On self-propulsion of $N$-sphere micro-robot

The aim of this paper is to describe the self-propulsion of a micro-robot (or micro-swimmer) consisting of $N$ spheres moving along a fixed line. The spheres are linked to each other by arms with the lengths changing periodically. For the derivation, we use the asymptotic procedure containing the two-timing method and a distinguished limit. We show that in the main approximation, the self-propulsion velocity appears as a linear combination of velocities of all possible triplets of spheres. Velocities and efficiencies of three-, four-, and five-swimmers are calculated. The paper is devoted to H.K.Moffatt, who initiated the author's interests in low-Reynolds-number fluid dynamics.