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]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[2]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467
[3]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[4]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275
[5]

Tian Ma, Shouhong Wang. Topological phase transition III: Solar surface eruptions and sunspots. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020350
[6]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349
[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[8]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[9]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451
[10]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443
[11]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018
[12]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051
[14]

Annegret Glitzky, Matthias Liero, Grigor Nika. Dimension reduction of thermistor models for large-area organic light-emitting diodes. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020460
[15]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166
[16]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076
[17]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321
[18]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[19]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273
[20]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences