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

Abstract

The stability against small disturbances of the plane laminar motion of an electrically conducting fluid between parallel plates in relative motion under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and non-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 wavenumber 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 there exists a stationary mode of instability in addition to a travelling-wave mode of instability, and that except for the case when P m is sufficiently small, the fluid flow becomes more unstable to the stationary mode as P m increases.