Inertial torque on a squirmer

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 ${\mathbf{T}}^{\prime }=-\frac{9}{8}{m}_f({\mathbf{v}}_s^{(0)}\wedge {\mathbf{v}}_g^{(0)})$ where ${\mathbf{v}}_s^{(0)}$ is the swimming velocity, ${\mathbf{v}}_g^{(0)}$ 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.

10 pages, 3 figures

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