Motion planning and motility maps for flagellar microswimmers

TL;DR

This paper explores how non-sinusoidal periodic actuation influences the motion of flagellar microswimmers, revealing mechanisms for directional control and curved trajectories through asymptotic analysis and motility maps.

Contribution

It extends previous sinusoidal actuation models to generic periodic inputs, demonstrating new control capabilities for microswimmer trajectories using velocity modulation.

Findings

Velocity modulation enables lateral translation.

Internal actuation can produce curved paths.

Asymptotic formulas match numerical simulations.

Abstract

We study two microswimmers consisting of a spherical rigid head and a passive elastic tail. In the first one the tail is clamped to the head, and the system oscillates under the action of an external torque. In the second one, head and tail are connected by a joint allowing the angle between them to vary periodically, as a result of an oscillating internal torque. Previous studies on these models were restricted to sinusoidal actuations, showing that the swimmers can propel while moving on average along a straight line, in the direction given by the symmetry axis around which beating takes place. We extend these results to motions produced by generic (non-sinusoidal) periodic actuations within the regime of small compliance of the tail. We find that modulation in the velocity of actuation can provide a mechanism to select different directions of motion. With velocity modulated inputs the externally actuated swimmer can translate laterally with respect to the symmetry axis of beating, while the internally actuated one is able to move along curved trajectories. The governing equations are analysed with an asymptotic perturbation scheme, providing explicit formulas whose results are expressed through motility maps. Asymptotic approximations are further validated by numerical simulations.