A novel diffuse-interface model and a fully-discrete maximum-principle-preserving energy-stable method for two-phase flow with surface tension and non-matching densities

Two well-established classes of the interface capturing models are the level-set and phase-field models. Level-set formulations satisfy the maximum principle for the density but are not energy-stable. On the other hand, phase-field models do satisfy the second law of thermodynamics but lack the maximum principle for the density. In this paper we derive a novel model for incompressible immiscible two-phase flow with non-matching densities and surface tension that is both energetically-stable and satisfies the maximum principle for the density. The model finds its place at the intersection of level-set and phase-field models. Its derivation is based on a diffusification of the incompressible two-phase Navier–Stokes equations with non-matching densities and surface tension and involves functional entropy variables. Additionally, we present an associated fully-discrete energy-stable method. Isogeometric analysis is used for the spatial discretization and the temporal-integration is performed with a new time-integration scheme that is a perturbation of the second-order midpoint scheme. The fully-discrete scheme is unconditionally energy-dissipative, pointwise divergence-free and satisfies the maximum principle for the density. Numerical examples in two and three dimensions verify the energetic-stability of the methodology.