KINETIC THEORY FOR GRANULAR
MIXTURES AT MODERATE
DENSITIES: SOME APPLICATIONS
Vicente Garzó
Departamento de Física, Universidad de
Extremadura, Badajoz, Spain
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
OUTLINE
1. Granular binary mixtures. Smooth inelastic hard spheres:
Enskog kinetic equation for moderately dense systems
2. Homogeneous cooling state
3. Navier-Stokes hydrodynamics: transport coefficients
4. Instabilities in granular dense mixtures
5. Conclusions
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
INTRODUCTION
Granular systems are constituted by macroscopic grains that
collide inelastically so that the total energy decreases with time.
Behaviour of granular systems under many conditions
exhibit a great similarity to ordinary fluids
Rapid flow conditions: hydrodynamic-like type equations.Good
example of a system which is inherently in non-equilibrium.
Dominant transfer of momentum and energy is through
binary inelastic collisions. Subtle modifications of
the usual macroscopic balance equations
To isolate collisional dissipation: idealized microscopic model
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
¤ = v¤ ¡ v¤
V 12
1
2
Smooth hard spheres
with inelastic collisions
V 12 = v 1 ¡ v 2
¤ ¢¾
b = ¡ ®V 12 ¢¾
b
V 12
Coefficient of restitution
0 < ®· 1
Direct collision
v 1¤ = v 1 ¡
1
b ¢V 12 ) ¾
b
( 1 + ®) ( ¾
2
v 2¤ = v 2 +
1
b ¢V 12 ) ¾
b
( 1 + ®) ( ¾
2
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Momentum conservation
v 1 + v 2 = v 1¤ + v 2¤
Collisional energy change
´
1 ³ ¤2
m
¤2
2
2
b)2
¢ E = m v1 + v2 ¡ v1 ¡ v2 = ¡
( 1 ¡ ®2 ) ( V 12 ¢¾
2
4
Very simple model that captures many properties of granular flows,
especially those associated with dissipation
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Real granular systems characterized by some degrees of
polidispersity in density and size:
Multicomponent granular systems
Mechanical parameters:
Direct collision:
f m i ; ¾i ; ®i j g;
i = 1; ¢¢¢; s
³
´
m
j
b ¢V 12 ) ¾
b
v 1¤ = v 1 ¡
1 + ®i j ( ¾
mi + mj
v 2¤ =
³
´
mi
b ¢V 12 ) ¾
b
v2 +
1 + ®i j ( ¾
mi + mj
´
1³
1 mi mj
¤2
¤2
2
2
b)2
(¢ E ) ij =
m i v1 + m j v2 ¡ m i v1 ¡ m j v2 = ¡
( 1 ¡ ®2ij ) ( V 12 ¢¾
2
2 mi + mj
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
REVISED ENSKOG KINETIC THEORY
S-multicomponent mixture of smooth hard spheres or disks
of masses mi, diameters i, and coefficients of restitution ij
At a kinetic level: fi(r1,v1;t)
³
´
¡ 1
@t + v 1 ¢r r 1 + m i F i ( r 1 ) ¢r v 1 f i ( r 1 ; v 1 ; t ) = Ci ( r 1 ; v 1 ; t )
Ci ( r 1 ; v 1 ; t ) =
Z
Xs
¾id¡j 1
j= 1
³
£
Z
dv 2
b£ (¾
b ¢g12 ) ( ¾
b ¢g12 )
d¾
®¡i j 2 f i j ( r 1 ; v 100; r 1 ¡
¾i j ; v 200; t ) ¡ f i j ( r 1 ; v 1 ; r 1 +
Two-particle distribution function
´
¾i j ; v 2 ; t )
V 12 ´ g12
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Collision rules:
where
´
¡ 1
b ¢g12 ) ¾
b
v 1 ¡ ¹ j i 1 + ®i j ( ¾
´
³
¡ 1
00
b
b ¢g12 ) ¾
v 2 = v 2 + ¹ i j 1 + ®i j ( ¾
v 100 =
³
ij=mi/(mi+mj)
Kinetic theory approach: velocity correlations are neglected
(molecular chaos hypotesis !!)
f i j ( r 1 ; v 1 ; r 2 ; v 2 ; t) !
