Many important physical processes occur at interfaces and they are often challenging to describe mathematically. In particular I am interested in the stability of viscous surface waves (see here for joint work with Ian Tice).
Ranging from local weather prediction to long-term climate models, the various scales of interest in geophysical flows require an assorted array of mathematical models. In collaboration with Leslie Smith and Sam Stechmann I have studied the role of moisture in atmospheric models. The incorporation of moisture means additional physical processes to consider as well as the introduction of phase boundaries (such as air-cloud interfaces). This leads to challenging problems involving free boundaries (see here) and a form of "nonlinear eigendecomposition" (see here).
Many everyday fluids, such as milk, blood, or liquid crystals (ubiquitous in the electronic displays that surround us) are so-called "complex fluids". This means that they are fluids in which microstructure is present (typically at the microscopic scale) which impacts the overall behaviour of the fluid. More precisely, in these examples, the microstructure corresponds to the fat molecules in milk, the hemoglobin in blood, and the constituting molecules of the liquid crystal themselves.
For a mathematician-friendly introduction to micropolar fluids (i.e. written in mathematical language rather than engineering parlance), see the first chapter of my doctoral thesis. In joint work with Ian Tice we studied the stability of rod-like and pancake-like microstructure (see here and here).