Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-18T19:22:44.443Z Has data issue: false hasContentIssue false
  • English
  • Français

Initiation of shape instabilities of free boundaries in planar Cauchy–Stefan problems

Published online by Cambridge University Press:  26 September 2008

Qiang Zhu ,
Anthony Peirce  and
John Chadam
Qiang Zhu
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
Anthony Peirce
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
John Chadam
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
Get access Rights & Permissions [Opens in a new window]

Abstract

The linearized shape stability of melting and solidifying fronts with surface tension is discussed in this paper by using asymptotic analysis. We show that the melting problem is always linearly stable regardless of the presence of surface tension, and that the solidification problem is linearly unstable without surface tension, but with surface tension it is linearly stable for those modes whose wave numbers lie outside a certain finite interval determined by the parameters of the problem. We also show that if the perturbed initial data is zero in the vicinity of the front, but otherwise quite general, it does not affect the stability. The present results complement those in Chadam & Ortoleva [4] which are only valid asymptotically for large time or equivalently for slow-moving interfaces. The theoretical results are verified numerically.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 4 , Issue 4 , December 1993 , pp. 419 - 436
Copyright
Copyright © Cambridge University Press 1993

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

[1]Mullins, W. W. & Sekerka, J. 1963 Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys. 34, 323–329.CrossRefGoogle Scholar
[2]Langer, J. S. 1980 Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52, 1–28.CrossRefGoogle Scholar
[3]Turland, B. D. & Peckover, R. S. 1980 The stability of planar melting fronts in two-phase thermal Stefan problems. IMA J. Appl. Math. 25, 1–5.CrossRefGoogle Scholar
[4]Chadam, J. & Ortoleva, P. 1983 The stabilizing effect of surface tension on the development of the free boundary in a planar, one-dimensional, Cauchy-Stefan problem. Ima J. Appl. Math. 30, 57–66.CrossRefGoogle Scholar
[5]Chadam, J., Baillon, J. C., Bertsch, M., Ortoleva, P. & Peletier, L. 1984 College de France Seminar, Vol. VI (Brezis, H. and Lions, J. L. (Eds)). Research Notes in Mathematics 109, 27–47, Pitman.Google Scholar
[6]Ricci, R. & Xie, W. 1991 On the stability of some solutions of the Stefan problem. Euro. J. Appl Math. 2, 1–15.CrossRefGoogle Scholar
[7]Dewynne, N., Howison, S. D., Ockendon, J. R. & Xie, W. 1989 Asymptotic behaviour of solutions to the Stefan problem with a kinetic condition at free boundary. J. Austral. Math. Soc., Ser. B, 31, 81–96.CrossRefGoogle Scholar
[8]Mahler, E. G., Schechter, R. S. & Wissler, E. H. 1968 Stability of a fluid layer with time-dependent density gradients. Phys. Fluids 11, 1901–1912.CrossRefGoogle Scholar
[9]Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Clarendon, Oxford.Google Scholar
[10]Rubinstein, L. 1982 Global stability of the Neumann solution of two-phase Stefan problems, IMA J. Appl. Math 28.CrossRefGoogle Scholar
[11]Ockendon, J. 1979 Linear and non-linear stability of a class of moving boundary problems. In: Proc. Seminar on Free Boundary Problems, Pavia, Italy.Google Scholar
[12]Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
[13]Zhu, Q., Peirce, A. & Chadam, J. A boundary integral numerical technique for free boundary problems (in preparation).Google Scholar