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Chinese Annals of Mathematics, Series B

, Volume 31, Issue 5, pp 743–758 | Cite as

Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows

  • Jie Shen
  • Xiaofeng Yang
Article

Abstract

Numerical approximations of Cahn-Hilliard phase-field model for the two-phase incompressible flows are considered in this paper. Several efficient and energy stable time discretization schemes for the coupled nonlinear Cahn-Hilliard phase-field system for both the matched density case and the variable density case are constructed, and are shown to satisfy discrete energy laws which are analogous to the continuous energy laws.

Keywords

Phase-field Two-phase flow Navier-Stokes Cahn-Hilliard Energy stable 

2000 MR Subject Classification

65M12 65M70 65P99 65Z05 76T99 

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References

  1. [1]
    Allen, S. M. and Cahn, J. W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater., 27, 1979, 1085–1095.CrossRefGoogle Scholar
  2. [2]
    Anderson, D. M., McFadden, G. B. and Wheeler, A. A., Diffuse-interface methods in fluid mechanics, 30, 1998, 139–165.MathSciNetGoogle Scholar
  3. [3]
    Becker, R., Feng, X. and Prohl, A., Finite element approximations of the Ericksen-Leslie model for nematic liquid crystal flow, SIAM J. Numer. Anal., 46(4), 2008, 1704–1731.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Caffarelli, L. A. and Muler, N. E., An L bound for solutions of the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 133(2), 1995, 129–144.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Cahn, J. W. and Hilliard, J. E., Free energy of a nonuniform system, I: Interfacial free energy, J. Chem. Phys., 28, 1958, 258–267.CrossRefGoogle Scholar
  6. [6]
    Condette, N., Melcher, C. and Süli, E., Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth, Math. Comp., to appear.Google Scholar
  7. [7]
    Feng, X., He, Y. and Liu, C., Analysis of finite element approximations of a phase field model for two-phase fluids (electronic), Math. Comp., 76(258), 2007, 539–571.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Guermond, J. L., Minev, P. and Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., 195, 2006, 6011–6045.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Guermond, J. L. and Quartapelle, L., A projection FEM for variable density incompressible flows, J. Comput. Phys., 165(1), 2000, 167–188.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Guermond, J. L. and Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation, J. Comput. Phys., 228(8), 2009, 2834–2846.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Gurtin, M. E., Polignone, D. and Vinals, J., Two-phase binary fluids and immiscible fluids described by an order parameter, Math. Models Methods Appl. Sci., 6(6), 1996, 815–831.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Jacqmin, D., Diffuse interface model for incompressible two-phase flows with large density ratios, J. Comput. Phys., 155(1), 2007, 96–127.CrossRefMathSciNetGoogle Scholar
  13. [13]
    Kessler, D., Nochetto, R. H. and Schmidt, A., A posteriori error control for the Allen-Cahn problem: circumventing Gronwall’s inequality, M2AN Math. Model. Numer. Anal., 38(1), 2004, 129–142.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Lin, P., Liu, C. and Zhang, H., An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics, J. Comput. Phys., 227(2), 2007, 1411–1427.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Liu, C. and Shen, J., A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D, 179(3–4), 2003, 211–228.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Lowengrub, J. and Truskinovsky, L., Quasi-incompressible Cahn-Hilliard fluids and topological transitions, R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci., 454(1978), 1998, 2617–2654.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Nochetto, R. and Pyo, J. H., The gauge-Uzawa finite element method part I: the Navier-Stokes equations, SIAM J. Numer. Anal., 43, 2005, 1043–1068.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    Prohl, A., Projection and quasi-compressibility methods for solving the incompressible Navier-Stokes equations, Advances in Numerical Mathematics, BG Teubner, Stuttgart, 1997.zbMATHGoogle Scholar
  19. [19]
    Pyo, J. and Shen, J., Gauge-uzawa methods for incompressible flows with variable density, J. Comput. Phys., 221, 2007, 181–197.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Rannacher, R., On Chorin’s projection method for the incompressible Navier-Stokes equations, Lecture Notes in Mathematics, 1530, Springer-Verlag, Berlin, 1991.Google Scholar
  21. [21]
    Shen, J., Efficient spectral-Galerkin method I. direct solvers for second- and fourth-order equations by using Legendre polynomials, SIAM J. Sci. Comput., 15, 1994, 1489–1505.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    Shen, J., On error estimates of projection methods for the Navier-Stokes equations: second-order schemes, Math. Comp, 65, 1996, 1039–1065.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    Shen, J. and Yang, X., A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities, SIAM J. Sci. Comput., 32, 2010, 1159–1179.CrossRefGoogle Scholar
  24. [24]
    Shen, J. and Yang, X., Numerical approximations of allen-cahn and cahn-hilliard equations, Discrete and Continuous Dynamical Systems, Series A, 28, 2010, 1669–1691.CrossRefGoogle Scholar
  25. [25]
    Walkington, N. J., Compactness properties of the DG and CG time stepping schemes for parabolic equations, SIAM J. Numer. Anal., 47(6), 2010, 4680–4710.CrossRefMathSciNetGoogle Scholar
  26. [26]
    Yue, P., Feng, J. J., Liu, C., et al., A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech, 515, 2004, 293–317.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Editorial Office of CAM (Fudan University) and Springer Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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