Simulations of problems with free boundaries and several phases still present great challenges. This is both because of the need to model the various physical phenomena that are important, and to devise efficient and accurate computational methods. One example of a generic free boundary problem is the growth of a crystal in a melt. Depending on the properties of the crystal, different, often very complicated, patterns and shapes will appear. A seemingly distant example is wetting and capillarity in liquids. In both these cases however, the interplay between heat and mass transfer around the free boundaries, and the surface energy, will determine the outcome of the process. A third example is thermocapillarity, where a fluid motion is driven by a gradient in surface tension along a free surface. One growing area where this type of phenomena are important is the micro fluid mechanics of microscopic devices for analysis and synthesis in chemistry and biological applications.

Diffuse interface methods, or often phase field methods, have been gaining popularity over recent years. In these methods, the interface is represented by introducing a field, the phase field, which typically has one constant value in each phase, and the interface is represented by the layers of rapid transition between these values. In many cases the model equations can be conveniently derived from basic thermodynamical considerations. Solidification, phase change, surface tension and wetting, etc, can be treated this way.

In this talk I will give a brief overview of various problems that I have been working on in recent years, such as surface tension driven thermocapillary flow, which has relevance for welding and crystal growth; fluid motion driven by wetting; dielectrophoretic separation; phase change problems and crystallization. Some effort will be spent on describing the treatment of free boundaries using diffuse interface methods.