HYDRODYNAMIC DIFFUSION OF A SPHERE SEDIMENTING THROUGH A DILUTE SUSPENSION OF NEUTRALLY BUOYANT SPHERES

The motion of a heavy sphere sedimenting through a dilute background s uspension of neutrally buoyant spheres is analysed for small Reynolds number and large Peclet number. For this particular problem, it is pos sible not only to calculate the mean velocity of the heavy particle, b ut also the variance of the velocity and the coefficient of hydrodynam ic diffusivity. Pairwise, hydrodynamic interactions between the heavy sphere and the background sphere are considered exactly using volume i ntegrals and a trajectory analysis. Explicit formulae are given for th e two limiting cases when the radius of the heavy sphere is much great er and much less than that of the background spheres, and numerical re sults are given for moderate size ratios. The mean velocity is relativ ely insensitive to the ratio of the radius of the background spheres t o that of the heavy sphere, unless this ratio is very large, whereas t he hydrodynamic diffusivity increases rapidly as the radius ratio is i ncreased. The predictions are in reasonable agreement with the results of falling-ball rheometry experiments.