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

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, Sl. We show that the first effect of fluid inertia on particle velocities for times much larger than the viscous time scales as rootSl 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.