Break-up of a falling drop containing dispersed particles

The general purpose of this paper is to investigate some consequences of the randomness of the velocities of interacting rigid particles falling under gravity through viscous fluid at small Reynolds number. Random velocities often imply diffusive transport of the particles, but particle diffusion of the conventional kind exists only when the length characteristic of the diffusion process is small compared with the distance over which the particle concentration is effectively uniform. When this condition is not satisfied, some alternative analytical description of the dispersion process is needed. Here we suppose that a dilute dispersion of sedimenting particles is bounded externally by pure fluid and enquire about the rate at which particles make outward random crossings of the (imaginary) boundary. If the particles are initially distributed with uniform concentration within a spherical boundary, we gain the convenience of approximately steady conditions with a velocity distribution like that in a falling spherical drop of pure liquid. However, randomness of the particle velocities causes some particles to make an outward crossing of the spherical boundary and to be carried round the boundary and thence downstream in a vertical ‘tail’. This is the nature of break-up of a falling cloud of particles. A numerical simulation of the motion of a number of interacting particles (maximum 320) assumed to act as Stokeslets confirms the validity of the above picture of the way in which particles leak away from a spherical cluster of particles. A dimensionally correct empirical relation for the rate at which particles are lost from the cluster involves a constant which is indeed found to depend only weakly on the various parameters occurring in the numerical simulation. According to this relation the rate at which particles are lost from the blob is proportional to the fall speed of an isolated particle and to the area of the blob boundary. Some photographs of a leaking tail of particles in figure 5 also provide support for the qualitative picture.