Liquid-metal flow in a thin conducting pipe near the end of a region of uniform magnetic field

Abstract

This paper treats the liquid-metal flow in a straight circular pipe with a thin metal wall. A strong magnetic field is applied by a magnet with parallel poles that end abruptly. In the plane midway between the magnet poles: (a) far upstream, the flow is uniform, fully developed in a uniform magnetic field; (b) as the flow enters the non-uniform magnetic field near the end of the magnet, the flow moves away from the central part of the pipe and becomes concentrated as two jets near the points where the magnetic field is tangential to the pipe wall; (c) further downstream where the magnetic field strength is O(c^⅙) compared with its value upstream, the flow migrates from these jets back towards a uniform flow distributed over the entire pipe cross-section. Here, c is the wall conductance ratio, which is assumed to be small. The analysis also applies to flow into the magnetic field, because inertial effects and induced magnetic fields are neglected. There are circulations of electric current in planes parallel to the magnet poles. These currents produce a pressure drop in addition to that for two fully developed flows in a uniform magnetic field and in no magnetic field, joined at the end of the magnet. This pressure drop is given by 0.9336 $\beta^{\frac{1}{3}}c^{\frac{2}{3}}\sigma$V₀B²₀L, where β is a measure of the magnetic field gradient with a minimum value of 2/π, σ is the liquid metal's electrical conductivity, V₀ is the average velocity, B₀ is the strength of the uniform magnetic field, and L is the inside radius of the pipe. This three-dimensional pressure drop is $O(c^{-\frac{1}{3}}L) $ times the pressure gradient for the fully developed flow in the uniform magnetic field.