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

Franck Boyer (1) , Flore Nabet (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

In this paper we propose a "Discrete Duality Finite Volume" method (DDFV for short) for the diffuse interface modelling of incompressible 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 define 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.

Mots clés: finite volume method; contact angle dynamics; dynamic boundary conditions; Cahn-Hilliard/Stokes model

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Franck Boyer, Flore Nabet. A DDFV method for a Cahn-Hilliard/Stokes phase field model with dynamic boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis, 2017, 51 (5), pp.1691-1731 ⟨10.1051/m2an/2016073⟩. ⟨hal-01249262v2⟩