Two-Phase Continuum Dynamics of Suspensions at Low Reynolds Number

Abstract

Many fluid flows central to biological and industrial interests are actually suspension flows, characterized by the coupled transport of solid particles and their suspending fluid medium. Yet, despite more than a century of earnest theoretical effort, a universally accepted theory of suspension fluid dynamics has remained elusive. Here, we present a mathematical framework which is used to derive a continuum-level description of a suspension of identical spherical particles at low Reynolds number. Our approach clarifies the methods required to model suspensions beyond the dilute limit and corrects inconsistencies in previously proposed suspension models, resulting in dynamical equations which are stable and may be expanded to any desired order of accuracy.The bulk of this work presents our two-phase low Reynolds number suspension framework and some elementary applications in detail. First, we derive our suspension equations of motion starting directly from Stokes’ equations and model the particles using a force singularity representation. Using spatial averaging, we derive suspension continuum equations of motion which take the form of a multipole expansion in terms of force moments on the suspended particles. Applying our equations to an isotropic suspension of identical rigid no-slip particles, we correct and extend results obtained separately by A. Einstein and G.K. Batchelor. We also use the two-particle Smoluchowski equation to calculate the normal stress differences in a dilute suspension subject to shear flow. Next, we show that our framework can be formulated equivalently through either spatial or ensemble averaging. Utilizing ensemble averaging, we develop an asymptotic procedure to derive effective continuum boundary conditions by analyzing particle-wall effects through a Green’s function approach. Surprisingly, we find that a continuum suspension may necessitatethe specification of an arbitrary boundary condition, having negligible impact on the dynamics of the bulk suspension away from the container walls. We propose a smooth extrapolation as a simple boundary condition choice, which allows a suspension to exhibit an apparent slip at the walls, although alternative choices are possible. Finally, we extend our original framework to allow for small spatial variations in the particle singularity strengths throughout the suspension. We find contributions to the bulk suspension stress and average particle velocity that may arise in nonuniform suspensions or flows with curvature. To investigate this and other effects, we develop a numerical algorithm which simulates a non-uniform continuum suspension in pressure-driven channel flow at low Reynolds number. We find that suspensions of neutrally-buoyant particles remain stable and evolve towards a uniform particle distribution except in the presence of normal stress differences. We also address potential numerical instabilities and their relation to the averaging methods we employ.