Review Lecture - The annulenes

Abstract

This paper is concerned with the parallel flow of conducting fluid along an insulating pipe of uniform cross-section perpendicular to which a uniform magnetic field. $\mathbf B_0$, is applied. The cross-section is supposed to have tangents parallel to $\mathbf{B}_0$ only at isolated points of its perimeter. The density, kinematic viscosity and electrical conductivity of the fluid are denoted by $\rho, \nu$ and $\sigma$, respectively. It is known (Shercliff 1962) that, in the limit $B_0 \rightarrow \infty$, the flow may be divided into three parts: (i) a Hartmann boundary layer, thickness $\sim(\rho\nu/\sigma)^\frac{1}{2}(B_0 \cos \theta)^{-1},$ at every point of the wall except those at which cos $\theta$ = 0, where $\theta$ is the angle between $\mathbf B_0$ and the normal to the wall at the point concerned, (ii) a 'mainstream', far from the walls, which is controlled by the Hartmann layers and in which a quasi-hydrostatic balance subsists between the Lorentz force and the applied pressure gradient driving the motion, and (iii) a complicated boundary-layer singularity, at each point of the wall at which $\cos \theta$ = 0, which is controlled by the flow in regions (i) and (ii). The solutions for regions (i) and (ii) can be obtained easily by Shercliff's methods. It is the purpose of this paper to elucidate region (iii). Here the boundary-layer thickness is O(M$^{-\frac{2}{3}}$) and extends round the periphery of the wall for a distance which is O(M$^{-\frac{1}{3}}$), where M is a Hartmann number, $B_0\mathscr{L}(\sigma/\rho\nu)^{\frac{1}{2}}$, based on a typical dimension, $\mathscr{L}$, of the pipe. The corresponding contribution to U, the mean flow down the duct, is of order $M^{-\frac{7}{3}}$. In fact, for a circular duct of radius a( = $\mathscr{L}$), the main case discussed in this paper, it contributes the final term to the following expression: $U = \frac{64}{3\pi} U_0\big[\frac{1}{M} - \frac{3\pi}{2M^2} + \frac{3.273}{M\frac{7}{3}}\big].$ Here U$_0$ is the mean flow in the absence of field.