Optimal Strokes for Low Reynolds Number Swimmers: An Example

Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers. This is the regime of interest for microorganisms and microor nano-robots. We focus in this paper on a simple yet representative example: the three-sphere swimmer of Najafi and Golestanian [16]. For this system, we show how to cast the problem of swimming in the language of control theory, prove global controllability (which implies that the three-sphere swimmer can indeed swim), and propose a numerical algorithm to compute optimal strokes (which turn out to be suitably defined sub–Riemannian geodesics). ∗Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay cedex, France, [email protected] †SISSA-International School for Advanced Studies, via Beirut 2-4, 34014 Trieste, Italy, [email protected] ‡Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay cedex, France, [email protected]