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

TL;DR

This paper introduces a thermodynamically consistent quasi-incompressible diffuse-interface model for two-phase flows with different densities and develops a novel linear energy-stable numerical scheme that ensures stability and mass conservation.

Contribution

It presents a new thermodynamically consistent model and a linear energy-stable scheme for simulating two-phase flows with density differences, enabling larger time steps and improved stability.

Findings

The scheme is the first linear method satisfying discrete energy dissipation for such flows.

Numerical experiments demonstrate stability with high density ratios and large time steps.

The model unifies existing approaches and is verified through droplet dynamics simulations.

Abstract

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 nonsolenoidal 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 break-up dynamics of droplets including pinching due to gravity.