Suppose I have a block of ice resting on an insulated surface in a closed room. Suppose the room also contains air at a temperature above the ice's melting point. I want to simulate the transient behavior of the ice melting and flowing (due to gravity) onto the insulated surface over time. Particularly, I'm interested in tracking the evolution of the interfaces between each phase (solid, liquid, and gas) over time.
I'm sure that the heat equation and navier stokes equations will be involved, but I'm not sure how to adjust them to account for the existence of three phases (solid ice, liquid water, and air) and how to model the transition between phases (e.g. equation of state). What are the complete set of equations that I need to model this situation? More importantly, what is the best approach to handle the evolution of the interfaces over time?