Stokes' second problem for magnetohydrodynamics flow in a Burgers' fluid: the cases γ = λ²/4 and γ>λ²/4

The present work is concerned with exact solutions of Stokes second problem for magnetohydrodynamics (MHD) flow of a Burgers' fluid. The fluid over a flat plate is assumed to be electrically conducting in the presence of a uniform magnetic field applied in outward transverse direction to the flow. The equations governing the flow are modeled and then solved using the Laplace transform technique. The expressions of velocity field and tangential stress are developed when the relaxation time satisfies the condition γ =  λ²/4 or γ> λ²/4. The obtained closed form solutions are presented in the form of simple or multiple integrals in terms of Bessel functions and terms with only Bessel functions. The numerical integration is performed and the graphical results are displayed for the involved flow parameters. It is found that the velocity decreases whereas the shear stress increases when the Hartmann number is increased. The solutions corresponding to the Stokes' first problem for hydrodynamic Burgers' fluids are obtained as limiting cases of the present solutions. Similar solutions for Stokes' second problem of hydrodynamic Burgers' fluids and those for Newtonian and Oldroyd-B fluids can also be obtained as limiting cases of these solutions.