Generalised Jeffery's equations for rapidly spinning particles. Part 1: Spheroids

TL;DR

This paper develops a multiscale framework to derive effective equations of motion for rapidly spinning active spheroids in shear flow, extending Jeffery's classical theory to include active spinning effects.

Contribution

It introduces a systematic derivation of emergent dynamics for active spheroids with rapid spinning, broadening Jeffery's equations to active, spinning particles.

Findings

Derived explicit closed-form expressions for effective shape of active spheroids.

Linked emergent dynamics to classical Jeffery's orbits.

Validated results with numerical examples.

Abstract

The observed behaviour of passive objects in simple flows can be surprisingly intricate, and is complicated further by object activity. Inspired by the motility of bacterial swimmers, in this two-part study we examine the three-dimensional motion of rigid active particles in shear Stokes flow, focusing on bodies that induce rapid rotation as part of their activity. Here, in Part 1, we develop a multiscale framework to investigate these emergent dynamics and apply it to simple spheroidal objects. In Part 2 (arXiv:2301.11032), we apply our framework to understand the emergent dynamics of more complex shapes; helicoidal objects with chirality. Via a multiple-scales asymptotic analysis for nonlinear systems, we systematically derive emergent equations of motion for long-term trajectories that explicitly account for the strong (leading-order) effects of fast spinning. Supported by numerical examples, we constructively link these effective dynamics to the well-known Jeffery's orbits for passive spheroids, deriving an explicit closed-form expression for the effective shape of the active particle, broadening the scope of Jeffery's seminal study to spinning spheroids.