A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions

¹ Université Toulouse 3 – Paul Sabatier, CNRS, Institut de Mathématiques de Toulouse, UMR 5129, 31062 Toulouse, France
franck.boyer@math.univ-toulouse.fr
² CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France
flore.nabet@polytechnique.edu
³ Team RAPSODI, Inria Lille – Nord Europe, 40 av. Halley, 59650 Villeneuve d’Ascq, France

Received: 31 December 2015
Revised: 3 August 2016
Accepted: 12 November 2016

Abstract

In this paper we propose a “Discrete Duality Finite Volume” method (DDFV for short) for the diffuse interface modelling of incompressible two-phase flows. This numerical method is, conservative, robust and is able to handle general geometries and meshes. The model we study couples the Cahn−Hilliard equation and the unsteady Stokes equation and is endowed with particular nonlinear boundary conditions called dynamic boundary conditions. To implement the scheme for this model we have to derive new discrete consistent DDFV operators that allows an energy stable coupling between both discrete equations. We are thus able to obtain the existence of a family of solutions satisfying a suitable energy inequality, even in the case where a first order time-splitting method between the two subsystems is used. We illustrate various properties of such a model with some numerical results.