Lateral migration in sheared suspensions: a case study of the 'diffusion' model

摘要

The 'diffusion' model for lateral migration flux has successfully been used to describe shear-induced particle migration in concentrated suspensions of non-Brownian particles subject to simple shear flows. Subsequent analyses, which included the linear momentum equation, attempted to embed this model into a more comprehensive framework that included general inhomogeneous shear flows and their concomitant pressure gradients. Based upon the latter, more general framework, the present paper presents a case study of a simple suspension flow that leads to a prediction that contradicts Darcy's law. The explicit example considered involves the steady, radial, low Reynolds number flow of a concentrated suspension of neutrally buoyant, non-Brownian spheres permanently confined within the annular space between two concentric spherical shells, each shell being permeable only to the interstitial fluid. As such, the annular domain contains a time-independent dispersion of spherical particles permanently confined within its boundaries, while the interstitial fluid flows past these fixed-in-space 'suspended' particles. The foregoing general model for this suspension flow consists of: (i) local mass conservation equations for both the fluid and suspended particles phases; (ii) an overall mass conservation equation for the confined particulate phase; (iii) the constitutive equation for the so-called 'diffusive' particle flux; and (iv) the linear momentum equation governing the local mass-average velocity. The analysis which follows examines the plausibility of the resulting predictions of the radial particle distribution within the annular space, as well as the direction of the radial pressure gradient (relative to the direction of the interstitial flow) required to maintain the steady fluid motion. Although the accepted radial migration/suspension flow equations predict a plausible spatial particle distribution in the annular region, they nevertheless predict the local pressure gradient to be invariant to the direction of the interstitial flow, and to depend upon the viscosity gradient-both conclusions being in conflict with Darcy's law for flow through a 'stationary' bed of particles, which would be expected to apply to our example problem. This predictive failure of the foregoing diffusion model suggests the need for a significant modification of the suspension-scale momentum equation, at least in circumstances where large particle concentration gradients exist.