Abstract
Using a different method, the balance equations of mass, momentum, and kinetic energy are derived for an arbitrary one-dimensional system of inelastic particles from its kinematic description. The failure of the hydrodynamic equations for such dissipative systems must be attributed to the inconsistency of constitutive relations with the microscopic structure of the system. Among the constitutive relations, the Fourier law of heat conduction is the most inappropriate.
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C. Vamoş, N. Suciu, A. Georgescu (1997), Hydrodynamic equations for one-dimensional systems of inelastic particles, Phys. Rev. E, 55(5), 6277-6280, doi: 10.1103/PhysRevE.55.6277
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Hydrodynamic equations for one-dimensional systems of inelastic particles
C. Vamos¸ and N. Suciu
‘‘T. Popoviciu’’ Institute of Numerical Analysis, P.O. Box 68, 3400 Cluj-Napoca 1, Romania
A. Georgescu
Institute of Applied Mathematics, P.O. Box 1-24, Ro-70700 Bucharest, Romania
~Received 9 December 1996!
Using a different method, the balance equations of mass, momentum, and kinetic energy are derived for an
arbitrary one-dimensional system of inelastic particles from its kinematic description. The failure of the hy-
drodynamic equations for such dissipative systems must be attributed to the inconsistency of constitutive
relations with the microscopic structure of the system. Among the constitutive relations, the Fourier law of heat
conduction is the most inappropriate. @S1063-651X~97!06505-7#
PACS number~s!: 47.50.1d, 05.20.Dd, 83.20.Di, 81.05.Rm
Recently, Du, Li, and Kadanoff @1# have shown that with
the introduction of a very small dissipation in the micro-
scopic dynamics, the usual hydrodynamics approach for
granular flows fails to give a correct picture for a many-
particle system. They have numerically simulated the one-
dimensional motion of N sizeless inelastic particles of iden-
tical mass, confined by two walls of infinite mass. If this
system is driven by collisions with the boundary walls, most
of the particles become squeezed in a small ‘‘clump’’ and
move with a smaller velocity than the other particles. This
result is independent of how the energy is pumped in at the
boundaries. Contrary to the prediction of the usual hydrody-
namic equations, the clump does not disappear as the dissi-
pation tends to zero. In fact, the particles in the clump are
squeezed into a smaller space and move with slower speed.
A similar phenomenon has been reported in the cooling of
inelastic particle systems @2–4#.
In this paper we investigate the reason for the hydrody-
namics breakdown for this corpuscular system. The hydro-
dynamic description of a continuum consists of two parts.
The first is a set of balance equations for local macroscopic
fields modeling fundamental physical quantities ~mass, mo-
mentum, energy, entropy, etc.!@5#. The balance equations
are postulated and have a general form and validity. In gen-
eral, the number of continuous fields is greater than the num-
ber of balance equations, so additional relations are needed
~e.g., the expression of the stress tensor for a specified ma-
terial!. In continuum mechanics, such relations are referred
to as ‘‘constitutive relations’’ @6# and represent the second
part of the hydrodynamic description. Thus the hydrody-
namic equations always consist of balance equations and
constitutive relations. The constitutive relations describe the
macroscopic properties of the material and are related to the
specific microscopic structure of the corpuscular system.
The balance equations are valid for any conserved physi-
cal quantity @5,7,8#. If the physical quantity is not conserved,
then the balance equations are supplemented with the supply
and production densities for the physical quantity, for in-
stance, the entropy production for irreversible processes in a
continuum. The same approach works for the kinetic energy
lost in inelastic collisions. Therefore, the contradiction be-
tween the numerical results in @1# and the solution of the
usual hydrodynamic equations cannot be associated with the
balance equations, but with constitutive relations inadequate
to the microscopic structure.
Du, Li, and Kadanoff @1# have used the constitutive rela-
tions for relatively dense granular materials in high shear
flow derived in @9,10#. In that case the flux of the kinetic
energy is proportional to the gradient of the kinetic tempera-
ture ~the Fourier law! and the dissipation rate of the kinetic
energy is proportional to the kinetic temperature to
3
2
power.
But the Fourier law is not valid for the corpuscular system in
@1#. Indeed, the fast particle running between the left wall
and the ‘‘clump’’ transports the energy gained from the col-
lision with the wall and loses it during the inelastic collision
with the neighboring particle. This nonvanishing energy flux
near the wall corresponds to a vanishing gradient of the ki-
netic temperature, because in this region only the fast par-
ticle can move and its average state does not depend on
position.
The numerical experiment in @1# contains only 100 par-
ticles and an unquestionable continuum macroscopic inter-
pretation of the results is difficult to obtain. In this paper we
use a simplified form of a more general approach valid for
three-dimensional systems as well @11#. A space-time coarse-
grained average for an arbitrary physical quantity attached to
the particles is defined. Its first-order partial derivatives are
almost everywhere continuous. For an arbitrary number of
particles, we prove that this space-time average satisfies a
relation of the same form as the balance equations. If the
temporal and spatial averaging intervals can be chosen such
that the discontinuities of the partial derivatives vanish or
become negligible, the usual continuous fields and balance
equations are obtained.
Our approach has the advantage that the macroscopic con-
tinuous description does not depend on the number of the
particles or the microscopic dynamical laws. The space-time
coarse-grained averages can be defined and satisfy a balance
equation even for a single particle. Also, we can analyze the
macroscopic properties of the corpuscular system without a
PHYSICAL REVIEW E MAY 1997 VOLUME 55, NUMBER 5
55 1063-651X/97/55~5!/6277~4!/$10.00 6277 © 1997 The American Physical Society
direct study of the dynamical system associated with the par-
ticles. For instance, Du, Li, and Kadanoff’s corpuscular sys-
tem does not verify the equipartition of energy, but we do
not need to find out the cause of this behavior. First we
derive the balance equations of mass, momentum, and ki-
netic energy for the same system as in @1#. Then we apply
our method to a very simple situation such that all the terms
in the balance equations can be explicitly determined. The
constitutive relations are directly tested and we prove that in
this case the Fourier law of heat conduction is not valid.
Consider N identical sizeless particles of mass m . We
study the evolution of the system during the time interval I
5@ 0,T # . The particles are confined within the spatial interval
@0,1# by collisions with two walls of infinite mass. The col-
lisions between particles are inelastic and are characterized
by the numerical parameter e 5(1 2r )/2. The restitution co-
efficient r is defined in terms of the particles velocities after
and before the collision by v
1
8 2v
2
8 52r ( v
1
2v
2
). The par-
ticles interact only when they collide and there are no exter-
nal forces acting on the particles, except that due to the col-
lisions with the walls. We assume that the system kinematics
is known, i.e., the position of each particle i <N is a given
function of time x
i
: I →@ 0,1# . When a collision occurs, the
corresponding velocities j
i
5x
˙
i
undergo jumps and between
collisions the motion is uniform. We suppose that the total
number of collisions during I is finite.
Let w
i
( t ), t PI , be the real function of time describing the
variation of an arbitrary physical quantity w attached to the
i th particle. In the following, w
i
will represent only the mass
( m ), the momentum ( m j
i
), and the kinetic energy
( m j
i
2
/2). Since the only variations of the velocity j
i
are the
jumps from one constant value to another, the temporal de-
rivative of w
i
identically vanishes w
˙
i
[0 almost everywhere.
Consider two real parameters 0 ,t ,T /2 and a .0 and de-
fine the function
^
w & ~ x , t ! 5
1
4 t a
(
i 51
N
E
t 2t
t 1t
G
i
~ x , t 8 ! dt 8 , ~1!
where
G
i
~ x , t ! 5w
i
~ t ! H „a 2u x
i
~ t ! 2x u …, ~2!
and H is the left continuous Heaviside function. A nonvan-
ishing contribution to ^w& is due only to particles lying in the
spatial interval ( x 2a , x 1a ) during the temporal interval ( t
2t , t 1t ). Therefore, ^
w & ( x , t ) characterizes the mean dis-
tribution of w about the point x and the time t . It is a coarse-
grained average over the space and time intervals defined by
a and t, i.e., the density of w. Obviously, ^w& also depends on
the parameters a and t, but we do not explicitly write this
dependence. The average ^w& is nonvanishing only if x P
( 2a ,11a ) and the integral interval in Eq. ~1! is contained in
I only if t P( t , T 2t ).
For a given x , the integrand ~2! is a continuous function,
except at a finite number of points where it has discontinui-
ties of jump type. Hence G
i
is Riemann integrable and the
partial derivative with respect to t of ^w& is
]
t
^
w & 5
1
4 t a
(
i 51
N
@ G
i
~ x , t 1t ! 2G
i
~ x , t 2t !# . ~3!
