General Solutions for Magnetohydrodynamic Unidirectional Motions of a Class of Fluids with Power-Law Dependence of Viscosity on Pressure Through a Planar Channel

An analytical study is conducted on unsteady, one-directional magnetohydrodynamic (MHD) flows of electrically conducting, incompressible, and viscous fluids, where the viscosity varies with pressure following a power-law relationship. The flow takes place within a planar channel and is driven by the lower plate, which moves along its own plane with an arbitrary, time-dependent speed. The effects of gravitational acceleration are also considered. General exact formulas are derived for both the dimensionless velocity of the fluid and the resulting non-zero shear stress. Moreover, these are the only general solutions for the MHD motions of the fluids considered, and they can produce precise solutions for any motion of this type for respective fluids. The proposed analytical method leads to simple forms of analytical solutions and can be useful in the study of other cases of fluids with viscosity depending on pressure. As an example, solutions related to the modified Stokes’ second problem are presented and confirmed through graphical validation. These solutions also help highlight the impact of the magnetic field on fluid dynamics and determine the time needed for the system to achieve a steady state. Graphical representations indicate that a steady state is reached more quickly and the fluid moves more slowly when a magnetic field is applied.