(505e) 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.