Vorticity generation in localized magnetic fields

Abstract

The generation of vorticity in localized magnetic fields is explored under the low magnetic Reynolds number approximation. Analytic solutions for a creeping flow past a magnetic point dipole are obtained through a perturbation method. Also, numerical calculations for the flow past a finite size magnetic dipole are presented. It is shown that vortical structures may become unstable when inertial effects are not negligible. Introduction. Non-uniformities are commonly used to generate vorticity in magnetohydrodynamic (MHD) flows. For instance, the use of solid obstacles in the flow under uniform magnetic fields has been widely explored both theoretically and experimentally to understand vortex formation and evolution (1, 2). Vorticity can also be created under uniform fields through the expansion or contraction of duct geometry. Another interesting possibility of vortex generation in the channel flow under uniform fields is due to inhomogeneities in the electrical conductivity of the walls, as has been demonstarted by Alpher et al. (3) and Buhler (4). It is also well known that vorticity can be generated due to fringing magnetic fields. In fact, most of the investigations of flows in fringing fields has been devoted to the analysis of duct flows with a field that varies in the streamwise direction as it approximately occurs at the entrance or exit of the poles of a magnet. In this case, current loops are elongated in the flow direction giving rise to streamwise current density components that produce Lorentz forces pointing towards the side walls. These forces are responsible for the creation of M-shape velocity profiles with high side-layer velocitites (5). The strong shear layers created by the non- uniform field remain confined by the side walls, which determine their evolution inside or outside the magnetic field. Actually, in channel flows a non-uniform field acts as an electromagnetic brake, as those commonly used in metallurgical applications. It is then possible to visualize a non-uniform field as an obstacle for the flow and rise the question of how the flow evolution would develop in the ab- sence of confining side walls. In experiments performed in a thin electrolytic layer with a localized moving magnetic field, Honji and Haraguchi (6, 7) demonstarted that generation of vorticity in inhomogeneous magnetic fields may lead to time- dependent flows with complex vortical structures. They used a shallow layer of salt water contained in a long tank with an electric current injected transversally to the tank axis. A permanent magnet located externally but close to the layer was moved at a constant velocity along the center line of the water tank. In the wake behind the region influenced by the field, different flow patterns including a wavy motion, symmetric vortex pairs and periodic vortex shedding were observed depending on the velocity of the magnets and the injected electric current (6, 7). New experimental evidence was recently obtained by Afanasyev and Korabel (8)