An unconditionally stable uncoupled scheme for a triphasic Cahn–Hilliard/Navier–Stokes model

We propose an original scheme for the time discretization of a triphasic Cahn–Hilliard/Navier–Stokes model. This scheme allows an uncoupled resolution of the discrete Cahn–Hilliard and Navier‐Stokes system, which is unconditionally stable and preserves, at the discrete level, the main properties of the continuous model. The existence of discrete solutions is proved, and a convergence study is performed in the case where the densities of the three phases are the same. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq. 2013

[1]  A. Chorin Numerical solution of the Navier-Stokes equations , 1968 .

[2]  K. Deimling Nonlinear functional analysis , 1985 .

[3]  J. Kan A second-order accurate pressure correction scheme for viscous incompressible flow , 1986 .

[4]  J. Simon Compact sets in the spaceLp(O,T; B) , 1986 .

[5]  Elaine S. Oran,et al.  Surface tension and viscosity with Lagrangian hydrodynamics on a triangular mesh. Memorandum report , 1988 .

[6]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[7]  Harald Garcke,et al.  Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix , 1997 .

[8]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[9]  D. Jacqmin Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[10]  Jean-Marie Seiler,et al.  MATERIAL EFFECTS ON MULTIPHASE PHENOMENA IN LATE PHASES OF SEVERE ACCIDENTS OF NUCLEAR REACTORS , 2000 .

[11]  David J. Torres,et al.  The point-set method: front-tracking without connectivity , 2000 .

[12]  L. Quartapelle,et al.  A projection FEM for variable density incompressible flows , 2000 .

[13]  Franck Boyer,et al.  A theoretical and numerical model for the study of incompressible mixture flows , 2002 .

[14]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[15]  Junseok Kim,et al.  CONSERVATIVE MULTIGRID METHODS FOR TERNARY CAHN-HILLIARD SYSTEMS ∗ , 2004 .

[16]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[17]  Pierre Fabrie,et al.  Eléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles , 2006 .

[18]  Franck Boyer,et al.  Study of a three component Cahn-Hilliard flow model , 2006 .

[19]  Xiaobing Feng,et al.  Fully Discrete Finite Element Approximations of the Navier-Stokes-Cahn-Hilliard Diffuse Interface Model for Two-Phase Fluid Flows , 2006, SIAM J. Numer. Anal..

[20]  Chun Liu,et al.  Convergence of Numerical Approximations of the Incompressible Navier-Stokes Equations with Variable Density and Viscosity , 2007, SIAM J. Numer. Anal..

[21]  David Kay,et al.  Finite element approximation of a Cahn−Hilliard−Navier−Stokes system , 2008 .

[22]  Marcus Herrmann,et al.  A balanced force refined level set grid method for two-phase flows on unstructured flow solver grids , 2008, J. Comput. Phys..

[23]  Franck Boyer,et al.  A local adaptive refinement method with multigrid preconditionning illustrated by multiphase flows simulations. , 2009 .

[24]  S. Zaleski,et al.  Numerical simulation of droplets, bubbles and waves: state of the art , 2009 .

[25]  Sébastian Minjeaud,et al.  Cahn–Hilliard/Navier–Stokes Model for the Simulation of Three-Phase Flows , 2010 .

[26]  Franck Boyer,et al.  Numerical schemes for a three component Cahn-Hilliard model , 2011 .