Available online at www.sciencedirect.com
Powder Technology 182 (2008) 130 – 136
www.elsevier.com/locate/powtec
Introduction to granular temperature ☆
Isaac Goldhirsch
School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel
Available online 14 December 2007
Abstract
The goal of this article is to provide a somewhat critical introduction to the concept of granular temperature and some of its applications. A
brief history of the concept is followed by a presentation of several of its major properties and implications thereof. A number of misconceptions
concerning this concept is presented and discussed. Certain open questions are described and some recent developments are briefly outlined.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Temperature; Granular matter; Granular gases; Kinetic theory
PACS: 05.20.Dd; 47.45.Ab; 45.70.Mg; 83.80.Fg; 62.20.Dc
1. Introduction
In his classic paper on Brownian motion [1] Einstein notes
that “... a dissolved molecule is differentiated from a suspended
body solely by its dimensions” and claims that the properties of
a collection of suspended macroscopic particles are basically the
same as those of suspended molecules. Put in modern terms, if
molecules are replaced by macroscopic particles, one can use all
the tools and concepts that pertain to molecular assemblies
(though quantum mechanics is not directly relevant in this case)
to obtain similar properties albeit on larger spatial and temporal
scales. It is for this reason that one may find it strange that the
concept of granular temperature as a measure of the velocity
fluctuations in a fluidized granular system (now known as a
granular gas) was put forward only as late as 1978 [2,3]. The
importance of energy fluctuations in granular gases was also
appreciated by a few others in the late seventies, cf .e.g., [4–7].
Perhaps the reason for this is that most prior work on granular
matter had focused on dense and nearly-static granular materials
(in particular, soils) where one has little a-priori reason to define
a (granular) temperature characterizing the random parts of the
velocities of the grains. The idea expressed in [2,3] was almost
immediately put to use (in the early eighties) in constructing
☆
Support from the Israel Science Foundation (ISF), grant. no. 689/04, the
German Israel Science Foundation (GIF), grant. no. 795/2003 and the US-Israel
Binational Science Foundation (BSF), grant. no. 2004391, is gratefully
acknowledged.
E-mail address: isaac@eng.tau.ac.il.
0032-5910/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.powtec.2007.12.002
kinetic-based theories for granular gases, whose states were
then referred to as “rapid granular flows”. Early work can be
found in [8,9]. Even some of the phenomenology of molecular
gases, like mean field and mean free path theories, was adapted
to granular matter [10,11]. A summary of these early results,
their successes and failures, can found in the review article [12].
Granular kinetic theory was perfected and made systematic in
the nineties of the twentieth century [13,14]. Some of the
differences between the pioneering approaches to granular
gases and the more modern treatments, as well as new insights
and results obtained from the latter are presented below, see also
the review paper [15].
While Einstein's theory of Brownian motion (and its
extensions) is now widely accepted, one must bear in mind that
Einstein studied an equilibrium system, and that Brownian
motion is a manifestation of equilibrium fluctuations. It is not apriori clear in general that a concept derived from equilibrium
thermodynamics or statistical mechanics is directly relevant to
granular assemblies, even granular gases. The reason is that unlike
molecules, grains collide inelastically (spontaneous retrieval of
the energy taken up by the grains can only happen on huge time
scales, not relevant to the present discussion) and therefore one
needs to pump energy into a granular gas in order to maintain it
fluidized. Consequently, granular gases are always in nonequilibrium states. Furthermore, grains undergo attrition,
breakup, coagulation and other processes, see e.g., [16], which
are rare for molecular gases at not-too-high temperatures.
Therefore it is not immediately evident that a characterization
of the motion of grains by a granular temperature (field) and the
I. Goldhirsch / Powder Technology 182 (2008) 130–136
related applications in kinetic and hydrodynamic theories are
useful. Indeed, some researchers expressed doubts concerning the
applicability of kinetic theory and hydrodynamics (both of which
employ the granular temperature as a basic entity) to granular
gases [17–19] whereas others ignored the granular temperature in
their theories, see e.g., [20]. One must realize though that in
addition to the physical and mathematical arguments (part of
which are presented below) which support the use of the granular
temperature and (the related) kinetic theories, the successes of the
kinetic and hydrodynamic theories of granular matter, even
beyond their nominal ranges of applicability [21–24], are
impressive and speak for themselves.
