Phase transitions of a phase field model

Previous Article
Long time behavior of some epidemic models
DCDS-B Home
This Issue
Next Article
Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology

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

Abstract
Related Papers

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.

Keywords: dynamic transition theory, center manifold reduction., Navier-Stokes/Cahn-Hilliard system, Phase field model.

Mathematics Subject Classification: Primary: 82B26, 82C26, 82D15, 37L10, 35Q3.

Citation: Honghu Liu. Phase transitions of a phase field model. Discrete & 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. doi: 10.1063/1.1744102. Google Scholar
[2]	C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, in, (1989), 35. Google Scholar
[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. doi: 10.1142/S0218202596000341. Google Scholar
[4]	P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, Rev. Mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar
[5]	C. Hsia, Bifurcation of binary systems with the Onsager mobility,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3406383. Google Scholar
[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. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar
[7]	T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences,", to appear., (). Google Scholar
[8]	T. Ma and S. Wang, "Bifurcation Theory and Applications,", vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, (2005). Google Scholar
[9]	T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Disc. Cont. Dyn. Sys. B, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741. Google Scholar
[10]	T. Ma and S. Wang, Phase separation of binary systems,, Physica A, 388 (2009), 4811. doi: 10.1016/j.physa.2009.07.044. Google Scholar
[11]	A. Miranville, Some generalizations of the Cahn-Hilliard equation,, Asymptotic Anal., 22 (2000), 235. Google Scholar
[12]	A. Novich-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation,, Physica D, 10 (1984), 277. doi: 10.1016/0167-2789(84)90180-5. Google Scholar
[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. Google Scholar

show all references

References:

[1]	J. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar
[2]	C. M. Elliott, The Cahn-Hilliard model for the kinetics of phase separation,, in, (1989), 35. Google Scholar
[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. doi: 10.1142/S0218202596000341. Google Scholar
[4]	P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, Rev. Mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar
[5]	C. Hsia, Bifurcation of binary systems with the Onsager mobility,, J. Math. Phys., 51 (2010). doi: 10.1063/1.3406383. Google Scholar
[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. doi: 10.1016/S0167-2789(03)00030-7. Google Scholar
[7]	T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences,", to appear., (). Google Scholar
[8]	T. Ma and S. Wang, "Bifurcation Theory and Applications,", vol. 53 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, (2005). Google Scholar
[9]	T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Disc. Cont. Dyn. Sys. B, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741. Google Scholar
[10]	T. Ma and S. Wang, Phase separation of binary systems,, Physica A, 388 (2009), 4811. doi: 10.1016/j.physa.2009.07.044. Google Scholar
[11]	A. Miranville, Some generalizations of the Cahn-Hilliard equation,, Asymptotic Anal., 22 (2000), 235. Google Scholar
[12]	A. Novich-Cohen and L. A. Segel, Nonlinear aspects of the Cahn-Hilliard equation,, Physica D, 10 (1984), 277. doi: 10.1016/0167-2789(84)90180-5. Google Scholar
[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. Google Scholar

[1]	Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423
[2]	Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
[3]	Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[4]	Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002
[5]	Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019040
[6]	T. Tachim Medjo. A Cahn-Hilliard-Navier-Stokes model with delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2663-2685. doi: 10.3934/dcdsb.2016067
[7]	Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419
[8]	Ariane Piovezan Entringer, José Luiz Boldrini. A phase field $\alpha$-Navier-Stokes vesicle-fluid interaction model: Existence and uniqueness of solutions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 397-422. doi: 10.3934/dcdsb.2015.20.397
[9]	Qiang Du, Manlin Li, Chun Liu. Analysis of a phase field Navier-Stokes vesicle-fluid interaction model. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 539-556. doi: 10.3934/dcdsb.2007.8.539
[10]	Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
[11]	T. Tachim Medjo. Robust control of a Cahn-Hilliard-Navier-Stokes model. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2075-2101. doi: 10.3934/cpaa.2016028
[12]	Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
[13]	Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[14]	Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[15]	Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[16]	Shixing Li, Dongming Yan. On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3077-3088. doi: 10.3934/dcdsb.2018301
[17]	Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132
[18]	Bo You. Global attractor of the Cahn-Hilliard-Navier-Stokes system with moving contact lines. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2283-2298. doi: 10.3934/cpaa.2019103
[19]	T. Tachim Medjo. The exponential behavior of a stochastic Cahn-Hilliard-Navier-Stokes model with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1117-1138. doi: 10.3934/cpaa.2019054
[20]	M. Hassan Farshbaf-Shaker, Harald Garcke. Thermodynamically consistent higher order phase field Navier-Stokes models with applications to biomembranes. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 371-389. doi: 10.3934/dcdss.2011.4.371

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences