficient due to stability and convergence issues. However, BVP4C’s computational intensity requires careful selection of initial guesses. 5. The results and discussion can be improved by including basic scientific reasons for the trend exhibited by various parameters on the flow fields. Author’s reply: The revised discussion section now offers a detailed scientific interpretation of velocity, temperature, and concentration trends, supported by physical laws and real-world applications. Each parameter's impact is analyzed with scientific reasoning and quantitative analysis , and comparisons with existing literature are included to validate the findings. 6. Include the future scope of the study after the conclusion section. Author’s reply: A dedicated section on future research directions follows the conclusion, outlining potential extensions of this study. These include examining unsteady flow conditions to capture transient effects, exploring various hybrid nanoparticle combinations to enhance heat transfer performance, and extending the model to incorporate non-Newtonian rheological properties relevant to biomedical and industrial applications. 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 Fatunmbi EO. Peer Review Report For: Effect of non-linear thermal radiation and Cattaneo-Christov heat and mass fluxes on the flow of Williamson hybrid nanofluid over a stretching porous sheet [version 1; peer review: 2 approved with reservations] . F1000Research 2025, 14 :210 ( https://doi.org/10.5256/f1000research.176670.r366913) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. 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