The transverse shear-induced liquid and particle tracer diffusivities of a dilute suspension of spheres undergoing a simple shear flow

Abstract

We study the shear-induced self-diffusion of both a liquid tracer and a tagged spherical particle along the directions perpendicular to the ambient flow in a dilute suspension of neutrally buoyant spheres undergoing a simple shearing motion in the absence of inertia and Brownian motion effects. The calculation of the liquid diffusivity requires the velocity of a fluid point under the influence of two spheres, which was determined via Lamb's series expansion; conversely, the calculation of the particle diffusivity involves the trajectories of three spheres, which were determined using far-field and near-field asymptotic expressions. The displacements of the liquid tracer and of the tagged sphere were then computed analytically when the spheres and the tracer are all far apart, and numerically for close encounters. After summing over all possible encounters, the leading terms of the lateral liquid diffusion coefficients, both within and normal to the plane of shear, were thereby found to be 0.12γac and 0.004γac, respectively, where γ is the applied shear rate, a the radius of the spheres and c their volume fraction. The analogous coefficients of the lateral particle diffusivity were found to be 0.11γac, and 0.005γac, respectively. Also, liquid and particle diffusivities in a monolayer, with the liquid tracer and all the particle centres lying on the same plane of shear, were found to be 0.067γyac, and 0.032γac, respectively, with c denoting the areal fraction occupied by the spheres on the plane.