Georgia Institute of Technology
Atlanta, Georgia, U. S. A.
Shear-induced migration in concentrated noncolloidal suspensions has been modeled by two essentially distinct approaches. One approach is phenomenological and is based upon kinematical arguments that nonuniformities in the shear rate and particle fraction both set up fluxes of particulates. The second, termed the suspension balance approach and employed in the work presented here, is based on dynamical arguments which focus attention upon the particle-induced non-Newtonian rheology of the mixture. This approach centers around the idea that particles migrate in order that the bulk non-Newtonian suspension satisfy the continuum stress balance. Within this approach based upon basic two-phase flow analysis, normal stresses are found to be responsible for the migration and thus our focus is on this aspect of the rheology.
A brief overview of the crucial nature of the particle stress, $% \mbox{\boldmath $ \Sigma $}_P$, and in particular of its normal components, for two-phase flow behavior will be given. The connection of the suspension balance approach to work in polymeric fluids in the past and the implications for two-phase flows in general will be demonstrated.
Results of our studies to determine the appropriate form for the stress constitutive law for concentrated suspension flows will be presented. The focus here will be upon bulk flow studies, with results of some microstructural analysis for support of the rheological modeling. For use in bulk flow modeling, the normal stresses for concentrated noncolloidal suspensions have been modeled as simply as possible. The dependence of the normal stresses upon the particle fraction is expressed as $\Sigma_{11} = -\eta_N \dot{\gamma}$ where $\eta_N$ is termed the ``normal stress viscosity'' and depends strongly upon particle fraction, much like the effective shear viscosity of the suspension. The anisotropy of the normal stresses is captured by taking $\lambda_2 \equiv\Sigma_{22}/\Sigma_{11}$ and $\lambda_3 \equiv \Sigma_{33}/\Sigma_{11}$ as independent of particle fraction but with $1 \ne \lambda_2\ne \lambda_3$. In these expressions 1, 2, and 3 denote the flow, gradient, and vorticity directions, respectively, of a rheometric shear flow. Analytical and numerical results for curvilinear rheometric flows demonstrate the importance of the normal stresses and agreement of the model with available experimental data is excellent. A simple nonlocal stress law whose form is based upon theoretical analysis of the stress propagation in Stokes flows allows extension of the approach to flows in which the shear rate, or its extension $\dot{\gamma} = (% \mbox{\boldmath $ e $} {\bf :}\mbox{\boldmath $ e $})^{1/2}$ where $% \mbox{\boldmath $ e $}$ is the rate of strain, goes to zero within the flow. Results determined using this stress law, which is convenient because it does not require the determination of further fields beyond those required in the local law, will be demonstrated for pressure-driven channel flow.