Optimizing Undulatory Swimming

Most organisms live in aqueous environments and propel themselves by swimming. A large subclass is micro-organisms that have slender rod-like shapes, e.g. sperm. These organisms propel themselves using undulations that follow certain waveforms depending on the type of desired motion. At these small lengths scales and slow velocities, water behaves as a viscous fluid: the Reynolds number is small and Resistive Force Theory is a good approximation. Recently RFT has been extended to non-traditional types of fluids, such as dense granular matter, in order to model sand-swimming of undulating animals and robots.

The optimal planar undulatory strategy for a swimming filament in a viscous fluid is the sawtooth waveform, which was identified by Lighthill. Although this result was intended for infinite-length filaments, it also is applicable to finite-length filaments where the number of undulations is large, $U \gtrsim 10$. However the sawtooth's sharp kinks limits the applicability of Lighthill's result in nature and engineering applications, and thus we consider planar waveforms which have constrained curvatures, $| {\mathcal C} | \le {\mathcal C}_\mathrm{max}$. This naturally leads to the dimensionless number, $N = {\mathcal C}_\mathrm{max} S/(2\pi)$, which we call the winding number. 

We find that a piece-wise constant curvature function is optimal, which we determine for a range of winding numbers. These results for viscous fluids also transfer to sand-swimming, albeit the optimal choice for parameters is different.