Planar Rotational Equilibria of Two Non-identical Microswimmers

TL;DR

This paper analyzes the planar motion and stability of two hydrodynamically coupled, non-identical micro-swimmers modeled as force dipoles, revealing conditions for stable rotational equilibria and quasi-periodic states at interfaces.

Contribution

It provides analytical solutions for circular orbits and stability analysis of non-identical microswimmers, including effects at stress-free interfaces, which advances understanding of their collective dynamics.

Findings

Stable rotational equilibria can occur for non-identical pushers or pullers at interfaces.

Stable quasi-periodic localized states are observed for pullers, arising from bifurcations.

All stable two-dimensional equilibria are unstable to three-dimensional perturbations.

Abstract

We study a planar motion of two hydrodynamically coupled non-identical micro-swimmers, each modelled as a force dipole with intrinsic self-propulsion. Using the method of images, we demonstrate that our results remain equally applicable at a stress-free liquid-gas interface as in the bulk of a fluid. A closed analytical form of circular periodic orbits for a pair of two pullers and a pair of two pushers is presented and their linear stability is determined with respect to two- and three-dimensional perturbations. A universal stability diagram of the orbits with respect to two-dimensional perturbations is constructed and it is shown that two non-identical pushers or two non-identical pullers moving at a stress-free interface may form a stable rotational equilibrium. For two non-identical pullers, we find stable quasi-periodic localized states, associated with the motion on a two-dimensional torus in the phase space. Stable tori are born from circular periodic orbits as the result of a torus bifurcation. All stable equilibria in two dimensions are shown to be monotonically unstable with respect to three-dimensional perturbations.