As is well known, the macroscopic velocity of a fluid, or its
velocity field, does not represent the velocities of the molecular
motions: whereas the speeds of the molecules are typically of
the order of the speed of sound in a fluid, the fluid may exhibit
much slower macroscopic (also known as hydrodynamic or
averaged or coarse-grained) speeds. The difference between the
velocity of a molecule that resides (more accurately, it's center
of mass does) at point r at time t and the value of the hydrodynamic velocity at that point is known as the peculiar or
fluctuating velocity of the molecule and is basically a random
entity (note that ‘random’ does not imply e.g., ‘directionally
isotropic’). Since by definition the average (whose precise
nature must be specified) of the peculiar velocity is zero, a good
measure of its magnitude is the average of the square of this
entity. This average is proportional to the local temperature of
the fluid (a field in its own right), by definition.
A similar argument can be used to justify the definition of a
granular temperature. One can define a velocity field as an
averaged or coarse-grained entity (see the paper by Serero et al in
this issue for formal definitions) and distinguish it from the actual
velocities of the individual grains. Let r i(t) denote the center of
mass position of grain i at time t and v i(t) its velocity. Let V(r,t)
represent the velocity field of the system one considers. The
macroscopic velocity field at the center of mass position of
particle i is given by V(ri(t),t). The fluctuating (or peculiar)
velocity of the grain is defined as u(ri(t),t) ≡ v i(t) − V(ri(t),t) (for a
more general definition that takes into account the finite resolution
involved in coarse-graining see the paper by Serero et al in this
issue). Clearly (as in the molecular case) the average of the
peculiar velocity vanishes. Therefore it is convenient to define the
average of the square of the peculiar velocity as a measure of the
velocity fluctuations. It is common to define the granular temperature for a monodisperse system as D1 bu2 N, where D denotes
the spatial dimension and bXN is the average of the entity X (over
space or time or both, or an ensemble; see more below). A slightly
different definition is usually employed for polydisperse systems
(see the paper by Serero et al).
The fact that one can define an entity such as the granular
temperature (defining objects is usually not difficult) does not
prove that this definition begets useful results; only such results
can provide a justification for the definition. Indeed this is the
case, as shown below and as this entire special issue hopefully
demonstrates.
Though temperature has intuitive and everyday significance,
∂S 1
its (equilibrium) thermodynamic definition is: T u ∂U
,
131
where S denotes the entropy of a system and U its energy (all
other independent thermodynamic variables being kept fixed).
Statistical mechanics shows that the thermodynamic temperature of a classical system is the same as that defined using the
peculiar velocity (up to a factor that includes the mass of a
particle and, when one wishes to measure the temperature in
degrees, another factor of Boltzmann's constant). The advantage of the statistical mechanical definition is that it makes
possible to define temperature as a statistical property without
the need to refer to a state of equilibrium. Moreover, one does
not even need to define an ensemble of states since one can
average u2 (as is often done in simulations) over (usually small)
volumes and time ranges instead. Therefore the temperature is
usually well defined even for far-from-equilibrium systems, but
it does not necessarily possess (all of) the same properties as the
equilibrium temperature.
The reason that the concept of temperature is so useful in
most molecular systems is not only the fact that it comprises an
important characterization of the velocity fluctuations, but the
strong scale separation that characterizes these systems. Even
for “large” temperature gradients, say 100 K/cm, the change of
temperature over a typical interatomic distance in a solid (about
3 Å) is 3 × 10− 6 K; in air under standard (STP) conditions the
corresponding change over a mean free path (about 0.1 μm) is
still only 10− 3 K. Therefore on scales of many interatomic
distances or mean free paths the temperature (as well as other
characterizations) can be considered to be nearly constant,
justifying the notion of local equilibrium. This way, a thermometer of macroscopic size can measure a meaningful temperature.
