This is an experimental and numerical study of dry, frictional powder flows in the quasi-static
and intermediate regimes using the simple geometry of the Couette device. We measure
normal and shear stresses on the shearing surface and propose a constitutive equation valid
in both the slow frictional, quasi-static and the intermediate (dense) collisional regimes of
flow. This constitutive equation is then used in a new, specially developed numerical scheme
to solve the continuum equations of motion and to obtain stress and velocity distributions in
the powder. While the measurements to obtain the constitutive equation are performed in a
concentric Couette device, the numerical scheme is used to predict the torque and stresses
in two additional geometries: an eccentric Couette device where the inner, rotating cylinder
is placed off-center with different eccentricities and a more complicated geometry where a
cylindrical body is introduced in the middle between the rotating and stationary cylinders and
obstructs part of the shearing gap. Further experiments are then conducted in the two new
geometries and the torque on the inner, rotating cylinder is measured and compared to the
numerical solution.
We find experimentally, that it is possible to measure normal stresses on the shearing wall of
the Couette device inside the granular layer and calculate the ratio of the average shear to
normal stress as a function of shear rate. It appears that the powder’s dynamic angle of
friction is reproduced by this ratio only at very low shear rates. As the shearing rate
increases, the ratio of the stresses also increases due to collisions between particles that
sustain loads in addition to dry friction that is prevalent at low shear rates. We show that a
modified Couette device with slow axial flow superimposed on the shearing motion induced
by the rotating cylinder can be used to determine the constants (“b” and “n”) of a yield
condition for any powdery material that is somewhat free flowing. The yield condition is valid
in both the quasi-static as well as the “intermediate” regime of flow and contains a term
characterizing “solid”-like behavior and an additional term that captures some “fluid”-like
properties at higher shear rates.
The paper also describes a new finite element solution, realized in the FEM solver
FleatFlow, of the generalized Navier-Stokes equations that uses, in addition to the yield
condition determined above, a generalized viscosity that describes a Newtonian fluid, a
Bingham Plastic, an incompressible frictional powder (Schaeffer solid) and a power-law fluid.
We use the numerical method to validate some experimental measurements and calculate
the torque in the Couette device in three different geometries: a concentric, two cylinder,
arrangement and two new geometries in which the cylinder is positioned eccentric in the Couette and one where an additional cylindrical object is placed into the shearing gap and
obstructs parts of it.