Dispersion of a ball settling through a quiescent neutrally buoyant suspension

Abstract

Individual falling balls were allowed to settle through otherwise quiescent well-mixed suspensions of non-colloidal neutrally buoyant spheres dispersed in a Newtonian liquid. Balls were tracked in three dimensions to determine the variances in their positions about a mean uniform vertical settling path. The primary experimental parameters investigated were the size of the falling ball and the volume fraction and size of the suspended particles. Unlike the horizontal variances, the vertical variances were found to be affected by short-time deterministic behaviour relating to the instantaneous local configurational arrangement of the suspended particles. For sufficiently long intervals between successive observations, the trajectories of the balls were observed to disperse about their mean settling paths in a random manner. This points to the existence of a Gaussian hydrodynamic dispersivity that characterizes the linear temporal growth of the variance in the position of a falling ball. The functional dependence of these horizontal and vertical dispersivities upon the parameters investigated was established.

The dispersivity dyadic was observed to be transversely isotropic with respect to the direction of gravity, with the vertical component at least 25 times larger than the horizontal component. The vertical dispersivity Dˆ_v (made dimensionless with the diameter of the suspended spheres and the mean settling velocity) was observed to decrease with increasing falling ball diameter, but to decrease less rapidly with concentration than theoretically predicted for very dilute suspensions; moreover, for falling balls equal in size to the suspended spheres, Dˆ_v increased linearly with increasing volume fraction ϕ of suspended solids.

In addition to the above experiments performed on suspensions of spheres, previously published settling-velocity data on the fall of balls through neutrally buoyant suspensions of rods possessing an aspect ratio of 20 were re-analysed, and vertical dispersivities calculated therefrom. (These data, taken by several of the present investigators in conjunction with other researchers, had only been grossly analysed in prior publications to extract the mean settling velocity of the ball, no attempt having been made at the time to extract dispersivity data too.) The resulting vertical dispersivities, when rendered dimensionless with the rod length and mean settling velocity, showed no statistically significant dependence upon the falling-ball diameter; moreover, all other things being equal, these dispersivities were observed to increase with increasing rod concentration.