A consistent and conservative Phase-Field method for multiphase incompressible flows

A consistent and conservative Phase-Field method, including both the model scheme, is developed for multiphase flows with an arbitrary number of immiscible incompressible fluid phases. The consistency mass conservation momentum transport are implemented to address issue physically coupling equation, which locates different phases, hydrodynamics. These two conditions provide ``optimal'' because (i) new equation resulting from them Galilean invariant implies kinetic energy conservation, regardless details (ii) failures satisfying second law thermodynamics or reduction flow only result same but not due equation. Physical interpretation their formulations first provided, general that obtained independent velocity summarized. Several novel techniques inherit physical properties after discretization, gradient-based phase selection procedure, method surface force, theorems preserve on discrete level. Equipped those techniques, a scheme present analyzed. Numerical applications demonstrate robust effective in studying complicated dynamics, especially large-density ratios.