A three-phase hydrodynamic phase field model with stabilization of the Cahn-Hilliard equation

A numerical implementation of a three-phase hydrodynamic phase-field model is presented, which combines the Navier-Stokes equations for incompressible fluid and the Cahn-Hilliard equation. The phase field model is highly capable for modeling multiphase media, but the numerical implementation of the phase field method has a number of computational difficulties associated with the stability of numerical schemes and, as a consequence, the need to choose a small time step, which complicates the calculations. The paper proposes a method for stabilizing the numerical algorithm by adding a fictitious relaxation derivative of the chemical potential, which reduces the spectral radius of the iteration matrix for the Cahn-Hilliard equation and allows a significant increase in the time step without losing stability. The implementation uses semi-implicit discretization and the projection method to calculate the velocity and pressure. Verification calculations are performed in a square domain with solid walls for a model fluid consisting of three phases. In static tests for different values of surface tension coefficients, equilibrium configurations were obtained in accordance with the expected contact angles at the junction of the three phases. In dynamic tests, it was shown that the model correctly reproduces the fall of a drop through the boundary of a two-layer system, as well as the collision of two particles of different phases moving in the third phase.