General Solutions for Some MHD Motions of Second-Grade Fluids between Parallel Plates Embedded in a Porous Medium

Abstract

General solutions are established for an initial boundary value problem by means of the integral transforms. They correspond to the isothermal MHD unidirectional motion of incompressible second-grade fluids between infinite horizontal parallel plates embedded in a porous medium. The fluid motion, which in some situations becomes symmetric with respect to the median plane, is generated by the two plates that apply time-dependent arbitrary shear stresses to the fluid. Closed-form expressions are established both for the fluid velocity and the corresponding non-trivial shear stress. Using an important remark regarding the governing equations of velocity and shear stress, exact general solutions are developed for similar motions of the same fluids when both plates move in their planes with arbitrary time-dependent velocities. The results that have been obtained here can generate exact solutions for any motion with the technical relevance of this type of incompressible second-grade fluids and their correctness being proved by comparing them with the numerical solution or with known results from the existing literature. Consequently, both motion problems of these fluids with shear stress or velocity on the boundary are completely solved.