Diffuse-Interface Two-Phase Flow Models with Different Densities: A New Quasi-Incompressible Form and a Linear Energy-Stable Method

摘要

While various phase-field models have recently appeared for two-phase fluidswith different densities, only some are known to be thermodynamicallyconsistent, and practical stable schemes for their numerical simulation arelacking. In this paper, we derive a new form of thermodynamically-consistentquasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for atwo-phase flow of incompressible fluids with different densities. Thederivation is based on mixture theory by invoking the second law ofthermodynamics and Coleman-Noll procedure. We also demonstrate that our modeland some of the existing models are equivalent and we provide a unificationbetween them. In addition, we develop a linear and energy-stabletime-integration scheme for the derived model. Such a linearly-implicit schemeis nontrivial, because it has to suitably deal with all nonlinear terms, inparticular those involving the density. Our proposed scheme is the first linearmethod for quasi-incompressible two-phase flows with nonsolenoidal velocitythat satisfies discrete energy dissipation independent of the time-step size,provided that the mixture density remains positive. The scheme also preservesmass. Numerical experiments verify the suitability of the scheme for two-phaseflow applications with high density ratios using large time steps byconsidering the coalescence and break-up dynamics of droplets includingpinching due to gravity.