Dipole interactions in doubly-periodic domains

TL;DR

This paper formulates and analyzes the dynamics of finite dipoles in doubly-periodic domains, revealing complex behaviors such as periodicity, collision, and stability of dipole lattices, with implications for understanding fish schooling.

Contribution

It introduces the equations of motion for finite dipoles in doubly-periodic domains and studies their dynamic behaviors and stability properties, extending previous models to periodic settings.

Findings

Single dipole exhibits periodic and aperiodic behaviors in doubly-periodic domains.

Two dipoles can collide, avoid collision, or synchronize passively.

Rectangular lattice is unstable, diamond lattice is linearly stable.

Abstract

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 uid was first proposed by Tchieu, Kanso & Newton in [1] 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 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 insights these models provide in the context of fish schooling.