Electrically driven cylindrical free shear flows under an axial uniform magnetic field

We consider a mathematical model of two-dimensional electrically driven laminar axisymmetric circular free shear flows in a cylindrical vessel under the action of an applied axial uniform magnetic field. The mathematical approach is based on the studies by J.C.R. Hunt and W.E. Williams (J. Fluid. Mech., 31, 705, 1968). We solve a system of stationary partial differential equations with two unknown functions of velocity and induced magnetic field. The flows are generated as a result of the interaction of the electric current injected into the liquid and the applied field using one or two pairs of concentric annular electrodes located apart on the end walls. Two lateral free shear layers and two Hartmann layers on the end walls and a quasi-potential flow core between them emerge when the Hartmann number Ha >> 1. As a result, almost all injected current passes through these layers. Depending on the direction of the current injection, coinciding or two counter flows between the end walls are realized. The Hartmann number varies in a range from 2 to 300. When a moderate magnetic field (Ha = 50) is reached, the flow rate and the induced magnetic field flux cease to depend on the magnitude of the applied field but depend on the injected electric current value. Increasing magnetic field leads only to inner restructuring of the flows. Redistributions of velocities and induced magnetic fields, electric current density versus Hartmann number are analyzed. Figs 18, Refs 21.