An Analytical Investigation of MHD Williamson Fluid Flow over an Inclined Stretching Sheet in a Porous Medium with Non-Uniform Internal Heat Generation and Mixed Convection
Keywords:
Williamson fluid, Mixed convection, Non-uniform internal heat generation, Legendre Collocation
Abstract
The complex interplay between heat transfer, fluid motion, and porous media permeability plays a pivotal role in numerous physical and engineering applications. In particular, the control of heat generation and absorption under mixed convection has garnered considerable attention due to its relevance in thermofluid systems embedded in permeable structures. This research presents a semi-analytical approach for solving the nonlinear Williamson fluid model, accounting for the combined effects of mixed convection, medium permeability, and non-uniform heat generation. Through similarity transformations, the governing partial differential equations are reduced to a system of ordinary differential equations, which are subsequently solved using Legendre polynomials as basis functions and Gauss-Lobatto collocation points. The resulting algebraic system is handled in Mathematica 11.0, with solution accuracy verified by comparison to results from the classical Runge-Kutta shooting technique. Numerical findings reveal that increasing the Grashof number enhances fluid velocity, while higher porosity intensifies thermal fields but suppresses flow due to increased resistance in the porous medium. Moreover, spatially varying heat generation induces steep thermal gradients, potentially leading to localized thermal stresses. The proposed methodology proves effective for analyzing complex nonlinear fluid dynamics, offering robust insights for applications in energy systems, geophysical flows, and thermal engineering.
References
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Rosli W., Mohamed M., Sarif N., Mohammad N., & Soid S. (2022). Blood conveying ferroparticle flow on a stagnation point over a stretching sheet: non-Newtonian Williamson hybrid ferrofluid. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 97(2), 175-185.
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Anagandula, S., & Reddy, K. S. (2024). Velocity and thermal slips impact on the Williamson fluid flow above a stretching sheet in the existence of radiation and inclined magnetic field. CFD Letters, 16(7), 118-135.
Alizadeh, R., Karimi, N., Arjmandzadeh, R., & Mehdizadeh, A. (2018). Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media. Journal of Thermal Analysis and Calorimetry, 135(1), 489-506.
Ibrahim, W., & Gamachu, D. (2019). Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet. Heat Transfer, 49(2), 800-822.
Maleque, K. (2020). Similarity requirements for mixed convective boundary layer flow over vertical curvilinear porous surfaces with heat generation or absorption. International Journal of Aerospace Engineering, 2020, 1-9.
Oderinu, R. A., Ayanbukola, F. J., Alao, S., Sanusi, B. A., & Oyeyinka, T. A. (2025). Mixed convective and nonuniform internal heat generation effect on hydromagnetic micropolar fluid flowing across a permeable stretchy wall: A numerical investigation. Heat Transfer, 2025(1).
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Akter, R. (2023). Unsteady Williamson fluid flow over exponentially stretching sheet in the presence of non-uniform heat generation or absorption and inclined magnetic field. Journal of Scientific Research, 15(3), 621-636.
Jyotshna, M., and Dhanalaxmi, V. (2022). Impact of activation energy and heat source or sink on 3D flow of Williamson nanofluid with GaN nanoparticles over a stretching sheet. European Journal of Mathematics and Statistics, 3(5), 16-29.
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Raju, C., Sandeep, N., Ali, M., & Nuhait, A. (2019). Heat and mass transfer in three-dimensional MHD Williamson-Casson fluid flow over a stretching surface with non-uniform heat source or sink. Thermal Science, 23(1), 281-293.
Swain, K., Parida, S., & Dash, G. (2018). Effects of non-uniform heat source or sink and viscous dissipation on MHD boundary layer flow of Williamson nanofluid through a porous medium. Defect and Diffusion Forum, 389, 110-127.
