Section: New Results

Energy-optimal strokes for multi-link micro-swimmers

Participants : Laetitia Giraldi, François Alouges [École Polytechnique] , Antonio Desimone [SISSA Trieste, Italy] , Yshar Or [Technion, Haifa, Israel] , Oren Wiezel [Technion, Haifa, Israel] .

In a common work that is presented in [33], submitted to New Journal of Physics, we consider a slender planar multi-link micro-swimmer ($N$ links, see Section 7.10), where the time derivatives of the angles defining the shape are taken as controls, and we are mostly interested in small-amplitude undulations about its straight configuration.

Based only on the leading order dynamics in that vicinity, the optimal stroke to achieve a given prescribed displacement in a given time period is then obtained as the largest eigenvalue solution of a constrained optimal control problem. Remarkably, the optimal stroke is an ellipse lying within a two-dimensional plane in the ($N$-1)-dimensional space of joint angles, where $N$ can be arbitrarily large. For large $N$, the optimal stroke is a traveling wave of bending, modulo edge effects.

We also solved, numerically, the fully non-linear optimal control problem for the cases $N=3$ (Purcell’s three-link swimmer) and $N=5$ showing that, as the prescribed displacement becomes small, the optimal solutions obtained using the small-amplitude assumption are recovered. We also show that, when the prescribed displacements become large, the picture is different. For $N=3$ we recover the non-convex planar loops already known from previous studies. For $N=5$ we obtain non-planar loops, raising the question of characterizing the geometry of complex high-dimensional loops.