Self-diffusion in sheared suspensions

Abstract

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either macroscopically quiescent or in linear flow at arbitrary Péclet number Pe = γ + ̄a2/2D, where γ + ̄is the shear rate, a is the particle radius, and D = kBT/6πηa is the diffusion coefficient of an isolated particle. Here, kB is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by kB T times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction φ = 4/3πa3n and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension it perturbs the average microstructure, and the long-time self-diffusion tensor, D∝s, is given by the sum of D0s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D0s and D∝s are anisotropic, in general, with the anisotropy of D0s due solely to that of the steady microstructure. The influence of flow upon D∝s is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations. The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(φPe3/2) for Pe ≪ 1 and φ ≪ 1, both with and without hydrodynamic interactions between the particles. The leading dependence upon flow of D0〉 is 0.22DφPeÊ, where Ê is the rate-of-strain tensor made dimensionless with γ + ̄. Regardless of whether or not the particles interact hydrodynamically, flow influences D∝〉 at O(φPe) and O(φPe3/2). In the absence of hydrodynamics, the leading correction is proportional to φPeDEâ. The correction of O(φPe3/2), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to φPe3/2DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê. and the second a non-isotropic tensor. At high φ a scaling theory based on the approach of Brady (1994) is used to approximate D∝s. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of P̄e = γ + ̄a2/2D0s(φ). At small P̄e the dependence on P̄e is the same as at low φ.