Taylor's swimming sheet: Analysis and improvement of the perturbation series

TL;DR

This paper extends G.I. Taylor's analysis of a swimming sheet by formalizing the perturbation series to higher orders, improving convergence, and validating results against numerical methods for larger wave amplitudes.

Contribution

It systematizes Taylor's perturbation series for the swimming sheet, enabling high-order calculations and improved convergence for larger amplitudes.

Findings

Series diverges at order-one amplitudes

Transformed series show better convergence

Results agree with numerical boundary integral methods

Abstract

In G.I. Taylor's historic paper on swimming microorganisms, a two dimensional sheet was proposed as a model for flagellated cells passing traveling waves as a means of locomotion. Using a perturbation series, Taylor computed swimming speeds up to fourth order in amplitude. Here we systematize that expansion so that it can be carried out formally to arbitrarily high order. The resultant series diverges for an order one value of the wave amplitude, but may be transformed into series with much improved convergence properties and which yield results comparing favorably to those obtained numerically via a boundary integral method for moderate and large values of the wave amplitudes.