On an Important Remark Concerning Some MHD Motions of Second-Grade Fluids through Porous Media and Its Applications

In this work it is proven that the governing equations for the fluid velocity and non-trivial shear stress corresponding to some isothermal MHD unidirectional motions of incompressible second-grade fluids through a porous medium have identical forms. This important remark is used to provide exact steady-state solutions for motions with shear stress on the boundary when similar solutions of some motions with velocity on the boundary are known. Closed-form expressions are provided both for the fluid velocity and the corresponding shear stress and Darcy's resistance. As a check of the results that are obtained here, the solutions corresponding to motions over an infinite flat plate are presented in different forms whose equivalence is graphically proven. In the case of the motions between infinite parallel plates, the fluid behavior is symmetric with respect to the median plane due to the boundary conditions.

Automated summary

The governing equations for the fluid velocity and shear stress in isothermal MHD unidirectional motions of incompressible second-grade fluids through porous medium have identical forms, allowing for exact steady-state solutions with shear stress on the boundary. Closed-form expressions for fluid velocity, shear stress, and Darcy's resistance are provided, with results verified through solutions for motions over an infinite flat plate presented in different forms. The fluid behavior in motions between infinite parallel plates is symmetric with respect to the median plane due to boundary conditions.