Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel

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Salih Al-Khafajy" }, { "@type": "Person", "name": "Amal Al-Hanaya" }, { "@type": "Person", "name": "Munirah Aali Alotaibi" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": " Background Magnetohydrodynamic oscillatory flows of non-Newtonian fluids in porous channels arise in many industrial and geophysical applications. Understanding the combined influence of variable viscosity, heat generation, and concentration is essential for accurate flow prediction. Methods A mathematical model for unsteady MHD oscillatory flow of a Bingham fluid through an inclined porous channel was formulated. The governing nonlinear partial differential equations for momentum, energy, and concentration were nondimensionalized and solved using the separation of variables technique. Numerical evaluation and graphical analysis were performed using Wolfram Mathematica. Results The results show that increasing heat generation and radiation parameters enhances fluid temperature and velocity, while higher magnetic and oscillation parameters suppress flow motion. Concentration was found to increase with higher oscillation frequency and Péclet number, whereas Schmidt and chemical reaction parameters reduced mass diffusion. Variable viscosity significantly amplified velocity compared to constant-viscosity cases. Conclusions The study demonstrates that temperature-dependent viscosity and yield-stress effects strongly control MHD oscillatory Bingham fluid flow in inclined porous channels. The results are relevant to engineering systems involving non-Newtonian transport with thermal and mass diffusion effects. " } { "@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/15-476/v1", "name": "Effects of variable viscosity, concentration and heat variation on..." } } ] } Home Browse Effects of variable viscosity, concentration and heat variation on... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Al-Khafajy DGS, Al-Hanaya A and Alotaibi MA. Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.12688/f1000research.172909.1 ) 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 Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] Dheia G. Salih Al-Khafajy 1 , Amal Al-Hanaya 2 , Munirah Aali Alotaibi https://orcid.org/0000-0001-8487-5389 2 Dheia G. Salih Al-Khafajy 1 , Amal Al-Hanaya 2 , Munirah Aali Alotaibi https://orcid.org/0000-0001-8487-5389 2 PUBLISHED 06 Apr 2026 Author details Author details 1 Department of Mathematics, University of Al-Qadisiyah, Diwaniya, Iraq 2 Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia Dheia G. Salih Al-Khafajy Roles: Conceptualization, Data Curation, Formal Analysis, Methodology, Project Administration, Software, Supervision, Validation, Writing – Original Draft Preparation, Writing – Review & Editing Amal Al-Hanaya Roles: Conceptualization, Data Curation, Formal Analysis, Methodology, Software, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Munirah Aali Alotaibi Roles: Data Curation, Formal Analysis, Funding Acquisition, Methodology, Project Administration, Writing – Original Draft Preparation, Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS Abstract Background Magnetohydrodynamic oscillatory flows of non-Newtonian fluids in porous channels arise in many industrial and geophysical applications. Understanding the combined influence of variable viscosity, heat generation, and concentration is essential for accurate flow prediction. Methods A mathematical model for unsteady MHD oscillatory flow of a Bingham fluid through an inclined porous channel was formulated. The governing nonlinear partial differential equations for momentum, energy, and concentration were nondimensionalized and solved using the separation of variables technique. Numerical evaluation and graphical analysis were performed using Wolfram Mathematica. Results The results show that increasing heat generation and radiation parameters enhances fluid temperature and velocity, while higher magnetic and oscillation parameters suppress flow motion. Concentration was found to increase with higher oscillation frequency and Péclet number, whereas Schmidt and chemical reaction parameters reduced mass diffusion. Variable viscosity significantly amplified velocity compared to constant-viscosity cases. Conclusions The study demonstrates that temperature-dependent viscosity and yield-stress effects strongly control MHD oscillatory Bingham fluid flow in inclined porous channels. The results are relevant to engineering systems involving non-Newtonian transport with thermal and mass diffusion effects. READ ALL READ LESS Keywords Bingham fluid, thermal transfer, concentration, magnetohydrodynamic (MHD), inclined porous channel Corresponding Author(s) Munirah Aali Alotaibi ( [email protected] ) Close Corresponding author: Munirah Aali Alotaibi Competing interests: No competing interests were disclosed. Grant information: This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R522), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Copyright: © 2026 Al-Khafajy DGS 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: Al-Khafajy DGS, Al-Hanaya A and Alotaibi MA. Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.12688/f1000research.172909.1 ) First published: 06 Apr 2026, 15 :476 ( https://doi.org/10.12688/f1000research.172909.1 ) Latest published: 06 Apr 2026, 15 :476 ( https://doi.org/10.12688/f1000research.172909.1 ) 1. Introduction A Bingham fluid is a category of non-Newtonian fluid that exhibits solid-like behavior at low-stress levels and flows as a viscous liquid under elevated stress levels. They demonstrate yield stress, indicating that the fluid remains static until the applied stress is above a specific critical threshold. Below this yield stress, the material exhibits rigid body behavior. Upon surpassing the yield stress, the flow behavior becomes linear, characterized by a constant viscosity. The linear correlation between stress and strain rate manifests similarly to that observed in Newtonian fluids. Typical instances encompass toothpaste, mayonnaise, and concrete. These materials necessitate a certain force to initiate flow. Bingham fluids are significant in numerous industrial and scientific applications, especially where materials necessitate exact regulation of flow characteristics. Bird et al. 1 examined the behaviour of a Bingham fluid within a hard circular tube. Kapur 2 introduced several mathematical models, such as Cais-son, the Bingham, and Herschel-Bulkley models. Rathy 3 examined the fluid dynamics of a Bingham fluid within a channel and an annulus featuring impermeable barriers. Vajravelu et al. 4 conducted an experimented on the flow of Bingham fluid in an annular tube with a porous wall. Ramakrishna et al 5 investigated the flow behaviour of a Bingham fluid on a permeable bed in an enclosed channel. Goverdhan 6 examined the flow of Bingham fluid in a porous