Combined effects of Joule heating and non-uniform heat source/sink on unsteady MHD mixed convective flow over a vertical stretching surface embedded in a Darcy-Forchheimer porous medium

This paper deals with an unsteady magnetohydrodynamics (MHD) heat and mass transfer for a viscous incompressible fluid through a vertical stretching surface embedded in a Darcy-Forchheimer porous medium in the presence of a non-uniform heat source/sink and first-order chemical reaction. The porous surface is subjected to a uniform transverse magnetic field. The influence of velocity, thermal, and concentration slip is also investigated. The governing equations are coupled non-linear partial differential equations, which have been converted via similarity transformation into a set of ordinary differential equations. The resultant system of non-linear ordinary differential equations has been solved numerically with the help of the “MATLAB” BVP4C Solver. Results are presented graphically to analyze the effects of various physical parameters discovered in the problem such as Hartmann number (M), Forchheimer number (Fr), Grashof number (Gr), solutal Grashof number (Gc), suction parameter (S), porosity parameter ($\tilde{\lambda }$), dimensionless velocity slip ($S_v$), Prandtl number (Pr), dimensionless thermal slip ($S_t$), space-dependent heat source/sink parameter (${\tilde{A}}_1^{\ast }$), temperature-dependent heat source/sink (${\tilde{B}}_1^{\ast }$), Eckert number (Ec), Schmidt number (Sc), chemical reaction parameter ($\gamma$), unsteadiness parameter (A), and dimensionless concentration slip ($S_c$) on non-dimensional velocity ${\tilde{\chi }}^{\prime }\left(\eta \right)$, temperature $\zeta \left(\eta \right)$, and concentration $\tilde{\varphi }\left(\eta \right)$ profiles. The influence of these parameters on skin-friction coefficient ($C_f^{\ast }$), Nusselt number ($N{u}_x^{\ast }$), and Sherwood number ($S{h}_x^{\ast }$) are expressed in tabular form. It is observed that an enhancement in Fr and $\tilde{\lambda }$ results in the declination of the velocity profile. There is an enhancement in temperature with an increment in the ${\tilde{A}}_1^{\ast }$ and ${\tilde{B}}_1^{\ast }$. The physical representation of flow characteristics that appeared in the problem is presented using various graphs to depict real-world applications in industrial and engineering operations. The results were compared to previous studies, revealing that the two are in good agreement. The novelty of the present investigation is: To interpret the combined effects of viscous dissipation and Joule heating on a vertical stretching surface embedded in a highly porous medium modeled using the Darcy-Forchheimer model. The findings could be valuable in understanding the flow of oil, gas, and water through an oil or gas field reservoir, as well as groundwater migration and filtering and purification procedures.