Theoretical modeling of capillary surfer interactions on a vibrating fluid bath

TL;DR

This paper develops a theoretical model for capillary surfers on vibrating fluid baths, explaining their interactions, bound states, and stability, aligning well with experimental observations and advancing understanding of active matter systems.

Contribution

The paper introduces a novel analytical model for capillary surfers, capturing their dynamics and interactions, and predicts quantized stable states consistent with experiments.

Findings

Model accurately predicts bound states of surfers.

Bound states are quantized on the capillary wavelength.

Stable and unstable equilibrium branches identified.

Abstract

We present and analyze a theoretical model for the dynamics and interactions of "capillary surfers," which are millimetric objects that self-propel while floating at the interface of a vibrating fluid bath. In our companion paper [1], we reported the results of an experimental investigation of the surfer system, which showed that surfer pairs may lock into one of seven bound states, and that larger collectives of surfers self-organize into coherent flocking states. Our theoretical model for the surfers' positional and orientational dynamics approximates a surfer as a pair of vertically oscillating point sources of weakly viscous gravity-capillary waves. We derive an analytical solution for the associated interfacial deformation and thus the hydrodynamic force exerted by one surfer on another. Our model recovers the bound states found in experiments and exhibits good quantitative agreement with experimental data. Moreover, a linear stability analysis shows that the bound states are quantized on the capillary wavelength, with stable branches of equilibria separated by unstable ones. Generally, our work shows that self-propelling objects coupled by interfacial flows constitute a promising platform for studying active matter systems in which both inertial and viscous effects are relevant.