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Phase transitions of a phase field model

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  • 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.
    Mathematics Subject Classification: Primary: 82B26, 82C26, 82D15, 37L10, 35Q35.


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  • [1]

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    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.


    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.


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    C. Hsia, Bifurcation of binary systems with the Onsager mobility, J. Math. Phys., 51 (2010), 063305.doi: 10.1063/1.3406383.


    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.


    T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences," to appear.


    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.


    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.


    T. Ma and S. Wang, Phase separation of binary systems, Physica A, 388 (2009), 4811-4817.doi: 10.1016/j.physa.2009.07.044.


    A. Miranville, Some generalizations of the Cahn-Hilliard equation, Asymptotic Anal., 22 (2000), 235-259.


    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.


    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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