The function ^w& depends on x through the instants u when
the i th particle enters or leaves the interval ( x 2a , x 1a ).
These instants are given by the zeros of the equations
x
i
~ u ! 2x 6a 50,
and using the implicit function theorem we obtain du / dx
51/j
i
( u ). If u P( t 2t , t 1t ), then u occurs as the integra-
tion limit in Eq. ~1! and the derivative of ^w& with respect to
x is
]
x
^
w & 5
1
4 t a
(
i 51
N
F
(
u PU
i
8
w
i
~ u !
j
i
~ u !
2
(
u PU
i
9
w
i
~ u !
j
i
~ u !
G
, ~4!
where U
i
8 ( U
i
9 ) is the set containing the instants when the
i th particle leaves ~enters! the interval ( x 2a , x 1a ) during
the interval ( t 2t , t 1t ). One can prove that the partial de-
rivatives ~3! and ~4! are almost everywhere continuous @11#.
Relation ~3! shows that ]
t
^
w & is related to the change of
G
i
from t 2t to t 1t . Since w
˙
i
[0 and H
˙
[0 almost every-
where, the changes of G
i
are only jumps. When the i th par-
ticle enters @leaves# the interval ( x 2a , x 1a ), the change of
G
i
is 1w
i
( u ) @ 2w
i
( u ) # . According to Eq. ~4!, the corre-
sponding part of ]
t
^
w & is equal to 2]
x
^
wj & . The change of
G
i
due to w
i
occurs when the particles collide inside the
interval ( x 2a , x 1a ). The collision part of ]
t
^
w & is
d
c
w 5
1
4 t a
(
i 51
N
(
s PV
i
@
w
i
~ s 20 ! 2w
i
~ s 10 !# , ~5!
where V
i
is the set containing the instants s when the i th
particle collides inside ( x 2a , x 1a ) during ( t 2t , t 1t ),
w
i
( s 10) is the limit to the left, and w
i
( s 20) is the limit to
the right. We deduce that the relation
]
t
^
w & 1]
x
^
wj & 5d
c
w ~6!
is always true. In the following we show that this identity is
the general form of the balance equations for the corpuscular
system considered.
First we apply the identity ~6! for mass, i.e., w
i
5m . The
space-time average ~1! becomes the mass density ^ m & . Since
the masses of the particles are identical, we have ^ m &
5mc , where c 5^ 1 & is the particle number density or the
concentration obtained from Eq. ~1!, for w
i
[1. The mean
velocity field v is defined by ^
j & 5c v if c 0 and is zero
otherwise. ~In @1# the velocity field is incorrectly defined, but
this oversight has not affected the hydrodynamic equations
for granular flows used in @1#.! The term d
c
w defined by Eq.
~5! vanishes because the collisions do not imply the variation
of the mass of particles or their number. Then relation ~6!
becomes
]
t
c 1]
x
~ c v ! 50, ~7!
which is the continuity equation.
For momentum, we choose w
i
5m j
i
and then ^
w &
5mc v . The second term on the left-hand side of Eq. ~6! can
6278 55 BRIEF REPORTS
be written as the sum of two terms, namely, ^ m j
2
& 5mc v
2
1m ^ ( j 2v )
2
& . The first term is due to the mean motion of
the particles and the second one can be interpreted as the
‘‘microscopic’’ flux of momentum. The momentum is con-
served when two particles collide such that d
c
j vanishes. But
the collision with the walls induces a variation of the particle
momentum and d
c
j is nonvanishing for x P( 2a , a ) l(1
2a ,11a ). In this case Eq. ~6! represents the balance equa-
tion of momentum
]
t
~ c v ! 1]
x
~ c v
2
! 1]
x
~ c u ! 5d
c
j . ~8!
Here we have introduced the kinetic temperature u 5^ ( j
2v )
2
& / c ~when c 0 !, known for the granular materials as
the granular or fluctuating temperature.
In a similar manner, for w
i
5m j
i
2
/2 we obtain from Eq.
~6! the balance equation of the kinetic energy
]
t
e 1]
x
~ v e ! 1]
x
~ 2 v c u ! 1]
x
q 5d
c
j
2
, ~9!
where e 5c ( v
2
1u ) is proportional to the density of the ki-
netic energy and q 5^ ( j 2v )
3
& is the microscopic flux of the
granular temperature. The collision term does not vanish be-
cause of the energy loss at inelastic collisions.