One should be reminded that local equilibrium is not equilibrium
since e.g., a system subject to a temperature gradient supports a
heat flux, unlike a state of equilibrium. This implies that the
distribution function of a system in local equilibrium cannot be
Maxwellian since the heat flux corresponding to a Maxwellian
distribution vanishes; however the distribution function can be
close to Maxwellian, thereby further justifying the notion of local
equilibrium.
Returning to granular systems, it turns out that they do not
usually possess the strong scale separation that characterizes
molecular systems, except in the case of nearly-elastic interactions. This fact has numerous consequences, one of which is the
sizeable normal stress differences in granular gases (see below).
Still, the use of granular temperature in the realm of granular
gases, and to some extent in dense granular fluids, has proven
highly successful [15].
Fluid temperatures can be measured using thermometers.
There is no known granular thermometer [see though the suggestion proposed in [25]] or direct way to measure the granular
temperature. The granular temperature of a system (or parts of it)
can be found either by detailed measurements of the grains’
velocities, numerically (in simulations) or experimentally (see
e.g., the paper by Wildman and Huntley in this issue) or indirectly
through its consequences. The fact that a granular thermometer is
not available cannot be taken as an argument against the concept
itself, since its consequences are measurable and important.
Another argument against the usefulness of the concept of
granular temperature is the fact that equipartition does not hold in
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I. Goldhirsch / Powder Technology 182 (2008) 130–136
granular systems. This fact is fully consistent with results obtained
from kinetic theory and is by itself not truly surprising since
equipartition is not supposed to be valid in any nonequilibrium
system. An interesting manifestation of the lack of equipartition is
the fact that the granular temperatures of the components of a
mixture do not equal each other (see the paper by Serero et al
in this issue). The rotational and translational temperatures of
granular gases that experience tangential restitution (all real
granular gases do) do not equal each other either. The only
relatively strong argument against the use of the concept of
granular temperature or kinetic theory is the fact that granular
gases (and solids, as a matter of fact) lack strong scale separation.
This issue is taken up below.
The structure of this paper is as follows. Section 2 relates
granular temperature to kinetic theory and mentions some of
the results of this theory. It also explains the difference between kinetic-based theories and systematic kinetic approaches.
Section 3 is devoted to the lack of strong scale separation and some
of its consequences. Section 4 describes one of the important
consequences of the inelasticity of the granular collisions, namely
the clustering phenomenon and presents some of the consequences
of this effect. It also presents the collapse phenomenon. Section 5 is
devoted to the description of some of the misconceptions that are
still abound concerning granular temperatures. Finally, Section 6
provides a brief summary and some additional comments, and
briefly outlines some recent developments and extensions of the
concept of granular temperature.
2. Granular temperature and kinetic theory
One of the main applications of the concept of granular
temperature is the construction of kinetic and (from them)
hydrodynamic theories for granular gases. In the pioneering
papers (as well as some recent ones) in the field, the pertinent
Boltzmann equation [26] was not directly employed. Instead, a
set of relations among the low order moments of the (single
particle) distribution function, f (r,v,t)(that represents the
number density of particles at point r and time t whose velocity
is v), known as the Enskog equations, see e.g., [8,9,27] and
references therein, was used (these equations can be obtained
from the Boltzmann equation, and, more generally from the
BBGKY hierarchy). Since the Enskog equations are not closed
(the equation for a given moment depends on higher order
moments), closure was obtained by conjecturing a form of the
distribution function, typically a Maxwellian multiplied by
terms that depend on the local gradients of the hydrodynamic
fields. These guesses missed certain terms that systematic
theories [13,14] obtain. For instance, when the gradients vanish
these guesses would entail the assumption that the distribution
function is Maxwellian. However, in the absence of gradients
(or forcing) a granular gas is in a “homogeneous cooling state”
(HCS) whose distribution is known not to be Maxwellian, see
[15,28] and references therein. The missed corrections are
important when the grain collisions are not close to elastic and
therefore the “old” theories are at best relevant in the nearelastic case. The above theories also account for finite density
effects by using a correction to the Boltzmann equation
proposed by Enskog. The hydrodynamic equations of motion
resulting from the application of these theories resemble the
Navier–Stokes equations with one (very important) difference:
the equation for the energy density (or granular temperature)
contains a “sink term” that represents the loss of energy due to the
inelasticity of the collisions (as a matter of fact, the form of these
equations can be shown to be very general, much beyond the
realm of validity of kinetic theory). This term is “responsible” for
the existence of steady granular shear flows (else the work by
shear would have “heated” the granular system indefinitely) and
for many other phenomena that characterize granular gases. The
agreement of the early theories with experiments is reasonable
given their approximate nature, see e.g., [12].
