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Diffuse Interface Methods for Multiple Phase Materials: An Energetic Variational Approach

  • J. Brannick (a1), C. Liu (a1), T. Qian (a2) and H. Sun (a1)


In this paper, we introduce a diffuse interface model for describing the dynamics of mixtures involving multiple (two or more) phases. The coupled hydrodynamical system is derived through an energetic variational approach. The total energy of the system includes the kinetic energy and the mixing (interfacial) energies. The least action principle (or the principle of virtual work) is applied to derive the conservative part of the dynamics, with a focus on the reversible part of the stress tensor arising from the mixing energies. The dissipative part of the dynamics is then introduced through a dissipation function in the energy law, in line with Onsager's principle of maximum dissipation. The final system, formed by a set of coupled time-dependent partial differential equations, reflects a balance among various conservative and dissipative forces and governs the evolution of velocity and phase fields. To demonstrate the applicability of the proposed model, a few two-dimensional simulations have been carried out, including (1) the force balance at the three-phase contact line in equilibrium, (2) a rising bubble penetrating a fluid-fluid interface, and (3) a solid particle falling in a binary fluid. The effects of slip at solid surface have been examined in connection with contact line motion and a pinch-off phenomenon.


Corresponding author

*Email addresses: (J. Brannick), (C. Liu), (T. Qian), (H. Sun)


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[1]Adler, J.H., Brannick, James J., Liu, Chun, Manteuffel, Thomas A., and Zikatanov, Ludmil. First-order system least squares and the energetic variational approach for two-phase flow. J. Comput. Physics, 230(17):66476663, 2011.
[2]Brezzi, F. and Fortin, M.Mixed and Hybrid Finite Element Methods. Number 15 in Computational Mathematics. Springer–Verlag, 1991.
[3]Bronsard, L., Gui, C., and Schatzman, M.A three-layered minimizer in ℝ2 for a varational problem with a symmetric three-well potential. Comm. Pure and Applied Math., 49(673), 1996.
[4]Cahn, J.W. and Hillard, J.E.Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28:258267, 1958.
[5]Ceniceros, H.D. and García-Cervera, C.J.A new approach for the numerical solution of diffusion equations with variable and degenerate mobility. Journal of Computational Physics, 246(0):110, 2013.
[6]Doi, M. and Edwards, S.F.The Theory of Polymer Dynamics. Oxford Science Publication, 1986.
[7]Du, Q., Liu, C., Ryham, R., and Wang, X.Modeling the spontaneous curvature effects in static cell membrane deformations by a phase field formulation. Communications on Pure and Applied Analysis, 4:537548, 2005.
[8]Du, Q., Liu, C., Ryham, R., and Wang, X.The phase field formulation of the willmore problem. Nonlinearity, 18:12491267, 2005.
[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(2):450468, 2004.
[10]Duez, C., Ybert, C., Clanet, C., and Bocquet, L.Making a splash with water repellency. Nature Physics, 3(180), 2007.
[11]Huang, W., Ren, Y., and Russell, R.D.Moving mesh partial differential equations (mm-pdes) based on the equidistribution principle. SIAM J. Numer. Anal., 31:709730, 1994.
[12]Lowengrub, J.S.Kim, J.S.Phase field modeling and simulation of three-phase flows. Interfaces Free Bound, 7:435466, 2005.
[13]Kelly, D.W., Gago, J.P., Zienkiewicz, O.C., and Babuska, I.A posteriori error analysis and adaptive proces in the finite element method: part i – error analysis. International Journal for Numerical Methods in Engineering, 19:15931619, 1983.
[14]Kim, J.Phase field computations for ternary fluid flows phase field computations for ternary fluid flows. Computer Methods in Applied Mechanics and Engineering, 196(45):47794788, 2007.
[15]Lamb, H.Hydrodynamics. Cambridge, 6th edition, 1932.
[16]Lei, Z., Liu, C., and Zhou, Y.Global solutions for incompressible viscoelastic fluids. Arch. Rational Mech. Anal., 188, 2008.
[17]Lin, F.H., Liu, C., and Zhang, P.On a micro-macro model for polymeric fluids near equilibrium. Comm. Pure Appl. Math., 59:129, 2005.
[18]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, 179:211228, 2003.
[19]Qian, T., Wang, X.P., and Sheng, P.Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E, 68(016306), 2003.
[20]Qian, T., Wang, X.P., and Sheng, P.A variational approach to the moving contact line hydrodynamics. J. Fluid Mech., 564:333360, 2006.
[21]Eisenstat, S.S., Elman, H., and Schultz, M.Variational iterative methods for nonsymmetric systems of linear equations. SIAM J. of Num. Anal., 20(2):345357, 1983.
[22]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. Computing, 32(1159), 2010.
[23]Sun, H. and Liu, C.On energetic variational approaches in modeling the nematic liquid crystal flows. DCDS-A, 23:455475, 2008.
[24]Turek, S.Efficient solvers for incompressible flow problems : an algorithmic and computational approach. Springer, 1999.
[25]Yue, P., Feng, J., Liu, C., and Shen, J.A diffuse-interface method for simulating two-phase flows of complex fluids. Journal of Fluid Mechanics, 515:293317, 2005.



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