One model for granular materials treats them as a collection of inelastic hard spheres. There are no attractive forces between particles but in each collision energy is lost. A range of behavior is observed in computational and analytic studies of such systems. One possibility is inelastic collapse, in which the particles undergo an infinite number of collisions in a finite time. Another area of interest is the relevance of granular hydrodynamics in the quasi-elastic limit.
For theoretical purposes, one can approximate granular materials as a
collection of hard spheres undergoing inelastic collisions. In our work,
these interactions are treated as instantaneous collisions in which the
energy loss occurs along the line between the centers of the particles and
the degree of inelasticity is parameterized by the coefficient of
restitution,
r, which takes on values between 0 and 1. Of course, for all r
values,
momentum is conserved. When r=1, the system is completely elastic, and a
collision corresponds to an exchange of velocities. When r=0, the system
is completely inelastic and particles collide and effectively "stick
together". For intermediate r values an array of rich behavior is
observed.
In many cases, only extensive computer simulations can give clues about
the complex evolution of a granular "gas" or "liquid"
made up of colliding hard spheres. There are, however, instances where first
analytical results have been obtained. Below we have more information on
a few research "nuggets" (
):
One area of focus is the quasi-elastic regime
,
with r very near but not equal to one. In this region, we can test the
application
of hydrodynamic theories to granular materials.
As r is decreased and the collisions become more and more
inelastic,
simple hydrodynamics clearly breaks down and new descriptions are being
sought. One particularly intriguing aspect in this regime is the occurrence
of the phenomenon of inelastic collapse.
Quasi-elastic collisionsWe first consider a one-dimensional system: A group of particles is confined
to move on a line between two walls (see picture below; the top row gives
the initial configuration, the rows below show the system as time progresses;
colors correspond to velocity [or kinetic energy, or granular temperature]
with "blue" for very small and "red" for large velocity).
The right wall is reflecting, the left wall is held at a fixed
"temperature".
Since the collisions between particles are constantly dissipating energy,
it is the energy input from this left wall that allows the system to reach
a steady state. From equations requiring the conservation of particle number,
momentum, and energy, a continuum hydrodynamic description of the system
can be derived. However, when simulations are done, the behavior is seen
to be nowhere near the theoretical predictions [see the
paper
by Du, Li, Kadanoff]. Instead, a practically stationary cluster of particles
forms against the reflecting wall (as far as possible from the heated [=left]
wall). If there are N particles in the system, N-1 will be in this cluster,
and the remaining particle will move rapidly back and forth between the
cluster and the heated wall.
While this system fails to substantiate the existence of granular hydrodynamics,
it does allow us to study the growth of the stationary cluster over time
and understand the mechanism that produces this cluster. (Incidently, the
wall is not a necessary part of this phenomenon; if both walls are heated,
the cluster will form in the middle of the system.) Using a Boltzmann treatment,
we can solve for the probability distribution for the velocity and density
of the fast moving particles (the ones that have not yet been absorbed in
the cluster). This allows us to predict how the cluster grows over time
as a function of the degree of inelasticity, the total number of particles
in the system and the type of boundary forcing at the heated wall [see the
paper by Grossman and Roman].
One can also consider the analogous system in two dimensions, as in the picture above. Here, development of a hydrodynamical theory is more difficult as the finite size of the particles must be taken into account. Nonetheless, for a system in a steady state, one can use the energy balance equation to write a second order differential equation that describes the density variation in the system as a function of distance from the heated wall. The key ingredients to this theory are expressions for the coefficient of thermal conductivity, the mean free path, and the pressure that are valid for a wide range of densities. When the predictions are compared with simulations, it is found that, in the quasi-elastic limit (1-r << 1), granular hydrodynamics provides a fairly accurate description of this system [see the paper by Grossman, Zhou and Ben-Naim]. We have also examined the breakdown of this model: what assumptions are violated as the system becomes more inelastic, i.e. as 1-r grows. In this regime, the belief that the system is locally at or near thermal equilibrium is no longer valid and the granular temperature at a given position is no longer well defined. While this means that the original theoretical description will not be relevant, the energy balance equation must still hold, and we can still make predictions about the way a characteristic velocity changes as a function of distance from the heated wall.
Inelastic CollapseFor intermediate r values the phenomenon known as inelastic collapse
can occur. Here a group of particles forms a cluster so tightly packed that
there occur an infinite number of collisions in a finite amount of time.
In one dimension, this means that the particles dissipate all of their energy,
as viewed in the center of mass frame. This situation is analogous to one
inelastic ball bouncing on an infinite plate. For many particle systems
(confined, periodic, or unbounded), there exists a critical r value that
separates the region where inelastic collapse is possible from the region
where it cannot occur. For groups of a few particles, the value of this
critical r can be calculated exactly, but for larger systems, it must be
calculated through simulations, although there do exit theoretical predictions
for how it behaves as the size of the system becomes infinite.
This phenomenon has been studied in several forms here at the University
of Chicago. The analytic work has focused mainly on three-particle
systems. In one dimension, three particles on a line or on a ring were
studied by mapping the system onto a two-dimensional billiard with unconventional
reflection laws. The importance of initial conditions in inelastic collapse
were examined. We also studied regions of quasi-periodic behavior of particles
on a ring when r is greater than the critical value. In this situation it
becomes clear that the behavior of the system in the limit that r goes to
1 is completely distinct from the behavior of the elastic system, in other
words r --gt; 1 is a singular limit. Finally, one possibility that is neglected
by the inelastic collision model is that of triple collisions. We therefore
analyzed possible ways to define the outcome of a triple collision in one
dimension.
In two dimensions, the basin of attraction for three particle inelastic
collapse is calculated as a function of the coefficient of restitution and
the final configuration of the particles. This work is then extended by
reworking the model to include particle rotation and frictional effects.
Many-particle simulations are done to emphasize the qualitatively different
outcomes that result from this alteration in the model. Specifically, the
simpler model predicts that inelastic collapse can only occur for r below
a critical r value that is distinct from 1 (although it approaches one as
the number of particles goes to infinity). When rotation is included, we
find that, even for three particles, collapse can occur at any r value other
than one, depending on the initial conditions and the friction characteristic
of the system. Another important difference is that for the basic model,
inelastic collapse occurs along linear structures with the two-dimensional
system. When rotation is added, the structures take on a zig-zag form imposed
over the long range linear structure. This is clearly seen in the picture
below. This image from a computer simulation
shows the location of particles participating in the inelastic collapse
(filled-in black disks). This simulation used periodic boundary conditions,
repeating the situation inside the central rectangle ad infinitum in all
four directions. The inset (the box in the upper right hand corner) show
the collapsing region for a simulation without rotation. Note the absence
of the zig-zag structure.
Papers on granular
hydrodynamics, inelastic collisions, and inelastic collapse: