A modified phase field method for the simulation of two-phase system in complex geometries

In this work, we propose a new numerical approximation method for the simulation of two-phase system in complex geometries. In this method, a novel formulation of the free energy is established according to the ternary phase field model and the model is derived by minimizing the total free energy of the system. By this method, a fixed phase field variable is employed to represent the profile of complex geometries and the bulk region of the two-phase system is extended to a regular domain that includes this phase. The contact angle boundary condition is imposed into the coefficient in the bulk of the model, which is determined implicitly by the surface tension coefficient of the system. We develop an unconditionally energy stable numerical scheme for the new phase field model. Moreover, we couple the phase field model with the incompressible Navier–Stokes equations to simulate the dynamic behavior of two-phase flows in complex geometries. Some numerical experiments including the two-phase system on a flat or curved substrate, two-phase flows over an undulated channel, and bubbles in porous media are given to show the capacity of the new method.