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

Franck Pigeonneau (1) , Elie Hachem (1) , Pierre Saramito (2)

1 CEMEF - Centre de Mise en Forme des Matériaux
2 EDP - Equations aux Dérivées Partielles

Two-phase flows driven by the interfacial dynamics are studied by tracking implicitly interfaces in the framework of the Cahn-Hilliard theory. The fluid dynamics is described by 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. The rates of convergence in space and time are evaluated leading to an optimal-order in error bounds in space and a second order in time with a backward differentiation formula at the second order. Numerical tests devoted to two-phase flows are provided on ellipsoidal droplet retraction, on the capillary rising of a liquid in a tube and on the wetting drop over a horizontal solid wall.

Mots clés: phase-field theory; discontinuous Galerkin; droplet retraction; finite element; two-phase flow; capillary rising

Domaines: Sciences de l'ingénieur [physics]; Milieux fluides et réactifs; Physique [physics]; Physique mathématique [math-ph]; Physique [physics]; Physique [physics]; Dynamique des Fluides [physics.flu-dyn]; Mathématiques [math]; Analyse numérique [math.NA]; Physique [physics]; Mécanique [physics]; Mécanique des fluides [physics.class-ph]

Franck Pigeonneau, Elie Hachem, Pierre Saramito. Discontinuous Galerkin finite element method applied to the coupled unsteady Stokes/Cahn-Hilliard equations. International Journal for Numerical Methods in Fluids, 2019, 90 (6), pp.267-295. ⟨10.1002/fld.4720⟩. ⟨hal-01773404v2⟩