Phase field modeling and simulation of three-phase flow on solid surfaces

Phase field models are widely used to describe the two-phase system. The evolution of the phase field variables is usually driven by the gradient flow of a total free energy functional. The generalization of the approach to an N phase (N≥3) system requires some extra consistency conditions on the free energy functional in order for the model to give physically relevant results. A projection approach is proposed for the derivation of a consistent free energy functional for the three-phase Cahn-Hilliard equations. The system is then coupled with the Navier-Stokes equations to describe the three-phase flow on solid surfaces with moving contact line. An energy stable scheme is developed for the three-phase flow system. The discrete energy law of the numerical scheme is proved which ensures the stability of the scheme. We also show some numerical results for the dynamics of triple junctions and four phase contact lines.