Modeling, analysis and simulation of capillary-scale active matter: from individual to collective dynamics

This project aims to understand the propulsion mechanism and interactions of vertically oscillating objects that float on a fluid interface. These newly discovered “surfers” have been shown to self-propel along a fluid bath, and to self-organize through their mutual wave field into moving clusters. However, the mechanisms by which they self-propel and interact have remained elusive. The insights gained from this research have the potential to impact the biology and engineering communities, as scientific attention has been given to the self-propulsion of both inanimate objects and living organisms at the liquid-gas interface. These observations have inspired the development of biomimetic robots that self-propel using similar mechanisms. Capillary-scale robots have many engineering applications including environmental monitoring, water remediation and cargo transport. The project will train undergraduate and graduate students in mathematical modeling and physical applied mathematics, giving them tools that can be broadly applied to problems arising in the natural sciences and engineering. From a mathematical perspective, the main challenge is that surfers generate interfacial waves, unlike so-called “dry” active matter systems whose constituents propel and self-organize solely through steric interactions. A consequence of the inertial wave-mediated coupling between constituents is that multiple interaction modes coexist for the same experimental parameters. To address this challenge, a three-pronged approach will be utilized to develop mathematical models capable of successfully describing the surfers’ dynamics. The first goal is to develop a new mathematical and numerical framework for modeling the waves and flows generated by a single surfer, and the dependence of both on the surfer’s geometry. The mathematical challenge is to prove the existence of and derive approximations to solutions to a linear elliptic boundary value problem with mixed boundary conditions. Doing so will uncover the self-propulsion mechanism of surfers. The second goal is to simulate and analyze the collective behavior of surfers. Continuum partial differential equations (PDE) models will be systematically derived and completed using closure conditions, which will be benchmarked against particle-based models. The third goal is to develop new data-driven PDE discovery techniques to learn the governing mean-field equations for collectives of surfers, directly from experimental video data provided by collaborators. These results will be compared with the models based on physical principles developed throughout the grant period. This program will pave the way for the adoption of machine learning tools in understanding interfacial active systems. Generally, the surfer system represents a versatile and accessible platform for the exploration of active matter, yet contains mathematical modeling and simulation challenges due to the effects of fluid inertia and waves. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.