Modern granular flow theory began with the work of Bagnold in 1954 [1]. In
his seminal experiments, 1-mm wax spheres suspended in a glycerin-water-alcohol
mixture were sheared in a coaxial cylinder rheometer. The rheometer was cleverly
designed to measure both the shear and normal forces applied to the walls. From the
experiments, Bagnold identified two distinct flow regimes: the macroviscous and the grain
inertia. These regimes can be distinguished using a quantity that is now referred to as
the Bagnold number, Ba= pd2 λ^{1/2}
λ^{2}/ µ , where ñ is the density
(generally taken as the value for the particle phase), d is the particle diameter, ë is a function defined by
Bagnold that depends on the solid fraction, ã is the imposed shear rate, and µ is the dynamic
viscosity. In the macroviscous regime, the normal and the tangential stresses
were linearly proportional to the liquid’s viscosity and the shear rate. In the
grain-inertia regime, the stresses were independent of the fluid viscosity, and instead
depended on the density, the square of the shear rate and the square of the particle diameter.
The dependence on the function ë also differed in the two regimes.

Despite its importance and the almost universal acceptance of these results,
there are still questions regarding the analyses and the conclusions. For example, the
stress behaviors predicted by Bagnold’s analyses also result from simple dimensional
analysis, if the assumption is made that in one regime the stresses are independent of the
particle (diameter dependence) and in the other regime the stresses are independent of
the fluid viscosity. More importantly, the interpretation of the experiments is
problematic because the experiments involved a suspension and only a single particle size was
studied. In a density-matched suspension, the inertia of the particle and that of the fluid
are indistinguishable; hence, the results for the scaling of the stresses may not be
applicable to situations in which the densities differ. Additional questions surround the
normal stress, which was measured by using a flexible membrane for the inner stationary
cylinder that would deflect in response to the applied pressure. The system was
insensitive to small pressure changes, and the membrane deflection may have
altered the gap size and the mean solid concentration.

The current work at Caltech focuses on experiments and simulations for conditions similar to those originally examined by Bagnold. The low-gravity
environment offers the opportunity to reexamine Bagnold’s experiments for
particles of varying densities. These experiments should provide an opportunity to separate
the effects of the inertia of both the fluid and solid phases on the shear and
normal stresses. We plan to conduct experiments using different particle sizes and density,
varying particle concentrations and shear rates that would extend the Bagnold number
well into the grain inertia regime. In addition, the work involves a computational
component that uses a combination of smooth-particle hydrodynamics and the discrete element
method to compute liquid-solid flows.

The experimental apparatus will consist of a coaxial shear cell with a rotating
outer cylinder. The inner cylinder will be mounted on a friction-free air
bearing and constrained from rotation by a load cell to measure the shear force. For the
flight instrument, pressure sensors will be inserted in the wall of the inner cylinder
to simultaneously measure the normal force [2,3]. The granular material and the
fluid will be contained in the annular region of the experiment. We have a prototype shear
cell already constructed that we are using as a basis for the design of the flight
instrument.

In addition to the experimental measurements, the project also involves a computational phase that uses a combination of the smoothed particle
hydrodynamics technique (SPH) and the discrete element method (DEM) to model flows containing
a viscous fluid and solid macroscopic particles. The two-dimensional numerical simulations are validated by comparing with experimental measurements for wake
size, drag coefficient and heat transfer for flow past a circular cylinder at Reynolds
numbers up to approximately 100. The comparisons demonstrate that the technique can be successfully used for incompressible flows at Reynolds numbers up to
approximately 100.

The central focus of the work, however, is in computing flows of liquid-solid
mixtures. Hence, the simulations were run for neutrally buoyant particles
contained between two shearing plates for different solid fractions, fluid viscosities and
shear rates. The simulations involved up to 18 solid particles sheared between two plates
separated by a distance of approximately 8 particle diameters. The no-slip boundary
condition was satisfied on the surface of the particles and the bounding walls. The tangential
force resulting from the presence of particles shows an increasing dependence on the
shear rate, from a linear regime for macro-viscous flows to a square dependence for grain
inertia flows. The normal force showed considerable variation with time, which is not
fully understood and is part ongoing numerical simulations. In addition, the
simulations indicated stability problems at higher shear rates; the instability has also
been observed in other SPH simulations. Hence, the focus of the current work involves the effect
of the smoothing function and other features of the simulation technique on the flow
parameters.

Hunt, M.L., Campbell, C.S., Brennen, C.E., Granular Material Flows with Interstitial Fluid Effects, Proceedings of the Fifth Microgravity Fluid Physics and Transport Phenomena Conference, NASA Glenn Research Center, Cleveland, OH, CP-2000-210470, pp. 1352-1354, August 9, 2000.