The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field

Abstract

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number R c, the critical wave number α c, and the critical wave speed c c are obtained for wide ranges of the magnetic Prandtl number P m and the Hartmann number M. It is found that except for the case when P m is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as P m increases.