The effects of rapid yawing on simple swimmer models and planar Jeffery's orbits

TL;DR

This paper investigates how rapid yawing oscillations influence the long-term motion of microswimmers, revealing biases in direction, changes in speed, and shape effects through asymptotic analysis and simulations.

Contribution

It provides a systematic asymptotic analysis of high-frequency yawing effects on microswimmer dynamics, including conditions for unbiased motion and shape modifications.

Findings

Rapid oscillations can bias the average swimming direction.

Yawing affects propulsion speed and hydrodynamic shape.

Theoretical results are validated by computational simulations.

Abstract

Over a sufficiently long period of time, or from an appropriate distance, the motion of many swimmers can appear smooth, with their trajectories appearing almost ballistic in nature and slowly varying in character. These long-time behaviours, however, often mask more complex dynamics, such as the side-to-side snakelike motion exhibited by spermatozoa as they swim, propelled by the frequent and periodic beating of their flagellum. Many models of motion neglect these effects in favour of smoother long-term behaviours, which are often of greater practical interest than the small-scale oscillatory motion. Whilst it may be tempting to ignore any yawing motion, simply assuming that any effects of rapid oscillations cancel out over a period, a precise quantification of the impacts of high-frequency yawing is lacking. In this study, we systematically evaluate the long-term effects of general high-frequency oscillations on translational and angular motion, cast in the context of microswimmers but applicable more generally. Via a multiple-scales asymptotic analysis, we show that rapid oscillations can cause a long-term bias in the average direction of progression. We identify sufficient conditions for an unbiased long-term effect of yawing, and we quantify how yawing modifies the speed of propulsion and the effective hydrodynamic shape when in shear flow. Furthermore, we investigate and justify the long-time validity of the derived leading-order solutions and, by direct computational simulation, we evidence the relevance of the presented results to a canonical microswimmer.