Effects of symmetry and hydrodynamics on the cohesion of groups of swimmers

When groups of inertial swimmers move together, hydrodynamic interactions play a key role in shaping their collective dynamics, including the cohesion of the group. To explore how these interactions influence group cohesion, we develop a three-dimensional, inviscid, far-field model of a swimmer. Focusing on symmetric triangular, diamond, and circular group arrangements, we investigate whether passive hydrodynamics alone can promote cohesive behavior, and what role symmetry of the group plays. While small symmetric (and even asymmetric) groups can be cohesive, larger groups typically are not, instead breaking apart into smaller, self-organized subgroups that are cohesive. Notably, we discover circular arrangements of swimmers that chase each other around a circle, resembling the milling behavior of natural fish schools; we call this hydrodynamic milling. Hydrodynamic milling is cohesive in the sense that it is a fixed point of a particular Poincar\'e map, but it is unstable, especially to asymmetric perturbations. Our findings suggest that while passive hydrodynamics alone cannot sustain large-scale cohesion indefinitely, controlling interactions between subgroups, or controlling the behavior of only the periphery of a large group, could potentially enable stable collective behavior with minimal active input.