Purcell's swimmers in pairs.

We investigate the effects of hydrodynamic interactions between microorganisms swimming at low Reynolds numbers, treating them as a control system. We employ Lie brackets analysis to examine the motion of two neighboring three-link swimmers interacting through the ambient fluid in which they propel themselves. Our analysis reveals that the hydrodynamic interaction has a dual consequence: on one hand, it diminishes the system's efficiency; on the other hand, it dictates that the two microswimmers must synchronize their motions to attain peak performance. Our findings are further corroborated by numerical simulations of the governing equations of motion.

The N -Link Swimmer in Three Dimensions: Controllability and Optimality Results.

The controllability of a fully three-dimensional N -link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal 2-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the 2-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all eight linearly independent directions in the combined configuration and shape space, leading to controllability; the swimmer can move from any starting configuration and shape to any target configuration and shape by operating on the two shape variables. The result is subsequently extended to the N -link swimmer. Finally, the minimal time optimal control problem and the minimization of the power expended are addressed and a qualitative description of the optimal strategies is provided.

Optimal design of Purcell's three-link swimmer.

In this paper we address the question of the optimal design for the Purcell three-link swimmer. More precisely, we investigate the best link length ratio which maximizes its displacement. The dynamics of the swimmer is expressed as an ordinary differential equation, 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%.