A consistent phase-field model for three-phase flows with cylindrical/spherical interfaces

A phase-field model for three-phase flows with cylindrical/spherical interfaces is established by combining the Navier-Stokes (NS), the continuity, and the energy equations, with an explicit form of curvature-dependent modified Allen-Cahn (AC) and Cahn-Hilliard (CH) equations. These modified AC and CH equations are proposed to solve the inconsistency of the phase-field method between flat and curved interfaces, which can result in "phase-vanishing" problems and the break of mass conservation during the phase-changing process. It is proved that the proposed model satisfies the energy dissipation law (energy stability). Then the icing process with three phases, i.e., air, water, and ice, is simulated on the surface of a cylinder and a sphere, respectively. It is demonstrated that the modification of the AC and CH equations remedies the inconsistency between flat and curved interfaces and the corresponding "phase-vanishing" problem. The evolution of the curved water-air and the water-ice interfaces are captured simultaneously, and the volume expansion during the solidification owing to the density difference between water and ice agrees with the theoretical results. A two-dimensional icing case with bubbles rising is simulated. The movement and deformation of bubbles, as well as the evolution of the interfaces, effectively illustrate the complex interactions between different phases in the icing process with phase changes.