Controllability of 3D low Reynolds number swimmers

¹ Both authors are with Institut Élie Cartan UMR 7502, Université de Lorraine, CNRS, INRIA, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France
jerome.loheac@univ-lorraine.fr
² INRIA Nancy Grand Est, Projet CORIDA, France
Authors both supported by ANR CISIFS Second author supported by ANR GAOS. alexandre.munnier@univ-lorraine.fr

Received: 27 February 2012
Revised: 3 December 2012

Abstract

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies on short time intervals, for a wide class of cost functionals.