Electrically driven steady flows in magnetohydrodynamics

The flow of electric currents between electrodes immersed in a conducting fluid permeated by a uniform magnetic field is considered. The current interacts with the field to generate a velocity field which modifies the steady current flow pattern. The magnetic Reynolds number of the motion is assumed small, and inertia forces are assumed negligible compared with either magnetic or viscous forces. Under these circumstances the flow pattern depends only on the Hartmann number M in addition to the boundary conditions imposed. When M → ∞, layers of discontinuity may develop in the fluid. Across these layers the tangential component of electric field is discontinuous, the normal component within the layer being of order δ −1 where δ is the layer thickness. Two particular geometries are discussed in detail. In the first, the magnetic field is perpendicular to the electrodes, and a layer of discontinuity, with the character of a non-spreading jet, develops as M → ∞. In the second, the magnetic field is parallel to the electrodes, and the discontinuities that develop in the limit are confined to the boundaries of the fluid.