Diffusion of Brownian particles in shear flows

Abstract

The coupling of Brownian displacements and shear-induced convection of spherical colloidal particles in dilute suspensions is examined using solutions of appropriate convective diffusion equations for the time-dependent probability density and also by calculation of relevant statistical quantities for an ensemble of diffusing particles from Langevin equations. Based on a fundamental solution for convective diffusion from a point in a general linear field, analytical expressions for the probability density fα(r; t) are given for the case of an arbitrary, two-dimensional linear flow field. The parameter α, which characterizes the flow, may range from − 1 (pure rotation), through zero (simple shear), to + 1 (pure elongation). The Langevin approach offers interesting insights into the physical mechanism of diffusive–convective coupling, and may also be used to obtain rigorous expressions for moments of the probability density appropriate to a particle diffusing in an unbounded quadratic (Poiseuille) flow. Preliminary experiments are described which qualitatively verify the theoretical predictions for Poiseuille flow, and which suggest a simple, direct method for measuring particle diffusivities. Finally the effect of bounding walls on convective diffusion is considered by means of Monte Carlo calculations. Results show that particle-wall interactions significantly affect the average behaviour of particles located initially within distances of a few particle radii of the wall, since the frictional force is no longer isotropic.

Nonlinear rheology of colloidal dispersions

“…The accelerated rate of mixing achieved by stirring a dilute dispersion is thus almost entirely attributable to local Taylor dispersion. We note also that analogous effects arising from flow-diffusion coupling can also be identified in other flow geometries, such as the practically relevant case of Poiseuille flow along a cylindrical tube [55].…”

“…Although non-interacting colloids represent a trivial case, it is nevertheless instructive to consider the meansquared-displacement (MSD), characterizing the diffusive particle motion, both parallel and orthogonal to the flow direction in simple shear [55]. In both the vorticity and shear gradient directions, flow has no influence and the equilibrium result is recovered, δz 2 = δy 2 = 2D 0 t, with D 0 the single particle diffusion coefficient.…”

“…A partial integration yields an alternative solution of (53) which is exactly equivalent to (55), but more suited to approximation…”

“…Both solutions (55) and (56) are valid for arbitrary flow geometries and time-dependence. The relation between the two formal solutions is analogous to the Heisenberg and Schrödinger pictures of quantum mechanics in which the time evolution of the system is attributed to either the wavefunction (equation (55)) or the operators representing physical observables (equation (56)).…”

Brownian motion in shear flow: Direct observation of anomalous diffusion

“…For a Brownian particle undergoing shear, a similar effect is also expected. The Brownian dynamics has been theoretically studied on the basis of a convective diffusion equation [2][3][4][5], which was used by Taylor, and a Langevin equation [5][6][7][8]. Both give the same mean-square displacement (MSD) in the flow direction x for a simple shear flow [4][5][6][7]:…”