Stability of Tollmien-Schlichting Modes in Magnetohydrodynamic Boundary Layer Flow in Porous Medium: Energy Budget Analysis

Linear temporal and spatial stability analyses of the magnetohydrodynamic boundary layer flow over a wedge embedded in a porous medium have been carried out to analyze the effects of pressure gradient, Hartmann and Darcy numbers. First, we determine the base state velocity profiles by imposing suitable similarity transformations on governing boundary layer equations and then find linear perturbed equations involving Reynolds number and disturbance wavenumber using Fourier modes. The Chebyshev spectral collocation method which provides insight into the complete structure of the eigenspectrum is used. The effect of Hartmann and Darcy numbers on the boundary layer is to stabilize the flow for all adverse pressure gradient parameters while only unstable modes are noticed in the absence of these effects. The noticed unstable modes are always part of the wall mode for which the phase speeds are found to approach zero. The eigenspectrum for all involved parameters has a balloon-like structure with the appearance of wall mode instability. The critical Reynolds number is found to be increasing for increasing pressure gradient, Hartmann and Darcy numbers. For an adverse pressure gradient, the energy budget shows that the energy production due to Reynolds stress dominates the viscous dissipation which results in destabilization of the flow, while the kinetic energy due to magnetic field and porous medium plays a role in stabilizing the flow. The physical dynamics behind these interesting modes are discussed.