Brownian Diffusion in a Dilute Dispersion of Droplets

An analytical study of Brownian motion in a dispersion of fluid drops is considered. The droplets, which are spherical and may differ in radius, are assumed to be close enough to interact hydrodynamically. Based on Einstein's description of Brownian motion that invokes an equilibrium and in which droplets are affected by a thermodynamic force, the Brownian diffusivities in two different situations are deduced. The first interaction concerns a homogeneous dilute suspension that is deformed locally. The relative diffusivity of two droplets with a given separation distance is derived from the mobility functions due to the low-Reynolds-number flow that arises because of two hydrodynamically interacting droplets. The second interaction concerns a suspension in which there is a concentration gradient of droplets. The thermodynamic force on each droplet in this case is shown to be equal to the gradient of the chemical potential of droplets, which brings the multidroplet excluded volume into the problem. For a determination of the average settling velocity of droplets falling through fluid under gravity, a theoretical result correct to the first order in volume fraction of the droplets is available. The diffusivity of the droplets is found to increase slowly as the concentration rises from zero. These results are generalized for an inhomogeneous suspension of several different species of droplet, and expressions for the diagonal and off-diagonal elements of the diffusivity matrix are obtained. The results, presented in simple closed forms, agree very well with the existing solutions for the limited cases of solid spheres. Moreover, the limiting diffusion situation of spherical gas bubbles is also considered. Copyright 1999 Academic Press.