The effect of Brownian motion on the bulk stress in a suspension of spherical particles

TLDR

Brownian motion tends to homogenize particle positions in a statistically homogeneous suspension, opposing the clustering induced by deformation. The authors model Brownian motion as coupled thermodynamic forces that both directly add to bulk stress and indirectly modify particle statistics, and for dilute rigid spheres both effects are O(φ²) with an explicit pairwise hydrodynamic expression, and the governing probability‑density equation is solved numerically for strong Brownian motion. The dilute suspension remains isotropic under any bulk flow, with an effective viscosity μ*(1+2.5φ+6.2φ²) to O(φ²), and weak Brownian motion yields a 7.6 coefficient for strain‑thickening in steady pure straining.

Abstract

The effect of Brownian motion of particles in a statistically homogeneous suspension is to tend to make uniform the joint probability density functions for the relative positions of particles, in opposition to the tendency of a deforming motion of the suspension to make some particle configurations more common. This smoothing process of Brownian motion can be represented by the action of coupled or interactive steady ‘thermodynamic’ forces on the particles, which have two effects relevant to the bulk stress in the suspension. Firstly, the system of thermodynamic forces on particles makes a direct contribution to the bulk stress; and, secondly, thermodynamic forces change the statistical properties of the relative positions of particles and so affect the bulk stress indirectly. These two effects are analysed for a suspension of rigid spherical particles. In the case of a dilute suspension both the direct and indirect contributions to the bulk stress due to Brownian motion are of order ø 2 , where ø([Lt ] 1) is the volume fraction of the particles, and an explicit expression for this leading approximation is constructed in terms of hydrodynamic interactions between pairs of particles. The differential equation representing the effects of the bulk deforming motion and the Brownian motion on the probability density of the separation vector of particle pairs in a dilute suspension is also investigated, and is solved numerically for the case of relatively strong Brownian motion. The suspension has approximately isotropic structure in this case, regardless of the nature of the bulk flow, and the effective viscosity representing the stress system to order ϕ 2 is found to be \[ \mu^{*} = \mu(1+2.5\phi + 6.2\phi^2). \] The value of the coefficient of ø 2 for steady pure straining motion in the case of weak Brownian motion is known to be 7[sdot ]6, which indicates a small degree of ‘strain thickening’ in the ø 2 -term.