Alao, S., Salawu, S. O., Oderinu, R., Oyewumi, A., & Akinola, E. (2024). Viscous heating and thermal gyration of magneto-micropolar fluid particles through an isothermal porous fixed channel with internal non-uniform heat generation: An analytical investigation. Results in Engineering, 23, 102474.
Jebreen, H. (2023). Solving fractional gas dynamics equation using Muntz-Legendre polynomials. Symmetry, 15(11), 2076.
Sanusi, B. A., Oderinu, R. A., Alao, S., Ayambukola, F. J., & Oyeyinka, T. A. (2024). Computational investigation of mixed convective flow of micropolar Casson fluid flowing through a stretchy porous channel with heat source. Adeleke University Journal of Science, 3(1).
Yousefi, A., Javadi, S., & Babolian, E. (2019). A computational approach for solving fractional integral equations based on Legendre collocation method. Mathematical Sciences, 13(3), 231-240.
Molla, H., & Saha, G. (2019). Numerical approximation of Fredholm integral equation of the second kind using Galerkin and collocation methods. Ganit: Journal of Bangladesh Mathematical Society, 38, 11-25.
Olabode, B. T., Kayode, S. J., Odeniyan-Fakudade, F. H., & Momoh, A. L. (2024). Numerical solution to singular boundary value problems (SBVPs) using modified linear multistep formulas (LMF). International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(1), 34-52.
Binta, B. A., Odekule, M. R., Umar, D., & Waniyos, H. U. (2023). Numerical solution of systems of integro differential equations using polynomial collocation method. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 9(2), 44-52.
Demir, D., Çınardal, T., Kürkçü, Ö., & Sezer, M. (2019). The Legendre matrix collocation approach for some nonlinear differential equations arising in physics and mechanics. European Journal of Science and Technology, 289-296.
Alao, S., Salawu, S., Oderinu, R., Oyewumi, A., & Akinola, E. (2024). Investigation of thermal radiation and viscous heating effects on the hydromagnetic reacting micropolar fluid species flowing past a stretchy plate in permeable media. International Journal of Thermofluids, 22, 100600.
Ayanbukola, F. J., Oderinu, R. A., Alao, S., Oyeyinka, T. A., & Sanusi, B. A. (2024). Theoretical and numerical investigation of thermal radiation in an MHD micropolar fluid flow within a porous channel. Adeleke University Journal of Science, 3(1).
Akolade, M. T., Adeosun, A. T., & Olabode, J. O. (2021). Influence of thermophysical features on MHD squeezed flow of dissipative Casson fluid with chemical and radiative effects. Journal of Applied and Computational Mechanics, 7(4), 1999-2009.
Wang, C. Y. (1989). Free convection on a vertical stretching surface. ZAMM: Journal of Applied Mathematics and Mechanics, 69(11), 418-420.
Gorla, R. S., & Sidawi, I. (1994). Free convection on a vertical stretching surface with suction and blowing. Applied Scientific Research, 52, 247-257.
Khan W. A. & Pop I. (2010). Boundary-layer flow of a nanofluid past a stretching sheet. International Journal of Heat and Mass Transfer, 53(11-12), 2477-2483.
Ali A., Ahammad N., Eldin S., Gamaoun F., Daradek Y., & Yassen M. (2022). MHD Williamson nanofluid flow in the rheology of thermal radiation, joule heating, and chemical reaction using the Levenberg-Marquardt neural network algorithm. Frontiers in Energy Research, 10.
Alzu'bi O., Alwawi F., Swalmeh M., Sulaiman I., Hamarsheh A., & Ibrahim M. (2022). Energy transfer through a magnetized Williamson hybrid nanofluid flowing around a spherical surface: numerical simulation. Mathematics, 10(20), 3823.
Alwawi F., Faqih F., Swalmeh M., & Ibrahim M. (2022). Combined convective energy transmission performance of Williamson hybrid nanofluid over a cylindrical shape with magnetic and radiation impressions. Mathematics, 10(17), 3191.
