Discontinuous Galerkin finite element method applied to the coupled Navier-Stokes/Cahn-Hilliard equations

Franck Pigeonneau (1) , Pierre Saramito (2)

1 SVI - Surface du Verre et Interfaces
2 EDP - Equations aux Dérivées Partielles

Two-phase flows driven by the interfacial dynamics is studied with a phase-field model to tract implicitly interfaces. The phase field obeys the Cahn-Hilliard equation. The fluid dynamics is described with the Stokes equations with an additional source term in the momentum equation taking into account the capillary forces. A discontinuous Galerkin finite element method is used to solve the coupled Stokes/Cahn-Hilliard equations. The Cahn-Hilliard equation is treated as a system of two coupled equations corresponding to the advection-diffusion equation for the phase field and a non-linear elliptic equation for the chemical potential. First, the variational formulation of the Cahn-Hilliard equation is presented. A numerical test is achieved showing the optimal-order in error bounds. Second, the variational formulation in discontinuous Galerkin finite element approach of the Stokes equations is recalled in which the same space of approximation is used for the velocity and the pressure with an adequate stabilization technique. Finally, numerical simulations describing the capillary rising in a tube is presented.

Mots clés: two-phase flows; Cahn-Hilliard equation; phase-field theory; discontinuous Galerkin finite element

Domaines: Equations aux dérivées partielles [math.AP]; Mécanique des fluides [physics.class-ph]

Franck Pigeonneau, Pierre Saramito. Discontinuous Galerkin finite element method applied to the coupled Navier-Stokes/Cahn-Hilliard equations. ICMF-2016 – 9th International Conference on Multiphase Flow, May 2016, Florence, Italy. ⟨hal-01323551⟩