Âi j ( r 1 ; r 2 j f n i g) f i ( r 1 ; v 1 ; t) f j ( r 2 ; v 2 ; t)
Spatial correlations (volume exclusion effects)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Bad news for the RET: Several MD simulations have shown that
velocity correlations become important as density increases
(McNamara&Luding, PRE (1998); Soto&Mareschal PRE (2001);
Pagonabarraga et al. PRE (2002);…..)
Good news for the RET: Good agreement at the level of macroscopic
properties for moderate densities and finite dissipation
(Simulations: Brey et al., PF (2000) ; Lutsko, PRE (2001);
Dahl et al., PRE (2002); Lois et al. PRE (2007);
Bannerman et al., PRE (2009);…..
Experiments: Yang et al., PRL (2002); PRE (2004);
Rericha et al., PRL (2002);…..)
RET is still a valuable theory for granular fluids for densities
beyond the Boltzmann limit and dissipation beyond
the quasielastic limit.
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
MACROSCOPIC BALANCE EQUATIONS
Z
Hydrodynamic fields
ni ( r ; t) =
dv f i ( r ; v ; t )
Z
1 X
U ( r; t) =
dv m i v f i ( r ; v ; t )
½( r ; t ) i
Z
1 X
mi
T ( r; t) =
dv ( v ¡ U ) 2 f i ( r ; v ; t )
n( r ; t ) i
d
Macroscopic equations are exact since they are obtained from the
first hierarchy equation (without the Enskog approximation)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Balance equation for the partial densities
D t n i + n i r ¢U + m ¡i 1 r ¢j i = 0
Balance equation for the flow velocity
½D t U¯ + r ° P° ¯ =
Xs
n i ( r ; t ) Fi ¯ ( r )
i= 1
Balance equation for the granular temperature
Xs F i ¢j i
d
d Xs r ¢j i
n ( D t + ³ ) T + P° ¯ r ° U¯ + r ¢q ¡ T
=
2
2 i= 1 mi
i= 1 mi
Cooling rate
D t ´ @t + v ¢r
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Z
Mass flux
j i ( r 1; t ) = m i
dv 1 V 1 f i ( r 1 ; v 1 ; t )
V=v-U
Pressure tensor
P° ¯ ( r 1 ; t ) = P°k¯ ( r 1 ; t ) + P°c¯ ( r 1 ; t )
Kinetic contribution
P°k¯ ( r 1 ; t ) =
Xs Z
Collisional contribution
dv 1 m i V1¯ V1° f i ( r 1 ; v 1 ; t )
i= 1
P°c¯ ( r 1 ; t )
=
Z
Z
Z
³
´
1 Xs
d
b £ (¾
b ¢g12 ) ( ¾
b ¢g12 ) 2
m j ¹ i j 1 + ®i j ¾i j dv 1 dv 2 d¾
2 i ;j = 1
b¯ ¾
b°
£¾
Z1
0
dxf i j ( r 1 ¡ x ¾i j ; v 1 ; r 1 + ( 1 ¡ x) ¾i j ; v 2 ; t ) :
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
q ( r 1 ; t ) = qk ( r 1 ; t ) + qc ( r 1 ; t )
Heat flux
qk
Xs Z
( r 1; t) =
i= 1
qc ( r 1 ; t ) =
1
dv 1 m i V12 V 1 f i ( r 1 ; v 1 ; t )
2
Z
Z
Z
´
1³
b £ (¾
b ¢g12 )
1 + ®i j m j ¹ i j ¾idj dv 1 dv 2 d¾
i ;j = 1 8
Xs
b ¢g12
£ (¾
b
£¾
Z1
0
)2
h³
´ ³
1 ¡ ®i j
´
´
b ¢g12 ) + 4 ¾
b ¢G i j
¹ j i ¡ ¹ ij ( ¾
dxf i j ( r 1 ¡ x ¾i j ; v 1 ; r 1 + ( 1 ¡ x ) ¾i j ; v 2 ; t ) ;
Gij= ijV1+ jiV2 is the center-of-mass velocity
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Cooling rate
³
=
Z
Z
Z
´
Xs ³
1
b
1 ¡ ®2ij m i ¹ j i ¾id¡j 1 dv 1 dv 2 d¾
2dnT i ;j = 1
b ¢g12 ) ( ¾
b ¢g12 ) 3 f i j ( r 1 ; v 1 ; r 1 + ¾i j ; v 2 ; t )
£ £ (¾
Balance equations become a closed set of hydrodynamic equations
for (ni,U,T) once the fluxes and the cooling rate are expressed as
functionals of (ni,U,T) (“constitutive relations”)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
CHAPMAN-ENSKOG NORMAL SOLUTION
Assumption: For long times (much longer than the mean free time)
and far away from boundaries (bulk region) the system reaches a
hydrodynamic regime.