channel. Narahari et al. 7 Investigate the movement of a Bingham fluid between two porous substrates, focusing on its unsteady behaviour. Recently, Murthy et al. 8 focused on analysing the flow of a Bingham fluid, which is not stable, in contact with a Newtonian fluid within two parallel plates. The objective is to determine the velocity field, mass flow rates, and interface velocity. Tsangaris et al. 9 conducted on the movement of a Bingham fluid between two porous walls that are parallel to each other. One wall moves steadily in the same direction as the other wall, which remains stationary. At the same time, there is a pressure difference throughout the length of the wall, and there is a flow of fluid across the walls due to their porous nature. Adnan and Abdulhadi 10 performed an investigation on the influence of an inclined magnetic field on the flow of incompressible Bingham plastic fluid within an inclined symmetric channel. The study also considered the effects of mass transfer and heat transfer. Slip circumstances are utilised for heat transmission and focus. Lakshminarayana et al. 11 examined the impact of wall slip circumstances, elasticity wall characteristics, and heat transfer on the movement of conducting Bingham fluid in an irregular channel using the presumptions of a lengthy wavelength and low Reynolds number. The current research by Mahabaleshwar et al. 12 investigates how radiation and chemical reactions affect the two-dimensional boundary layer flow of a bi-viscous Bingham fluid on a thermosolutal Marangoni boundary that is accompanied by a magnetic field and thermal source or sink. A mathematical model is developed using the Navier-Stokes equations to represent the physical flow problem. In order to convert these nonlinear PDEs into a system of nonlinear ODEs, they employed a similarity transformation. The study conducted by Basavarajappa et al. 13 examines the multilayer flow of a bi-viscous Bingham fluid within a vertical slab with a hybrid nanofluid, using the nonlinear Boussinesq approximation. Heat transport in Bingham fluids entails intricate interplay between the fluid’s rheological features and thermal aspects. The existence of a yield stress, which regulates flow commencement, profoundly influences heat transfer processes. Vradis et al. 14 numerically resolved the concurrent evolution of thermal fields and hydrodynamic at the entry region of a cylindrical pipe for a non-Newtonian Bingham-type fluid by employing the completely elliptic mathematical models of continuity, momentum, and energy. Mustafa et al. 15 conducted a heat transfer simulation for the swirling flow of a Bingham fluid constrained by a permeable rotation disk. The influence of concentration and temperature variations on magnetohydrodynamic (MHD) oscillatory flow in a porous media has numerous practical implications in the fields of engineering, industry, medical research, and issues related to the extraction and transportation of petroleum. Hamza et al. 16 conducted a study on the impact of slip condition, radiative heat transfer, and transverse magnetic field on the unsteady flow of a conducting optical thin fluid via a channel equipped with a porous media. Khudair and Al-Khafajy 17 proposed a heat transfer model for MHD oscillating flow of Williamson fluid over a porous plate, considering two different forms of flow. Al-Aridhee and Al-Khafajy 18 examined the impact of mass and heat transfer on the peristaltic movement of MHD flow of a non-Newtonian Jeffrey fluid via a cylindrical porous media channel. The investigation focuses on the flow within a wave frame of reference that is travelling at the velocity of the wave. Al-Khafajy and Labban 19 conducted a study on the combined impact of concentration and thermo-diffusion on the fluctuating flow of an incompressible Carreau fluid via an angled permeable channel. Liu et al. 20 investigate the steady flow and thermal transfer of Bingham fluid over a spinning disk of limited radius with radially varying thickness in the boundary layer. Eldabe et al. 21 examined the flow of non-Newtonian Bingham blood fluid down an irregular conduit. The fluid exhibits electrical conductivity, and an external uniform magnetic field is added to this motion. Heat and mass transmission are considered, leading to an examination of the Dufour and Soret effects. Al-Khafajy and Mohammed 22 examined a mathematical model elucidating the effects of thermal transfer on the oscillating flow of Bingham fluid with changing viscosity in a porous channel within the context of magnetohydrodynamics. Salahuddin et al. 23 performed a study analyzing numerical behavior utilizing the Adams-Bashforth predictor-corrector method of numerical analysis for the Williamson fluid model, considering variable viscosity, natural convection, and an angled magnetic field. Additionally, thermal radiation, Joule heating, and heat source/sink effects are incorporated into the thermal considerations. Akram et al. 24 studied the peristaltic flow of Bingham fluids under an inclined magnetic field, while Humnekar and Darbhasayanam 25 investigated variable-viscosity nanofluid flow in inclined porous media. These recent studies highlight the ongoing need for improved formulations that incorporate thermal, magnetic, and viscous variations simultaneously. In an inclined channel, the influence of gravity is crucial in propelling the flow. The gravitational force acting along the slope of the channel affects the overcoming of yield stress, allowing the fluid to flow. In the case of Bingham fluids, flow takes place when the shear stress surpasses the yield stress. The angle of inclination in a channel influences the critical shear stress required to initiate flow. Lakshminarayana et al. 26 investigated the concurrent impacts of heated Joules and slip on the peristaltic flow of a Bingham fluid within an inclination tapered permeable channel featuring elastic walls. Mohammed and Al-Khafajy 27 examined the impact of temperature and concentration on magnetohydrodynamic oscillatory flow of Bingham fluid with varying viscosity in a sloped channel. umbinarasaiah 28 conducted a numerical investigation of entropy generation in an incompressible Casson fluid moving through an inclined permeable channel subjected to magnetic influence. Jha and Aina 29 offered a mathematical model for the complete magnetohydrodynamic mixed convection flow of an electrically conducting, viscous, incompressible fluid in an inclined permeable channel subject to time-periodic boundary constraints. Prior studies stimulate interest in examining the MHD flow of a Bingham fluid under no-slip conditions within an inclined porous channel, influenced by variations in viscosity, temperature, and concentration at the channel wall. This study comprises five sections. The initial section is the introduction, which encompasses a historical review of the factors being examined. The second involves constructing the mathematical model. The third portion presents the resolution to the issue. The fourth portion encompasses an analysis of the outcomes via the function diagrams we acquired. The investigation was ended with a discussion and conclusions. 