Spherical particle motion in harmonic stokes flows

An analytical study is presented on the periodic motion of a small particle in a viscous fluid. Previous analytical and numerical contributions to the understanding of particle motion in time-dependent flows are reviewed. Previous works in this field addressed the long-term behavior of the particle (stationary solution) as opposed to the problem treated in this study, which includes initial transient effects. Also presented are the relative scaling of the virtual mass, Stokes, and history drag forces for the long-term solution following the fractional calculus theory. The general solution of the particle momentum equation for unsteady Stokes flows is particularized to a harmonic background fluid velocity. The effect of a nonzero initial relative velocity is studied, and a discussion on the relevance of the results to the Lagrangian simulation of turbulent multiphase flows is presented. The present theoretical results are relevant to the transient motion of small particles in viscous flows such as the ones found in crystal growth processes, fluid markers, and dilute turbulent multiphase flows in general.