Furthermore, as the relative veloci

furthermore, as the relative velocities induced by the shear rate
are only in the direction perpendicular to the velocity gradient,
this introduces anisotropies in the angular distribution of
collisions about a particle (as measured by campbell and
brennen [37]). interestingly, the collisional anisotropy was
included in the earliest, albeit incomplete, rapid flow models,
.savage and jeffrey [44] and jenkins and savage [51], which
only considered contact stresses. to include streaming stresses,
required modifying the velocity distribution function, which
proved intractable if the collisional anisotropy was included. as
the theories predict that s ≈ 1 (fig. 10), they are not selfconsistent
in that their predictions conflict with their implicit
assumptions.goldhirsch [52] cites sela and goldhirsch's [47]
comparison with normal stress difference data as evidence that
this effect is unimportant; but this argument is not applicable as
the comparison is done at ν = 0, the only point where the sela.
and goldhirsch calculation is valid. there, fig. 10 shows s ≈ 0
(t = ∞) so, of course γd «t1 / 2, and, while there may be
.collisional anisotropy due to the anisotropic granular temperature,
there will be no shear-induced collisional anisotropy
under the conditions of the sela and goldhirsch analysis.
finally, at the heart of all kinetic theories is the
assumption of boltzmann's "stosszahlansatz" or. molecular
chaos, that there are no correlations in the velocities or
positions of colliding particles.this is troubling because
common granular flows occur at such large concentrations
that any given particle will interact many times with its
neighbors and it is likely their velocities will be strongly
correlated. in addition, the aforementioned microstructures
[37,38] correlate the relative positions of particles. thus true
molecular chaos is unlikely in real granular systems although
.it is difficult to estimate the degree of error introduced by this
assumption.
in 1990, i wrote a review article on the field of rapid granular
flows [53]. the article ended with a list of "pressing concerns"
designed to push the field towards more realistic systems and it
is worth a paragraph to comment on the progress of the last 15
years. the concerns were: material properties,.microstructure,
non-spherical particles, non-uniform particle size and segregation,
interstitial fluid effects and solid / fluid behavior of
granular systems. however, it should have been obvious, even
in 1990, that the first 3 topics would be almost intractable, either
because they complicated the collision integrals from which the
constitutive properties are derived or because they violate the
.assumptions of molecular chaos. for example, even simple
properties such as a stick-slip surface friction make a
discontinuity in the collision integrals; as a result, friction is
only approximately incorporated in rapid-flow theories
through a tangential coefficient of restitution. also, friction
dissipates energy and as discussed above, if the energy
dissipation is large enough,.it may be possible to accurately
assess the velocity distribution function. like the collisional
anisotropy, the development of internal microstructure affects
the contact angle between particles and it is difficult to include
in the kinetic theories, partially because of the complications to
the collision integrals and partially because it violates the
stosszahlansatz.non-round particle shapes bring the particle
.

Furthermore, as the relative velocities induced by the shear rate
are only in the direction perpendicular to the velocity gradient,
this introduces anisotropies in the angular distribution of
collisions about a particle (as measured by Campbell and
Brennen [37]). Interestingly, the collisional anisotropy was
included in the earliest, albeit incomplete, rapid flow models,
Savage and Jeffrey [44] and Jenkins and Savage [51], which
only considered contact stresses. To include streaming stresses,
required modifying the velocity distribution function, which
proved intractable if the collisional anisotropy was included. As
the theories predict that S≈1 (Fig. 10), they are not selfconsistent
in that their predictions conflict with their implicit
assumptions. Goldhirsch [52] cites Sela and Goldhirsch's [47]
comparison with normal stress difference data as evidence that
this effect is unimportant; but this argument is not applicable as
the comparison is done at ν=0, the only point where the Sela
and Goldhirsch calculation is valid. There, Fig. 10 shows S≈0
(T=∞) so, of course γd≪T1/2, and, while there may be
collisional anisotropy due to the anisotropic granular temperature,
there will be no shear-induced collisional anisotropy
under the conditions of the Sela and Goldhirsch analysis.
Finally, at the heart of all kinetic theories is the
assumption of Boltzmann's "Stosszahlansatz" or molecular
chaos, that there are no correlations in the velocities or
positions of colliding particles. This is troubling because
common granular flows occur at such large concentrations
that any given particle will interact many times with its
neighbors and it is likely their velocities will be strongly
correlated. In addition, the aforementioned microstructures
[37,38] correlate the relative positions of particles. Thus true
molecular chaos is unlikely in real granular systems although
it is difficult to estimate the degree of error introduced by this
assumption.
In 1990, I wrote a review article on the field of rapid granular
flows [53]. The article ended with a list of "Pressing Concerns"
designed to push the field towards more realistic systems and it
is worth a paragraph to comment on the progress of the last 15
years. The concerns were: Material properties, Microstructure,
Non-spherical particles, Non-uniform particle size and segregation,
Interstitial fluid effects and Solid/Fluid behavior of
granular systems. However, it should have been obvious, even
in 1990, that the first 3 topics would be almost intractable, either
because they complicated the collision integrals from which the
constitutive properties are derived or because they violate the
assumptions of molecular chaos. For example, even simple
properties such as a stick–slip surface friction make a
discontinuity in the collision integrals; as a result, friction is
only approximately incorporated in Rapid-Flow theories
through a tangential coefficient of restitution. Also, friction
dissipates energy and as discussed above, if the energy
dissipation is large enough, it may be possible to accurately
assess the velocity distribution function. Like the collisional
anisotropy, the development of internal microstructure affects
the contact angle between particles and it is difficult to include
in the kinetic theories, partially because of the complications to
the collision integrals and partially because it violates the
Stosszahlansatz. Non-round particle shapes bring the particle