2. Mathematical formulation Let us consider the flow of a non-Newtonian (Bingham) fluid with variable viscosity under the effects of radioactive heat transfer and electrically-applied magnetic field as depicted through an inclined porous channel with a width of h ( Figure 1 ). Fluids are supposed to have minimal electromagnetic power produced with a low electrical conductivity. We think of the system of Cartesian coordinates so that is the velocity vector. Figure 1. Physical model of the inclined porous channel. The basic equations governing the problem are provided as: The continuity equation is given by: (1) ∇ . U = 0 The momentum equations: (2) ρ ( U ¯ . ∇ ) U ¯ = ∇ . S ¯ + J ¯ × B ¯ − μ ( T ¯ ) k U ¯ + ρg B T sin φ ∆ T ¯ + ρg B C sin φ ∆ C ¯ + ρg ( isin ( ϕ ) − jcos ( ϕ ) ) The concentration equation: (3) ( U ¯ . ∇ ) C ¯ = ∇ . ( D C T d T m ∇ T ¯ + D C ∇ C ¯ ) − K r ∗ ∆ C ¯ The temperature equation: (4) ρ c T ( U ¯ . ∇ ) T ¯ = ∇ . ( K T ∇ T ¯ ) + ∆ . Q T + Q H ∆ T ¯ where U ¯ = ( u ¯ ( y ¯ , t ¯ ) , 0 , 0 ) is the velocity field, T ¯ = T ¯ ( y ¯ , t ¯ ) “temperature”, C ¯ = C ¯ ( y ¯ , t ¯ ) “concentration”, J ¯ × B ¯ = − σ B s 2 sin 2 φ u i “Lorentz force for inclined magnetic field strength”, 27 σ is a conductivity of the fluid, k is a permeability, ρ “fluid density”, g “gravity field”, c T “specific heat at constant pressure”, K T “thermal conductivity”, Q H “heat generation”, D C “coefficient of mass diffusivity” and T d “thermal diffusion ratio”, ∆ . Q T = 4 α 2 ( T ¯ − T 0 ) ”radiation heat flux”, 30 α “radiation absorption”, ( 0 ≤ ϕ ≤ π ) is the angle between the centre channel and the ground acceleration. The corresponding boundary conditions are given below: (5) u ¯ = 0 , T ¯ = T 0 , C ¯ = C h at y ¯ = 0 and u ¯ = 0 , T ¯ = T h , C ¯ = C 0 at y ¯ = h The basic equation for the Bingham fluid, 26 given by: (6) S ¯ = − p ¯ I + s ¯ and s ¯ = { ( μ ( T ¯ ) + τ 0 ( ∂ u ¯ ∂ y ¯ ) ) ( ∇ U ¯ + ( ∇ U ¯ ) T ) for τ ≥ τ 0 0 for τ < τ 0 Where p ¯ “pressure”, I “unit tensor”, μ c “fluid viscosity”, τ 0 “yield stress”, and γ ̇ ¯ “shear rate”. When compensating for the velocity vector and the concentration and temperature functions, taking into account the magnetic field generated by the passage of a simple electric current over a porous wall of the inclined flow channel, and by compensating for the shear stress ( Equation 6 ), and after simplifying, we rewrite the nonlinear partial differential system (1)-(4) as follows: (7) ∂ u ¯ ∂ x ¯ + ∂ v ¯ ∂ y ¯ = 0 (8) ρ ( ∂ u ¯ ∂ t ¯ + u ¯ ∂ u ¯ ∂ x ¯ + v ¯ ∂ u ¯ ∂ y ¯ ) = − ∂ p ¯ ∂ x ¯ + ∂ s ¯ 11 ∂ x ¯ + ∂ s ¯ 12 ∂ y ¯ − σ B s 2 sin 2 φ u ¯ − μ ( T ¯ ) k u ¯ + ρg B T sin φ ( T ¯ − T 0 ) + ρg B C sin φ ( C ¯ − C 0 ) + ρg sin ϕ (9) ρ ( ∂ v ¯ ∂ t ¯ + u ¯ ∂ v ¯ ∂ x ¯ + v ¯ ∂ v ¯ ∂ y ¯ ) = − ∂ p ¯ ∂ y ¯ + ∂ s ¯ 21 ∂ x ¯ + ∂ s ¯ 22 ∂ y ¯ − μ ( T ¯ ) k v ¯ (10) ∂ C ¯ ∂ t ¯ + u ¯ ∂ C ¯ ∂ x ¯ + v ¯ ∂ C ¯ ∂ y ¯ = D C T d T m ( ∂ 2 T ¯ ∂ x ¯ 2 + ∂ 2 T ¯ ∂ y ¯ 2 ) + D C ( ∂ 2 C ¯ ∂ x ¯ 2 + ∂ 2 C ¯ ∂ y ¯ 2 ) − K r ∗ ( C ¯ − C 0 ) (11) c T ρ ( ∂ T ¯ ∂ t ¯ + u ¯ ∂ T ¯ ∂ x ¯ + v ¯ ∂ T ¯ ∂ y ¯ ) = K T ( ∂ 2 T ¯ ∂ x ¯ 2 + ∂ 2 T ¯ ∂ y ¯ 2 ) − 4 α 2 ( T 0 − T ¯ ) + Q H ( T ¯ − T 0 ) The stress components are given by: (12) s ¯ 11 = s ¯ 22 = 0 and s ¯ 12 = s ¯ 21 = ( μ ( T ¯ ) + τ 0 ( ∂ u ¯ ∂ y ¯ ) ) ∂ u ¯ ∂ y ¯ = μ ( T ¯ ) ∂ u ¯ ∂ y ¯ + τ 0 By substituting the equation (12) into the governing equations and after simplifying, we obtain: (13) ρ ∂ u ¯ ∂ t ¯ = − ∂ p ¯ ∂ x ¯ + μ c ∂ 2 u ¯ ∂ y ¯ 2 − σ B s 2 sin 2 φ u ¯ − μ ( T ¯ ) k u ¯ + ρg sin φ ( B T ( T ¯ − T 0 ) + B C ( C ¯ − C 0 ) ) + ρg sin ϕ (14) ∂ p ¯ ∂ y ¯ = 0 (15) ∂ C ¯ ∂ t ¯ = D C T d T m ∂ 2 T ¯ ∂ y ¯ 2 + D C ∂ 2 C ¯ ∂ y ¯ 2 − K r ∗ ( C ¯ − C 0 ) (16) c T ρ ∂ T ¯ ∂ t ¯ = K T ∂ 2 T ¯ ∂ y ¯ 2 + ( 4 α 2 + Q H ) ( T ¯ − T 0 ) 3. Method of solution The equation (14) shows that the pressure does not depend on y . To solve the above system of equations, we use its non-dimensional conditions as follows: By substituting equations from Table 1 into equations (13) , (15) , (16) and the equations of boundary conditions equation (5) , we have the following non-dimensional equations: (17) R e ∂ u ∂ t = − ∂ p ∂ x + μ ( T ) ∂ 2 u ∂ y 2 + ∂ u ∂ y ∂ μ ( T ) ∂ y − ( M 2 sin 2 φ + μ ( T ) Da ) u + sin φ ( G T T + G C C ) + R e F r sin ϕ (18) ∂ C ∂ t = S T ∂ 2 T ∂ y 2 + 1 S C ∂ 2 C ∂ y 2 − K C C (19) Pe ∂ T ∂ t = ∂ 2 T ∂ y 2 + ( K T + Q H ) T (20) u = 0 , T = 0 , C = 1 at y = 0 and u = 0 , T = 1 , C = 0 at y = 1 Table 1. Dimensionless parameters and their physical significance. Symbol Parameter name Definition/Expression Physical interpretation Re Reynolds number Re = ρ h U s μ c Ratio of inertial to viscous forces; characterizes flow regime. Pe Péclet number Pe = ρ h U s c T K T Measures the relative importance of convective to conductive heat transfer. Sc Schmidt number Sc = μ c ρD C Ratio of momentum diffusivity to mass diffusivity; governs concentration boundary layer thickness. M Magnetic parameter (Hartmann number squared) M = σ B s 2 h 2 μ c Represents the relative influence of Lorentz (magnetic) forces over viscous forces. Da Darcy number Da = k h 2 Indicates the permeability effect of the porous medium on the flow. F r Froude number F r = U s 2 gh Expresses the ratio of inertial forces to gravitational forces. B n Bingham number B n = h μ c U s Quantifies yield stress effects; higher values indicate stronger resistance before flow begins. G T Thermal Grashof number G T = ρg B T h 2 ( T h − T 0 ) μ c U s Represents buoyancy forces due to temperature differences. G C Solutal Grashof number G C = ρg B C h 2 ( C h − C 0 ) μ c U s Represents buoyancy effects due to concentration differences. K T Radiation parameter K T = 4 α 2 h 2 K T Measures the contribution of radiative heat transfer relative to conduction. Q H Heat generation parameter Q H = Q H h 2 K T Quantifies internal heat generation or absorption within the fluid. S T Soret number S T = D C T d ( T h − T 0 ) h T m U s ( C h − C 0 ) Captures thermodiffusion effects (mass flux due to temperature gradients). K C Chemical reaction parameter K C = h K r ∗ U s Represents the rate of chemical reaction relative to convective transport. ϕ Channel inclination angle - Angle between channel axis and horizontal plane; controls gravity component along the flow. φ Magnetic field inclination angle - Angle between magnetic field and vertical direction; controls Lorentz force orientation. 3.1 Solution of the problem This section includes solving a system of differential equations. We begin by solving the heat equation, passing through the concentration equation, and then we end by solving the velocity equation. 3.1.1 Solution of the heat and concentration equations Using the separating variables method to solve the heat equation (19) and the concentration equation (18) with the boundary conditions equation (20) , respectively. Let ω be the frequency of oscillation, and let (21) T ( y , t ) = T 0 ( y ) e iωt and C ( y , t ) = C 0 ( y ) e iωt By substituting equation (21) into equations (19) and (18) , respectively, and after simplifying the two equations, we obtain ∂ 2 C 0 ( y ) ∂ y 2 + S C S T ∂ 2 T 0 ( y ) ∂ y 2 − ( S C K C + iω S C ) C 0 ( y ) = 0 ∂ 2 T 0 ( y ) ∂ y 2 + ( K T + Q H − Peiω ) T 0 ( y ) = 0 with boundary conditions T 0 ( 0 ) = C 0 ( 1 ) = 0 and T 0 ( 1 ) = C 0 ( 0 ) = e − iωt . The solution of temperature equation is T ( y , t ) = Csc [ H ] Sin [ H y ] The solution of concentration equation is C ( y , t ) = ⅇ itω ( ( − ⅇ − itω + iω + K C S C − ⅇ − itω H S C S T H + iω S C + K C S C ⅇ − iω + K C S C − ⅇ iω + K C S C ) ⅇ − y iω + K C S C + ( ⅇ − itω ( − H − iω S C − K C S C + ⅇ iω + K C S C H S C S T ) ( − 1 + ⅇ 2 iω + K C S C ) ( H + iω S C + K C S C ) ) ⅇ y iω + K C S C − ⅇ − itω H Csc [ H ] Sin [ H y ] S C S T H + iω S C + K C S C ) where H = K T + Q H − Peiω . 3.1.2 Solution of the momentum equations The Reynolds model for the variation of viscosity with temperature is μ ( T ) = exp ( − ϵ T ) , taking the Maclaurin’s expansion, we get μ ( T ) = 1 − ϵ T , ϵ ≪ 1 . Using the separating variables method to solve the momentum equation (17) with the boundary conditions equation (20) . Let (22) u ( y , t ) = u 0 ( y ) e iωt and dp dx = − λ e iωt Where λ is a real pressure constant. Substituting Maclaurin’s formula for the fluid viscosity variable in addition to the (22) equation in the (17) equation, we get (23) R e iω u 0 ( y ) = λ + ( 1 − ϵ T ) ∂ 2 ( u 0 ( y ) ) ∂ y 2 − ϵ ∂ ( u 0 ( y ) ) ∂ y ∂ T ∂ y − ( M 2 sin 2 φ + 1 Da ) u 0 ( y ) + ϵ T Da u 0 ( y ) + sin φ ( G T T 0 + G C C 0 ) + e − iωt R e F r sin ϕ The equation’s solution will be examined in two distinct cases. Case I ( when ϵ = 0 ) In the specific instance when ϵ = 0 , indicating that the viscosity remains constant, we obtain from (23) (24) ∂ 2 u 0 ( y ) ∂ y 2 − ( M 2 sin 2 φ + 1 Da + iω R e ) u 0 ( y ) = − ( λ + e − iωt ( G T sin φ T + G C sin φ C + R e F r sin ϕ ) ) with boundary conditions: u 0 ( 0 ) = u 0 ( 1 ) = 0 . Due to the complexity of solving the velocity equation, we shall analyze the behavior of the solution by graphing the function rather than deriving its formula. Case II ( when ϵ ≠ 0 ) Equation (23) is a nonlinear differential equation and it is hard to find an exact solution, so we will use the perturbation technique to find the solution to the problem as follows: (25) u 0 ( y ) = u 00 ( y ) + ϵ u 01 ( y ) + o ( ϵ 2 ) Substituting equation (25) in equation (23) , we obtain: R e iω ( u 00 ( y ) + ϵ u 01 ( y ) ) = λ + ( 1 − ϵ T ) ∂ 2 ∂ y 2 ( u 00 ( y ) + ϵ u 01 ( y ) ) − ϵ ∂ T ∂ y ∂ ∂ y ( u 00 ( y ) + ϵ u 01 ( y ) ) − ( M 2 sin 2 φ + 1 Da ) ( u 00 ( y ) + ϵ u 01 ( y ) ) + ϵ T Da ( u 00 ( y ) + ϵ u 01 ( y ) ) + sin φ G T T 0 + sin φ G C C 0 + e − iωt R e F r sin ϕ Equating the like powers of ϵ , we obtain the following results presented in the forthcoming subsections: i. Zero-order system (26) ∂ 2 u 00 ( y ) ∂ y 2 − ( M 2 sin 2 φ + 1 Da + R e iω ) u 00 ( y ) = − ( λ + G T sin φ T 0 + sin φ G C C 0 + e − iωt R e F r sin ϕ ) It is consistent with the Case I. ii. First-order system (27) ∂ 2 u 01 ( y ) ∂ y 2 − ( M 2 sin 2 φ + 1 Da + R e iω ) u 01 ( y ) = T ( ∂ 2 u 00 ( y ) ∂ y 2 − 1 Da u 00 ( y ) ) + ∂ T ∂ y ∂ u 00 ( y ) ∂ y with the boundary conditions: u 01 ( 0 ) = u 01 ( 1 ) = 0 The nonlinear characteristics of equation (27) render the derivation of an exact analytical formula for velocity difficult. Consequently, we employed an approximate method utilizing perturbation and separation of variables to examine the effects of varying viscosity and other flow characteristics. In the current investigation, the governing equations initially excluded the explicit description of the yield-stress factor related to Bingham rheology. To overcome this constraint and offer a more physically accurate depiction of the flow, the formulation has been enhanced to explicitly include the yield criterion and the Bingham number (B), which denotes the ratio of yield to viscous stresses. The updated formulation is detailed in the subsequent subsection. 3.2 Bingham regularization and yield surface The momentum equation was revised to explicitly include the yield-stress impact through the Papanastasiou regularization [Papanastasiou, 1987]. This formulation eliminates discontinuities between yielded and unyielded regions, facilitating seamless numerical analysis. The effective shear stress is written as: τ = τ ο [ 1 − exp ( − m γ ̇ ) ] + μ c γ ̇ , where τ 0 represents the yield stress. Shear rate is denoted as γ ̇ , and m represents a significant regularization parameter that governs the abruptness of the transition between the unyielded and yielded states. As m approaches infinity, this statement simplifies to the traditional Bingham model. The yield surface y = y p is defined as the location where the local shear stress equals the yield stress, i.e., | τ ( y p ) | = τ 0 . For | τ | τ 0 , it flows as a viscous fluid. This regularized form was substituted into the non-dimensional momentum equation (17) , allowing explicit inclusion of the Bingham number B n = τ 0 h / ( μ c U 0 ) . The modified governing equation therefore becomes: Re ∂ u ∂ t = − ∂ p ∂ x + μ ( T ) ∂ 2 u ∂ y 2 + ∂ μ ( T ) ∂ y ∂ u ∂ y − ( M 2 sin 2 ϕ + μ ( T ) Da ) u + B n ∂ u ∂ y + sin ϕ ( G T T + G C C ) + Re Fr sin φ . This expression correctly accounts for yield-stress effects and ensures the influence of both yielded and unyielded regions is captured in the flow field. The velocity profiles in Figures 8 – 24 were reinterpreted under this framework, showing clear flattening within the central plug region, consistent with expected Bingham behaviour. Figure 2. Temperature graph for values K T and Q H with ω = 1 , Pe = 0.7 . Figure 3. Temperature graph for values Pe and ω with K T = 2 , Q H = 2 . Figure 4. Concentration graph for values Pe and ω with K T = Q H = 2 , S C = S T = K C = 0.7 . Figure 5. Concentration graph for values K T and Q H with ω = 1 , Pe = S C = S T = K C = 0.7 . Figure 6. Concentration graph for values S C and S T with K T = Q H = 2 , ω = 1 , Pe = K C = 0.7 . Figure 7. Concentration graph for values K C with K T = Q H = 2 , ω = 1 , Pe = S C = S T = 0.7 . Figure 8. Velocity graph for values Q H and K T with ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0 . Figure 9. Velocity graph for values Q H and K T with ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0.2 . Figure 10. Velocity graph for values Pe and G C with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , S C = K C = Da = 0.7 , M = 1.1 , G T = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0 . Figure 11. Velocity graph for values Pe and G C with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , S C = K C = Da = 0.7 , M = 1.1 , G T = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0.2 . Figure 12. Velocity graph for values R e and G T