The leading effect of fluid inertia on the motion of rigid bodies at low Reynolds number

Abstract

We investigate the influence of fluid inertia on the motion of a finite assemblage of solid spherical particles in slowly changing uniform flow at small Reynolds number, $Re$, and moderate Strouhal number, $\hbox{\it Sl}$. We show that the first effect of fluid inertia on particle velocities for times much larger than the viscous time scales as $\sqrt{\hbox{\it Sl\,Re}}$ given that the Stokeslet associated with the disturbance flow field changes with time. Our theory predicts that the correction to the particle motion from that predicted by the zero-$Re$ theory has the form of a Basset integral. As a particular example, we calculate the Basset integral for the case of two unequal particles approaching (receding) with a constant velocity along the line of their centres. On the other hand, when the Stokeslet strength is independent of time, the first effect of fluid inertia reduces to a higher order of magnitude and scales as $Re$. This condition is fulfilled, for example, in the classical problem of sedimentation of particles in a constant gravity field.