Solid-liquid phase changes with different densities
Authors:
Michel Frémond and Elisabetta Rocca
Journal:
Quart. Appl. Math. 66 (2008), 609-632
MSC (2000):
Primary 80A22, 34A34, 74G25
DOI:
https://doi.org/10.1090/S0033-569X-08-01100-0
Published electronically:
September 10, 2008
MathSciNet review:
2465138
Full-text PDF Free Access
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Abstract: In this paper we present a new thermodynamically consistent phase transition model describing the evolution of a liquid substance, e.g., water, in a rigid container $\Omega$ when we freeze the container. Since the density $\varrho _{2}$ of ice with volume fraction $\beta _{2}$ is lower than the density $\varrho _{1}$ of water with volume fraction $\beta _{1}$, experiments, for instance the freezing of a glass bottle filled with water, show that the water pressure increases up to the rupture of the bottle. When the container is not impermeable, freezing may produce a nonhomogeneous material, for instance water, ice or sorbet. Here we describe a general class of phase transition processes, including this example as a particular case. Moreover, we study the resulting nonlinear and singular PDE system from the analytical viewpoint, recovering existence of a global (in time) weak solution and also uniqueness for some particular choices of the nonlinear functions involved.
References
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References
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- Barbu V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden (1976). MR 0390843 (52:11666)
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- Bonetti E., Colli P., Fabrizio M., Gilardi G.: Global solution to a singular integro-differential system related to the entropy balance, Nonlinear Anal. 66, 1949–1979 (2007). MR 2304972
- Bonetti E., Colli P., Fabrizio M., Gilardi G.: Modelling and long-time behaviour of an entropy balance and linear thermal memory model for phase transitions, Discrete Contin. Dyn. Syst. Ser. B. 6, 1001–1026 (2006). MR 2224868 (2007c:80006)
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- Bonetti E., Frémond M., Rocca E.: A new dual approach for a class of phase transitions with memory: Existence and long-time behaviour of solutions, J. Math. Pures Appl. 88, 455–481 (2007). MR 2369878
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Additional Information
Michel Frémond
Affiliation:
Dipartimento di Ingegneria Civile, Università di Roma “Tor Vergata”, Via del Politecnico, 1, I-00133 Roma, Italy
Address at time of publication:
CMLA, ENS Cachan -Département de Mécanique, ENSTA, Paris
Email:
fremond@cmla.ens-cachan.fr
Elisabetta Rocca
Affiliation:
Dipartimento di Matematica, Università di Milano, Via Saldini, 50, I-20133 Milano, Italy
Email:
rocca@mat.unimi.it
Keywords:
Phase transitions with voids,
singular and nonlinear PDE system,
global existence of solutions.
Received by editor(s):
February 26, 2007
Published electronically:
September 10, 2008
Article copyright:
© Copyright 2008
Brown University