Numerical Strategies for Stroke Optimization of Axisymmetric Microswimmers∗

We propose a computational method to solve optimal swimming problems, based on the boundary integral formulation of the hydrodynamic interaction between swimmer and surrounding fluid and direct constrained minimization of the energy consumed by the swimmer. We apply our method to axisymmetric model examples. We consider a classical model swimmer (the three-sphere swimmer of Golestanian et al.) as well as a novel axisymmetric swimmer inspired by the observation of biological micro-organisms. ∗Preprint SISSA 33-2009-M. †Corresponding Author. Email: Luca Heltai ; Tel.: +39 040-3787449; Fax: +39 040-3787528 1 ar X iv :0 90 6. 45 02 v2 [ m at h. N A ] 1 7 Ju l 2 00 9 1 Swimming at low Reynolds Numbers Swimming is the ability to advance within a fluid by performing cyclic shape changes, in the absence of external propulsive forces. One of the main difficulties of swimming at small scales is given by the time-reversal property of Stokes flows, which describe hydrodynamics at low Reynolds numbers. The physical implication of this mathematical property is that simple swimming strategies used in nature at larger scales, for example by scallops, where the same shape change is executed forward and backward at different velocities to achieve propulsion in one direction, do not work for micro-, or nano-swimmers. At this scale, swimmers have to undergo non-reciprocal deformations in order to achieve propulsion. Also known as the ScallopTheorem, this fact is discussed by Purcell in [1], where a simple mechanical device that can indeed swim at low Reynolds numbers, the three-link swimmer, is proposed. A mathematical statement of the scallop theorem and its proof can be found in [2]. More recently several other simple swimmers have been presented, for example, in Golestanian and Najafi [3] and Avron et al. [4]. A mathematical approach to the problem of finding an optimal stroke has been proposed by Alouges et al. [5], where it is shown how to formulate and solve numerically the problem of finding optimal strokes for low Reynolds number swimmers by focusing on the three-sphere swimmer of Najafi and Golestanian [3] (a simple, yet representative example). The analysis carried out in [5] shows how to address quantitatively swimming as the problem of controlling shape in order to produce a net displacement at the end of one stroke. By casting it in the language of control theory, the problem of swimming reduces to the controllability of the system, and the search of optimal strokes to an optimal control problem leading to the computation of suitable sub-Riemannian geodesics. The use of numerical algorithms to find optimal strokes can lead to dramatic improvements. For the three-sphere swimmer, one can achieve an increase of efficiency exceeding 300% with respect to more naive proposals [5]. The simplicity of three-sphere swimmer, which is a system with three degrees of freedom, enables one to carry out the analysis in full detail. The study of biologically relevant swimmers, however, requires more abstract mathematical tools and more efficient numerical algorithms. The aim of this paper is to introduce further tools to overcome some of the computational limits of the numerical method used in [5], opening up the possibility to treat more