There are two main systematic approaches to the kinetics of
granular gases. The first is based on the observation that in the
limit of elastic collisions and when the gradients vanish, the
distribution function of the grain velocities is Maxwellian,
corresponding to a state of (local) equilibrium. This limit is not
singular [13] and therefore one can expand the solution in two
small parameters around this state of equilibrium: the Knudsen
number (a non-dimensional measure of gradients, defined as the
ratio of the mean free path and the scale over which the
hydrodynamic fields vary) and the degree of inelasticity, defined
as ϵ ≡ 1 − e2, where e is the coefficient of normal restitution (when
more complex interaction are allowed, the definition of the degree
of inelasticity needs to be modified; also, additional small
parameters appear when tangential restitution is allowed). This
theory is restricted to the case of nearly-elastic collisions. The
second approach [14] is based on an expansion in the Knudsen
number (or gradients) without any other small parameters. The
zeroth order in the latter expansion is the HCS, for any value of the
coefficient (or coefficients) of restitution. The respective expansions in both methods are extensions of the classical Chapman–
Enskog expansion of kinetic theory [26]. Both approaches
produce constitutive relations, which agree with each other in
the common domain of validity (nearly-elastic collisions).
Accurate results for the transport coefficients for granular gases
have been recently obtained [28] on the basis of the method
proposed in [14].
Among the novel results obtained using the systematic
approaches we would like to mention the discovery of a new
contribution to the heat flux, which is proportional to the
gradient of the number density (of course, the Fourier term,
which is proportional to the gradient of the granular temperature
is there as well). This term has been shown to strongly affect
e.g., the temperature profile in a vertically vibrated granular bed
[29], predicting a nonmonotonic dependence of the temperature
on the height. Another important result [13] is that the normal
stress differences observed in simulations and experiments
[12,15] stem from contributions of the Burnett terms, i.e. terms
in the constitutive relations which are of second-order in the
gradients and therefore beyond the Navier–Stokes level of
description. The latter result quantitatively agrees with simulations. The reason that the Burnett terms are of importance for
granular gases is discussed below in Section (3).
While the pioneering studies of granular kinetics did employ
the Enskog correction, the first systematic studies did not. This
I. Goldhirsch / Powder Technology 182 (2008) 130–136
seems to have led some people to believe that the systematic
studies are only good for very low volume fractions. This was
rectified by papers in which moderately dense granular gases
were studied [30] on the basis of the Enskog corrected
Boltzmann equation. Another approach to the study of the
dynamics of moderately dense granular gases involves the
derivation of corresponding Green–Kubo relations [31,32],
whose validity is not limited in density (see also the article by
Brey and Prados in this issue). This problem is currently under
active research. Yet another method to extend the validity of
kinetic theory for granular gases is to go to higher orders in the
BBGKY hierarchy and include the “ring contributions” [33]
that account for events in which a particle recollides with
another particle after the other particle collides with at least one
more (third) particle. These events are known to contribute to
the buildup of precollisional correlations.
