Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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
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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.


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  • 1. Baiocchi C.: Sulle equazioni differenziali astratte lineari del primo e del secondo ordine negli spazi di Hilbert, Ann. Mat. Pura Appl. (4), 76, 233)-304 (1967). MR 0223697 (36:6745)
  • 2. Barbu V.: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden (1976). MR 0390843 (52:11666)
  • 3. Bonetti E.: Modelling phase transitions via an entropy equation: Long-time behaviour of the solutions, Dissipative phase transitions, 21-42, Ser. Adv. Math. Appl. Sci., 71, World Sci. Publ., Hackensack, NJ (2006). MR 2223371 (2006m:35361)
  • 4. 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
  • 5. 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)
  • 6. Bonetti E., Colli P., Frémond M.: A phase field model with thermal memory governed by the entropy balance, Math. Models Methods Appl. Sci. 13, no. 11, 1565-1588 (2003). MR 2024463 (2004k:80007)
  • 7. Bonetti E., Frémond M.: A phase transition model with the entropy balance, Math. Meth. Appl. Sci. 26, 539-556 (2003). MR 1967321 (2004a:80008)
  • 8. 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
  • 9. Bonetti E., Rocca E.: Global existence and long-time time behaviour for a singular integro-differential phase-field system, Commun. Pure Appl. Anal. 6, 367-387 (2007). MR 2289826
  • 10. Brezis H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Studies 5 North-Holland, Amsterdam (1973). MR 0348562 (50:1060)
  • 11. Brokate M., Sprekels J.: Hysteresis and Phase Transitions, Appl. Math. Sci. 121, Springer, New York (1996). MR 1411908 (97g:35127)
  • 12. Colli P., Gilardi G., Grasselli M., Schimperna G.: Global existence for the conserved phase field model with memory and quadratic nonlinearity, Port. Math. (N.S.) 58, no. 2, 159-170 (2001). MR 1836260 (2002g:35112)
  • 13. Colli P., Sprekels J.: Positivity of temperature in the general Frémond model for shape memory alloys, Contin. Mech. Thermodyn. 5, no. 4, 255-264 (1993). MR 1247344 (94j:73009)
  • 14. Darcy H.: Les fontaines publiques de la ville de Dijon, V. Dalmont Paris (1856).
  • 15. Frémond M.: Non-smooth thermomechanics, Springer-Verlag, Berlin (2002). MR 1885252 (2003g:74004)
  • 16. Frémond M., Rocca E.: Well-posedness of a phase transition model with the possibility of voids, Math. Models Methods Appl. Sci. 16, no. 4, 559-586 (2006). MR 2218214 (2007k:80007)
  • 17. Germain P.: Cours de mécanique des milieux continus, Masson, Paris (1973). MR 036854 (51:4782)
  • 18. Lions J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris (1969). MR 0259693 (41:4326)
  • 19. Lions J.L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, Berlin (1972). MR 0350177 (50:2670)
  • 20. Moreau J.J.: Fonctionnelles convexes, Collège de France (1966) and Dipartimento di Ingegneria Civile Università di Roma Tor Vergata (2003).
  • 21. Simon J.: Compact sets in the space $ L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146, 65-96 (1987). MR 916688 (89c:46055)
  • 22. Visintin A.: Models of Phase Transitions, Progress in Nonlinear Differential Equations and their Applications 28, Birkhäuser, Boston (1996). MR 1423808 (98a:80006)

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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

DOI: https://doi.org/10.1090/S0033-569X-08-01100-0
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

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