April  2011, 4(2): 391-407. doi: 10.3934/dcdss.2011.4.391

Phase separation in a gravity field

1. 

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1, Czech Republic

2. 

WIAS Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany, Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano

3. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D–10117 Berlin

Received  May 2009 Revised  August 2009 Published  November 2010

We prove here well-posedness and convergence to equilibria for the solution trajectories associated with a model for solidification of the liquid content of a rigid container in a gravity field. We observe that the gravity effects, which can be neglected without considerable changes of the process on finite time intervals, have a substantial influence on the long time behavior of the evolution system. Without gravity, we find a temperature interval, in which all phase distributions with a prescribed total liquid content are admissible equilibria, while, under the influence of gravity, the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.
Citation: Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391
References:
[1]

M. Brokate, J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer, New York, 1996.

[2]

P. Colli, M. Frémond and A Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math., 48 (1990), 31-47.

[3]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63.

[4]

M. Frémond, "Non-Smooth Thermo-Mechanics," Springer-Verlag, Berlin, 2002.

[5]

M. Frémond and E. Rocca, Well-posedness of a phase transition model with the possibility of voids, Math. Models Methods Appl. Sci., 16 (2006), 559-586. doi: 10.1142/S0218202506001261.

[6]

M. Frémond and E. Rocca, Solid liquid phase changes with different densities, Q. Appl. Math., 66 (2008), 609-632.

[7]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, Berlin, 1986.

[8]

P. Krejčí, Hysteresis operators-A new approach to evolution differential inequalities, Comment. Math. Univ. Carolinae, 33 (1989), 525-536.

[9]

P. Krejčí, E. Rocca and J. Sprekels, A bottle in a freezer, SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X.

[10]

P. Krejčí, Elastoplastic reaction of a container to water freezing, Mathematica Bohemica, to appear (2010).

[11]

P. Krejčí, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal., 34 (2002), 409-434. doi: 10.1137/S0036141001387604.

[12]

P. Krejčí, J. Sprekels and U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl., 13 (2003), 695-712.

[13]

E. Rocca and R. Rossi, A nonlinear degenerating PDE system modelling phase transitions in thermoviscoelastic materials, J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006.

[14]

E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5.

[15]

A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser Boston, 1996.

[16]

H. Y. Erbil, "Surface Chemistry of Solid and Liquid Interfaces," Blackwell Publishing, John Wiley & Sons, 2006.

show all references

References:
[1]

M. Brokate, J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer, New York, 1996.

[2]

P. Colli, M. Frémond and A Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math., 48 (1990), 31-47.

[3]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63.

[4]

M. Frémond, "Non-Smooth Thermo-Mechanics," Springer-Verlag, Berlin, 2002.

[5]

M. Frémond and E. Rocca, Well-posedness of a phase transition model with the possibility of voids, Math. Models Methods Appl. Sci., 16 (2006), 559-586. doi: 10.1142/S0218202506001261.

[6]

M. Frémond and E. Rocca, Solid liquid phase changes with different densities, Q. Appl. Math., 66 (2008), 609-632.

[7]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, Berlin, 1986.

[8]

P. Krejčí, Hysteresis operators-A new approach to evolution differential inequalities, Comment. Math. Univ. Carolinae, 33 (1989), 525-536.

[9]

P. Krejčí, E. Rocca and J. Sprekels, A bottle in a freezer, SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X.

[10]

P. Krejčí, Elastoplastic reaction of a container to water freezing, Mathematica Bohemica, to appear (2010).

[11]

P. Krejčí, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal., 34 (2002), 409-434. doi: 10.1137/S0036141001387604.

[12]

P. Krejčí, J. Sprekels and U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl., 13 (2003), 695-712.

[13]

E. Rocca and R. Rossi, A nonlinear degenerating PDE system modelling phase transitions in thermoviscoelastic materials, J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006.

[14]

E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5.

[15]

A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser Boston, 1996.

[16]

H. Y. Erbil, "Surface Chemistry of Solid and Liquid Interfaces," Blackwell Publishing, John Wiley & Sons, 2006.

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