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 ( 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 (-1)-dimensional space of joint angles, where can be arbitrarily large. For large , 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 (Purcell’s three-link swimmer) and 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 we recover the non-convex planar loops already known from previous studies. For we obtain non-planar loops, raising the question of characterizing the geometry of complex high-dimensional loops.