Time integration for diffuse interface models for two-phase flow

We propose a variant of the $\theta$-scheme for diffuse interface models for two-phase flow, together with three new linearization techniques for the surface tension. These involve either additional stabilizing force terms, or a fully implicit coupling of the Navier-Stokes and Cahn-Hilliard equation. In the common case that the equations for interface and flow are coupled explicitly, we find a time step restriction which is very different to other two-phase flow models and in particular is independent of the grid size. We also show that the proposed stabilization techniques can lift this time step restriction. Even more pronounced is the performance of the proposed fully implicit scheme which is stable for arbitrarily large time steps. We demonstrate in a Taylor flow application that this superior coupling between flow and interface equation can render diffuse interface models even computationally cheaper and faster than sharp interface models

Diffuse-interface two-phase flow models with different densities: a new quasi-incompressible form and a linear energy-stable method

While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable schemes for their numerical simulation are lacking. In this paper, we derive a new form of thermodynamically-consistent quasi-incompressible diffuse-interface Navier–Stokes–Cahn–Hilliard model for a two-phase flow of incompressible fluids with different densities. The derivation is based on mixture theory by invoking the second law of thermodynamics and Coleman–Noll procedure. We also demonstrate that our model and some of the existing models are equivalent and we provide a unification between them. In addition, we develop a linear and energy-stable time-integration scheme for the derived model. Such a linearly-implicit scheme is nontrivial, because it has to suitably deal with all nonlinear terms, in particular those involving the density. Our proposed scheme is the first linear method for quasi-incompressible two-phase flows with non-solenoidal velocity that satisfies discrete energy dissipation independent of the time-step size, provided that the mixture density remains positive. The scheme also preserves mass. Numerical experiments verify the suitability of the scheme for two-phase flow applications with high density ratios using large time steps by considering the coalescence and breakup dynamics of droplets including pinching due to gravity

Thermodynamically Consistent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is also generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the two latter cases the classical Gibbs-Thomson equation has to be modified to include kinetic terms. Finally, we show that all sharp interface models fulfill natural energy inequalities.Comment: 34 page