A Hybrid Level–set–cahn–hilliard Model for Two-phase Flow

We present a hybrid level–set–Cahn–Hilliard model for the simulation of two phase flow in micro structures. Our method combines accurate contact line dynamics in the Cahn–Hilliard equation with the computationally cheaper level set method in regions where not the full Cahn–Hilliard dynamics are necessary to describe the flow. This is realized by employing the Cahn–Hilliard model close to rigid boundaries, together with a level set method away from boundaries. We use a level set formulation with smoothed color function so that the interface representation in both regions is the same. The combined model is coupled to the incompressible Navier–Stokes equations, where the surface tension acts as a forcing term, calculated by the stress form in the Cahn–Hilliard model and a continuous surface tension representation for the level set part. A continuous function is used as a switch between the two models, retaining mass conservation of the overall method. The gains of the method are twofold — the equations to be solved in the interior are structurally easier than the Cahn–Hilliard equation and, more importantly, the level set model can accurately model two-phase flow already at coarser meshes. For oil expulsion by inflowing water we are able to obtain results of comparable quality when using less elements than with a Cahn–Hilliard model on the whole domain.