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

Published online by Cambridge University Press:  28 March 2006

J. C. R. Hunt
Affiliation:
Central Electricity Research Laboratories
Now at the Department of Thermal Engineering, Cornell University, Ithaca, New York.
S. Leibovich
Affiliation:
University College London
Seconded to School of Engineering Science, University of Warwick, Coventry, Warwickshire.

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 ] ftfb, 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 Rm [Lt ] 1, where Rm 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.

Type
Research Article
Copyright
© 1967 Cambridge University Press

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