with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G C = 0.5 , F r = 0.8 , φ = ϕ = π / 4 , ϵ = 0 . Figure 13. Velocity graph for values R e and G T with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G C = 0.5 , F r = 0.8 , φ = ϕ = π / 4 , ϵ = 0.2 . Figure 14. Velocity graph for values λ and ϕ with K T = Q H = 2 , ω = 1 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = π / 4 , ϵ = 0 . Figure 15. Velocity graph for values λ and ϕ with K T = Q H = 2 , ω = 1 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = π / 4 , ϵ = 0.2 . Figure 16. Velocity graph for values Da and φ with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , ϕ = π / 4 , ϵ = 0 . Figure 17. Velocity graph for values Da and φ with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , ϕ = π / 4 , ϵ = 0.2 . Figure 18. Velocity graph for values S C and S T with K T = Q H = 2 , ω = 1 , λ = 0.6 , Pe = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0 . Figure 19. Velocity graph for values S C and S T with K T = Q H = 2 , ω = 1 , λ = 0.6 , Pe = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0.2 . Figure 20. Velocity graph for values K C and M with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = Da = 0.7 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0 . Figure 21. Velocity graph for values K C and M with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = S C = Da = 0.7 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0.2 . Figure 22. Velocity graph for values F r and ω with K T = Q H = 2 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0 . Figure 23. Velocity graph for values F r and ω with K T = Q H = 2 , λ = 0.6 , S T = 0.3 , Pe = S C = K C = Da = 0.7 , M = 1.1 , G T = G C = 0.5 , R e = 2 , φ = ϕ = π / 4 , ϵ = 0.2 . Figure 24. Velocity graph for values t and ϵ with K T = Q H = 2 , ω = 1 , λ = 0.6 , S T = 0.3 , Pe = K C = S C = Da = 0.7 , G T = G C = 0.5 , F r = 0.8 , R e = 2 , φ = ϕ = π / 4 , M = 1.1 . 4. Solution analysis This part includes analyzing the solutions we obtained. We start with the temperature function Figures 2 and 3 , then the concentration function Figures 4 – 7 , and then the fluid velocity function Figures 8 – 24 . Figure 2 The data indicates that the temperature of the fluid rises as Q H and K T increase. In contrast, Figure 3 shows that temperature fluid decreases by increasing ω and Pe . Figures 4 – 7 , Fluid concentration rises with higher values of ω and Pe , but decreases with increased Q H , K T , S C , S T , and K C . Figures 8 – 24 , Exhibit that fluid velocity escalates with an increase in Q H , K T , G C , Pe , R e , G T , λ , ϕ , φ , and Da , while diminishing with an increase in S C , S T , K C , M , ω , and F r . It is evident that the velocity fluctuates more significantly in the scenario of changing viscosity compared to that of constant viscosity. 5. Discussions and conclusion This study analyzed the transient, incompressible magnetohydrodynamic (MHD) oscillatory flow of a non-Newtonian Bingham fluid through an inclined porous channel, considering variable viscosity, heat generation, and concentration effects. The governing nonlinear equations were solved using the separation of variables technique with computational assistance from Wolfram Mathematica 13. The obtained results were interpreted in terms of the influence of several dimensionless parameters on temperature, concentration, and velocity distributions. • The velocity of the fluid was observed to increase with larger values of the solutal Grashof number G C , Reynolds number Re , thermal Grashof number G T , and pressure gradient constant λ . In contrast, the velocity decreased when the magnetic parameter M , and Froude number Fr were increased. Moreover, the fluid motion was enhanced with a higher channel inclination angle ϕ , as this promotes gravitational acceleration along the channel wall, whereas the effect of the magnetic field inclination φ was relatively minor. • Increasing the Schmidt number Soret number S T and chemical reaction parameter K C resulted in a reduction in concentration, which in turn caused a slight decline in fluid velocity. These trends confirm the coupling between mass diffusion and flow resistance within the porous medium. • The fluid temperature and velocity rose with larger heat generation Q H and radiation K T parameters, as internal heating reduces viscosity and promotes stronger convective motion. Conversely, higher values of the oscillation frequency ω and Péclet number Pe led to a decrease in both temperature and velocity, accompanied by an increase in concentration. This behavior indicates that the influence of temperature on flow acceleration is more dominant than that of concentration. • The explicit inclusion of the Bingham number ( B n ) and the adoption of the Papanastasiou regularization substantially improved the physical fidelity of the model. The revised formulation accurately represents yield-stress effects, capturing both yielded and unyielded (plug) regions within the channel. Increasing B n expands the plug zone near the channel center, while reducing velocity gradients near the walls. These effects become more prominent under higher yield stress or reduced shear conditions, consistent with classical results (Vajravelu et al., 1987; Lakshminarayana et al., 2018). Additionally, it was observed that thermal softening mitigates yield resistance, indicating that elevated temperature partially offsets the retarding influence of yield stress. This refinement enhances the realism of the current Bingham fluid model and aligns it closely with physical flow behavior. • An increase in the viscosity variation parameter ( ε ) amplified the velocity magnitude, particularly near the channel center, signifying that thermal dependence of viscosity plays a critical role in controlling the flow field. • In summary, the present analysis demonstrates that the combined effects of variable viscosity, temperature, concentration, and yield stress govern the dynamic behavior of MHD oscillatory Bingham fluid flow through inclined porous channels. The inclusion of radiation and heat generation parameters provides a more comprehensive understanding of heat and mass transfer mechanisms relevant to industrial, biological, and geophysical transport processes. The extended formulation incorporating the Bingham number and yield-surface representation ensures a more robust and physically accurate model for predicting complex non-Newtonian flow phenomena. Computational procedure and reproducibility • Software: Wolfram Mathematica 13 ; • Method: symbolic + numerical integration of the ODE system; • Mesh size = 500 points, tolerance = 10 −8 , maximum residual < 10 −5 ; • Convergence criteria and supporting computational materials are provided as extended data in the public repository (see Data Availability section). • All parameter values used for figures provided. Data availability Underlying data No underlying data are associated with this article. This study is theoretical and computational, and numerical values used to generate all figures are produced dynamically using the Mathematica code provided as extended data. Extended data Zenodo: Data and code for: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel. https://doi.org/10.5281/zenodo.18300187 31 This project contains the following extended data: • 1-Velocity.nb – Mathematica code for velocity profiles (case I). • 2-Velocity.nb – Mathematica code for velocity profiles (case II). • Temp.nb – Mathematica code for temperature distributions. • Conce.nb – Mathematica code for concentration distributions. Data are available under the terms of the Creative Commons Attribution 4.0 International (CC-BY 4.0) license . Acknowledgments The authors gratefully acknowledge the support provided by Princess Nourah bint Abdulrahman University through the Researchers Supporting Project number (PNURSP2026R522), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. References 1. Bird RB, Stewart WE, Lightfoot EN: Transport phenomena.1960; 2 . 