The distribution of velocities of a concentric sphere in a dilute dispersion of spheres sedimenting in a spherical container

We consider the motion of identical spheres in an incompressible fluid which is bounded by a spherical container wall. The test sphere is momentarily concentric with the container and the centres of the other spheres are distributed in the accessible region according to a Poisson process. We derive the third and fourth polyadic central moments of velocity and examine their components. These moments, together with the mean and covariance which are known from earlier work, provide a good description of the theoretical initial distribution. Estimates of the hard-sphere equilibrium distribution are obtained from computer simulations using a fixed number of hard spheres. Simulated values agreed closely with the theoretical, indicating that the Poisson assumption, which greatly simplifies the analysis, has little effect on the results. In small containers, the vertical component of velocity is skewed downward and is platykurtic. In large containers, the distributions of all components approach normality.