On the unsteady forces during the motion of a sedimenting particle

Abstract

We consider the unsteady motion of a sedimenting rigid spherical particle in order to examine the relative strength of the hydrodynamical forces acting on particles in fluid flows. The relative strength of the forces on all stages of the particle motion is a major concern for closing constitutive equations describing the more complex motion of particulate flows such as fluidised beds. The formulation results in a first order nonlinear integro-differential equation in terms of the instantaneous velocity of the sphere. This equation is made dimensionless and the particle Reynolds number and the fluid–particle density ratio are identified as the relevant physical parameters describing the particle motion. We obtain analytical solutions for the limits of small density ratios and small Reynolds number. In addition, a numerical solution is used for arbitrary values of the density ratio. The results show that the motion of spherical particles is significantly affected by the unsteady drag dominated by the memory Basset force on the early stages of the motion and on the approach to the steady state (terminal velocity). The present calculations indicate that the unsteady hydrodynamic drags might become of the same order of magnitude of the dominant viscous drag for flows with moderate particle–fluid density ratio. Therefore, unsteady drags should be taken into account on modelling multiphase particulate flows with moderate density ratio.