This is an area of great technological importance in many fields, including

Such processes are modeled by nonlinear (systems of) partial
differential equations.
The characteristic feature is the presense of unknown boundaries
(separating the phases) which evolve in time, whence they belong
to the class of ** moving boundary problems**.
The prototype of such problems is the

The objective is to predict the evolution of the phases and of the strongly and nonlinearly coupled temperature, concentration and/or velocity fields.

The numerical simulations are computationally intensive and parallel computation (using message passing) is often necessary.

My favored approach is to formulate the problems globally as systems of conservation laws valid in a mathematically weak sense throughout the material (irrespectively of phase), together with appropriate equations of state expressing the thermochemistry of the phases. With such formulations, it becomes possible to numerically simulate the entire process effectively.

For more information, see the book

** Mathematical Modeling of Melting and Freezing Processes**

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