Here is a list of research questions I worked on: (most recent first, largely)

What kinds of simplification in the dynamics can result when a physical process causes one or more physical quantities to become large, possible arbitrarily large, relative to other quantities present? Examples I have thought about are the surface-tension driven breakup of a liquid drop, the dewetting of a thin liquid film. One of the most common simplification is that the dynamics becomes scale-invariant, and forgets all depdence of initial and boundary conditions. As a consequence, breakup dynamics at different times prior to breakup can be collapsed onto a single dynamics via an appropriate dynamic rescaling of the coordinate axes. Recently, via a combination of experiments (Itai Cohen & Sidney R. Nagel), simulation (Pankaj Doshi & Osman Basaran Purdue) and theory (with Peter Howell & Mike Siegel), we showed that scale-invariat dynamics is NOT THE ONLY POSSIBLE DYNAMICS NEAR BREAKUP. The breakup of a water drop in viscous oil proceeds via a very different process because the interior of the water drop is essentially static. During breakup, the neck of the drop simply collapes uniformly inward, with a complete imprint of initial conditions. Here is a Science article on this exceptional breakup dynamics.

What physical mechanism is responsible for the surprising rheology of dilute polymer-clay solutions? These shake-gels solidify into a "gel" when severely shaken. When left at rest, the solidified "gel" relaxes back into a clear liquid, which flows when gently tilted.

What happens to the dynamics of fluid flow, mixing and the transport of solutes, drops and bubbles when we scale pipes down to microfluidic channels?