A problem that has not been yet satisfactorily addressed in
the kinetic theories of granular gases is that of the “molecular
chaos” assumption. The only assumption underlying the derivation of the Boltzmann equation for (molecular or granular) gases
is that the velocities of particles about to collide are not correlated. This assumption has been validated in simulations of
elastically colliding particles at low density but inelastically
colliding particles do not possess this property (except for the
near elastic case), cf. e.g., [34]. One of the basic reasons for this
difference is that in inelastic collisions the normal component of
the relative velocity of the colliding particles is decreased. The
Enskog correction accounts for part of these correlations but not
all (e.g., their directional dependence). The successes of the
existing granular flow models suggest that in most cases these
correlations do not strongly affect the hydrodynamic constitutive
relations, but the matter deserves further study. The mentioned
derivations of Green–Kubo relations and studies of higher order
equations in the BBGKY hierarchy should contribute to the
elucidation of the role of precollisional correlations.
3. Lack of strong scale separation in granular gases
In molecular fluids under non-extreme conditions the mean
free paths are much smaller than the scales on which gradients
change (small Knudsen numbers) and the mean free times are
much smaller than typical macroscopic times scales. It turns out
that in granular gases (as a matter of fact, also in solids) there is
no strong scale separation, except in the near-elastic case [35].
This fact and some of its consequences seem to underlie a
certain confusion in part of the community, and has led to
“hasty” conclusions that the concept of granular temperature as
well as kinetic theory may not be relevant to granular gases. The
reason for the lack of strong scale separation is presented
immediately below.
Consider a simply sheared (monodisperse) granular gas, i.e.
one whose density and granular temperature are constant and
whose velocity field is given by: V(r,t) = γyx̂, where γ denotes
the shear rate and x̂ is a unit vector. In the absence of gravity (for
simplicity) the only ‘input’ parameter in this problem that has
dimensions of time is γ− 1. The only relevant length scale is the
mean free path ℓ (which equals nr1T , where n is the number
133
density and σ T the total collisional cross section). It follows
from dimensional analysis that the granular temperature, T,
must satisfy: T∝γ2ℓ2. Next, assume that the collisions are
characterized by a fixed coefficient of normal restitution, e. The
degree of inelasticity is defined by ϵ ≡ 1 − e2. Clearly, in the
absence of inelasticity the granular temperature steadily increases
because of the shear work. Therefore the steady state temperature
diverges2 when
ϵ is set to zero. One can therefore assume that
2
T ¼ C g ℓ , where C is a constant. Phenomenological models [10]
as well as accurate kinetic theoretical studies [15] corroborate this
result and determine C to be about 1 in two dimensions and 3 in
three dimensions, i.e. O(1). The mean free time, τ, is given by the
ratio of the mean free path and the thermal speed: s ¼ pℓffiffiTffi. The
macroscopic time scale in this problem is the inverse shear rate.
Therefore the ratio of the microscopic time scale, τ, and the
s
macroscopic time scale is: 1=g
¼ sg ¼ pℓγffiffiTffi. Substituting the above
pffi
s
formula for T one obtains: 1=g
¼ pffiffiffi. Thus, unless the collisions
C
are nearly-elastic, the mean free time is of the same order as the
macroscopic time scale, though smaller than unity. Furthermore,
the change of velocity over a distance that equals the mean free
path in the y direction is γℓ, i.e. of the order of the thermal speed,
hence the gradient of the velocity is “large” compared to the
pffi
ℓ ∂Vx
molecular case: p∂yffiffiTffi ¼ pffiffiCffi.
