Electrically driven free shear flows in a duct under a transverse uniform magnetic field

Abstract

A mathematical model of two-dimensional electrically driven laminar plane free shear flows in a straight duct under the action of an applied spanwise uniform magnetic field is considered. The mathematical approach is like that used in the research of Hunt and Williams (J. Fluid. Mech., 31, 705, 1968) and Kolesnikov and Kalis (Magnetohydrodynamics, vol. 57, 2021, no. 2). A system of stationary partial differential equations with two unknown functions of velocity and induced magnetic field is solved. The electric current is injected into the liquid by means of two couples of linear electrodes located vis-à-vis on opposite duct walls, perpendicular to the magnetic field. Three cases are considered. One pair of electrodes is current supplied and, depending on the direction of electric current injection on the electrode pair, two coinciding or two counter flows are also driven. At Hartmann numbers Ha >> 1, quasi-potential cores are formed in these flows, bounded by lateral Shercliff free boundary layers parallel to the field and two Hartmann layers on the walls perpendicular to the field. As a result, almost all of the injected current passes through these layers. An increase of the magnetic field leads only to an internal rearrangement of the potential cores of the flows. The Hartmann number varies in the range from 1 to 100. Figs 15, Refs 11.