Exact solutions for unidirectional magnetohydrodynamics flow of non-newtonian fluid in a porous medium with and without rotation

Abstract

In this work, the physical problems dealing with unidirectional magnetohydrodynamic (MHD) flows of some viscoelastic fluids in a porous medium and rotating frame are investigated. By using modified Darcy’s law, the corresponding equations governing the flow are modelled. Employing Fourier sine transform, new results in terms of the exact solutions of the modelled equations are generated for the problems of constantly accelerating and oscillatory MHD flows of second grade fluid in a porous space, accelerated MHD flows of an Oldroyd–B fluid in a porous medium and rotating frame, accelerating rotating MHD flow of second grade fluid in a porous space, accelerated MHD flow of Maxwell fluid in a porous medium and rotating frame, accelerated rotating MHD flow of generalized Burgers’ fluid in a porous medium. The Fourier sine and Laplace transforms are then utilized to obtain new exact solutions by solving analytically the Stokes’ first problem for two types of MHD fluids, namely the second grade fluid and Maxwell fluid in a porous medium and rotating frame. The new explicit solutions for the corresponding velocity fields are obtained for constant accelerated, variable accelerated and constant velocity flows for each problem mentioned above. The well – known solutions for Newtonian fluid in a porous medium in the cases mentioned above, are significantly shown to appear as the limiting cases of the present analysis. Finally, the effects of the material parameters (i.e. rotation, MHD and porous) on the velocity fields are demonstrated via graphical illustrations. These graphs generally show that: (i) by increasing the rotation parameter, this would lead to a decrease in the real part of the velocity profile; however for the magnitude of imaginary part of the velocity profile, it is found to be quite the opposite, (ii) when MHD parameter increases, the real and imaginary parts of the velocity profile decrease, and (iii) by increasing the porous parameter, both parts of the velocity profile increase