Relations ~7! – ~9! are of the same form as the usual bal-
ance equations, but they have a wider validity. We have not
imposed any restriction to the number of particles N , which
can be very small. Therefore, these relations also hold for
simple mechanical systems not possessing statistical proper-
ties characteristic to thermodynamic systems, but they pre-
serve the discontinuous nature of the initial corpuscular de-
scription. The first-order partial derivatives ~3! and ~4! have
discontinuous variations when a collision occurs or when a
particle enters or leaves the spatial interval ( x 2a , x 1a ). For
certain corpuscular systems, we can choose a and t such that
these discontinuous variations vanish or are negligible, and
the smoothness specific to the continuous fields is obtained.
When t →0, definition ~1! is reduced to an instantaneous
space average. In this case ^w& coincides with the usual mac-
roscopic fields if at an arbitrary time t the particles in ( x
2a , x 1a ) form a near-equilibrium thermodynamic system.
If the local equilibrium holds, then there is an a
min
such that
for a .a
min
this condition is satisfied. Hence the usual bal-
ance equations are particular cases of the relations ~7! – ~9!.
Definition ~1! also contains a time average. The time in-
tegral in Eq. ~1! is identical to that used in the formulation of
the ergodic hypothesis. Therefore, for a nonvanishing value
of t, more microstates of the corpuscular system are included
in the average ^w& and the smoothness of ^w& is improved in
comparison to an instantaneous space average. The upper
limit of t is given by the requirement that the time averaging
should not distort the time evolution of the local fields ^w&. If
the corpuscular system is in a steady state, we can take
t →‘ and then the space averaging can be reduced even to a
point ( a →0).
Relations ~6! – ~9! are either identities or equations, ac-
cording to the available information on the microscopic
structure. If the motion of each particle is explicitly known,
then Eqs. ~6! – ~9! are simple identities containing only
known functions. Otherwise, they become the balance equa-
tions for the averages ^w&, which now are unknown func-
tions, and to obtain a solvable problem, the constitutive re-
lations are needed. To obtain some information on the
inadequate constitutive relations in @1#, we consider a very
simple case of Du, Li, and Kadanoff’s corpuscular system
such that the motion of the particles is known and the aver-
ages ^w& can be determined.
The system contains only two particles. The simplest pe-
riodic motion occurs if the following restrictive conditions
are imposed: 0 ,e ,
1
3
; the left wall returns the first particle
with constant velocity v
0
51; the collision of the second
particle with the right wall is perfectly elastic. The collisions
of the two particles always take place at the same point x
c
5
1
2
2e
2
/(1 22 e 2e
2
) and the velocity of the first particle
after the collision is v
1
8 5(1 23 e )/(1 1e ). The velocity
modulus of the second particle is the same u v
2
u 5(1
2e )/(1 1e ) before and after the collision. The period of the
motion is (1 2e
2
)/(1 22 e 2e
2
).
Usually, in continuum mechanics, the spatial average is
taken over a volume microscopic in comparison to the mac-
roscopic scale of the continuous phenomenon. For our ap-
proach this corresponds to a →0. First we take the average
~1! over a spatial interval much smaller than the system
length, i.e., a !1, and then we study the limit case. For a
more condensed description we distinguish five spatial re-
gions. Each of the two boundary regions ( 2a , a ) and (1
2a ,11a ) contains one wall and the particles can move only
within a subdomain of the averaging spatial interval ( x
2a , x 1a ). In the transition region ( x
c
2a , x
c
1a ) both par-
ticles contribute to the average ~1!. The transition region and
the boundary regions bound two ‘‘uniform’’ regions ( a , x
c
2a ) and ( x
c
1a ,12a ).
This is a simple mechanical system and the averages ^w&
could present discontinuous variations. The periodicity of the
motion allows one to completely eliminate these discontinui-
ties, although the local equilibrium assumption is not satis-
fied. For a fixed, the necessary smoothness is obtained if 2t
is equal to the period of the motion. Then Eq. ~3! implies that
all the temporal derivatives vanish and we could say that the
system is in a ‘‘steady’’ state.
We estimate the quantities in the balance equations ~7! –
~9!. The mean velocity field identically vanishes v [0 be-
cause ^
j & [0. In the uniform regions the concentration c
takes two constant values c
1
5(1 2e )/ t (1 23 e ) and c
2
5(1 1e )/ t (1 2e ). In the boundary and transition regions
c has a linear variation between these constant values and the
exterior null value. The granular temperature u has a linear
variation only in the transition region between the constant
values u
1
5(1 23 e )/(1 1e ) and u
2
5(1 2e )
2
/(1 1e )
2
. We
notice that u has the same value in a uniform region and the
adjacent boundary region.
