Unsteady motion of nearly-spherical particles in viscous fluids

An understanding of how non-spherical particles move in viscous fluids is critical to many applications. The motion of such particles is often studied using the unsteady Stokes equations. Zhang & Stone, J. Fluid Mech. 367, 329-358 (1998) reported an asymptotic theory for nearly-spherical particles, to first order in particle non-sphericity. Importantly, some key physical phenomena are absent at this order, including (i) the coupling between the torque experiences by the particle and its linear translation, (ii) the force it experiences and its rotation, and (iii) the effect of non-sphericity on the orientation-averages of these forces and torques. Here, we accommodate these phenomena through derivation of an asymptotic theory correct to second-order in particle non-sphericity, for the force and torque acting on the particle in a general unsteady Stokes flow. The derived analytical formulae apply to particles of arbitrary shape, giving the leading order theory for the above mentioned phenomena. Several example nearly-spherical particles are considered including a spheroid, a`pear-shaped' particle and a 'spiked' particle. We report independenent simulations of the Navier-Stokes equations that validate the theory.