A phase-field method for two-phase fluid flow in arbitrary domains

We present a practical and efficient numerical method for two-phase flow simulations in arbitrary domains. A simple mathematical equation for the two-phase fluid in arbitrary domains is used. The proposed method is based on the ternary Cahn–Hilliard (CH) system coupled with the modified Navier–Stokes (NS) equation. A complex domain is expressed by the third phase in the ternary system. When we solve the ternary CH system, we only solve the equation for one phase with the fixed third phase. The second phase value is obtained from the local mass conservation. A no-slip condition was imposed for the velocity on the arbitrary domain boundaries. The proposed method uses an unconditionally gradient stable type scheme (Y. Li et al., 2016) for the convective CH equation and a projection method for the modified NS equation. Various numerical experiments, such as the droplet movement in a cavity flow, the Rayleigh–Taylor instability in a porous media, droplet passing through a triangle-shaped solid structure, and droplet movement in a circuitous channel, were performed to demonstrate that the proposed method can deal with the two-phase flow in arbitrary domains.