Collective surfing of two self-propelled swimmers at liquid-air interface aided by self-induced Marangoni flow

We report a closed analytical form of two types of linearly unstable rotational equilibria for a planar motion of two hydrodynamically coupled, inertia-free, pusher swimmers in the bulk of a fluid and at a planar, stress-free, fluid interface. Both types correspond to a periodic motion of pushers along circular trajectories with a constant angular velocity in such a way that the distance between the pushers and their relative orientation remain constant. The first orbit type represents the motion along a common circle, when each pusher makes a 35.2 degrees angle with their relative position vector. In the second type of orbiting, the pushers move along circles of different radii while the orientation vectors of the pushers make 20.90 degrees and 110.90 degrees angles with their relative position vector. The first orbit type is monotonically unstable and the second orbit type is oscillatorily unstable. Next we show that both types of equilibria can be stabilized by self-induced Marangoni flow, generated by two pushers bound to move along a planar liquid-air interface. Neglecting inertia, we couple the motion of pushers with the advection-diffusion equation for the concentration of an insoluble surfactant, produced by the swimmers The surfactant is assumed to homogeneously decompose at a constant rate. Numerical simulations in the regime of nonzero Peclet number reveal the existence of stable periodic orbits that are directly linked to the unstable equilibria found analytically in the absence of the Marangoni flow.