Two-dimensional incompressible magneto-hydrodynamic flow across an elliptical solenoid

Abstract

An analysis has been made of the two-dimensional flow of an incompressible constant-conductivity fluid through an elliptically shaped solenoid containing a constant magnetic field directed normal to the flow plane. The effect of both Hall current and ion slip has been included in the generalized Ohm's law used for the fluid. The analysis is based on a perturbation procedure in two parameters, one being the magnetic Reynolds number Rm and the other the ratio S of magnetic force per unit area to dynamic pressure. Calculations have been carried to the first order in each parameter, and closed-form analytic expressions have been obtained for the force and moment on the solenoid, the current density, stream function, magnetic field and other pertinent physical quantities.It was found that, to the zeroth order, there is a force but no moment on the solenoid. To the first order in S, where the flow field is modified but the magnetic field is not, there is a moment and a force, the latter being anti-parallel to the zeroth order force. To the first order in Rm, where the magnetic field is modified but the flow field is not, there is a moment but no force. Thus, to the first order the lift to drag ratio is the same as in the zeroth order. Graphs which illustrate some of the effects of angle of attack, fineness ratio of the ellipse, Hall current and ion slip, on the forces and moments are presented.