Gireesha B., Sindhu S., Sowmya G., & Felicita A. (2020). Magnetohydrodynamic flow of Williamson fluid in a microchannel for both horizontal and inclined loci with wall shear properties. Heat Transfer, 50(2), 1428-1442.
Rosli W., Mohamed M., Sarif N., Mohammad N., & Soid S. (2022). Blood conveying ferroparticle flow on a stagnation point over a stretching sheet: non-Newtonian Williamson hybrid ferrofluid. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 97(2), 175-185.
Ahmed, K., Akbar, T., & Muhammad, T. (2021). Physical aspects of homogeneous-heterogeneous reactions on MHD Williamson fluid flow across a nonlinear stretching curved surface together with convective boundary conditions. Mathematical Problems in Engineering, 2021, 1-13.
Anagandula, S., & Reddy, K. S. (2024). Velocity and thermal slips impact on the Williamson fluid flow above a stretching sheet in the existence of radiation and inclined magnetic field. CFD Letters, 16(7), 118-135.
Alizadeh, R., Karimi, N., Arjmandzadeh, R., & Mehdizadeh, A. (2018). Mixed convection and thermodynamic irreversibilities in MHD nanofluid stagnation-point flows over a cylinder embedded in porous media. Journal of Thermal Analysis and Calorimetry, 135(1), 489-506.
Ibrahim, W., & Gamachu, D. (2019). Finite element method solution of mixed convection flow of Williamson nanofluid past a radially stretching sheet. Heat Transfer, 49(2), 800-822.
Maleque, K. (2020). Similarity requirements for mixed convective boundary layer flow over vertical curvilinear porous surfaces with heat generation or absorption. International Journal of Aerospace Engineering, 2020, 1-9.
Oderinu, R. A., Ayanbukola, F. J., Alao, S., Sanusi, B. A., & Oyeyinka, T. A. (2025). Mixed convective and nonuniform internal heat generation effect on hydromagnetic micropolar fluid flowing across a permeable stretchy wall: A numerical investigation. Heat Transfer, 2025(1).
Panezai, A., Rehman, A., Sheikh, N., Iqbal, S., Ahmed, I., Iqbal, M., & Zulfiqar, M. (2019). Mixed convective magnetohydrodynamic heat transfer flow of Williamson fluid over a porous wedge. American Journal of Mathematical and Computer Modelling, 4(3), 66.
Raji, N., Shahabudin, N., Awang, N., Ilias, M., & Ishak, S. (2023). Aligned magnetohydrodynamics mixed convection on various base fluids with carbon nanotubes over an inclined plate. CFD Letters, 15(6), 12-25.
Akter, R. (2023). Unsteady Williamson fluid flow over exponentially stretching sheet in the presence of non-uniform heat generation or absorption and inclined magnetic field. Journal of Scientific Research, 15(3), 621-636.
Jyotshna, M., and Dhanalaxmi, V. (2022). Impact of activation energy and heat source or sink on 3D flow of Williamson nanofluid with GaN nanoparticles over a stretching sheet. European Journal of Mathematics and Statistics, 3(5), 16-29.
Alao, S., Oderinu, R. A., Sanusi, B. A., Oyeyinka, T. A., & Ayambukola, F. J. (2025). Computational investigation of magnetohydrodynamics Casson micropolar fluid flowing past a permeable linearly stretchy wall with heat source. Journal of the Nigerian Society of Physical Sciences, 7(3), 2577.
Kumar, G., Reddy, K., Konijeti, R., & Reddy, M. (2018). Non-uniform heat source or sink and Joule heating effects on chemically radiative MHD mixed convective flow of micropolar fluid over a stretching sheet in porous medium. Defect and Diffusion Forum, 388, 281-302.
Raju, C., Sandeep, N., Ali, M., & Nuhait, A. (2019). Heat and mass transfer in three-dimensional MHD Williamson-Casson fluid flow over a stretching surface with non-uniform heat source or sink. Thermal Science, 23(1), 281-293.
