An Analytic Study of the Reversal of Hartmann Flows by Rotating Magnetic Fields

The effects of a background uniform rotating magnetic field acting in a conducting fluid with a parallel flow are studied analytically. The stationary version with a transversal magnetic field is well known as generating Hartmann boundary layers. The Lorentz force includes now one term depending on the rotation speed and the distance to the boundary wall. As one intuitively expects, the rotation of magnetic field lines pushes backwards or forwards the flow. One consequence is that near the wall the flow will eventually reverse its direction, provided the rate of rotation and/or the magnetic field are large enough. The configuration could also describe a fixed magnetic field and a rotating flow. 1. Introduction In [1], a generalization of the Hartmann flow was announced. The Hartmann flow is a parallel one influenced by a strong transversal magnetic field [2, 3] and has been studied intensively for its role in many aspects of plasma physics, such as liquid metal pumping [4, 5], plasma convection [6, 7], and geophysics [8]. We now will allow the background magnetic field to rotate while keeping its spatial uniformity. Our aim in this paper is to analyze in depth the action of this field upon the flow through the Lorentz force, emphasizing the possibility of flow reversal at some time near the walls confining the fluid. Given the Galilean invariance of the relevant equations, we could change to a rotating reference frame and interpret the problem as consisting in a fixed magnetic field and a rotating flow; this configuration could perhaps be more easily constructed and have wider applications. Let us begin by recalling briefly the MHD equations appropriate for our configuration. The flow will be assumed two dimensional, and the background magnetic field will be uniform in space but rotating in time at a rate given by the angle : Although this field will create a secondary one , it is assumed that all the terms in may be neglected when an analogous one for is present; thus, , . This does not hold for the current density, as , . The approximate induction equation becomes where is the flow velocity and the resistivity. By substituting by its value in (1.1), uncurling (1.2) to find and taking it to the Lorents force , we may write the momentum equation as where is the conductivity, the viscosity, and is the sum of the pressure and the gauge obtained by uncurling (1.2). Detailed calculations may be found in [1]. If in analogy with the classical Hartmann flow we assume a horizontal flow , (1.3) reduces to where includes all the conservative forces upon the