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
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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.
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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
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© Copyright 1997
American Mathematical Society