Flow of power-law fluids in fixed beds of cylinders or spheres

An ensemble average of the equations of motion for a Newtonian fluid over particle configurations in a dilute fixed bed of spheres or cylinders yields Brinkman’s equations of motion, where the disturbance velocity produced by a test particle is influenced by the Newtonian fluid stress and a body force representing the linear drag on the surrounding particles. We consider a similar analysis for a power-law fluid where the stress $\boldsymbol{\tau} $ is related to the rate of strain $ \mathbisf{e} $ by $\boldsymbol{\tau} = 2m \mathop{ \vert \mathbisf{e} \vert }\nolimits ^{n\ensuremath{-} 1} \mathbisf{e} $, where $m$ and $n$ are constants. In this case, the ensemble-averaged momentum equation includes a body force resulting from the nonlinear drag exerted on the surrounding particles, a power-law stress associated with the disturbance velocity of the test particle, and a stress term that is linear with respect to the test particle’s disturbance velocity. The latter term results from the interaction of the test particle’s velocity disturbance with the random straining motions produced by the neighbouring particles and is important only in shear-thickening fluids where the velocity disturbances of the particles are long-ranged. The solutions to these equations using scaling analyses for dilute beds and numerical simulations using the finite element method are presented. We show that the drag force acting on a particle in a fixed bed can be written as a function of a particle-concentration-dependent length scale at which the fluid velocity disturbance produced by a particle is modified by hydrodynamic interactions with its neighbours. This is also true of the drag on a particle in a periodic array where the length scale is the lattice spacing. The effects of particle interactions on the drag in dilute arrays (periodic or random) of cylinders and spheres in shear-thickening fluids is dramatic, where it arrests the algebraic growth of the disturbance velocity with radial position when $n\geq 1$ for cylinders and $n\geq 2$ for spheres. For concentrated random arrays of particles, we adopt an effective medium theory in which the drag force per unit volume in the medium surrounding a test particle is assumed to be proportional to the local volume fraction of the neighbouring particles, which is derived from the hard-particle packing. The predictions of the averaged equations of motion are validated by comparison with simulations of randomly distributed hydrodynamically interacting cylinders.