The Optimal Swimming Sheet

TL;DR

This paper numerically determines the optimal waveform for a two-dimensional swimming sheet, achieving higher efficiency than traditional sine waves, with implications for understanding microscopic propulsion mechanisms.

Contribution

It introduces a numerically derived, large-amplitude optimal waveform for the swimming sheet that surpasses previous models in efficiency and differs from classical shapes.

Findings

Optimal waveform is a symmetric regularized cusp.

Achieves 25% higher efficiency than sine-wave models.

Optimal shape differs from 3D flagella and small-amplitude predictions.

Abstract

Propulsion at microscopic scales is often achieved through propagating traveling waves along hair-like organelles called flagella. Taylor's two-dimensional swimming sheet model is frequently used to provide insight into problems of flagellar propulsion. We derive numerically the large-amplitude waveform of the two-dimensional swimming sheet that yields optimum hydrodynamic efficiency; the ratio of the squared swimming speed to the rate-of-working of the sheet against the fluid. Using the boundary element method, we show the optimal waveform is a front-back symmetric regularized cusp that is 25% more efficient than the optimal sine-wave. This optimal two-dimensional shape is smooth, qualitatively different from the kinked form of Lighthill's optimal three-dimensional flagellum, not predicted by small-amplitude theory, and different from the smooth circular-arc-like shape of active elastic filaments.