Mean field control of thin film droplet dynamics

Interfacial instabilities in volatile liquid thin films on a hydrophobic substrate can lead to complex droplet dynamics such as droplet merging, splitting, and transport. We study a class of mean field control formulations for these droplet dynamics. An optimal control problem is designed by formulating the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. As an example, we consider a thin volatile liquid film laden with an active suspension, with control achieved through its activity field. Numerical examples, including droplet transport, bead-up/spreading, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism. This is joint work with Guosheng Fu (Notre Dame), Will Pazner (PSU), and Wuchen Li (USC).