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In this paper we address the question of the optimal design for the Purcell 3-link swimmer. More precisely we investigate the best link length ratio which maximizes its displacement. The dynamics of the swimmer is expressed as an ODE, using the Resistive Force Theory. Among a set of optimal strategies of deformation (strokes), we provide an asymptotic estimate of the displacement for small deformations, from which we derive the optimal link ratio. Numerical simulations are in good agreement with this theoretical estimate, and also cover larger amplitudes of deformation. Compared with the classical design of the Purcell swimmer, we observe a gain in displacement of roughly 60%.

The study of self-propulsion at microscopic scale is attracting increasing attention in the recent literature both because of its intrinsic biological interest, and for the possible implications on the design of bio-inspired artificial replicas reproducing the functionalities of biological systems (see for instance [1–4]). At this scale, inertia forces are negligible compared to the viscous ones ie low Reynolds number, calling for different swimming strategies than at greater scales. Thus, we assume that the surrounding fluid is governed by Stokes equations which implies that hydrodynamic forces and torques are linear with respect to the swimmer’s velocity. In the case of planar flagellar propulsion, the Resistive Force Theory (RFT) provides a simple and concise way to compute a local approximation of hydrodynamic forces and Newton laws (see [5]). The resulting equations can be written as a system of linear ODEs (see [6–8]). In this paper we focus on one of the first example of micro …