Flows from Bins: New Results - Selected Topics of Computational and Experimental Fluid Mechanics - page 170

of depth. The origin of this behavior is in the static friction between the grains and
the walls of the container. Due to this friction, the container walls can support part
of the weight of the material.
The validity and possible shortcomings of the simple theory underlying the orig-
inal Janssen model have been extensively tested experimentally and numerically.
Janssen himself carried out experiments to measure the pressure on the base of a silo,
and apparently found good agreement with his theoretical results. However, careful
laboratory observations have revealed serious difficulties in the measurements of this
vertical pressure, with the results depending on the method of filling and even on the
jamming of the displacement of the piston employed to measure the pressure. Precise
and reproducible pressure profiles have been measured in some carefully designed
experiments, and molecular dynamics simulations which rely on the notion that
the grains have settled to a final state (Landry et al. 2003 ) have reproduced these
measurements. A comparison shows that the predictions of Janssen model are rea-
sonably good for most of the granular material but degrade in a region around the
top of the column where the pressure varies nearly linearly with vertical distance.
This discrepancy has led to modifications of Janssen's model. Among these, a two-
parameter modification, which suppresses friction with the wall in an upper slice of
the column, improves the agreement with the experimental and numerical vertical
pressure profiles in that region.
Here we explore the main changes between the pressure distributions and traction
forces due to the use of Janssen's model and the modified Janssen's model. In a recent
work we have shown that the measurements of the traction force that the granular
material exerts on the wall overcomes the difficulties often encountered in the direct
measurements of the vertical pressure on a piston, which are due to jamming of the
piston displacement induced by the compression force. Measurements of the traction
force give further evidence of the existence of a region not predicted by the original
Janssen model.

5.1 Calculation of the Pressure and Traction Force in the Silo

Consider a cylindrical vertical tube of radius r 0 filled with a dry granular material to
a certain height H . Take a horizontal section of the tube at a depth z in the granular
material. According to Janssen's model, the force exerted per unit area of this section
by the material above it on the material below is a vertical pressure p z ( z ) .This

pressure is not equal to the horizontal pressure p r ( of the material on the wall of

the tube at the same depth, but the two pressures are linearly related by Janssen ( 1895 )

z )

p r ( z ) = Kp z ( z ), (12)

where K , called the Janssen parameter, is a dimensionless constant that characterizes
the conversion of vertical stress into horizontal stress within the granulate.