Linear stability of parallel flow of liquid metal in a rectangular duct driven by a constant pressure gradient under the influence of a uniform magnetic field

Linear stability of the plane Poiseuille flow of liquid metal under a uniform magnetic field is numerically studied. The liquid metal is driven by a constant pressure gradient in a rectangular duct. The induced magnetic field and the Joule heating are neglected in this analysis. The governing equations are the continuity of mass, momentum equation, Ohm's law and conservation of electric charge. The solution of basic state is well known as the Hartmann flow in the limit of the large aspect ratio of the cross-section of the duct. It is supposed that disturbance has periodicity along the direction of the basic flow. The linear stability of the basic flow that depends only on the Hartmann number is analyzed by a finite difference method discretized by a fourth order central difference scheme together with the use of HSMAC algorithm. We obtained the phase velocity of Tollmien-Schlichting wave and the Reynolds number as the neutral stability state when input parameters such as the aspect ratio, the Hartmann number and the wavenumber were given. It is predicted that the critical Reynolds number at the onset of instability is about 4.0 x 10(5) and the wavenumber is about 1.15 when the Hartmann number is 5.