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Edge pinch instability of oblate liquid metal drops in a transverse AC magnetic field

Published online by Cambridge University Press:  03 May 2011

JĀNIS PRIEDE
JĀNIS PRIEDE*
Affiliation:
Applied Mathematics Research Centre, Coventry University, Priory Street, Coventry CV1 5FB, UK
*
†Email address for correspondence: [email protected]
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Abstract

This paper considers the stability of liquid metal drops subject to a high-frequency AC magnetic field. An energy variation principle is derived in terms of the surface integral of the scalar magnetic potential. This principle is applied to a thin perfectly conducting liquid disk, which is used to model the drops constrained in a horizontal gap between two parallel insulating plates. Firstly, the stability of a circular disk is analysed with respect to small-amplitude harmonic edge perturbations. Analytical solution shows that the edge deformations with the azimuthal wavenumbers m = 2, 3, 4, . . . start to develop as the magnetic Bond number exceeds the critical threshold Bmc = 3π(m + 1)/2. The most unstable is m = 2 mode, which corresponds to an elliptical deformation. Secondly, strongly deformed equilibrium shapes are modelled numerically by minimising the associated energy in combination with the solution of a surface integral equation for the scalar magnetic potential on an unstructured triangular mesh. The edge instability is found to result in the equilibrium shapes of either two- or threefold rotational symmetry depending on the magnetic field strength and the initial perturbation. The shapes of higher rotational symmetries are unstable and fall back to one of these two basic states. The developed method is both efficient and accurate enough for modelling of strongly deformed drop shapes.

JFM classification

Interfacial Flows (free surface): Fingering instability Mathematical Foundations: Variational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 676 , 10 June 2011 , pp. 218 - 236
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Abramowitz, A. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.Google Scholar
Baptiste, L., van Landschoot, N., Gleijm, G., Priede, J., Schade van Westrum, J., Velthuis, H. & Kim, T.-Y. 2007 Electromagnetic levitation: A new technology for high rate physical vapour deposition of coatings onto metallic strip. Surf. Coating Technol. 202, 1189–1193.CrossRefGoogle Scholar
Batchelor, G. K. 1973 Three-dimensional flow fields extending to infinity. In An Introduction to Fluid Dynamics. sec. 2.9, p. 121. Cambridge.Google Scholar
Chandrasekhar, S. 1961 General variational principle. In Hydrodynamic and Hydromagnetic Stability, chap. 14. Oxford.Google Scholar
Conrath, M., Kocourek, V. & Karcher, Ch. 2006 Behavior of a liquid metal disc in a magnetic field of a circular current loop. In Fifth International Symposium on Electromagnetic Processing of Materials (EPM 2006), Sendai, Japan, pp. 210–213.Google Scholar
Conrath, M. 2007 Dynamics of liquid metal drops influenced by electromagnetic fields. PhD Thesis, Ilmenau University of Technology, Germany, chap. 5, pp. 54–58.Google Scholar
Cowper, G. R. 1973 Gaussian quadrature formulas for triangles. Intl J. Numer. Meth. Engng 7, 405–408.CrossRefGoogle Scholar
Fautrelle, Y., Sneyd, A. & Etay, J. 2007 Effect of AC magnetic fields on free surfaces. In Magnetohydrodynamics – Historical Evolution and Trends (ed. Molokov, S., Moreau, R. & Moffatt, H. K.), pp. 345–355. Springer.CrossRefGoogle Scholar
Hinaje, M., Vinsard, G. & Dufour, S. 2006 a Determination of stable shapes of a thin liquid metal layer using a boundary integral method. J. Phys. D: Appl. Phys. 39, 1244–1248.CrossRefGoogle Scholar
Hinaje, M., Vinsard, G. & Dufour, S. 2006 b Analytical modelling of a thin liquid metal layer submitted to an ac magnetic field. J. Phys. D: Appl. Phys. 39, 2641–2646.CrossRefGoogle Scholar
Kocourek, V., Karcher, Ch., Conrath, M. & Schulze, D. 2006 Stability of liquid metal drops affected by a high-frequency magnetic field. Phys. Rev. E 74, 026303 (1–7).CrossRefGoogle ScholarPubMed
Li, L.-W., Kang, X.-K. & Leong, M.-S. 2002 Spheroidal Wave Functions in Electromagnetic Theory, pp. 13–17. Wiley.Google Scholar
Mohring, J.-U., Karcher, C. & Schulze, D. 2005 Dynamic behavior of a liquid metal interface under the influence of a high-frequency magnetic field. Phys. Rev. E 71, 047301 (1–4).CrossRefGoogle ScholarPubMed
Perrier, D., Fautrelle, Y. & Etay, J. 2003 Free surface deformations of a liquid metal drop submitted to a middlefrequency AC magnetic field. In Fourth International Conference on Electromagnetic Processing of Materials (EPM 2003), (ed. Asai, S., Fautrelle, Y., Gillon, P. & Durand, F.), Lyon, France, pp. 279–282.Google Scholar
Press, W. H., Teukolsky, S. A. & Vetterling, W. T. 1996 Directions set (Powell's) methods in multidimensions. In Numerical Recipes in Fortran 90: The Art of Parallel Scientific Computing, sec. 10.5.Google Scholar
Priede, J., Etay, J. & Fautrelle, Y. 2006 Edge pinch instability of liquid metal sheet in a transverse high-frequency ac magnetic field. Phys. Rev. E 73, 066303 (1–10).CrossRefGoogle Scholar
Sneyd, A. D. & Moffatt, H. K. 1982 Fluid dynamical aspects of the levitation melting. J. Fluid Mech. 117, 45–70.CrossRefGoogle Scholar
Wolfram, S. 1996 The Mathematica Book, 3rd edn. Wolfram Media.Google Scholar