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Parametric resonance in low-frequency magnetic stirring

Published online by Cambridge University Press:  26 April 2006

J. M. Galpin ,
Y. Fautrelle  and
A. D. Sneyd
J. M. Galpin
Affiliation:
INPG-MADYLAM, B.P. 95X, 38402 St Martin d'Hères Cedex, France Present address: IRSID, Station d'Essais, 57210 Maizières-les-Metz, France.
Y. Fautrelle
Affiliation:
INPG-MADYLAM, B.P. 95X, 38402 St Martin d'Hères Cedex, France
A. D. Sneyd
Affiliation:
University of Waikato, Private Bag, Hamilton, New Zealand
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Abstract

This paper analyses the effect of an alternating magnetic field of low frequency ω on a cylindrical tank of liquid metal. Previous work with higher-frequency fields has focused attention on the mean recirculating motion, but in the low-frequency limit periodic motion and surface waves become important. We show that a system of forced standing axisymmetric waves of frequency 2ω is established, and that the growth of non-axisymmetric modes is governed by a coupled system of Mathieu-type equations. The stability regions associated with this system are discussed and it is shown that the most easily excited transition to a non-axisymmetric mode is subharmonic, with frequency ω. Comparison with experiment shows that the theory gives qualitatively correct predictions.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 239 , June 1992 , pp. 409 - 427
Copyright
© 1992 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. E., 1965 Handbook of Mathematical Functions. Dover.
Arnol'D, V. I.: 1973 Ordinary Differential equations. MIT Press.
Benjamin, T. B. & Ursell, F., 1954 The stability of a plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Briskman, V. A. & Shaidurov, G. F., 1968 Parametric instability of a fluid surface in an alternating electric field. Sov. Phys. Dokl. 13, 540–542.Google Scholar
Ciliberto, S. & Gollub, J. P., 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Davidson, P. A., Hunt, J. C. R. & Moros, A. 1988 Turbulent recirculating flows in liquid metal MHD. Proc. 5th Beer-Sheva Seminar on MHD flows and turbulence. In Liquid Metal Flow: Magnetohydrodynamics and Applications (ed. H. Branover, M. Mond & Y. Unger). Prog. in Astron. and Aeron., vol. 111, pp. 400–420. AIAA.
Faraday, M.: 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
Fautrelle, Y. R.: 1981 Analytical and numerical aspects of the electromagnetic stirring induced by alternating magnetic fields. J. Fluid Mech. 102, 405–430.Google Scholar
Galpin, J. M. & Fautrelle, Y., 1992 Liquid-metal flows induced by low-frequency alternating magnetic fields. J. Fluid Mech. 239, 383–408 (referred to herein as GF).Google Scholar
Gel'Fand, I. M. & Lidskii, V. B. 1958 On the structure of the regions of stability of linear canonical systems of differential equations. Am. Math. Soc. Transl. 8 (2), 143–182.Google Scholar
Jordan, D. W. & Smith, P., 1987 Nonlinear Ordinary Differential Equations. Oxford University Press.
Mikelson, Y. Y., Jakovitch, A. T. & Pavlov, S. I., 1978 Numerical investigation of averaged MHD flow in cylindrical regions with the adoption of working hypotheses for turbulent stresses. Magnit. Girodin. 14, 51–58.Google Scholar
Miles, J. W.: 1984 Resonantly forced surface waves in a circular cylinder. J. Fluid Mech. 149, 15–31.Google Scholar
Miles, J. W. & Henderson, D., 1990 Parametrically forced surface waves. Ann. Rev. Fluid Mech. 22, 143–165.Google Scholar
Moffatt, H. K.: 1984 High frequency excitation of liquid metal systems. In Proc. IUTAM Symp. on Metallurgical Applications of MHD, 1982, Cambridge UK, pp. 180–189. London: The Metals Society.
Moore, D. J. & Hunt, J. C. R. 1984 Flow, turbulence and unsteadiness in coreless induction furnaces. In Proc. IUTAM Symp. on Metallurgical Applications of MHD, 1982, Cambridge UK, pp. 93–107. London: The Metals Society.
Nayfeh, A. H. & Mook, D. T., 1979 Nonlinear Oscillations. Wiley.
Sneyd, A.: 1971 Generation of fluid motion in a circular cylinder by an unsteady applied magnetic field. J. Fluid Mech. 49, 817–827.Google Scholar
Sneyd, A.: 1979 Fluid flow induced by a rapidly alternating or rotating magnetic field. J. Fluid Mech. 92, 35–51.Google Scholar
Taberlet, E. & Fautrelle, Y., 1985 Turbulent stirring in an experimental induction furnace. J. Fluid Mech. 159, 409–431.Google Scholar
Tarapore, E. & Evans, J. W., 1976 Fluid velocities in induction melting furnaces, part 1: theory and laboratory experiments. Metall. Trans. B 7, 343–351.Google Scholar
Trakas, C., Tabeling, P. & Chabrerie, J. P., 1984 Etude expérimentale du brassage turbulent dans le four à induction. J. Mec. Theor. Appl. 3, 345–370.Google Scholar