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The Quarterly Journal of Mechanics and Applied Mathematics 1961 14(2):173-195; doi:10.1093/qjmam/14.2.173
© 1961 by Oxford University Press
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ON THE MAGNETOHYDRODYNAMICS OF ASYMMETRIC SURFACE WAVES IN A DISH OF SHALLOW CONDUCTING LIQUID

L. E. FRAENKEL

( Aeronautics Department, Imperial College London )

This paper describes small disturbances of an inviscid, electrically conducting liquid resting in a circular dish made of insulating material, and subject to a vertical magnetic field. The depth of the liquid is assumed to be much smaller than the diameter of the dish or than a typical wavelength. The response to an initial disturbance is investigated by means of approximate equations which are an extension of the linear ‘shallow-water’ theory. The mathematical problem is essentially an eigenvalue problem for a fourth-order partial differential equation with mixed space-time derivatives. It is found that, as a result of dissipation by Joule heating, all the modes of the solution are damped, but the initial rate of decay of the lowest asymmetric mode is only about one-sixth that of the axisymmetrio modes. Wave profiles and current lines for a particular asymmetric motion are shown.


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