Advected phase-field method for bounded solution of the Cahn–Hilliard Navier–Stokes equations

Phase-field methods based on the Cahn–Hilliard (CH) equation coupled to the incompressible Navier–Stokes equation are becoming increasingly popular for interface resolving numerical simulations of two-phase flows of immiscible fluids. One major limitation of this approach, however, is that the volume of each phase is not inherently preserved. This is associated with the phase-discriminating order parameter, which in the course of the simulation remains in general not within its initial physical bounds. This shortcoming relates to the fact that the CH equation with standard Ginzburg–Landau chemical potential has no volume-preserving stationary solution for interfaces with uniform (non-zero) curvature. In this paper, a curvature-dependent chemical potential is proposed which allows for bounded stationary solutions of the CH equation for drops/bubbles exhibiting uniform curvature. Numerical solutions of the coupled Cahn–Hilliard Navier–Stokes equations show that the proposed chemical potential significantly improves boundedness and phase volume conservation over the standard one.