The lack of strong scale separation demonstrated above is of
very general nature in granular materials and has numerous
consequences, part of which are mentioned below. First, since
the (non-dimensionalized by the mean free path) gradients of
the fields are typically not as tiny as in molecular gases, one
may have to consider higher order terms in the gradient expansion. Returning to the example of simple shear flow, the ratio of
the stress component, σxx, and the pressure, p (proportional to
the trace of the stress tensor) is dimensionless, and must be a
function of the dimensionless shear rate, pgℓffiffiTffi. Since the stress is
invariant to the transformation
γ→−γ
(and assuming analyticity)
2 2
one obtains: rxx ¼ p 1 þ Cx gTℓ , to leadingnonvanishing
order
g2 ℓ 2
in γ, where Cx is a number. Similarly, ryy ¼ p 1 Cx T , since
the average stress is the pressure. Note that the corrections are of
second order in the gradients, hence they pertain to the Burnett
order. The value of Cx is of the order of unity, e.g., for two
dimensional disks Cx = 0.679 (see [15] and refs therein). Therefore: rrxxyy ¼
22
1þCx g Tℓ
22
1Cx g Tℓ
2 2
. In air at STP conditions and a shear rate of 1 s− 1
the ratio gTℓ is of the order of 10−19 and thus the normal stress can
justifiably be considered to be isotropic. In contrast, in a granular
gas one obtains
(after substituting the above formula for T ):
pffi
pffiffi
1þC
x
rxx
ffi
pC
ryy ¼ 1Cx pffiffi , an O(1) entity that is typically significantly different
C
from unity. Indeed, normal stress differences have been observed
in flowing granular materials and the above formula has been
verified against simulations [15]. A curious consequence of the
lack of scale separation is the finding (O. Herbst and I.G.,
unpublished) that granular shear flows have a sizeable streamwise
heat flux (of the order of the spanwise heat flux), again – a Burnett
contribution. This result has been quantitatively corroborated by
molecular dynamics simulations.
134
I. Goldhirsch / Powder Technology 182 (2008) 130–136
The Burnett coefficients for molecular fluids of finite density
(see [15] and references therein) are known to diverge (the
higher order corrections to Navier–Stokes are non-analytic in
the gradients) and even if one ignores this fact (it is ignored in
the gas dynamic community that manages to use Burnett order
hydrodynamics to model shocks) the Burnett equations are ill
posed and do not have physically based boundary conditions.
Some of these problems can be overcome (see the work by Jin and
Slemrod cited in [15]) whereas others are still open. Whether the
same problems afflict granular gases is yet unknown. Accepting
the proven relevance of the Burnett contributions and the lack of
strong scale separation as the reason for their importance, one may
ask if even higher order terms in the gradient expansion (the next
order after Burnett is known as super-Burnett) are important.
Indeed such a case exists [36], but usually Burnett order hydrodynamics suffices for the description of the dynamics of granular
gases, and often even Navier–Stokes hydrodynamics does well.
While systems in which scale separation is weak or nonexistent are known (e.g., turbulent fluids, most rheological
systems), their description often requires the use of scaledependent transport coefficients (such as the “eddy viscosity” of
turbulent flows). Furthermore, the fields themselves (including
the granular temperature) can be scale dependent, since they
may significantly change on the scale of a mean free path. An
analysis of this problem as it pertains to granular gases can be
found in [37]. It seems that the problems that emanate from the
lack of scale separation in granular gases are not as severe as
those encountered in e.g., fluid mechanical turbulence. The
reason is that granular gases do possess scale separation, weak
as it may be. The description of steady states of granular gases
may require, as mentioned, the use of corrections to the Navier–
Stokes constitutive relations but this is merely a solvable technical problem. More serious problems may be encountered in
the realm of the transient dynamics of granular gases, since in
the latter case, due to the weak scale separation, the velocity
distribution function may not “have sufficient time” to saturate
to its hydrodynamic value; in particular, the hydrodynamic
constitutive relations may not saturate. Consider, for instance,
the heat flux. When there is strong scale separation (e.g, when
the collisions are nearly-elastic) the heat flux induced by a
temperature gradient is given by the corresponding hydrodynamic constitutive relation, since even when the external conditions (e.g., imposed gradients) change in time, the collisions are
sufficiently frequent to establish a local distribution function
that depends on the external parameters on time scales that are
much shorter than the (typical) macroscopic time scales. When
the temperature gradient changes “too rapidly” the heat flux
cannot follow the change and need not be given by the hydrodynamic constitutive relation. One way of overcoming this
problem is to take the heat flux as a dynamical variable (same for
the stress tensor), as is done in some Grad-based theories, see e.g.,
[38] and references therein, or use an even more extended set of
hydrodynamic fields. In this case one needs to modify the
Chapman–Enskog expansion since the unresolved degrees of
freedom (e.g., peculiar velocity fluctuations) are assumed to be
enslaved to the extended (rather than the original) set of hydrodynamic fields.