Furthermore, as the relative velocities induced by the shear rate
are only in the direction perpendicular to the velocity gradient,
this introduces anisotropies in the angular distribution of
about a particle collisions (as measured by Campbell and
Brennen [37 ).Interestingly, the collisional
anisotropy was included in the earliest, albeit incomplete, rapid flow models,
and Savage Jeffrey [44 Jenkins and Savage] and [51, which only considered
contact stresses. To stresses include streaming,
required modifying the velocity distribution function, which
proved intractable if the collisional anisotropy was included. As
theories predict that the S ≈ 1 (Fig. 10), they are not selfconsistent
in their predictions that conflict with their
implicit assumptions.52 Goldhirsch [] cites Sela and Goldhirsch's [47 ]
comparison with normal stress difference
data as evidence that this effect is unimportant; but this argument is not applicable as the comparison is done at ν
= 0, the only point where the calculation is valid Sela
and Goldhirsch.There, Fig. 10 shows S ≈ 0
(T=∞) so, of course γ ≪ T D 1/2, and, while there may be
due to the collisional anisotropy anisotropic granular temperature,
there will be no shear-induced collisional anisotropy Sela
under the conditions of the analysis and Goldhirsch.
Finally, at the heart of all kinetic theories is the assumption of
Boltzmann Stosszahlansatz" or "S"
molecular chaos, that there are no correlations in the
positions or velocities of colliding particles.
This is troubling because common granular flows occur at such large concentrations
that any given particle will interact many times with its neighbors and
it is likely their velocities will be strongly correlated
. In addition, the aforementioned microstructures
[37,] 38 correlate the relative positions of particles. Thus
true molecular chaos is unlikely in real granular systems although
It is difficult to estimate the degree of error introduced by this assumption
.
In 1990, I wrote a review article on the field of rapid granular flows
[53 . The article ended with a list of "Pressing Concerns"
designed to push the field towards
more realistic systems and it is worth a paragraph to comment on the progress of the last 15 years
. The concerns were: Material properties,Microstructure,
Non - spherical particles, Non - uniform particle size and segregation,
Interstitial fluid effects and Solid/Fluid
behavior of granular systems. However, it should have been obvious, even
in 1990, that the first 3 topics would be almost intractable, either
because they complicated the collision integrals
from which the constitutive properties are derived or because they violate the
assumptions of molecular chaos. For example, even simple properties such as a stick
-
slip surface friction make a discontinuity in the collision integrals; as a result, friction is only approximately incorporated in Rapid
-
Flow theories through a tangential coefficient of restitution.Also,
friction dissipates energy and as discussed above, if the energy dissipation
is large enough,It may be possible to accurately assess the velocity distribution function
.
collisional Like the anisotropy, the development of the internal microstructure affects
contact angle between particles and it is difficult to include in the
kinetic theories, partially because of the complications to
the collision integrals and partially because it violates the
Stosszahlansatz.Non - round particle shapes bring the particle