Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-23T11:54:11.537Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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