On the long-time persistence of hydrodynamic memory.

Abstract

The Basset-Boussinesq-Oseen (BBO) equation correctly describes the nonuniform motion of a spherical particle at a low Reynolds number. It contains an integral term with a singular kernel which accounts for the diffusion of vorticity around the particle throughout its entire history. However, if there are any departures in either rigidity or shape from a solid sphere, besides the integral force with a singular kernel, the Basset history force, we should add a second history force with a non-singular kernel, related to the shape or composition of the particle. In this work, we introduce a fractional generalized Basset-Boussinesq-Oseen equation which includes both history terms as fractional derivatives. Using the Laplace transform, an integral representation of the solution is obtained. For a driven single particle, the solution shows that memory effects persist indefinitely under rather general driving conditions.