Rheological effects of nonuniform particle distributions in dilute suspensions

Abstract

A continuum mixture theory for the slow flow of a dilute suspension of solid particles in a viscous fluid is outlined. The momentum exchange between the fluid and disperse particulate phases accounts for buoyancy, drag and lift forces, and the additional viscous transport associated with the presence of the particles in the fluid. Explicit constitutive equations are posed that specialize the drag to include Stokes and Faxén forces, and the lift to accommodate the ‘‘slip–shear’’ force identified by Saffman and the ‘‘disturbance–shear’’ and ‘‘disturbance–curvature’’ forces treated by Ho and Leal. The additional viscous transport is fixed by comparison with the well‐known Einstein ‘‘effective viscosity’’ correction. Finally, the pressure difference between the phases is specified by a constitutive equation that accounts for Brownian diffusion, local inertia, and bulk viscous effects. Plane Poiseuille flow is examined to illustrate the features of the model. Both approximate analytical solutions and numerical solutions for the full, coupled, nonlinear governing equations show nonuniform distributions for the particles across the channel. This affects the apparent viscosity of the mixture according to whether the particles concentrate in regions of lesser or greater mean shear rate. Finally, the inhomogeneous particle distributions cause the mixture to appear non‐Newtonian, because the form of the distribution (and, consequently, the apparent viscosity) is rate‐dependent.