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: 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".
- Vasilios Alexiades, Rapid freezing of dilute alloys, IMA J. Appl. Math. 30 (1983), no. 1, 61–79. MR 711103, DOI https://doi.org/10.1093/imamat/30.1.67
- Vasilios Alexiades and John R. Cannon, Free boundary problems in solidification of alloys, SIAM J. Math. Anal. 11 (1980), no. 2, 254–264. MR 559867, DOI https://doi.org/10.1137/0511025
A. Bermudez and C. Saguez, Etude Numerique d’un Probleme de solidification d’un alliage, INRIA Report, 1981
B. Chalmers, Principles of Solidification, Wiley, 1964
S. H. Cho and J. E. Sunderland, Heat conduction problems with melting or freezing, J. Heat Transfer, 91C, 421–426 (1969)
J. Christian, The theory of transformations in metals and alloys, Pergamon, Oxford, 1965
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)
- C. M. Elliott and J. R. Ockendon, Weak and variational methods for moving boundary problems, Research Notes in Mathematics, vol. 59, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 650455
- A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I, J. Math. Anal. Appl. 57 (1977), no. 3, 694–723. MR 487016, DOI https://doi.org/10.1016/0022-247X%2877%2990256-6
G. J. Fix, Numerical methods for alloy solidification problems, pp. 109–128 in [22]
A. A. Lacey. J. R. Ockendon and A. B. Tayler, Modelling mushy regions, IMA J. Appl. Math. (to appear)
- A. A. Lacey and M. Shillor, The existence and stability of regions with superheating in the classical two-phase one-dimensional Stefan problem with heat sources, IMA J. Appl. Math. 30 (1983), no. 2, 215–230. MR 714070, DOI https://doi.org/10.1093/imamat/30.2.215
J. S. Langer, Instabilities and pattern formation in crystal growth, Review Mod. Physics, 52, 1–28 (1980)
G. H. Meyer, A numerical method for the solidification of a binary alloy, Int. J. Heat Mass Transfer, 24, 778–781 (1981)
J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford, 1975
F. Rosenberger, Fundamentals of crystal growth, I, Springer-Verlag, Berlin, 1979
- L. I. Rubenšteĭn, The Stefan problem, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by A. D. Solomon; Translations of Mathematical Monographs, Vol. 27. MR 0351348
- A. D. Solomon, D. G. Wilson, and V. Alexiades, Explicit solutions to phase change problems, Quart. Appl. Math. 41 (1983/84), no. 2, 237–243. MR 719507, DOI https://doi.org/10.1090/S0033-569X-1983-0719507-5
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.
A. B. Tayler, The mathematical formulation of Stefan problems, pp. 120–137 in [14].
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)
R. Trivedi, Theory of dendritic growth during the directional solidification of binary alloys, Journal of Crystal Growth, 49, 219–232 (1980)
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)
- D. G. Wilson, A. D. Solomon, and V. Alexiades, A model of binary alloy solidification, Internat. J. Numer. Methods Engrg. 20 (1984), no. 6, 1067–1084. MR 748963, DOI https://doi.org/10.1002/nme.1620200609
- David George Wilson, Alan D. Solomon, and Paul T. Boggs (eds.), Moving boundary problems, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0466887
M. Woods, Thermodynamics of fluid systems, Oxford Univ. Press, 1975
T. W. Clyne, Numerical modeling of directional solidification of metallic alloys, Metal science 16, 441–450 (1982)
- Stephan Luckhaus and Augusto Visintin, Phase transition in multicomponent systems, Manuscripta Math. 43 (1983), no. 2-3, 261–288. MR 707047, DOI https://doi.org/10.1007/BF01165833
V. Alexiades, Rapid freezing of dilute alloys, IMA J. Applied Math., 30, 67–79 (1983)
V. Alexiades and J. R. Cannon, Free boundary problems in alloy solidification, SIAM J. Math. Analysis, 11, 254–264 (1980)
A. Bermudez and C. Saguez, Etude Numerique d’un Probleme de solidification d’un alliage, INRIA Report, 1981
B. Chalmers, Principles of Solidification, Wiley, 1964
S. H. Cho and J. E. Sunderland, Heat conduction problems with melting or freezing, J. Heat Transfer, 91C, 421–426 (1969)
J. Christian, The theory of transformations in metals and alloys, Pergamon, Oxford, 1965
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)
C. M. Elliot and J. R. Ockendon, Weak and variational methods for moving boundary problems, Pitman, Boston, 1982
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)
G. J. Fix, Numerical methods for alloy solidification problems, pp. 109–128 in [22]
A. A. Lacey. J. R. Ockendon and A. B. Tayler, Modelling mushy regions, IMA J. Appl. Math. (to appear)
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)
J. S. Langer, Instabilities and pattern formation in crystal growth, Review Mod. Physics, 52, 1–28 (1980)
G. H. Meyer, A numerical method for the solidification of a binary alloy, Int. J. Heat Mass Transfer, 24, 778–781 (1981)
J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford, 1975
F. Rosenberger, Fundamentals of crystal growth, I, Springer-Verlag, Berlin, 1979
L. Rubinstein, The Stefan problem, AMS Transl. 27, Amer. Math. Society, Providence, 1971
A. D. Solomon, D. G. Wilson and V. Alexiades, Explicit solutions to phase change problems, Quarterly of Appl. Math., 41 (1983)
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.
A. B. Tayler, The mathematical formulation of Stefan problems, pp. 120–137 in [14].
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)
R. Trivedi, Theory of dendritic growth during the directional solidification of binary alloys, Journal of Crystal Growth, 49, 219–232 (1980)
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)
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)
D. G. Wilson, A. D. Solomon and P. T. Boggs, editors, Moving boundary problems, Academic Press, 1978
M. Woods, Thermodynamics of fluid systems, Oxford Univ. Press, 1975
T. W. Clyne, Numerical modeling of directional solidification of metallic alloys, Metal science 16, 441–450 (1982)
S. Luckhaus and A. Visintin, Phase transition in multicomponent systems, Manuscripta Math. 43, 261–288 (1983)
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© Copyright 1985
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