Another consequence of the weak scale separation becomes
evident when one analyzes numerical simulations. As there
need not be a significant plateau of scales for which the various
fields are nearly scale-independent one may obtain results that
are manifestly scale dependent [37]. This again does not pose a
problem for the study of steady states, since then one can use
small coarse graining volumes and large coarse graining times
to obtain steady (and convergent) statistics. However, the
description of transients requires special care and the establishment of connections with descriptions by ensembles is less
trivial and not satisfactorily solved. It is possible that one may
need to extend granular hydrodynamics in this case by using
stochastic rather then deterministic equations (in addition to the
extension of the set of fields). All of this really reinforces the
notion that granular gases are rheological fluids (but not in the
extreme sense of the term).
4. Clustering and collapse in granular gases
The appearance of density microstructures in granular gases
has been known since 1991 [12,15]. The clustering mechanism
proposed in [39] has been verified in simulations, detailed
theories and experiments [15]. This mechanism, often referred
to as “collisional cooling”, is explained next. Consider a mass
density fluctuation in a granular gas (as a many-body system, a
granular gas experiences fluctuations of every density). In a
region where the density is higher than in its ambient the rate of
collisions (proportional to the square of the number density) is
larger. Since the collisions are inelastic the energy density (or
granular temperature) decreases in this region at a relatively
rapid rate, causing a corresponding decrease in the pressure. The
result is that mass flows from the relatively dilute ambient into
the dense region further increasing its density until the process
is stopped by other mechanisms. Clustering has been observed
in granular shear flows, vibrated and other granular systems.
One of the consequences of the clustering mechanism is that
(at least) two different temperatures can coexist in the same
system (in stark contrast to molecular systems where the temperature is usually a continuous field, except for the case of
shocks). Numerous other consequences of clustering are
mentioned in [15] including a curious phenomenon coined the
“Maxwell demon” of granular gases. Since some clusters may
be quite small in size they may be sub-resolution objects in
coarse grained theories and contribute to the temperature and
density fluctuations.
A related but distinct phenomenon that characterizes (ideal)
hard grains is that of collapse. Consider e.g., a ball dropped
from rest at height h above a floor and let e be the coefficient
of restitution between the ball and the floor. As shown in
practically any elementary text on mechanics, the ball will reach
a maximal height e2h after its collision with the floor. The
sequence of heights, hn (the maximal height after hitting the
floor n times) satisfies hn = e2hn − 1. Similarly the time that
elapses between the instant at which the height of the ball is hn
and that in which it is hn + 1 satisfies the recursion relation:
τn +P
1 = eτn. Therefore the total time it takes the ball to fully relax
s0
is: l
n¼0 sn ¼ 1e. Since the coefficient of restitution is smaller
I. Goldhirsch / Powder Technology 182 (2008) 130–136
than unity this time is finite but the number of collisions of the
ball with the floor is infinite. A similar collapse phenomenon
can happen in a granular gas (see [15] and references therein).
This fact leads to some problems in simulations of hard (but not
soft) sphere gases. The collapse phenomenon can be preceded
by clustering [15] and it is one of the reasons that Kadanoff
expressed doubts about kinetic descriptions of granular matter.
As mentioned, collapse does not truly occur in soft sphere
systems and it basically does not occur in forced three dimensional granular gases either. Therefore it does not give rise to a
serious problem in the application of granular kinetics.
5. Some misconceptions concerning granular temperature
The following comprise a few prevalent misconceptions
concerning the granular temperature:
(1) The velocity distribution needs to be normal or Maxwellian
for the granular temperature to “make sense”. This is not
true since the distribution is only Maxwellian in equilibrium. As shown above, granular gases are always in nonequilibrium states. Furthermore, both measured velocity
distributions and calculated ones (e.g., in [28]) are not
Maxwellian.
