Decoupled energy stable schemes for a phase field model of three-phase incompressible viscous fluid flow

Abstract

We develop a numerical approximation for a hydrodynamic phase-field model of three immiscible, incompressible viscous fluid phases. The model is derived from a generalized Onsager's principle following an energetic variational formulation and is consisted of the momentum transport equation and the coupled phase transport equations. It conserves the volume of each phase and warrants the total energy dissipation in time. Its numerical approximation is given by an easy-to-implement, semi-discrete scheme, leading to a set of linear, decoupled elliptic equations at each time step which can be solved efficiently using fast solvers. We prove that the scheme is energy stable. Mesh refinement tests and three numerical examples of three-phase viscous fluid flows in 3D are presented to benchmark the effectiveness of the model as well as the efficiency of the numerical scheme.