A study of unsteady forces at low Reynolds number: a strong interaction theory for the coaxial settling of three or more spheres

Abstract

Unsteady multiparticle creeping motions are complicated by the appearance of Basset, virtual mass and acceleration forces and by the difficulty of calculating fluid-particle interactions for three or more closely spaced particles. The present theoretical and experimental investigation explores the importance of each of these complicating features by examining in detail the gravitational-hydrodynamical interaction between three or more spheres falling along a common axis. The strong interaction theory developed to describe this motion accurately satisfies the viscous boundary conditions along the surface of each sphere and includes all the unsteady force terms in the equations of motion for the spheres. The experimental measurements for the three-sphere chain are in excellent agreement with theoretical predictions provided the Basset force is retained in the dynamic force balance. These results indicate, in general, that the Basset force is the most important unsteady force in gravitational flows at low Reynolds numbers in which the flow configuration is slowly changing due to fluid-particle interactions. The unsteady theory for small but finite Reynolds numbers shows that the departures in particle spacings, due to the integrated effect of the Basset force, from those predicted by quasi-steady zero Reynolds number theory grow as $ for large times and are of the order of the particle dimensions if the duration of the interaction is of 0(Re«1a/C/t). Here Rero is based on the terminal settling velocity U t and radius a of the sphere. This condition is satisfied in most sedimentation problems of interest. Virtual mass and particle acceleration forces on the other hand, are of negligible importance except during a short-lived initial transient period. An intriguing new feature of the three-sphere motion for large times was discovered. One finds that there is a critical initial spacing criterion which determines whether the two leading spheres in the chain will asymptotically approach a zero or a finite fluid gap as time goes to infinity. Numerical solutions for longer chains show that there is a tendency for the leading third of the chain to break up into doublets and triplets whereas the spheres in the latter third of the chain tend to space out separately.