Phase-field models for free-boundary problems

Phase-field models are very attractive in view of their numerical simplicity. With only a few lines of code, one can model complex physical situations such as dendritic growth. From this point of view, they constitute very interesting tools for teaching purposes at graduate level. The main difficulty with these models is in their formulation, which incorporates the physical ingredients in a subtle way. We discuss these approaches on the basis of two examples: dendritic growth and multiphase flows. (Some figures in this article are in colour only in the electronic version)