Mixed Convection and Permeability Effects on Magnetohydrodynamic Williamson Fluid Flow Over an Inclined Stretchy Surface With Radiation Influence: Analytical Investigation

Abstract

The study introduces a computational method that combines Legendre polynomials with Gauss–Lobatto points to solve nonlinear coupled differential equations, focusing on the Williamson fluid model under the influence of mixed convection and permeability with mixed boundary conditions. The nonlinear governing equations were transformed from partial differential equations (PDEs) into ordinary differential equations (ODEs) using appropriate similarity transformations. By employing Legendre polynomials as trial functions and collocating the residual equations at Gauss–Lobatto points, the system was solved using mathematical software. The technique was validated by comparing the obtained solutions with existing literature, showing excellent agreement. The computed Nusselt numbers for $Pr=0.7$, 2, and 7 in this study were 0.4544, 0.9114, and 1.8954, respectively, compared to 0.4539, 0.9114, and 1.8954 reported in previous studies. The average discrepancy across comparisons was approximately 0.00018, demonstrating the accuracy of the proposed approach. Further validation was performed using the fourth-order Runge–Kutta method via the shooting technique, reinforcing the reliability of the results. The study's findings revealed that increasing the Grashof number ($Gr$) and modified Grashof number ($Gm$) enhanced skin friction, whereas higher values of the magnetic parameter ($M$), suction parameter ($f_w$), and porosity ($K_p$) reduced it. Additionally, greater porosity ($K_p$) and a higher Eckert number ($Ec$) contributed to an increase in the Nusselt number. Conversely, an increase in the Weissenberg number ($We$), Prandtl number ($Pr$), and radiation parameter ($R$) resulted in a decline in the Nusselt number. Graphical analysis indicated that an increase in $Gr$ enhanced velocity, whereas higher $K_p$ increased temperature but reduced fluid velocity. This approach provides an efficient and precise solution for complex nonlinear equations, with broader potential applications in fluid dynamics, including industrial processing, biomedical engineering, and aerospace engineering.