# American Institute of Mathematical Sciences

October  2011, 16(3): 883-894. doi: 10.3934/dcdsb.2011.16.883

## Phase transitions of a phase field model

 1 Department of Mathematics, Indiana University, Bloomington, IN 47405, United States

Received  July 2010 Revised  February 2011 Published  June 2011

We consider a phase field model for the mixture of two viscous incompressible uids with the same density. The model leads to a coupled Navier-Stokes/Cahn-Hilliard system. We explore the dynamics of the system near the critical point via a dynamic phase transition theory developed recently by Ma and Wang [7, 8]. Our analysis shows qualitatively the same phase transition result as the purely dissipative Cahn-Hilliard equation, which implies that the hydrodynamics does not play a role in the phase transition process of binary systems. This is different from the sharp interface situation, where numerical studies (see e.g. [3, 6]) suggest quite different behaviors between these two models.
Citation: Honghu Liu. Phase transitions of a phase field model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883
##### References:
 [1] J. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [2] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in "Mathematical Models for Phase Change Problems (Internat. Ser. Numer. Math. 88)" (eds. J. F. Rodriques), Birkhäuser-Verlag, Basel, Switzerland, (1989), 35-73. [3] M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341. [4] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. doi: 10.1103/RevModPhys.49.435. [5] C. Hsia, Bifurcation of binary systems with the Onsager mobility, J. Math. Phys., 51 (2010), 063305. doi: 10.1063/1.3406383. [6] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7. [7] T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences,", to appear., (). [8] T. Ma and S. Wang, "Bifurcation Theory and Applications," vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. [9] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems, Disc. Cont. Dyn. Sys. B, 11 (2009), 741-784. doi: 10.3934/dcdsb.2009.11.741. [10] T. Ma and S. Wang, Phase separation of binary systems, Physica A, 388 (2009), 4811-4817. doi: 10.1016/j.physa.2009.07.044. [11] A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259. [12] A. Novich-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5. [13] X. P. Wang and Y. G. Wang, The sharp interface limit of a phase field model for moving contact line problem, Methods Appl. Anal., 14 (2007), 287-284.

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##### References:
 [1] J. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102. [2] C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation, in "Mathematical Models for Phase Change Problems (Internat. Ser. Numer. Math. 88)" (eds. J. F. Rodriques), Birkhäuser-Verlag, Basel, Switzerland, (1989), 35-73. [3] M. E. Gurtin, D. Polignone and J. Viñals, Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6 (1996), 815-831. doi: 10.1142/S0218202596000341. [4] P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys., 49 (1977), 435-479. doi: 10.1103/RevModPhys.49.435. [5] C. Hsia, Bifurcation of binary systems with the Onsager mobility, J. Math. Phys., 51 (2010), 063305. doi: 10.1063/1.3406383. [6] C. Liu and J. Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179 (2003), 211-228. doi: 10.1016/S0167-2789(03)00030-7. [7] T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences,", to appear., (). [8] T. Ma and S. Wang, "Bifurcation Theory and Applications," vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. [9] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems, Disc. Cont. Dyn. Sys. B, 11 (2009), 741-784. doi: 10.3934/dcdsb.2009.11.741. [10] T. Ma and S. Wang, Phase separation of binary systems, Physica A, 388 (2009), 4811-4817. doi: 10.1016/j.physa.2009.07.044. [11] A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259. [12] A. Novich-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation, Physica D, 10 (1984), 277-298. doi: 10.1016/0167-2789(84)90180-5. [13] X. P. Wang and Y. G. Wang, The sharp interface limit of a phase field model for moving contact line problem, Methods Appl. Anal., 14 (2007), 287-284.
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