Magnetohydrodynamic flow in channels of variable cross-section with strong transverse magnetic fields

Abstract

This paper is an analysis of the steady incompressible, two-dimensional flow of conducting fluids through ducts of arbitrarily varying cross-section under the action of a strong, uniform, transverse, magnetic field. More precisely, the flow is such that the velocity is given by ${\bf u} = (u_x(\tilde{x}, \tilde{y}), u_y(\tilde{x}, \tilde{y}), 0)$, the position of the duct walls by $\tilde{y} = f_t(\tilde{x}), f_b(\tilde{x})$ and $\tilde{z} = \pm b $, where b [Gt ] f_t−f_b, and the magnetic field by B₀ = (0, B₀, O). It is assumed that the magnetic field is strong enough to satisfy the conditions that the interaction parameter, N(=M²/R) [Gt ] 1, where M is the Hartmann number and R is the Reynolds number, and also that M [Gt ] 1 and R_m [Lt ] 1, where R_m is the magnetic Reynolds number.

We examine the flow in three separate regions:

1. the ‘core’ region in which the pressure gradient is balanced by electromagnetic forces;

2. Hartmann boundary layers where electromagnetic forces are balanced by viscous forces;

3. thin layers parallel to the magnetic field in which electromagnetic forces, inertial forces, and the pressure gradient balance each other. These layers which have thickness $O(N^{-\frac{1}{3}})$ occur where the slope of the duct wall changes abruptly.

By expanding the solution as a series in descending powers of N we calculate the velocity distribution for regions (i) and (ii) for finite values of N attainable in the laboratory.