Free-boundary problems in hydrodynamics arise in a number of situations in science and technology, and our work in fluid dynamics has focused on exploring these problems using a combination of asymptotic methods, numerical simulations, and experiments.
One theme is capillarity. We predicted the unusual statics and dynamics of non-wetting drops that roll rather than slide, and experiments confirmed the prediction, and have led to a growing literature on the subject, with recent work finally showing that our predictions for the scaling laws and the nature of the fluid velocity are qualitatively correct. In coating flows, two basic papers on the experimental and theoretical aspects of the flow and patterns inside a rotating cylinder, the analog of the Taylor-Couette problem for free surfaces. Other contributions include a simple criterion for the stable existence of 4-phase coexistence; a series of papers on capillary interactions at a fluid interface seen in such instances as the “Cheerios effect” in cereal bowls, the properties of scum – a particulate layer on a fluid interface – with theory and experiments for how it buckles and cracks. Most recently, we used many of these ideas to provide a simple scaling theory for some experimental observations of how arrays of fluid-immersed slender pillars come together as they are driven together by surface tension to form Medusa-like braidsconfine, and complemented this with a complete theoretical and computational study of capillary coalescence.
A second theme is the dynamics of thin liquid filaments, sheets, and shells – generalizing the Stokes-Rayleigh analogy between elastic solids and viscous fluids to free boundary problems by providing a simple physical and geometric way of deriving results in one field in terms of the other. This has allowed us to explain the fluid rope trick – how a stream of honey coils on toast, or how cake batter folds back and forth, the dynamics of viscous catenaries, the rippling of burst bubbles, and most recently, the wrinkling of liquid sheets – using theory, experiment and computation.We have also generalized these geometric theories to include the effects of complex rheology. These results are relevant for a range of issues in industrial processing and geological fluid mechanics, and we used this to understand aspects of the structure of island arcs on earth.
In a third theme, fluid-structure interaction, our work led to an explanation of flag flutter using physical arguments that were corroborated by a calculation on how energy is transferred to the flag from the fluid using a 1:1 resonance mechanism. Later, we proposed a general theory building on this idea that unifies an entire range of fluid-elastic instabilities in viscous and inviscid situations, as well as in confined and unconfined flows.
Interfaces and boundaries are distinguished by crises, chaos, and creativity. Instabilities are often nucleated at boundaries, as are new phases, and interfaces often have a Janus-like life of their own, unable to completely forget one or the other material that they separate! Unusual capillary phenomena (of which there seems to be no dearth, two centuries after they were first quantified by Th. Young and P.S. Laplace) have been of long-standing interest to us. We got started by thinking about flows that involve the deposition of thin films of liquid onto surfaces as a result of external forces associated with inertia, viscosity, gravity, for example in a horizontally rotating cylinder.
We have also studied the unusual statics and dynamics of non-wetting droplets, and uncovered a solution in biology for making perfect non-wetting droplets have been used by insects for more than 200 million years as a means of keeping themselves and their environment clean ! We continue to be interested in various applied aspects of non-wetting droplets in physiology and chemical physics. More recently, we have been looking at elastocapillarity, the physics of soft objects at fluid interfaces, revisiting the problem of capillary rise between soft sheets, the way in which particulate materials aggregate, break, and buckle at interfaces, and how filaments fold and self-pack themselves when placed at interfaces.