Steady-State Solutions for MHD Motions of Burgers’ Fluids through Porous Media with Differential Expressions of Shear on Boundary and Applications

Abstract

Steady-state solutions for two mixed initial-boundary value problems are provided. They describe isothermal MHD steady-state motions of incompressible Burgers’ fluids over an infinite flat plate embedded in a porous medium when differential expressions of shear stress are given on a part of the boundary. The fluid is electrically conductive under the influence of a uniform transverse magnetic field. For the validation of the results, the expressions of the obtained solutions are presented in different forms and their equivalence is graphically proved. All of the obtained results could easily be particularized to give exact solutions for the incompressible Oldroyd-B, Maxwell, second-grade, and Newtonian fluids that were performing similar motions. For illustration, the solutions corresponding to Newtonian fluids are provided. In addition, as an application, the velocity fields were used to determine the time required to reach the steady or permanent state for distinct values of magnetic and porous parameters. We found that this time declined with increasing values of the magnetic or porous parameters. Consequently, the steady state for such motions of Burgers’ fluids was earlier reached in the presence of a magnetic field or porous medium.