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until tilted to a certain critical angle, upon which the
particles loose themselves from their neighbors and tum-
ble down the inclined face. We sleepily observe that the
grain aggregates must transport themselves with no help
from the surrounding fluid, in this case, air. We deduce by
observation that aggregates of particles may either be at
rest in a stable fashion or else they flow downslope like a
fluid. How does this behavior come about?
4.11.2
Static properties of grains
In order to simplify the initial problem, we assume, as did
Reynolds, that the particles in question are perfectly
round spheres. We are thus dealing with macroscopic par-
ticles of a size too large to exhibit mutual attraction or
repulsion due to surface energies, as envisaged for atoms.
While at rest a mass of such particles must support itself
against gravity at the myriad of contact points between
individual grains (Fig. 4.58). We can imagine two end-
members for geometrical arrangement, the ordering or
packing , of such spheres. The maximum possible close
packing would place the spheres in cannon ball fashion,
each fitting snugly within the depression formed by the
array of neighbors below and above. By way of contrast,
the minimum possible close packing would be a more ide-
alized arrangement, difficult to obtain in practice, but
nevertheless possible, where each sphere rests exactly
above or below adjacent spheres. The reader may recog-
nize these packing arrangements as similar to those
revealed by x-ray analysis of the arrangement of atoms in
certain crystalline solids, the former termed rhombohedral
and the latter cubic .
Using these simple end-member models for ideal pack-
ing we can define an important static property of granular
aggregates. This is solid concentration, C , or fractional
packing density . Its inverse is (1
4.11.1
Reynolds again
As so often in this text we follow the pioneering footsteps
(literally damp footsteps in this case) of Reynolds, who pre-
sented basic observations and hypotheses on the problem
in 1885. Reynolds pointed out that ideal, rigid, smooth
particles had long been used to explain the dynamics of
matter and that more recently they formed the physical
basis for the kinetic theory of gases and explanations for dif-
fusion. He pointed out, however, that the natural behavior
of masses of rigid particles, exemplified as he strode over a
damp sandy beach, had a unique property not possessed by
fluids or continuous solids that “consists in a definite
change of bulk, consequent on a definite change of shape
or distortional strain, any disturbance whatever causing a
change of volume.” Reynolds' walks across newly exposed
but still water-saturated beach sand: “When the falling tide
leaves the sand firm, as the foot falls on it the sand whitens,
or appears momentarily to dry round the foot . . . the pres-
sure of the foot causing dilatation...the surface of the
water . . . lowered below that of the sand.” Let us develop
Reynolds' concept in our own way.
C ), defining the inter-
granular concentration, P , termed porosity or void fraction .
To calculate C we take the total volume of space occupied
by the grain aggregate as a whole, as for example in some
real or imaginary container of known volume, and express
the fraction of its space occupied by the solid grains alone.
(a)
(b)
Rhombohedral (cannonball)
y 1
(c)
y 2 , line of contact points
for cubic packing
Cubic
Grain layer
lifts up by
d = y 2 - y 1
y 1
Fig. 4.58 (a) Mode of granular packing epitomized by this stable pyramid of cannonballs. (b) and (c) Any displacement from condition (b) to
(c) must involve a dilatation of magnitude,
d
y 2 - y 1 .
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