The role of the Marangoni effect in the formation of rotational equilibria in the system of hydrodynamically coupled micro-swimmers.

We study the motion of hydrodynamically coupled identical and non-identical micro-swimmers, in the bulk of a viscous fluid and at a stress-free liquid-air interface. Each swimmer is modeled in the far-field as a pusher or a puller with an intrinsic self-propulsion. Surfactant released by living bacteria such as Bacillus subtilis, Flavobacteria, Pseudomonas Aeruginosa, etc., alters the surface tension of the liquid-air interface thus generating the Marangoni flow in the surrounding fluid. Two identical pushers moving along a liquid-air interface may form an unstable rotational equilibrium orbit, whereby each pusher follows a circular path. The presence of the Marangoni flow is shown to stabilize the orbit, leading to the formation of three different types of rotational equilibria. For a pair of non-identical pushers or pullers in the absence of the Marangoni flow, we found circular and quasi-periodic localized states associated with the motion on a 2D torus. All planar equilibria are shown to be unstable with respect to three-dimensional perturbations. Finally, in the case of four identical pushers in the absence of the Marangoni flow, we observe a rich dynamical behavior, including the formation of two circular orbits with a slow drift and subsequent destruction.