Article Contents
Article Contents

# Phase separation in a gravity field

• 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.
Mathematics Subject Classification: Primary: 80A22; Secondary: 74C05, 35K50.

 Citation:

•  [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.