(2) The distribution function needs to be near-Maxwellian for
the kinetic approach to be valid. This is not correct either. It
has been shown in [28] that one could use kinetic theory to
obtain accurate results even when the distribution function
is far from Maxwellian or when one expands around a
Maxwellian corresponding to a “wrong” temperature (i.e.
one that does not equal the actual temperature at the
considered position).
(3) The “correct” temperature of a granular fluid is really a
tensor and the temperature is usually anisotropic. This
misconception seems to have its roots in a model [40] for
the distribution of the granular peculiar velocities of the
1
form e2uKu where K is a matrix, following a similar usage
by astrophysicists. It is easy to see that (since the kinetic
stress tensor σkin is proportional to K−1) all that this statement says is that the kinetic stress is a tensor and it is usually
anisotropic. This anisotropy is obtained from kinetic theories
without the need to model the distribution function as a
“generalized Maxwellian” and it does not imply that one
cannot employ the “usual” scalar granular temperature even
when the stress tensor is “very anisotropic”.
(4) Some kinetic theories account for equipartition whereas
others don't. This is true for the old kinetic theories in
which the form of the distribution functions was conjectured. It is not true for the modern systematic kinetic
approaches in which the distribution function is derived, as
explained above.
(5) The concept of granular temperature or kinetic theory
does not apply to dense granular gases. Not true. The
Enskog–Boltzmann equation has been used to obtain
hydrodynamic constitutive relations for moderately dense
granular gases and these have been successfully applied
even to “very dense” granular gases [22–24].
135
(6) Fluctuations in granular systems are large and there is a
lack of scale separation, therefore describing them using
kinetic theory is useless. Scale separation is weak in
granular gases but it does exist. Therefore Burnett
contributions are important, as explained above, but this
does not disqualify the kinetic approach.
(7) “Granular hydrodynamics does not describe some
experiments/simulations”. Though no theory comes
with a warranty, there seems to be a “rich man's problem”
in this field, namely that due to the fact that the grains are
macroscopic and one can often measure the dynamics of
each and every grain, hydrodynamic theories are often
expected to describe scales that they are not designed to
model or resolve. There are though (surprising) examples
in which granular hydrodynamics provides good descriptions of systems that are only five grains deep (see e.g. the
paper by Forterre and Pouliquen cited in [15]).
In addition the reader should be warned that many (usually
approximate) statistical approaches to granular gases call
themselves kinetic theoretical. While there is nothing wrong
with attempts to derive simplified theories (mostly of the mean
field variety), the possible failings of some of these models
should not be used as arguments against kinetic theory.
6. Conclusion
The granular temperature comprises an important characterization of granular fluids and is central to their kinetic and hydrodynamic descriptions. Due to the lack of strong scale separation
one needs to go beyond the Navier–Stokes level of description
and this raises certain problems that require further study. A good
description of the transient dynamics of granular fluids is yet to be
developed though it is possible that the presently available
hydrodynamic equations are good enough for the description of (at
least) part of the transient dynamics. An improved description of
transient granular dynamics will most likely require the extension
of the set of hydrodynamic fields. This problem clearly needs
further study as well. The description of dense granular fluids is far
from complete and requires, among other things, a better
elucidation of the nature of precollisional correlations. The
possibility that one may need to employ stochastic equations to
describe granular flows has been raised. The issue of granular
fluids in which persistent contacts are prevalent (and the
interactions are not collisional) has not been discussed here.
This is an important problem of much current interest.
Finally, it is worth mentioning the existence of relatively recent
theories that are based on a proposal by Edwards and co-workers,
which define an effective temperature for static granular assemblies
(see the paper by Serero et al in this issue and references therein).
While these theories have not been related heretofore to the now
classical granular temperature (which is kinetic in nature), experiments indicate that the Edwards temperature can be used in fluctuation–dissipation relations, thereby providing a link between this
new temperature and granular dynamics. Whether this approach
will lead to a uniform description of granular dynamics and/or a
universal definition of granular temperature remains to be seen.
136
I. Goldhirsch / Powder Technology 182 (2008) 130–136
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