Planar Rotational Equilibria of Two Nonidentical Microswimmers

Abstract

We derive a closed analytical form of planar rotational equilibria that exist in the three-dimensional motion of two hydrodynamically coupled nonidentical microswimmers, each modeled as a force dipole with intrinsic self-propulsion. Using the method of images for zero Reynolds number flows near interfaces, we demonstrate that our results remain equally applicable at a stress-free liquid–gas interface as in the bulk of a fluid. For a pair of two pullers and a pair of two pushers the linear stability of the equilibria is analyzed 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 nonidentical pushers or two nonidentical pullers moving at a stress-free interface may form a stable rotational equilibrium. For two nonidentical 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 the 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.