# What system of pde's models ice melt flow?

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?

• I like the idea, but asking for "the complete set of equations" may be a tough one to answer! – hazzey Jan 5 '16 at 2:15
• @hazzey: Agreed. It is an active area of research, but i just need a good starting place to simulate upon which i can hopefully verify and validate. – Paul Jan 5 '16 at 3:20

## 2 Answers

If you have only one substance (e.g., only water/ice) and neglect the fluid motion, this is called a Stefan problem, and the basic equations are known. Wikipedia provides them in 1D, although they appear to be normalized (unit latent heat and conductivity, for example). Here's a paper with more general conditions.

For higher dimensions and including fluid motion, you'll still need the Navier-Stokes equations and thermal energy equation, along with some interface conditions that simply generalize those discussed for the 1D Stefan problem. If you want the solid phase to move as well, you should include the Cauchy equation. (Unfortunately, due to time constraints I am not able to generalize the interface conditions at this time, but I am sure this answer will help you get started.)

Tracking the interfaces is still a research problem. Common numerical methods include the volume of fluid method or the level set method. While I have taken a CFD course which covered the basics of these two methods, I am not really qualified to make a particular recommendation. I get the impression that which method you choose depends on what you value, e.g., some methods may conserve mass, while others might not but have other attractive properties.

You may want to consider analyzing this problem in two parts; first model the melting of the ice with no fluid flow (the Stefan problem that Ben Trettel identified), then model the fluid flow separately using the Navier-Stokes equations with the ice melting results as a boundary condition for the inlet flow of water. I'm assuming that you're only interested in the flow of water over the insulated surface, and not the flow down the sides of the ice cube.

The ice melting and the thin-film flow of the water are two specialized types of problems, and trying to capture both of them in the same analysis might be difficult. You'd have to choose a mesh structure, solution methods, time step, etc. that works for both problems. By separating into two analyses, you get to optimize the approach for each one.