Geodesic
A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions
Boyer, Franck1  ; Nabet, Flore2, 3 
1 Université Toulouse 3 – Paul Sabatier, CNRS, Institut de Mathématiques de Toulouse, UMR 5129, 31062 Toulouse, France
2 CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France
3 Team RAPSODI, Inria Lille – Nord Europe, 40 av. Halley, 59650 Villeneuve d’Ascq, France
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1691-1731

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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.

DOI : 10.1051/m2an/2016073
Classification : 35K55, 65M08, 65M12, 76D07, 76M12, 76T10
Keywords: Cahn–Hilliard/Stokes model, dynamic boundary conditions, contact angle dynamics, finite volume method

Boyer, Franck 1 ; Nabet, Flore 2, 3

1 Université Toulouse 3 – Paul Sabatier, CNRS, Institut de Mathématiques de Toulouse, UMR 5129, 31062 Toulouse, France
2 CMAP, Ecole polytechnique, CNRS, Université Paris-Saclay, 91128, Palaiseau, France
3 Team RAPSODI, Inria Lille – Nord Europe, 40 av. Halley, 59650 Villeneuve d’Ascq, France 
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     title = {A {DDFV} method for a {Cahn\ensuremath{-}Hilliard/Stokes} phase field model with dynamic boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1691--1731},
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Boyer, Franck; Nabet, Flore. A DDFV method for a Cahn−Hilliard/Stokes phase field model with dynamic boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1691-1731. doi: 10.1051/m2an/2016073

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