Energy stable and mass conservative numerical method for a generalized hydrodynamic phase-field model with different densities

Abstract

A general hydrodynamic phase-field model for two-phase flows with general average velocity and variable densities is rigorously derived using thermodynamic laws and Onsager's variational principle. The pressure is naturally involved in both the Cahn-Hilliard equation and Navier-Stokes equation. The proposed model includes two famous phase-field models as its special cases. The model admits a natural energy dissipation law. A semi-implicit, totally linear, and energy stable numerical scheme is proposed for the model, which uses an intermediate velocity involving all driving forces, including surface tension, pressure, and gravity. The tight coupling relationship between pressure and velocity is decoupled. Another advantage of the proposed scheme is that the intermediate velocity allows us to preserve the mass conservation, and consequently, there is no need to impose any mass balance equation in the Navier-Stokes equation as usual. The discrete energy dissipation law is proved rigorously. Several numerical examples are simulated to demonstrate that the proposed method can preserve the energy stability and total mass conservation for complex hydrodynamical flow problems with large density contrasts and gravity.