Vortex arrays as emergent collective phenomena for circle swimmers

Abstract

Collective properties of many rodlike circle swimmers are explored by computer simulations in two spatial dimensions. In the model considered, the center of mass of a single swimmer moves on a circle with radius $R$. Therefore, the model provides an interpolation between an interacting self-propelled-rod model for linear swimmers ($R\ensuremath{\rightarrow}\ensuremath{\infty}$) and that of interacting passive rotors ($R=0$). We map out the state diagram for various swimmer densities and radii $R$. For increasing density, the dilute state is followed by vortices consisting of single particles (singlet-vortex state), where neighboring particles are perpendicularly oriented, and vortices of swimmer pairs (doublet-vortex state). The vortices exhibit strong structural ordering on an array. At higher densities, a slowed rotor fluid with a significant degree of mutual rotation hindrance occurs. The single-particle vortex structure becomes unstable above a threshold in the circling radius $R$, while pair vortices are stable only for intermediate radii $R$. A simple theory is proposed to predict the topology of the state diagram. Our results are verifiable for bacterial and artificial rodlike circle swimmers.