On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers

The existing model equations governing the accelerated motion of a spherical particle are examined and their predictions compared with the results of the numerical solution of the full Navier–Stokes equations for unsteady, axisymmetric flow around a freely moving sphere injected into an initially stationary or oscillating fluid. The comparison for the particle Reynolds number in the range of 2 to 150 and the particle to fluid density ratio in the range of 5 to 200 indicates that the existing equations deviate considerably from the Navier–Stokes equations. As a result, we propose a new equation for the particle motion and demonstrate its superiority to the existing equations over a range of Reynolds numbers (from 2 to 150) and particle to fluid density ratios (from 5 to 200). The history terms in the new equation account for the effects of large relative acceleration or deceleration of the particle and the initial relative velocity between the fluid and the particle. We also examine the temporal structure of the near wake of the unsteady, axisymmetric flow around a freely moving sphere injected into an initially stagnant fluid. As the sphere decelerates, the recirculation eddy size grows monotonically even though the instantaneous Reynolds number of the sphere decreases.