Conditions of compatibility for the solid-liquid interface
Authors:
F. Baldoni and K. R. Rajagopal
Journal:
Quart. Appl. Math. 55 (1997), 401-420
MSC:
Primary 80A22; Secondary 73B30
DOI:
https://doi.org/10.1090/qam/1466140
MathSciNet review:
MR1466140
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The seminal theory of singular surfaces propounded by Hadamard and Thomas is examined within the context of the dynamics of a solid-liquid interface. It is shown that most of the hypotheses upon which Clapeyron’s equation is based can be weakened and two generalized versions of it are derived: with and without curvature effects. The remaining part of the paper is mainly focused on the interface conditions for the classical Stefan problem. The counterpart of Clapeyron’s equation for such a problem will give an explicit expression for the supercooling temperature without recourse to linearization procedures. Furthermore, a decay law for the latent heat of melting is given which shows, in an explicit way, its complex dependence upon the curvature and the normal speed of the interface. Finally, a transport equation for the interface temperature is derived and a qualitative solution of a simplified version of it is given for the particular case in which the jump in the Helmoltz free energies of the bulk phases is a conserved quantity throughout the field.
J. W. Gibbs, Scientific Papers, Dover, New York, 1961
A. B. Pippard, Elements of Classical Thermodynamics, Cambridge University Press, Cambridge, 1966
J. Hadamard, Leçons sur la propagation des ondes et les équations de l’hydrodynamique, Hermann, Paris, 1903
G. M. Mavrovouniotis and H. Brenner, A Micromechanical Investigation of Interfacial Transport Processes. I. Interfacial Conservation Equation, Philos. Trans. Royal Soc. (345) 1675, 165–207 (1993)
- C. Truesdell and R. Toupin, The classical field theories, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226–793; appendix, pp. 794–858. With an appendix on tensor fields by J. L. Ericksen. MR 0118005
- Tracy Y. Thomas, Plastic flow and fracture in solids, Mathematics in Science and Engineering, Vol. 2, Academic Press, New York-London, 1961. MR 0127630
- W. Kosiński, Field singularities and wave analysis in continuum mechanics, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; PWN—Polish Scientific Publishers, Warsaw, 1986. Translated from the Polish by Kosiński and M. Doliński; Translation edited by R. W. Ogden. MR 872699
E. A. Guggenheim, Thermodynamics: an advanced treatment for chemists and physicists, North-Holland, Amsterdam, 1986
- J. J. Stoker, Differential geometry, Pure and Applied Mathematics, Vol. XX, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1969. MR 0240727
- P. Chadwick and B. Powdrill, Singular surfaces in linear thermoelasticity, Internat. J. Engrg. Sci. 3 (1965), 561–595 (English, with French, German, Italian and Russian summaries). MR 0189349, DOI https://doi.org/10.1016/0020-7225%2865%2990009-1
- Tracy Y. Thomas, Concepts from tensor analysis and differential geometry, Mathematics in Science and Engineering, Vol. 1, Academic Press, New York-London, 1961. MR 0120584
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360
J. W. Gibbs, Scientific Papers, Dover, New York, 1961
A. B. Pippard, Elements of Classical Thermodynamics, Cambridge University Press, Cambridge, 1966
J. Hadamard, Leçons sur la propagation des ondes et les équations de l’hydrodynamique, Hermann, Paris, 1903
G. M. Mavrovouniotis and H. Brenner, A Micromechanical Investigation of Interfacial Transport Processes. I. Interfacial Conservation Equation, Philos. Trans. Royal Soc. (345) 1675, 165–207 (1993)
C. Truesdell and R. Toupin, The Classical Field Theories, Handbuch der Physik, vol. 3/III, Springer-Verlag, Berlin, 1964
T. Y. Thomas, Plastic flow and fracture in solids, Academic Press, New York, 1961
W. Kosinski, Field singularities and wave analysis in continuum mechanics, Ellis-Horwood Limited, London, 1986
E. A. Guggenheim, Thermodynamics: an advanced treatment for chemists and physicists, North-Holland, Amsterdam, 1986
J. J. Stoker, Differential Geometry, Wiley-Interscience, New York, 1969
P. Chadwick and B. Powdrill, Singular surfaces in linear thermoelasticity, Internat. J. Engrg. Sci. 3, 561–595 (1965)
T. Y. Thomas, Concepts from Tensor Analysis and Differential Geometry, Academic Press, New York, 1961
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Wiley-Interscience, New York, 1962
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
80A22,
73B30
Retrieve articles in all journals
with MSC:
80A22,
73B30
Additional Information
Article copyright:
© Copyright 1997
American Mathematical Society
