Three-dimensional MHD duct flows with strong transverse magnetic fields Part 1. Obstacles in a constant area channel

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Abstract
This paper is an analysis of incompressible three-dimensional flows of electrically conducting fluids under the action of transverse magnetic fields which are assumed to be sufficiently strong that the interaction parameter N (= M2/R) [Gt ] 1, where M is the Hartmann number and R is the Reynolds number. We also assume that R [Gt ] 1 and Rm (magnetic Reynolds number) [Lt ] 1, so that experimental verification of the theory may be possible.The main results are: (i) when a thick body is placed in a parallel-sided channel with non-conducting walls the flow over it is highly dependent on the conductivity of the body, in a surprising way. If the body is non-conducting, there is no flow within that cylinder which circumscribes the body and is parallel to the magnetic field; outside the cylinder the flow is plane and potential and enters or leaves the surface shear layer of this cylinder at right angles. If the body is conducting, flow over it is possible and is of a different nature outside and inside the cylinder. (ii) When a non-conducting flat plate is placed in such a channel no blocking of the flow occurs. If the plate is elongated in the flow direction, the flow over it becomes identical to that calculated by Hasimoto (1960) and, if elongated at right angles to the flow, becomes identical to that calculated by Dix (1963).Of particular interest in our analysis are the two types of layer which occur in these flows, the first being the Hartmann boundary layer, which is shown to have a controlling influence on the vorticity of the core flow in three-dimensional situations analogous to that of the Eckman layer in rotating-fluid flows. The second type, the free shear layer at the circumscribing cylinder, is of interest because of its internal structure and effect on the external flow.