Ensemble Averaged Equations of Motion for Power-Law Fluid Flow in Random Fixed Beds of Cylinders and Spheres

Darcy's law for the flow of a Newtonian fluid through a porous medium balances the mean pressure gradient with a body force representing the drag exerted on the fixed solid material, yielding a linear relationship between the mean fluid velocity and the pressure drop.  A similar force balance applies to non-Newtonian fluid flows in porous media, but the nonlinear dependence of the stress on the strain rate in shear thinning or thickening fluids results in a nonlinear relationship between the pressure gradient and mean fluid velocity.  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.  A self-consistent calculation of this disturbance velocity and the drag on the test particle yields the permeability of the bed.  We consider a similar analysis for a power law fluid where the stress s is related to the rate of strain g by s=m|g|^n-1g.   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 neighboring particles.  It can be interpreted as arising from the fact that the effective viscosity of the fluid in the far field is influenced more by the shearing motion in the bulk medium than by the disturbance of the test particle.  We explore the solutions to these equations using scaling analyses for dilute beds and numerical simulations using the finite element method.  In Newtonian flow through random arrays of cylinders, the logarithmic divergence of the disturbance velocity of a cylinder is removed by the Brinkman screening associated with the drag on the surrounding particles leading to a well defined average drag on the cylinders.  The effects of particle interactions on the drag in dilute arrays of cylinders and spheres in shear thickening fluids is even more dramatic, where it arrests the algebraic growth of the disturbance velocity with radial position when n>1 for cylinders and n>2 for spheres.   For concentrated fixed beds, 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 neighboring particles which is derived from the hard-particle packing.