In the hydrodynamic equations used by Du, Li,
and Kadanoff @1#, the other quantities are given by constitu-
tive relations: d
c
j 50, q 52]
x
( C
1
u
3/2
), and d
c
j
2
5
2C
2
e c
2
u
3/2
with C
1
and C
2
numerical constants. In our
case, they can be calculated from the microscopic motion of
the corpuscular system. In the left uniform region, q has the
constant value q
1
54 e (1 2e )/ t (1 1e )
2
. In the transition re-
gion and the left boundary region it has a linear variation,
while for x .x
c
1a it vanishes. The momentum is not con-
served when the particles collide with the walls, such that in
the boundary regions d
c
j 56(1 2e )/2a t (1 1e ). For x
55 6279 BRIEF REPORTS
P(a,12a ), we have d
c
j 50. The kinetic energy is conserved
for the elastic collision with the right wall and the nonvan-
ishing values of the corresponding collision term are d
c
j
2
562 e (1 2e )/ a t (1 1e )
2
if x P( 2a , a ) or x P( x
c
2a , x
c
1a ). It is easy to test that the balance equations ~7! – ~9! are
identically satisfied, while all the constitutive relations for
granular materials used in @1# are incorrect.
The limit a →0 simplifies the above continuous descrip-
tion. The boundary regions disappear and the domain of defi-
nition of the continuous fields becomes ~0,1!. The transition
region is reduced to the collision point x
c
separating the two
uniform regions. In these uniform regions, all the terms of
the balance equations ~7! – ~9! are equal to zero. However, the
values of the continuous fields correspond to a state that is
analogous to that described in @1#. Indeed, there are two dis-
tinct regions with different concentrations and temperatures.
Because c
1
.c
2
and u
1
,u
2
, the ‘‘denser’’ and ‘‘colder’’
clump is composed by the first particle near the left wall. In
@1# the clump occurred near the right wall. This difference
appears because in our case, after the collision, the first par-
ticle loses kinetic energy, while the kinetic energy of the
second particle is conserved.
Now we can study the validity of the constitutive relations
in @1#. The first, d
c
j 50, is identically satisfied. But the con-
stitutive relations concerning the granular temperature u re-
main inappropriate. In particular, the proportionality of the
kinetic-energy flux to the temperature gradient, i.e., the Fou-
rier law, is contradicted. Indeed, although u is constant for
x P(0,x
c
), q 5q
1
is nonvanishing. This flux really exists be-
cause the first particle transports the energy gained from the
collision with the left wall and loses it during the inelastic
collision with the second particle. The energy dissipation is
described by d
c
j
2
, which is proportional to the Dirac func-
tion 2d ( x 2x
c
). Therefore, in @1# the constitutive relations
for q and d
c
j
2
must be modified. The validity of other con-
stitutive relations can be tested by applying our method to
the results of numerical simulations.
@1# Y. Du, H. Li, and L. P. Kadanoff, Phys. Rev. Lett. 74, 1268
~1995!.
@2# I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70, 1619
~1993!.
@3# S. McNamara and W. R. Young, Phys. Fluids A 5, 34
~1993!.
@4# S. McNamara and W. R. Young, Phys. Rev. E 50, R28 ~1994!.
@5# L. D. Landau and E. M. Lifshitz, Fluid Mechanics ~Pergamon,
New York, 1959!.
@6# C. Truesdell and R. A. Toupin, in The Classical Field Theo-
ries, edited by S. Flugge, Handbuch der Physik Vol. III
~Springer, Berlin, 1960!, Pt. 1.
@7# D. J. Evans and G. P. Morriss, Statistical Mechanics of Non-
equilibrium Liquids ~Academic, London, 1990!.
@8# I. Mu¨ller, Thermodynamics ~Pitman, Boston, 1985!.
@9# P. K. Haff, J. Fluid Mech. 134, 401 ~1983!.
@10# J. T. Jenkins and S. B. Savage, J. Fluid Mech. 130, 187 ~1983!.
@11# C. Vamos¸, A. Georgescu, and N. Suciu, Studii s¸i Cerceta˘ri
Matematice 48, 115 ~1996!; C. Vamos¸, A. Georgescu, N.
Suciu, and I. Turcu, Physica A 227, 81 ~1996!.
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