Travelling waves are hydrodynamically optimal for long-wavelength flagella

TL;DR

This paper demonstrates that for long-wavelength flagella, travelling wave kinematics optimize propulsion efficiency, with waves traveling at constant speed and always propelling opposite to the wave direction.

Contribution

It introduces a variational framework showing that optimal flagellar motion for propulsion involves travelling waves with constant speed, extending understanding of microorganism locomotion.

Findings

Travelling waves maximize propulsive force for fixed energy dissipation.

Optimal waves travel with constant speed, possibly on curved paths.

Propulsion direction is opposite to wave travel.

Abstract

Swimming eukaryotic microorganisms such as spermatozoa, algae and ciliates self-propel in viscous fluids using travelling wave-like deformations of slender appendages called flagella. Waves are predominant because Purcell's scallop theorem precludes time-reversible kinematics for locomotion. Using the calculus of variations on a periodic long-wavelength model of flagellar swimming, we show that the planar flagellar kinematics maximising the time-averaged propulsive force for a fixed amount of energy dissipated in the surrounding fluid correspond for all times to waves travelling with constant speed, potentially on a curved centreline, with propulsion always in the direction opposite to the wave.