Inertial torque on a squirmer

TL;DR

This paper demonstrates that active particles like squirmers experience inertial torques during settling, which influence their orientation relative to gravity, with stable upward swimming and unstable downward swimming.

Contribution

It reveals that inertial torques affect active particles' orientation during settling, extending understanding from passive spheroids to active swimmers.

Findings

Active particles experience inertial torque during settling.

The torque aligns the swimmer against gravity, favoring upward orientation.

Stable upward swimming and unstable downward swimming are demonstrated.

Abstract

A small spheroid settling in a quiescent fluid experiences an inertial torque that aligns it so that it settles with its broad side first. Here we show that an active particle experiences such a torque too, as it settles in a fluid at rest. For a spherical squirmer, the torque is $\boldsymbol{T}^\prime = -{\tfrac{9}{8}} m_f (\boldsymbol{v}_s^{(0)} \wedge \boldsymbol{v}_g^{(0)})$ where $\boldsymbol{v}_s^{(0)}$ is the swimming velocity, $\boldsymbol{v}_g^{(0)}$ is the settling velocity in the Stokes approximation, and $m_f$ is the equivalent fluid mass. This torque aligns the swimming direction against gravity: swimming up is stable, swimming down is unstable.