Permanent Solutions for MHD Motions of Generalized Burgers’ Fluids Adjacent to an Unbounded Plate Subjected to Oscillatory Shear Stresses

Abstract

Closed-form expressions have been obtained to characterize the non-dimensional velocity and corresponding non-trivial shear stress in the context of two magnetohydrodynamic (MHD) motions exhibited by incompressible generalized Burgers’ fluids. These motions occur over an infinite plate, which subjects the fluid to oscillatory shear stresses. The obtained solutions represent the first exact analytical solutions for MHD motions of such fluids under the condition of shear stress prescribed along the boundary. The establishment of these solutions relies upon the utilization of a perfect symmetry existing between the governing equations of fluid velocity and shear stress. To validate the results, a comprehensive analysis has been undertaken using two distinct methods. This validation process is further substantiated through graphical representation, demonstrating the congruence between the obtained solutions. Additionally, the convergence of the initial solutions, obtained through numerical techniques, towards their corresponding permanent counterparts has been visually established. This graphical depiction not only substantiates the accuracy of the solutions but also provides insights into the temporal evolution of the system toward its permanent state. An insight to characterize the non-dimensional shear stresses in the context of two values of the magnetic parameter is to identify that the permanent state is reached at an earlier time and the absolute magnitude of fluid velocity is reduced in the presence of an applied magnetic field.

Keywords

• generalized Burgers’ fluids
• permanent solutions
• shear stresses on the boundary