Reply to ‘‘Comment on ‘Model kinetic equation for low-density granular flow’ ’’
J. J. Brey and F. Moreno
Fı ´sica Teo ´rica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain
James W. Dufty
Department of Physics, University of Florida, Gainesville, Florida 32611
Received 22 July 1997; revised manuscript received 10 November 1997
The use of simple relaxation kinetic models for granular media is defended see preceding Comment by
Goldshtein and Shapiro, Phys. Rev. E 57, 6210 1998. S1063-651X9811004-8
PACS numbers: 05.20.Dd
A molecular gas at low density is well described by the
Boltzmann equation, but its complexity prohibits a transpar-
ent characterization of its solutions or their properties. His-
torically, kinetic models have been used to provide access to
such solutions and their context. Model kinetic equations are
obtained by replacing the Boltzmann operator J f , f with a
simpler, more tractable form that preserves its most impor-
tant features e.g., conservation laws. Within such con-
straints there is the flexibility to emphasize simplicity or ac-
curacy, depending on the objectives of its use. Most
knowledge of transport outside linear response and for
boundary driven states has been obtained in this way. In 1
we extended this approach of kinetic modelling to explore
the nature of solutions to the Boltzmann equation for inelas-
tic collisions. Our objective was to address fundamental
questions associated with the derivation of fluid dynamics
from a more fundamental basis in a kinetic theory. Such
questions arise because the energy is no longer a conserved
hydrodynamic field, and the reference state about which spa-
tial variations are measured is not local equilibrium, but
rather an unknown time dependent cooling state the homo-
geneous cooling state HCS. In short, we addressed the ques-
tion of how inelasticity affects the derivation, form, and va-
lidity of fluid dynamics equations. From the chosen kinetic
equation an exact solution was obtained for the HCS state,
and the exact solution for small inhomogeneities was ob-
tained through first order in the spatial gradients. The char-
acterization in the Comment on our recent paper 1 by
Goldshtein and Shapiro 2 of our treatment as ‘‘incorrect’’
and ‘‘inconsistent’’ certainly cannot apply to this analysis.
Instead, they question the general validity of using a relax-
ation kinetic model for granular flow. Their primary basis for
this position is that the kinetic model used in Ref. 1 pre-
dicted a homogeneous solution with a divergent fourth mo-
ment, while a calculation based on the Boltzmann equation
indicates it is finite. In the following, we show below that
this feature of the Boltzmann equation is reproduced by the
kinetic model equation if the two adjustable constants are
chosen to match the corresponding viscosity and cooling rate
for the Boltzmann equation. Consequently, the only substan-
tive argument against the kinetic model is removed.
Our chosen model kinetic equation for the distribution
function f ( r, v, t ) has the form
t
+v• f r, v, t =J f , f →- f r, v, t - f
0
r, v, t | f ,
1
where the right side is an approximate representation of the
Boltzmann collision operator for competition between scat-
tering into and out of the velocity state, v. It depends on two
free quantities: an effective collision frequency and a func-
tion f
0
( r, v, t | f ), which is a functional of the distribution
function through the constraints that the model kinetic equa-
tion yield exactly the same balance equations as the Boltz-
mann equation:
d v
1
v
1
2
m v
2
f r, v, t - f
0
r, v, t | f
=
0
0
1 -
2
w /
. 2
The term on the right side of this equation proportional to
(1 -
2
) represents the energy loss due to the inelastic colli-
sions, where is the restitution coefficient and w is a bilinear
functional known from the Boltzmann equation. A primary
effect of inelastic collisions is the violation of detailed bal-
ance, implying that there is no longer an evolution toward a
local Maxwellian. This violation of detailed balance is as-
sured for the kinetic model by constraint 2, and the model
kinetic equation agrees exactly with the Boltzmann equation
in the subspace spanned by (1,v, v
2
). This defines a class of
model kinetic equations, since the constraints do not
uniquely determine f
0
( r, v, t | f ). The choice in Ref. 1 is a
Gaussian with parameters determined by Eq. 2,
f
0
r, v, t | f =n r, t
m
2 k
B
T r, t
3/2
e
-m[ v-u r, t ]/2k
B
T r, t
, 3
1 -c 1 -
2
, c =
2 w
3 nk
B
T
. 4
The functions n ( r, t ), T ( r, t ), and u( r, t ) are the local den-
sity, temperature, and flow velocity which are defined as for
a normal gas via moments of f . The appearance of the factor
PHYSICAL REVIEW E MAY 1998 VOLUME 57, NUMBER 5
57 1063-651X/98/575/62122/$15.00 6212 © 1998 The American Physical Society