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Numerical Approximations for Allen-Cahn Type Phase Field Model of Two-Phase Incompressible Fluids with Moving Contact Lines

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical problems in dynamical systems

Published online by Cambridge University Press:  07 February 2017

Lina Ma ,
Rui Chen ,
Xiaofeng Yang  and
Hui Zhang
Lina Ma*
Affiliation:
Department of Mathematics, Penn State University, State College, PA 16802, USA
Rui Chen*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, P.R. China
Xiaofeng Yang*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA; Beijing Institute for Scientific and Engineering Computing, Beijing University of Technology, Beijing, 100124, P.R. China
Hui Zhang*
Affiliation:
Shool of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, P.R. China
*
*Corresponding author. Email addresses:linama@psu.edu (L. Ma), ruichenbnu@gmail.com (R. Chen), xfyang@math.sc.edu (X. Yang), hzhang@bnu.edu.cn (H. Zhang)
*Corresponding author. Email addresses:linama@psu.edu (L. Ma), ruichenbnu@gmail.com (R. Chen), xfyang@math.sc.edu (X. Yang), hzhang@bnu.edu.cn (H. Zhang)
*Corresponding author. Email addresses:linama@psu.edu (L. Ma), ruichenbnu@gmail.com (R. Chen), xfyang@math.sc.edu (X. Yang), hzhang@bnu.edu.cn (H. Zhang)
*Corresponding author. Email addresses:linama@psu.edu (L. Ma), ruichenbnu@gmail.com (R. Chen), xfyang@math.sc.edu (X. Yang), hzhang@bnu.edu.cn (H. Zhang)
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Abstract

In this paper, we present some efficient numerical schemes to solve a two-phase hydrodynamics coupled phase field model with moving contact line boundary conditions. The model is a nonlinear coupling system, which consists the Navier-Stokes equations with the general Navier Boundary conditions or degenerated Navier Boundary conditions, and the Allen-Cahn type phase field equations with dynamical contact line boundary condition or static contact line boundary condition. The proposed schemes are linear and unconditionally energy stable, where the energy stabilities are proved rigorously. Various numerical tests are performed to show the accuracy and efficiency thereafter.

Keywords

Phase-fieldtwo-phase flowNavier-Stokescontact linesstability

MSC classification

Secondary: 65M12: Stability and convergence of numerical methods 65M70: Spectral, collocation and related methods 65P99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 867 - 889
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Boyer, F. and Minjeaud, S.. Numerical schemes for a three component Cahn-Hilliard model. ESAIM. Mathematical Modelling and Numerical Analysis, 45(4):697–738, 2011.CrossRefGoogle Scholar
[2] Bray, A.J.. Theory of phase-ordering kinetics. Advances in Physics, 43(3):357–459, 1994.CrossRefGoogle Scholar
[3] Cahn, J.W.. Free energy of a nonuniform system. II. thermodynamic basis. Journal of Chemical Physics, 30(5), May 1959.CrossRefGoogle Scholar
[4] Cahn, J. W. and Hilliard, J. E.. Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys., 28:258–267, 1958.CrossRefGoogle Scholar
[5] Chen, L. Q. and Wang, Y.. The continuum field approach to modeling microstructural evolution. JOM, 48:13–18, 1996.CrossRefGoogle Scholar
[6] Chen, R., Ji, G., Yang, X., and Zhang, H.. Decoupled energy stable schemes for phase-field vesicle membrane model. Journal of Computational Physics, 302:509–523, 2015.CrossRefGoogle Scholar
[7] Condette, N., Melcher, C., and Suli, E.. Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth. Math. Comp., 80:205–223, 2011.CrossRefGoogle Scholar
[8] Du, Q., Liu, C., and Wang, X.. A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. Journal of Computational Physics, 198:450–468, 2004.CrossRefGoogle Scholar
[9] Du, Q., Liu, C., and Wang, X.. A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. Journal of Computational Physics, 198:450–468, 2004.CrossRefGoogle Scholar
[10] Gao, M. and Wang, X.. A gradient stable scheme for a phase field model for the moving contact line problem. Journal of Computational Physics, 231(4):1372–1386, February 2012.CrossRefGoogle Scholar
[11] Kessler, D., Nochetto, R. H., and Schmidt, A.. A posteriori error control for the Allen-Cahn problem: circumventing Gronwall's inequality. Mathematical Modeling and Numerical Analysis, 38:129–142, 2004.CrossRefGoogle Scholar
[12] Kim, J.. Phase-field models for multi-component fluid flows. Comm. Comput. Phys, 12(3):613–661, 2012.CrossRefGoogle Scholar
[13] 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):211–228, 2003.CrossRefGoogle Scholar
[14] Liu, C., Shen, J., and Yang, X.. Decoupled energy stable schemes for a phase field model of two phase incompressible flows with variable density. Journal of Scientific Computing, 62:601–622, 2015.CrossRefGoogle Scholar
[15] Lowengrub, J. S., Ratz, A., and Voigt, A.. Phase field modeling of the dynamics of multicomponent vesicles spinodal decomposition coarsening budding and fission. Physical Review E, 79(3), 2009.CrossRefGoogle ScholarPubMed
[16] Miehe, C., Hofacker, M., and Welschinger, F.. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Computer Methods in Applied Mechanics and Engineering, 199:2765–2778, 2010.CrossRefGoogle Scholar
[17] Minjeaud, S.. An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model. Commun. Comput. Phys., 29:584–618, 2013.Google Scholar
[18] Qian, T. Z., Wang, X., and Sheng, P.. Power-law slip profile of the moving contact line in two-phase immiscible flows. Physical Review Letters, 93:094501, 2004.CrossRefGoogle ScholarPubMed
[19] Qian, T-Z., Wang, X., and Sheng, P.. Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E, 68:016306, 2003.CrossRefGoogle ScholarPubMed
[20] Qian, T-Z., Wang, X., and Sheng, P.. Molecular hydrodynamics of the moving contact line in two phase immiscible flows. Comm. in Comput. Phys., 1(1):1–52, 2006.Google Scholar
[21] Qian, T-Z., Wang, X., and Sheng, P.. A variational approach to moving contact line hydrodynamics. Journal of Fluid Mechanics, 564:336–360, 2006.CrossRefGoogle Scholar
[22] Shen, J. and Yang, X.. Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. DCDS, Series A, 28:1169–1691, 2010.Google Scholar
[23] Shen, J. and Yang, X.. Decoupled energy stable schemes for phase field models of two phase incompressible flows. SIAM Journal of Numerical Analysis, 53(1):279–296, 2015.CrossRefGoogle Scholar
[24] Shen, J., Yang, X., and Yu, H.. Efficient energy stable numerical schemes for a phase field moving contact line model. Journal of Computational Physics, 284:617–630, 2015.CrossRefGoogle Scholar
[25] Spatschek, R., Brener, E., and Karma, A.. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits. Philosophical Magazine, 91:75–95, 2010.CrossRefGoogle Scholar
[26] Wang, X. and Du, Q.. Modelling and simulations of multi-component lipid membranes and open membranes via diffuse interface approaches. Journal of Mathematical Biology, 56:347–371, 2008.CrossRefGoogle ScholarPubMed