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

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 u = (uₓ(x̃, ỹ), uy(x̃, ỹ), 0) , the position of the duct walls by ỹ = fₜ(x̃), fb(x̃) and z̃ = ± b , where b [Gt ] f t − f b , and the magnetic field by B 0 = (0, B 0 , O). It is assumed that the magnetic field is strong enough to satisfy the conditions that the interaction parameter, N (= M 2 / 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: the ‘core’ region in which the pressure gradient is balanced by electromagnetic forces; Hartmann boundary layers where electromagnetic forces are balanced by viscous forces; 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-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.