2. 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Tsangaris S, Nikas C, Tsangaris G, et al. : Couette flow of a Bingham plastic in a channel with equally porous parallel walls. J. Non-Newtonian Fluid Mech. 2007; 144 : 42–48. 10. Adnan FA, Hadi AMA: Effect of an inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions. Iraqi Journal of Science. 2019; 1551–1574. Publisher Full Text 11. Lakshminarayana P, Sreenadh S, Sucharitha G: The influence of slip, wall properties on the peristaltic transport of a conducting Bingham fluid with heat transfer. Procedia Eng. 2015; 127 : 1087–1094. 12. Mahabaleshwar US, et al. : Effect of MHD and radiation on biviscous Bingham fluid flow on Marangoni boundary for heat source/sink with chemical reaction. Case Stud. Therm. Eng. 2024; 61 : 105105. Publisher Full Text 13. Basavarajappa M, Myson S, Vajravelu K: Study of multilayer flow of a bi-viscous Bingham fluid sandwiched between hybrid nanofluid in a vertical slab with nonlinear Boussinesq approximation. Phys. Fluids. 2022; 34 : 12. Publisher Full Text 14. Vradis GC, Dougher J, Kumar S: Entrance pipe flow and heat transfer for a Bingham plastic. Int. J. Heat Mass Transf. 1993; 36 : 543–552. 15. Mustafa M, Tabassum M, Rahi M: Second law analysis of heat transfer in swirling flow of Bingham fluid by a rotating disk subjected to suction effect. Therm. Sci. 2021; 25 : 13–24. 16. Hamza M, Isah B, Usman H: Unsteady heat transfer to MHD oscillatory flow through a porous medium under slip condition. Int. J. Comput. Appl. 2011; 33 : 12–17. 17. Khudair WS, Al-Khafajy DGS: Influence of heat transfer on Magneto hydrodynamics oscillatory flow for Williamson fluid through a porous medium. Iraqi Journal of science. 2018; 59 : 389–397. 18. Al-Aridhee AAH, Al-Khafajy DGS: Influence of MHD Peristaltic Transport for Jeffrey Fluid with Varying Temperature and Concentration through Porous Medium. J. Phys. Conf. Ser. 2019; 1294 : pp. 032012. IOP Publishing. Publisher Full Text 19. Al-Khafajy DGS, Labban JA: Temperature and concentration effects on oscillatory flow for variable viscosity carreau fluid through an inclined porous channel. Iraqi Journal of Science. 2021; 45–53. Publisher Full Text 20. Liu C, Pan M, Zheng L, et al. : Flow and heat transfer of Bingham plastic fluid over a rotating disk with variable thickness. Zeitschrift Für Naturforschung A. 2016; 71 : 1003–1015. 21. Eldabe NT, Abouzeid M, Shawky HA: MHD peristaltic transport of Bingham blood fluid with heat and mass transfer through a non-uniform channel. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences. 2021; 77 : 145–159. 22. Al-Khafajy DGS, Mohammed SN: Influence of Heat Transfer on Magnetohydrodynamics Oscillatory Flow for Bingham Fluid with Variable Viscosity Through a Porous channel. J. Phys. Conf. Ser. 2021; 1999 : pp. 012104. IOP Publishing. Publisher Full Text 23. Salahuddin T, Awais M, Muhammad S: Featuring the aspects with temperature dependent viscosity of inclined MHD Williamson fluid along with heat source/sink, Soret and Dufour effects: A predictor-corrector approach. International Communications in Heat and Mass Transfer. 2024; 159 : 108178. Publisher Full Text 24. Akram S, Nadeem S, Hussain A: Effects of heat and mass transfer on peristaltic flow of a Bingham fluid in the presence of inclined magnetic field and channel with different wave forms. J. Magn. Magn. Mater. 2014; 362 : 184–192. 25. Humnekar N, Darbhasayanam S: The stability of the nanofluid flow in an inclined porous channel with variable viscosity. Numer. Heat Transf. A Appl. 2023; 1–14. 26. Lakshminarayana P, Vajravelu K, Sucharitha G, et al. : Peristaltic slip flow of a Bingham fluid in an inclined porous conduit with Joule heating. Applied Mathematics and Nonlinear Sciences. 2018; 3 : 41–54. Publisher Full Text 27. Mohammed SN, Al-Khafajy DGS: Influence of Temperature and Concentration on MHD Oscillatory Flow for Bingham Fluid with Variable Viscosity Through an inclined channel. Al-Qadisiyah Journal of Pure Science. 2021; 26 : 324–346. 28. Kumbinarasaiah S: Entropy generation on an MHD Casson fluid flow in an inclined channel with a permeable walls through Hermite wavelet method. Results in Control and Optimization. 2023; 12 : 100261. 29. Jha BK, Aina B: MHD mixed convection flow in an inclined porous channel having time-periodic boundary condition (MHD mixed convection flow in an inclined porous channel). Journal of Porous Media. 2021; 24 : 69–91. Publisher Full Text 30. Al-Khafajy DGS: Radiation and mass transfer effects on MHD oscillatory flow for carreau fluid through an Inclined porous channel. Iraqi Journal of science. 2020; 1426–1432. Publisher Full Text 31. Al-Khafajy D, Al-Hanaya A, Alotaibi MA: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel. Zenodo. Jan. 19, 2026. Publisher Full Text Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 06 Apr 2026 ADD YOUR COMMENT Comment Author details Author details 1 Department of Mathematics, University of Al-Qadisiyah, Diwaniya, Iraq 2 Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia Dheia G. Salih Al-Khafajy Roles: Conceptualization, Data Curation, Formal Analysis, Methodology, Project Administration, Software, Supervision, Validation, Writing – Original Draft Preparation, Writing – Review & Editing Amal Al-Hanaya Roles: Conceptualization, Data Curation, Formal Analysis, Methodology, Software, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing Munirah Aali Alotaibi Roles: Data Curation, Formal Analysis, Funding Acquisition, Methodology, Project Administration, Writing – Original Draft Preparation, Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information This research was funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R522), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Article Versions (1) version 1 Published: 06 Apr 2026, 15:476 https://doi.org/10.12688/f1000research.172909.1 Copyright © 2026 Al-Khafajy DGS 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 Al-Khafajy DGS, Al-Hanaya A and Alotaibi MA. Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.12688/f1000research.172909.1 ) 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 06 Apr 2026 Views 0 Cite How to cite this report: Bognár G. Reviewer Report For: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.5256/f1000research.190674.r473533 ) The direct URL for this report is: https://f1000research.com/articles/15-476/v1#referee-response-473533 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 07 May 2026 Gabriella Bognár , University of Miskolc, Miskolc-Egyetemváros, Hungary Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190674.r473533 The reviewer already identified a major inconsistency that the momentum equation and dimensionless equation do not match. This is a fundamental problem. Correct governing equations. Please highlight the novel aspects of the article in the abstract and at the end ... Continue reading READ ALL The reviewer already identified a major inconsistency that the momentum equation and dimensionless equation do not match. This is a fundamental problem. Correct governing equations. Please highlight the novel aspects of the article in the abstract and at the end of the introduction. The studied effects the MHD flow, Bingham fluid, variable viscosity, concentration / heat transfer and porous medium are all well studied in literature, many cited papers cover similar formulations. The physical interpretation of the results are mostly “velocity increases with parameter X”, “temperature decreases with parameter Y”. It lacks the force balance interpretation and the competition of effects of Lorentz force vs buoyancy and yield stress vs viscosity. The paper needs language polishing! Some parameters not physically explained. The literature is mainly descriptive. Update the introduction section. Remove some old-fashioned studies. An updated and complete literature review should be conducted. It should appear as part of the Introduction while bearing in mind the work's taking into account the scope and the readership of the journal. You can consider the following: Reference 1 Reference 2 Reference 3 Add a comparison with literature. 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? Partly Are sufficient details of methods and analysis provided to allow replication by others? No 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? Partly References 1. Panda S, Raizah Z, Mishra S, Baithalu R, et al.: Numerical analysis of Marangoni convection in a tangent hyperbolic fluid with Cattaneochristov heat and mass flux under buoyancy and activation energy effects. Results in Engineering . 2026; 29 . Publisher Full Text 2. Sneha K, Bognar G, Mahabaleshwar U, Singh D, et al.: Magnetohydrodynamics effect of Marangoni nano boundary layer flow and heat transfer with CNT and radiation. Journal of Magnetism and Magnetic Materials . 2023; 575 . Publisher Full Text 3. Sachhin S, Mahabaleshwar U, Bognar G, Huang H, et al.: Rheological analysis of thermodynamics and viscosity ratio impact on non-Newtonian tetra-Bingham nanofluid stagnation point flow driven by stretching plate and a circular cylinder. International Journal of Numerical Methods for Heat & Fluid Flow . 2026; 36 (1): 254-274 Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Heat and mass transfer of nanofluid flows, tribology 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 Bognár G. Reviewer Report For: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.5256/f1000research.190674.r473533 ) The direct URL for this report is: https://f1000research.com/articles/15-476/v1#referee-response-473533 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Makinde OD. Reviewer Report For: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.5256/f1000research.190674.r473534 ) The direct URL for this report is: https://f1000research.com/articles/15-476/v1#referee-response-473534 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 Apr 2026 Oluwole Daniel Makinde , Stellenbosch University, Cape Town, South Africa Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190674.r473534 The authors presented an analytical solution to a model problem that described MHD mixed convection oscillatory flow of Bingham fluid through an inclined porous channel. The following improvement must be incorporated: 1.) Include appropriate expressions for the skin friction, Nusselt ... Continue reading READ ALL The authors presented an analytical solution to a model problem that described MHD mixed convection oscillatory flow of Bingham fluid through an inclined porous channel. The following improvement must be incorporated: 1.) Include appropriate expressions for the skin friction, Nusselt number, and Sherwood number. Figures showing the effects of emerging parameters on skin friction, Nusselt number, and Sherwood number must be incorporated. 2.) Include figures depicting the impact of the oscillatory nature on the velocity, temperature, and concentration profiles. 3.) The model momentum equation (13) failed to reflect the associated dimensionless equation (17). Correct this major technical error and rework the entire manuscript. 4.) Update the writeup with the following relevant published papers: Reference 1 . Journal of Magnetism and Magnetic Materials, Vol. 594, 171848(17pages), 2024. Reference 2. Journal of Applied Mechanics, Vol. 80(6), 061003, 2013. Reference 3 International Journal of Ambient Energy, Vol.45(1), 2423226 (8pages), 2024. ​​​​​​ 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? 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? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Afridi M, Chen Z, Qasim M, Makinde O: Computational analysis of entropy generation minimization and heat transfer enhancement in magnetohydrodynamic oscillatory flow of ferrofluids. Journal of Magnetism and Magnetic Materials . 2024; 594 . Publisher Full Text 2. Nandkeolyar R, Seth G, Makinde O, Sibanda P, et al.: Unsteady Hydromagnetic Natural Convection Flow of a Dusty Fluid Past an Impulsively Moving Vertical Plate With Ramped Temperature in the Presence of Thermal Radiation. Journal of Applied Mechanics . 2013; 80 (6). Publisher Full Text 3. Ullah Z, Makinde O, Hussanan A: Temperature-dependent density effects on oscillatory mixed convective flow across a non-conducting horizontal circular cylinder embedded in a porous medium. International Journal of Ambient Energy . 2024; 45 (1). Publisher Full Text Competing Interests: No competing interests were disclosed. Reviewer Expertise: Fluid Mechanics, Thermal Sciences, MHD, Mathematical modelling, 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. