Within the realm of fluid mechanics, I have worked on a range of problems. In one project, I developed a model for a wave instability beneath levitating millimetric drops, as was observed experimentally in the Tagawa Lab in Tokyo; at the millimetre scale, surface tension forces are important. In later work, I investigated electromagnetic control of fluid flows in Hele-Shaw cells, where a simple and elegant mathematical description is possible. Using such forcing, I recently generalized the class of traditionally realizable Hele-Shaw flows – see the opening pages of Van Dyke’s famous “An Album of Fluid Motion”– both theoretically and experimentally. Notably, these circulatory potential flows extend the existing analogy between Hele-Shaw flows and electrostatic problems. In an ongoing projects, I am working with Dr. Keaton Burns to develop an efficient numerical scheme for treating advection-diffusion problems; our method leverages some interesting complex variables transformations which date back to Boussinesq.

I have also applied complex analysis tools in the context of heat transfer. For example, I proved a theorem which qualified a recent conjecture in conductive heat transder, in collaboration with Professor John Lienhard of the Department of Mechanical Engineering, MIT. We continue to collaborate on a related class of problems. Note also that the advection-diffusion work mentioned under the header fluid mechanics, in collaboration with Dr. Keaton Burns, applies to heat transfer (by choosing Temperature, T, to be the passive scalar of interest).

During my time in Paris, I became interested in floating sea-ice problems after learning about experiments being performed in the lab groups of Stéphane Perrard and Antonin Eddi at ESPCI in Paris. I became interested in their lab-scale experiments, inspired by their recent expedition in Northern Canada. I worked on theoretical modeling to successfully collapse data from their lab-scale experiments.

Probably stemming from my background in Physics, I generally enjoy exploring and developing analogies between physical systems, especially when the analogy can be made precise mathematically. From the Hele-Shaw analogy which connects a class of Laplace problems (from electrostatics to potential flow), to the well-known analogy between Stokes flow and elasticity, analogies are extremely powerful as they often allow one to experimentally probe a given theory in an ostensibly unrelated (and easier to handle) physical system. In particular, my work on electromagnetic Hele-Shaw cells extends the analogy between potential flows and electrostatic systems. My PhD advisor, John Bush, is particularly interested in the relatively new analogy between walking droplets atop a vibrating bath and single-particle quantum dynamics (Hydrodynamic Quantum Analogues). In this exciting new field, the development of a formal mathematical analogy is still in progress; however, many promising connections have been demonstrated to date (in my view, the most interesting connection so far is the quantum corral analogy). In current and future work, I hope to contribute to the mathematical development of the Hydrodynamic-Quantum analogy.

Details coming soon 🙂