Interpreting the synchronisation of driven colloidal oscillators via the mean pair interaction

Abstract

To explore the generic features of synchronisation facilitated by hydrodynamic interactions dominated by viscous forces, we investigate small systems of rowers; rowers are a highly simplified model for motile cilia, with each cilium approximated by a rigid sphere driven through a geometrically updated force. We introduce a new framework to analyse rowers, in which we average the pair-wise interaction of rowers by converting to the natural phase. In doing so the function describing the interaction becomes continuous, dramatically simplifying the system and allowing standard dynamical system techniques to be applied. Through inspection of phase portraits, we capture the broad features of rowers driven by power law forces and quantify their coupling strength. This approach is not limited to power law potentials, and can be applied to any monotonically increasing function. When implemented in systems with more rowers, the phase portraits show more diverse behaviour. By exploring the phase space, small systems of rowers can be designed to demonstrate specific features.