Swain, K., Parida, S., & Dash, G. (2018). Effects of non-uniform heat source or sink and viscous dissipation on MHD boundary layer flow of Williamson nanofluid through a porous medium. Defect and Diffusion Forum, 389, 110-127.
Alao, S., Salawu, S. O., Oderinu, R., Oyewumi, A., & Akinola, E. (2024). Viscous heating and thermal gyration of magneto-micropolar fluid particles through an isothermal porous fixed channel with internal non-uniform heat generation: An analytical investigation. Results in Engineering, 23, 102474.
Jebreen, H. (2023). Solving fractional gas dynamics equation using Muntz-Legendre polynomials. Symmetry, 15(11), 2076.
Sanusi, B. A., Oderinu, R. A., Alao, S., Ayambukola, F. J., & Oyeyinka, T. A. (2024). Computational investigation of mixed convective flow of micropolar Casson fluid flowing through a stretchy porous channel with heat source. Adeleke University Journal of Science, 3(1).
Yousefi, A., Javadi, S., & Babolian, E. (2019). A computational approach for solving fractional integral equations based on Legendre collocation method. Mathematical Sciences, 13(3), 231-240.
Molla, H., & Saha, G. (2019). Numerical approximation of Fredholm integral equation of the second kind using Galerkin and collocation methods. Ganit: Journal of Bangladesh Mathematical Society, 38, 11-25.
Olabode, B. T., Kayode, S. J., Odeniyan-Fakudade, F. H., & Momoh, A. L. (2024). Numerical solution to singular boundary value problems (SBVPs) using modified linear multistep formulas (LMF). International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(1), 34-52.
Binta, B. A., Odekule, M. R., Umar, D., & Waniyos, H. U. (2023). Numerical solution of systems of integro differential equations using polynomial collocation method. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 9(2), 44-52.
Demir, D., Çınardal, T., Kürkçü, Ö., & Sezer, M. (2019). The Legendre matrix collocation approach for some nonlinear differential equations arising in physics and mechanics. European Journal of Science and Technology, 289-296.
Alao, S., Salawu, S., Oderinu, R., Oyewumi, A., & Akinola, E. (2024). Investigation of thermal radiation and viscous heating effects on the hydromagnetic reacting micropolar fluid species flowing past a stretchy plate in permeable media. International Journal of Thermofluids, 22, 100600.
Ayanbukola, F. J., Oderinu, R. A., Alao, S., Oyeyinka, T. A., & Sanusi, B. A. (2024). Theoretical and numerical investigation of thermal radiation in an MHD micropolar fluid flow within a porous channel. Adeleke University Journal of Science, 3(1).
Akolade, M. T., Adeosun, A. T., & Olabode, J. O. (2021). Influence of thermophysical features on MHD squeezed flow of dissipative Casson fluid with chemical and radiative effects. Journal of Applied and Computational Mechanics, 7(4), 1999-2009.
Wang, C. Y. (1989). Free convection on a vertical stretching surface. ZAMM: Journal of Applied Mathematics and Mechanics, 69(11), 418-420.
Gorla, R. S., & Sidawi, I. (1994). Free convection on a vertical stretching surface with suction and blowing. Applied Scientific Research, 52, 247-257.
Khan W. A. & Pop I. (2010). Boundary-layer flow of a nanofluid past a stretching sheet. International Journal of Heat and Mass Transfer, 53(11-12), 2477-2483.
Published
2026-03-03
How to Cite
Oyeyinka, T., Oderinu, R., Alao, S., Sanusi, B., & Ayanbukola, F. (2026). An Analytical Investigation of MHD Williamson Fluid Flow over an Inclined Stretching Sheet in a Porous Medium with Non-Uniform Internal Heat Generation and Mixed Convection. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 11(4), 123 - 137. Retrieved from https://ijmso.unilag.edu.ng/article/view/2920
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