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Mullins–Sekerka stability analysis for melting-freezing waves in helium

Published online by Cambridge University Press:  26 September 2008

Joseph D. Fehribach
Joseph D. Fehribach
Affiliation:
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA and Institute for Mathematcis & its Applications, University of Minnesota, Minneapolis, MN 55455, USA (email: [email protected])
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Abstract

This paper considers the stability of melt-solid interfaces to eigenfunction perturbations for a system of equations which describe the melting and freezing of helium. The analysis is carried out in both planar and spherical geometries. The principal results are that when the melt is freezing, under certain far-field conditions, the interface is stable in the sense of Mullins and Sekerka. On the other hand, when the solid is melting (at least when the melting is sufficiently fast), the interface is unstable. In some circumstances these instabilities are oscillatory, with amplitude and growth rate increasing with surface tension and frequency. The last section considers the original problem of Mullins and Sekerka in the present notation.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 1 , March 1994 , pp. 21 - 37
Copyright
Copyright © Cambridge University Press 1994

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References

Andreev, A. F. & Parshin, A. YA. 1978 Equilibrium shape and oscillations of the surface of quantum crystals. Sou. Phys. JETP 48, 763–766.Google Scholar
Castaing, B., Balibar, S. & Laroche, C. 1980 Mobilitée àa 1 MHz du font de fusion de 4He. J. Physique 41, 897–903.CrossRefGoogle Scholar
Chadam, J., Howison, S. D. & Ortoleva, P. 1987 Existence and stability for spherical crystals growing in a supersaturated solution. IMA J. Appl. Math. 39, 1–15.CrossRefGoogle Scholar
Chadam, J., Hoff, D., Merino, E., Ortoleva, P. & Sen, A. 1986 Reactive infiltration instabilities. IMA J. Appl. Math. 36, 207–221.CrossRefGoogle Scholar
Chen, X. & Reitich, F. 1992 Local existence and uniqueness of solutions of the Stefan problem with surface tension and kinetic undercooling. J. Math. Anal. Appl. 164, 350–362.CrossRefGoogle Scholar
Coriell, S. R. & McFadden, G. B. 1992 Morphological Stability. In Handbook of Crystal Growth, Vol. I (Hurle, D. T. J., ed.), North-Holland.Google Scholar
Duchon, J. & Robert, R. 1984 Evolution d'une interface par capillaritée et diffusion de volume, 1. Existence locale en temps. Analyse non linéeaire, 1, 361–378.CrossRefGoogle Scholar
Gurtin, M. E. 1990 A mechanical theory for crystallization of a rigid solid in a liquid melt; melting-freezing waves. Arch. Rat. Mech. Anal. 110, 287–312.CrossRefGoogle Scholar
Huber, T. E. & Maris, H. J. 1981 Capillary effects on the phonon transmission between liquid and solid helium. Phys. Rev. Lett. 47, 1907–1910.CrossRefGoogle Scholar
Keshishev, K. O., Parshin, A. YA. & Babkin, A. V. 1979 Experimental detection of crystallization waves in He4. JETP Lett. 30, 56–59.Google Scholar
Lebedev, N. N. 1972 Special Functions and Their Applications. Dover.Google Scholar
Luckhaus, S. 1990 Solutions for the two-phase Stefan problem with the Gibbs-Thompson law for the melting temperature. Euro. J. Appl. Math. 1, 101–111.CrossRefGoogle Scholar
Maris, H. J. & Andreev, A. F. 1987 The surface of crystalline helium-4. Phys. Today, February, 25–30.Google Scholar
McFadden, G. B., Boisvert, R. F. & Coriell, S. R. 1987 Nonplanar interface morphologies during unidirectional solidification of a binary alloy. II. Three-dimensional computations. J. Crystal Growth 84, 371–388.CrossRefGoogle Scholar
Mullins, W. W. & Sekerka, R. F. 1963 Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323–329.CrossRefGoogle Scholar
Ockendon, J. R. 1980 Linear and nonlinear stability of a class of moving boundary problems. In Paiva Seminar on Free Boundary Problems (Magenes, E., ed.), Inst. Naz. di Alta Mat., Rome.Google Scholar
O'Neill, B. 1966 Elementary Differential Geometry. Academic Press.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of fluid into a porous medium or Hele-Shaw cell. In Proc. Roy. Soc. London A 245, 312–329.Google Scholar
Zhu, Q., Pierce, A. & Chadam, J. 1992 Initiation of shape instabilities of free boundaries in planar Cauchy-Stefan problems. Euro. J. Appl. Math. (to appear).CrossRefGoogle Scholar