Temperature and moving boundary in two-phase freezing due to an axisymmetric cold spot
Author:
S. C. Gupta
Journal:
Quart. Appl. Math. 45 (1987), 205-222
MSC:
Primary 35R35; Secondary 80A20
DOI:
https://doi.org/10.1090/qam/895094
MathSciNet review:
895094
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Abstract: Short time analytic solution of the problem of two-phase freezing due to an axisymmetric cold spot is presented. The melt could be superheated and it occupies an infinite region bounded internally by a cylinder of finite radius. Although the method of solution is valid for various other types of boundary conditions, the results in the paper are given for prescribed flux which could be time and space dependent. The method of solution is simple and straightforward and consists of assuming fictitious initial temperatures in some fictitious extensions of the region originally occupied by the melt. The spread of the solidification is much faster along the surface of the cylinder than along the interior of the cylinder and the spread along the surface always depends on material parameters. Several interesting results can be deduced as particular cases of the general results.
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R. I. Pedroso and G. A. Damoto, Perturbation solutions for spherical solutions of saturated liquids, ASME J. Heat Transfer 95, 42–46 (1973)
T. R. Goodman, The heat balance integral—further considerations and refinements, ASME J. Heat Transfer 83, 81–85 (1961)
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S. C. Gupta, Two-dimensional solidification in a cylindrical mold with imperfect mold contact, Internat. J. Engrg. Sci. 23, 901–913 (1985)
B. H. Kear, E. M. Breinan, and L. E. Greenwald, Laser glazing—a new process for production and control of rapidly chilled metallurgical micro-structures, Metals Technology 6, 121–129 (1979)
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J. C. Meuhlbauer and J. E. Sunderland, Heat conduction with freezing or melting, Appl. Mech. Reviews 18, 951–959 (1965)
J. R. Ockendon and W. R. Hodgkins, Moving boundary problems in heat flow and diffusion, Clarendon Press, Oxford, 1975
D. G. Wilson, A. D. Solomon, and P. T. Boggs, eds., Moving boundary problems, Academic Press, 1978
D. L. Sikarskie and B. A. Boley, The solution of a class of two-dimensional melting and solidification problems, Internat. J. Solids and Structures 1, 207–234 (1965)
B. A. Boley, An applied overview of moving boundary problems, in Moving Boundary Problems, edited by D. G. Wilson, A. D. Solomon, and P. T. Boogs, pp. 205–231, Academic Press, 1978
H. P. Yagoda and B. A. Boley, Starting solution of a slab under plane or axisymmetric hot spot, Quart. J. Mech. Appl. Math. 23, 225–246 (1970)
B. A. Boley and H. P. Yagoda, The starting solution for two-dimensional heat conduction problems with change of phase, Quart. Appl. Math. 27, 223–246 (1969)
B. A. Boley and H. P. Yagoda, The three-dimensional starting solution for a melting slab, Proc. Roy. Soc. London Ser A 323, 89–110 (1971)
M. A. Boles and M. N. Oz̈isik, Exact solution for freezing in cylindrical symmetry, ASME J. Heat Transfer 105, 401–403 (1983)
L. N. Tao, The Stefan problem with arbitrary initial and boundary conditions, Quart. Appl. Math. 36, 223–233 (1978)
R. I. Pedroso and G. A. Damoto, Perturbation solutions for spherical solutions of saturated liquids, ASME J. Heat Transfer 95, 42–46 (1973)
T. R. Goodman, The heat balance integral—further considerations and refinements, ASME J. Heat Transfer 83, 81–85 (1961)
S. C. Gupta, Two-dimensional heat conduction with phase change in a semi-infinite mold, Internat. J. Engrg. Sci. 19, 137–146 (1981)
S. C. Gupta, Two-dimensional solidification in a cylindrical mold with imperfect mold contact, Internat. J. Engrg. Sci. 23, 901–913 (1985)
B. H. Kear, E. M. Breinan, and L. E. Greenwald, Laser glazing—a new process for production and control of rapidly chilled metallurgical micro-structures, Metals Technology 6, 121–129 (1979)
B. A. Boley, A method of heat conduction analysis of melting and solidification problems, J. Math. Phys. 40, 300–313 (1961)
H. S. Carslaw and J. C. Jaeger, Conduction of heat in solids, Clarendon Press, Oxford, 1959
B. A. Boley, Proceedings of the eleventh international congress of applied mechanics, Munich, 586–596, 1964
M. Abramovitz and I. A. Stegun (eds.), Handbook of mathematical functions, Dover, 1972
H. Jeffreys, Asymptotic approximations, Clarendon Press, Oxford, 1962
R. Vellingiri, Solutions of some radially symmetric problems of heat conduction with phase change, Ph.D. Thesis, Indian Institute of Science, Bangalore, 1983
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Article copyright:
© Copyright 1987
American Mathematical Society
