Dipole Interactions in Doubly Periodic Domains

We consider the interactions of finite dipoles in a doubly periodic domain. A finite dipole is a pair of equal and opposite strength point vortices separated by a finite distance. The dynamics of multiple finite dipoles in an unbounded inviscid fluid was first proposed by Tchieu, Kanso, and Newton in Tchieu et al. (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 468(2146):3006-3026, 2012) as a model that captures the far-field hydrodynamic interactions in fish schools. In this paper, we formulate the equations of motion governing the dynamics of finite dipoles in a doubly periodic domain. We show that a single dipole in a doubly periodic domain exhibits periodic and aperiodic behavior, in contrast to a single dipole in an unbounded domain. In the case of two dipoles in a doubly periodic domain, we identify a number of interesting trajectories including collision, collision avoidance, and passive synchronization of the dipoles. We then examine two types of dipole lattices: rectangular and diamond. We verify that these lattices are in a state of relative equilibrium and show that the rectangular lattice is unstable, while the diamond lattice is linearly stable for a range of perturbations. We conclude by commenting on the insight these models provide in the context of fish schooling.