Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T19:46:33.881Z Has data issue: false hasContentIssue false
  • English
  • Français

A Stefan problem in a Bridgman crystal grower

Published online by Cambridge University Press:  26 September 2008

P. den Decker ,
R. van der Hout ,
C. J. Van Duijn  and
L. A. Peletier
P. den Decker
Affiliation:
Akzo Research Laboratories Arnhem, Applied Mathematics Department, PB 9300, 6800 SB Arnhem, The Netherlands
R. van der Hout
Affiliation:
Akzo Research Laboratories Arnhem, Applied Mathematics Department, PB 9300, 6800 SB Arnhem, The Netherlands
C. J. Van Duijn
Affiliation:
Department of Mathematics, Delft University of Technology, PB 5031, 2600 GA Delft, The Netherlands and Mathematical Institute, Leiden University, PB 9512, 2300 RA Leiden, The Netherlands
L. A. Peletier
Affiliation:
Mathematical Institute, Leiden University, PB 9512, 2300 RA Leiden, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

We discuss a one-dimensional model for a Bridgman crystal grower, where the removal of heat is described by an internal heat sink. A consequence is the apparent existence of mushy regions for relatively large velocities of the cooling machine; these mushy regions are an artefact of the one-dimensional approximation. We show that for some types of cooling profiles there exists a critical speed for the existence of mushy regions, whereas for different cooling profiles no such critical speed exists. The presence of a mushy region may indicate a strong curvature of the liquid/solid interface in the real situation.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 3 , June 1995 , pp. 191 - 199
Copyright
Copyright © Cambridge University Press 1995

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

[BMP]Bertsch, M., de Mottoni, P. & Peletier, L. A. 1984 Degenerate diffusion and the Stefan problem. Nonlinear Anal. TMA 8, 1311–1336.CrossRefGoogle Scholar
[CM1]Comparini, E. & Manni, C. 1990 On a model for the temperature distribution in bimetallic strips. Meccanica 25, 115–123.CrossRefGoogle Scholar
[CM2]Comparini, E. & Manni, C. 1991 An approximation of the thermal field in a continuous casting process of a thin metal layer. Math. Meth. Appl. Sci. 14, 217–226.CrossRefGoogle Scholar
[C]Crank, J. 1984 Free and Moving Boundary Problems. Clarendon Press, Oxford.Google Scholar
[F]Filippov, A. F. 1988 Differential Equations with Discontinuous Righthand Sides. Kluwer, Dordrecht.CrossRefGoogle Scholar
[H]van der Hout, R. 1990 A travelling wave solution for a one-dimensional Stefan problem. Nonlinear Analysis TMA 14, 427–439.CrossRefGoogle Scholar
[M]Meirmanov, A. M. 1992 The Stefan Problem. Walter de Gruyter, Berlin.CrossRefGoogle Scholar
[R]Rubinstein, L. I. 1971 The Stefan problem. Trans. Math. Monographs 27, Amer. Math. Soc., Providence, RI.Google Scholar
[SS]Sewell, H. J. & Sewell, M. J. 1987 Mathematical modelling of the quench region in the continuous casting of a bimetallic strip. Math. Eng. in Industry 1, 289–312.Google Scholar
[V]van Veldhuizen, M. Private communication.Google Scholar