Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-21T10:51:18.877Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  28 May 2015

J. Brannick ,
C. Liu ,
T. Qian  and
H. Sun
J. Brannick*
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
C. Liu
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
T. Qian
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
H. Sun
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA
*
*Email addresses: brannick@psu.edu (J. Brannick), liu@math.psu.edu (C. Liu), maqian@ust.hk (T. Qian), sun@math.psu.edu (H. Sun)
Get access

Abstract

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.

Keywords

65F1065N2265N55multiphase flowenergetic variational approach
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 2 , May 2015 , pp. 220 - 236
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[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):6647–6663, 2011.CrossRefGoogle Scholar
[2]Brezzi, F. and Fortin, M.Mixed and Hybrid Finite Element Methods. Number 15 in Computational Mathematics. Springer–Verlag, 1991.CrossRefGoogle Scholar
[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.Google Scholar
[4]Cahn, J.W. and Hillard, J.E.Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys., 28:258–267, 1958.CrossRefGoogle Scholar
[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):1 – 10, 2013.CrossRefGoogle Scholar
[6]Doi, M. and Edwards, S.F.The Theory of Polymer Dynamics. Oxford Science Publication, 1986.Google Scholar
[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, 4:537 – 548, 2005.CrossRefGoogle Scholar
[8]Du, Q., Liu, C., Ryham, R., and Wang, X.The phase field formulation of the willmore problem. Nonlinearity, 18:1249 – 1267, 2005.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(2):450–468, 2004.CrossRefGoogle Scholar
[10]Duez, C., Ybert, C., Clanet, C., and Bocquet, L.Making a splash with water repellency. Nature Physics, 3(180), 2007.CrossRefGoogle Scholar
[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., 31:709–730, 1994.CrossRefGoogle Scholar
[12]Lowengrub, J.S.Kim, J.S.Phase field modeling and simulation of three-phase flows. Interfaces Free Bound, 7:435–466, 2005.Google Scholar
[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, 19:1593–1619, 1983.CrossRefGoogle Scholar
[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):4779–4788, 2007.CrossRefGoogle Scholar
[15]Lamb, H.Hydrodynamics. Cambridge, 6th edition, 1932.Google Scholar
[16]Lei, Z., Liu, C., and Zhou, Y.Global solutions for incompressible viscoelastic fluids. Arch. Rational Mech. Anal., 188, 2008.CrossRefGoogle Scholar
[17]Lin, F.H., Liu, C., and Zhang, P.On a micro-macro model for polymeric fluids near equilibrium. Comm. Pure Appl. Math., 59:1–29, 2005.Google Scholar
[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, 179:211–228, 2003.CrossRefGoogle Scholar
[19]Qian, T., Wang, X.P., and Sheng, P.Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E, 68(016306), 2003.CrossRefGoogle ScholarPubMed
[20]Qian, T., Wang, X.P., and Sheng, P.A variational approach to the moving contact line hydrodynamics. J. Fluid Mech., 564:333 – 360, 2006.CrossRefGoogle Scholar
[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):345–357, 1983.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[23]Sun, H. and Liu, C.On energetic variational approaches in modeling the nematic liquid crystal flows. DCDS-A, 23:455–475, 2008.CrossRefGoogle Scholar
[24]Turek, S.Efficient solvers for incompressible flow problems : an algorithmic and computational approach. Springer, 1999.CrossRefGoogle Scholar
[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, 515:293–317, 2005.CrossRefGoogle Scholar