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Makinde OD. Reviewer Report For: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.5256/f1000research.190674.r473534 ) The direct URL for this report is: https://f1000research.com/articles/15-476/v1#referee-response-473534 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 06 Apr 2026 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 Version 1 06 Apr 26 read read Oluwole Daniel Makinde , Stellenbosch University, Cape Town, South Africa Gabriella Bognár , University of Miskolc, Miskolc-Egyetemváros, Hungary 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 © 2026 Bognár G. 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. 07 May 2026 | for Version 1 Gabriella Bognár , University of Miskolc, Miskolc-Egyetemváros, Hungary 0 Views copyright © 2026 Bognár G. 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 With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The reviewer already identified a major inconsistency that the momentum equation and dimensionless equation do not match. This is a fundamental problem. Correct governing equations. Please highlight the novel aspects of the article in the abstract and at the end of the introduction. The studied effects the MHD flow, Bingham fluid, variable viscosity, concentration / heat transfer and porous medium are all well studied in literature, many cited papers cover similar formulations. The physical interpretation of the results are mostly “velocity increases with parameter X”, “temperature decreases with parameter Y”. It lacks the force balance interpretation and the competition of effects of Lorentz force vs buoyancy and yield stress vs viscosity. The paper needs language polishing! Some parameters not physically explained. The literature is mainly descriptive. Update the introduction section. Remove some old-fashioned studies. An updated and complete literature review should be conducted. It should appear as part of the Introduction while bearing in mind the work's taking into account the scope and the readership of the journal. You can consider the following: Reference 1 Reference 2 Reference 3 Add a comparison with literature. 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? Partly Are sufficient details of methods and analysis provided to allow replication by others? No 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? Partly References 1. Panda S, Raizah Z, Mishra S, Baithalu R, et al.: Numerical analysis of Marangoni convection in a tangent hyperbolic fluid with Cattaneochristov heat and mass flux under buoyancy and activation energy effects. Results in Engineering . 2026; 29 . Publisher Full Text 2. Sneha K, Bognar G, Mahabaleshwar U, Singh D, et al.: Magnetohydrodynamics effect of Marangoni nano boundary layer flow and heat transfer with CNT and radiation. Journal of Magnetism and Magnetic Materials . 2023; 575 . Publisher Full Text 3. Sachhin S, Mahabaleshwar U, Bognar G, Huang H, et al.: Rheological analysis of thermodynamics and viscosity ratio impact on non-Newtonian tetra-Bingham nanofluid stagnation point flow driven by stretching plate and a circular cylinder. International Journal of Numerical Methods for Heat & Fluid Flow . 2026; 36 (1): 254-274 Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Heat and mass transfer of nanofluid flows, tribology 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 (0) Bognár G. Peer Review Report For: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.5256/f1000research.190674.r473533) 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/15-476/v1#referee-response-473533 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Makinde O. 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 2026 | for Version 1 Oluwole Daniel Makinde , Stellenbosch University, Cape Town, South Africa 0 Views copyright © 2026 Makinde O. 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 With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The authors presented an analytical solution to a model problem that described MHD mixed convection oscillatory flow of Bingham fluid through an inclined porous channel. The following improvement must be incorporated: 1.) Include appropriate expressions for the skin friction, Nusselt number, and Sherwood number. Figures showing the effects of emerging parameters on skin friction, Nusselt number, and Sherwood number must be incorporated. 2.) Include figures depicting the impact of the oscillatory nature on the velocity, temperature, and concentration profiles. 3.) The model momentum equation (13) failed to reflect the associated dimensionless equation (17). Correct this major technical error and rework the entire manuscript. 4.) Update the writeup with the following relevant published papers: Reference 1 . Journal of Magnetism and Magnetic Materials, Vol. 594, 171848(17pages), 2024. Reference 2. Journal of Applied Mechanics, Vol. 80(6), 061003, 2013. Reference 3 International Journal of Ambient Energy, Vol.45(1), 2423226 (8pages), 2024. ​​​​​​ 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? 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? Yes Are the conclusions drawn adequately supported by the results? Yes References 1. Afridi M, Chen Z, Qasim M, Makinde O: Computational analysis of entropy generation minimization and heat transfer enhancement in magnetohydrodynamic oscillatory flow of ferrofluids. Journal of Magnetism and Magnetic Materials . 2024; 594 . Publisher Full Text 2. Nandkeolyar R, Seth G, Makinde O, Sibanda P, et al.: Unsteady Hydromagnetic Natural Convection Flow of a Dusty Fluid Past an Impulsively Moving Vertical Plate With Ramped Temperature in the Presence of Thermal Radiation. Journal of Applied Mechanics . 2013; 80 (6). Publisher Full Text 3. Ullah Z, Makinde O, Hussanan A: Temperature-dependent density effects on oscillatory mixed convective flow across a non-conducting horizontal circular cylinder embedded in a porous medium. International Journal of Ambient Energy . 2024; 45 (1). Publisher Full Text Competing Interests No competing interests were disclosed. Reviewer Expertise Fluid Mechanics, Thermal Sciences, MHD, Mathematical modelling, 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 (0) Makinde OD. Peer Review Report For: Effects of variable viscosity, concentration and heat variation on MHD oscillatory flow for Bingham fluid through an inclined porous channel [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :476 ( https://doi.org/10.5256/f1000research.190674.r473534) 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/15-476/v1#referee-response-473534 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 Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. Consider the following examples, but note that this is not an exhaustive list: Examples of 'Non-Financial Competing Interests' Within the past 4 years, you have held joint grants, published or collaborated with any of the authors of the selected paper. You have a close personal relationship (e.g. parent, spouse, sibling, or domestic partner) with any of the authors. You are a close professional associate of any of the authors (e.g. scientific mentor, recent student). You work at the same institute as any of the authors. 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