Group invariant solutions for the unsteady magnetohydrodynamic flow of a fourth grade fluid in a porous medium

Abstract

The e ects of non-Newtonian uids are investigated by means of two appropri- ate models studying a third and fourth grade uid respectively. The geometry of both these models is described by the unsteady unidirectional ow of an in-compressible uid over an in nite at rigid plate within a porous medium. The uid is electrically conducting in the presence of a uniform applied magnetic eld that occurs in the normal direction to the ow. The classical Lie symmetry approach is undertaken in order to construct group invariant solutions to the governing higher-order non-linear partial dif-ferential equations. A three-dimensional Lie algebra is acquired for both uid ow problems. In each case, the invariant solution corresponding to the non-travelling wave type is considered to be the most signi cant solution for the uid ow model under investigation since it directly incorporates the magnetic eld term. A numerical solution to the governing partial di erential equation is produced and a comparison is made with the results obtained from the analytical ap-proach. Finally, a graphical analysis is carried out with the purpose of observing the e ects of the emerging physical parameters. In particular, a study is carried out to examine the in uences of the magnetic eld parameter and the non-Newtonian fluid parameters.