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$\boldsymbol {T}^\prime = -{\frac {9}{8}} m_f (\boldsymbol {v}_s^{(0)} \wedge \boldsymbol {v}_g^{(0)})$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.