Macroscopic global modeling of binary alloy solidification processes
Authors:
V. Alexiades, D. G. Wilson and A. D. Solomon
Journal:
Quart. Appl. Math. 43 (1985), 143-158
MSC:
Primary 80A20
DOI:
https://doi.org/10.1090/qam/793522
MathSciNet review:
793522
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Abstract | References | Similar Articles | Additional Information
Abstract: A macroscopic mathematical model is constructed describing the evolution of the phases of a binary alloy or mixture undergoing solidification under the action of simultaneous conduction of heat and diffusion of solute. The formulation is global, in the form of a pair of conservation laws valid over the whole region occupied by the alloy in a weak (distributional) sense. Thus it is especially convenient for numerical solution since it does not require tracking of the interface, which, in fact, may develop into a ``mushy zone".
- [1] V. Alexiades, Rapid freezing of dilute alloys, IMA J. Applied Math., 30, 67-79 (1983) MR 711103
- [2] V. Alexiades and J. R. Cannon, Free boundary problems in alloy solidification, SIAM J. Math. Analysis, 11, 254-264 (1980) MR 559867
- [3] A. Bermudez and C. Saguez, Etude Numerique d'un Probleme de solidification d'un alliage, INRIA Report, 1981
- [4] B. Chalmers, Principles of Solidification, Wiley, 1964
- [5] S. H. Cho and J. E. Sunderland, Heat conduction problems with melting or freezing, J. Heat Transfer, 91C, 421-426 (1969)
- [6] J. Christian, The theory of transformations in metals and alloys, Pergamon, Oxford, 1965
- [7] A. B. Crowley and J. R. Ockendon, On the numerical solution of an alloy solidification problem, Int. J. Heat Mass Transfer, 22, 941-947 (1979)
- [8] C. M. Elliot and J. R. Ockendon, Weak and variational methods for moving boundary problems, Pitman, Boston, 1982 MR 650455
- [9] A. Fasano and M. Primicerio, General free boundary problems for the heat equation, I, II, III, J. Math. Analysis Appl. 57, 694-723 (1977); 56, 209-231 (1977); 59, 1-14 (1977) MR 0487016
- [10] G. J. Fix, Numerical methods for alloy solidification problems, pp. 109-128 in [22]
- [11] A. A. Lacey. J. R. Ockendon and A. B. Tayler, Modelling mushy regions, IMA J. Appl. Math. (to appear)
- [12] A. A. Lacey and M. Shillor, The existence and stability of regions with super heating in the classical two-phase one-dimensional Stefan problems with heat sources, IMA J. Appl. Math. (to appear) MR 714070
- [13] J. S. Langer, Instabilities and pattern formation in crystal growth, Review Mod. Physics, 52, 1-28 (1980)
- [14] G. H. Meyer, A numerical method for the solidification of a binary alloy, Int. J. Heat Mass Transfer, 24, 778-781 (1981)
- [15] J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford, 1975
- [16] F. Rosenberger, Fundamentals of crystal growth, I, Springer-Verlag, Berlin, 1979
- [17] L. Rubinstein, The Stefan problem, AMS Transl. 27, Amer. Math. Society, Providence, 1971 MR 0351348
- [18] A. D. Solomon, D. G. Wilson and V. Alexiades, Explicit solutions to phase change problems, Quarterly of Appl. Math., 41 (1983) MR 719507
- [19] A. D. Solomon, D. G. Wilson and V. Alexiades, A numerical simulation of a binary alloy solidification process, SIAM J. Scient. Stat. Comp., to appear.
- [20] A. B. Tayler, The mathematical formulation of Stefan problems, pp. 120-137 in [14].
- [21] R. H. Tien and G. E. Geiger, A heat transfer analysis of the solidification of a binary entectic system, J. Heat Transfer, 89C, 230-234 (1967)
- [22] R. Trivedi, Theory of dendritic growth during the directional solidification of binary alloys, Journal of Crystal Growth, 49, 219-232 (1980)
- [23] D. G. Wilson, A. D. Solomon and V. Alexiades, A shortcoming of the explicit solution for the binary alloy solidification problem, Letters in Heat and Mass Transfer, 9, 421-428 (1982)
- [24] D. G. Wilson, A. D. Solomon and V. Alexiades, A model of binary alloy solidification, Int. J. Numer. Methods. Eng., 20(6), 1067-1085 (1984) MR 748963
- [25] D. G. Wilson, A. D. Solomon and P. T. Boggs, editors, Moving boundary problems, Academic Press, 1978 MR 0466887
- [26] M. Woods, Thermodynamics of fluid systems, Oxford Univ. Press, 1975
- [27] T. W. Clyne, Numerical modeling of directional solidification of metallic alloys, Metal science 16, 441-450 (1982)
- [28] S. Luckhaus and A. Visintin, Phase transition in multicomponent systems, Manuscripta Math. 43, 261-288 (1983) MR 707047
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DOI:
https://doi.org/10.1090/qam/793522
Article copyright:
© Copyright 1985
American Mathematical Society