Jeffery's orbits and microswimmers in flows: A theoretical review

TL;DR

This paper provides a comprehensive theoretical review of Jeffery's equations and their extensions, explaining the orientation dynamics of microswimmers in flows, with insights into their stable, unstable, and chaotic behaviors.

Contribution

It offers a detailed theoretical foundation for Jeffery's orbits and extends the equations to more complex shapes and flow conditions, enhancing understanding of microswimmer dynamics.

Findings

Jeffery's equations model microswimmer orientation in flows.

Stable, unstable, and chaotic microswimmer behaviors are characterized.

Extensions to complex shapes and deformation are discussed.

Abstract

In this review, we provide a theoretical introduction to Jeffery's equations for the orientation dynamics of an axisymmetric object in a flow at low Reynolds number, and review recent theoretical extensions and applications to the motions of self-propelled particles, so-called microswimmers, in external flows. Bacteria colonize human organs and medical devices even with flowing fluid, microalgae occasionally cause huge harmful toxic blooms in lakes and oceans, and recent artificial microrobots can migrate in flows generated in well-designed microfluidic chambers. The Jeffery equations, a simple set of ordinary differential equation, provide a useful building block in modeling, analyzing, and understanding these microswimmer dynamics in a flow current, in particular when incorporating the impact of the swimmer shape since the equations contain a shape parameter as a single scalar, known as the Bretherton parameter. The particle orientation forms a closed orbit when situated in a simple shear, and this non-uniform periodic rotational motion, referred to as Jeffery's orbits, is due to a constant of motion in the non-linear equation. After providing a theoretical introduction to microswimmer hydrodynamics and a derivation of the Jeffery equations, we discuss possible extensions to more general shapes, including those with rapid deformation. In the latter part of this review, simple mathematical models of microswimmers in different types of flow fields are described, with a focus on constants of motion and their relation to periodic motions in phase space, together with a breakdown of degenerate orbits, to discuss the stable, unstable, and chaotic dynamics of the system. The discussion in this paper will provide a comprehensive theoretical foundation for Jeffery's orbits and will be useful to understand the motions of microswimmers under various flows.