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A shallow-liquid theory in magnetohydrodynamics

Published online by Cambridge University Press:  28 March 2006

L. E. Fraenkel
L. E. Fraenkel
Affiliation:
Aeronautics Department, Imperial College, London
Most of this work was done while the author was on leave of absence at the Guggenheim Aeronautical Laboratory, California Institute of Technology.
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Abstract

The non-linear and linear ‘shallow-water’ theories, which describe long gravity waves on the free surface of an inviscid liquid, are extended to the case of an electrically conducting liquid on a horizontal bottom, in the presence of a vertical magnetic field. The dish holding the liquid, and the medium outside it, are assumed to be non-conducting. The approximate equations are based on a small ratio of depth to wavelength, on the properties of mercury, and on a moderate magnetic field strength. These equations have a ‘magneto-hydraulic’ character, for in the shallow liquid layer the horizontal fluid velocity and current density are independent of the vertical co-ordinate.

Some explicit solutions of the linear equations are obtained for plane flows and for axi-symmetric flows in which the velocity vector lies in a vertical, meridional plane. The amplitudes of waves in a dish, and the amplitudes behind wave fronts progressing into undisturbed liquid, are found to be exponentially damped, the mechanical energy associated with a disturbance being dissipated by Joule heating.

The approximate non-linear equations for plane flow are studied by means of characteristic variables, and it appears that, because of the magnetic damping effect, there is less qualitative difference between solutions of the non-linear and linear approximate equations at large times than is the case when the magnetic field is absent. In particular, the characteristic curves depart only a finite distance from their ‘undisturbed positions’.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 7 , Issue 1 , January 1960 , pp. 81 - 107
Copyright
© 1960 Cambridge University Press

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References

Fox, P. A. 1955 Perturbation theory of wave propagation based on the method of characteristics. J. Math. Phys. 34, 133.Google Scholar
Fraenkel, L. E. 1959 A cylindrical sound pulse in a rotating gas. J. Fluid Mech. 5, 637.Google Scholar
Friedrichs, K. O. 1948 Formation and decay of shock waves. Commun. Pure Appl. Math. 1, 211.Google Scholar
Hartmann, J. 1937 Hg-Dynamics I. Theory of the laminar flow of an electrically conductive liquid in a homogeneous magnetic field. Math.-fys. Medd. 15, no. 6Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lehnert, B. 1952 On the behaviour of an electrically conductive liquid in a magnetic field. Arkiv för Fysik, 5, 69.Google Scholar
Lighthill, M. J. 1949 A technique for rendering approximate solutions to physical problems uniformly valid. Phil. Mag. (7) 40, 1179.Google Scholar
Lighthill, M. J. 1955 Article in General Theory of High Speed Aerodynamics (ed. W. R. Sears), section E. Princeton University Press.Google Scholar
Lin, C. C. 1954 On a peturbation theory based on the method of characteristics. J. Math. Phys. 33, 117.Google Scholar
Longuet-Higgins, M. S. 1958 Review of Stoker's ‘Water Waves’. J. Fluid Mech. 4, 435.Google Scholar
Lundquist, S. 1952 Studies in magneto-hydrodynamics. Arkiv för Fysik, 5, 297.Google Scholar
Meyer, R. E. 1948 The method of characteristics for problems of compressible flow involving two independent variables, part II. Quart. J. Mech. Appl. Math. 1, 451.Google Scholar
Stewartson, K. 1957 Magnetohydrodynamics of a finite rotating disc. Quart. J. Mech. Appl. Math. 10, 137.Google Scholar
Stoker, J. J. 1957 Water Waves. New York: Interscience.Google Scholar
Ursell, F. 1953 The long-wave paradox in the theory of gravity waves. Proc. Camb. Phil. Soc. 49, 685.Google Scholar
Van Der Pol, B. & Bremmer, H. 1955 Operational Calculus. Cambridge University Press.Google Scholar
Watson, G. N. 1944 Theory of Bessel Functions. Cambridge University Press.Google Scholar
Whitham, G. B. 1952 The flow pattern of a supersonic projectile. Commun. Pure Appl. Math. 5, 30.Google Scholar