American Institute of Mathematical Sciences

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

Phase transitions of a phase field model

Honghu Liu 1,

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.
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]

Quan Wang, Dongming Yan. On the stability and transition of the Cahn-Hilliard/Allen-Cahn system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020024
[3]

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
[4]

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
[5]

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
[6]

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
[7]

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
[8]

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
[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[19]

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
[20]

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

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences