Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 137

expansion. The expansion, Reynolds' dilatation , of granu-
lar masses under shear, requires energy to be expended
because in effect we are having to increase the solid layer's
potential energy by a small amount proportional to

Bagnold originally proposed that the dispersive stress is
greatest close to the basal shear plane of the granular flow
and that there, large particles exerted a higher stress (to
the square of diameter). Hence these larger particles move
upward through the flow boundary layer to equalize stress
gradients. However, a second hypothesis, termed kinetic
filtering, says that small grains simply filter through the
voids left momentarily below larger jostling grains until
they rest close to the shear plane; the larger grains must
therefore simply rise as a consequence. A simple test for
the rival hypotheses is to shear grains of equal size but con-
trasting density, since

d .
Some force, an inertial one via Reynolds' descending foot,
is required to do this. A gravity force may be more
directly imagined using a variant of Leonardo's friction
experiment (Section 3.9), as an initially horizontal solid
body free to move rests on another fixed solid body. As
the contact between the bodies is gradually steepened a
critical energy threshold is exceeded, at a slope angle
termed the angle of static friction or initial yield ,

i . Here
the block moves downslope as the roughnesses making up
the contact surface dilatate. In the case of a loose aggre-
gate, the grains flow downslope until they accumulate as a
lower pile whose slope angle is now less than the initial
slope threshold that caused the flow to occur in the first
place. This lower slope angle, termed the angle of resid-
ual friction or shear,

also depends upon grain density.
It is observed that sometimes the densest grains do
indeed rise to the flow surface. Further experiments
with naturally varying grain density and size reveals vari-
able patterns of grain segregation depending on size and
density of grains and the frequency of vibration. The dis-
persive stress hypothesis is only partly confirmed by such

less than the ini-
tial angle of yield for natural sand grains. The value
i r gives the dilatational rotation required for shear
and flow. Some more details on the often rather compli-
cated controls on natural sand frictional behavior are
given in Cookie 17.

r , is usually 5-15

4.11.4 Simple collisional dynamics of granular flows

Once in motion a granular flow comprises a multitude of
grains kept in motion above a basal shear plane. An equi-
librium must be set up such that the weight force of the
grains is resisted by an equal and opposite force,

, arising
from the transfer of normal grain momentum onto the
shear plane. This concept of dispersive normal stress pro-
posed by Bagnold (Cookie 18) is analogous to the transfer
of molecular momentum against a containing wall of a ves-
sel envisaged in the kinetic theory of gases (Section 4.18).
Such normal stresses have been used to explain the fre-
quent occurrence of upward-increasing grain size, in the
deposits of granular flows (see below). Marked downslope
variations in sorting and grain size also develop sponta-
neously (Fig. 4.60): larger grains are carried further than
smaller grains because they have the largest kinetic energy.
This leads to lateral (downslope) segregation of grain size.
More interestingly, when the larger grains have higher val-
ues of

, the mixture spontaneously stratifies as the smaller
grains halt first and the larger grains form an upslope-
ascending grain layer above them.
The phenomenon is popularly framed in granular
physics as the “Brazil nut problem,” or “why do Brazil
nuts rise to the top of shaken Muesli?” (Fig. 4.61).

Fig. 4.63 Sand avalanches on the steep leeside slope of a desert
dune. Here, repeated failure has occurred at the top of the dune
face: the sand has flowed downslope as a granular fluid, “stick-slip”
shearing internally to produce the observed pressure-ridges as it
does so. Shear along internal failure planes causes acoustic energy
signals to propagate, hence the “singing of the sands” that haunted
early desert explorers.