Normal solution
f i ( r ; v ; t) = f i ( v jf n i ( r ; t) g; U ( r ; t) ; T ( r ; t) g)
In some situations, gradients are controlled by boundary or initial
conditions. Small spatial gradients:
( 0)
fi = fi
( 1)
+ ²f i
+ ¢¢¢
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Some controversy about the possibility of going from kinetic
theory to hydrodynamics by using the CE method
The time scale for T is set by the cooling rate instead of spatial
gradients. This new time scale, T is much faster than in the
usual hydrodynamic scale. Some hydrodynamic excitations
decay much slower than T
For large inelasticity (-1 small), perhaps there were NO time scale
separation between hydrodynamic and kinetic excitations:
NO AGING to hydrodynamics!!
I assume the validity of a hydrodynamic description and compare with
computer simulations
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
HOMOGENEOUS COOLING STATE
(zeroth-order approximation)
Spatially homogeneous isotropic states
@t f i ( v; t ) =
X
J i j [vjf i ( t ) ; f j ( t ) ]
j
mi
ni T i =
d
Partial temperatures
Granular temperature
T =
X
Z
dv v 2 f i ( v )
x i Ti ;
x i = n i =n
i
Cooling rates for Ti
³ i = ¡ @t ln Ti ;
³ = T¡ 1
X
x i Ti ³ i
i
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Assumption
Hydrodynamic or normal state: all the time dependence of vdf occurs
only through the temperature T(t)
f i ( v; t ) = n i v0¡ d( t ) © i ( c) ;
c´
v
v0 ( t )
v0 ( t ) 2 / T ( t )
Consequence: temperature ratios must be constant
(independent of time)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Binary mixture: =T1/T2
HCS condition:
@t ln ° = ³ 2 ¡ ³ 1
Elastic collisions: ³ 1 = ³ 2 = 0;
³1 = ³2
T1 = T2 = T
Equipartition theorem for classical statistical mechanics
What happens if the collisions are inelastic ?
° 6
= 1
VG&Dufty PRE 60, 5706 (1999)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Time evolution of temperature ratio . Comparison with
Monte Carlo simulations
Montanero&VG, Granular Matter 4, 17 (2002)
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IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Comparison with molecular dynamics (MD) simulations
Dahl, Hrenya,VG& Dufty, PRE 66, 041301 (2002)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Breakdown of energy equipartition
Computer simulation studies: Barrat&Trizac GM 4, 57 (2002);
Krouskop&Talbot, PRE 68, 021304 (2003); Wang et al. PRE 68,
031301 (2003); Brey et al. PRE 73, 031301 (2006); Schroter et al.
PRE 74, 011307 (2006);…….
Real experiments: Wildman&Parker, PRL 88, 064301 (2002);
Feitosa&Menon, PRL 88, 198301 (2002).
All these results confirm this new feature in granular mixtures !!
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
NAVIER-STOKES HYDRODYNAMIC EQUATIONS
Previous studies: Jenkins et al. JAM 1987; PF 1989; Zamankhan,
PRE 1995; Arnarson et al. PF 1998; PF 1999; PF 2004;
Serero et al. JFM 2006
Limited to the quasielastic limit. They are based on the
energy equipartition assumption
Our motivation: Determination of the transport coefficients by using
a kinetic theory which takes into account the effect of
temperature differences on them. NO limitation to the
degree of dissipation
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Constitutive equations (binary mixture)
m 21 n
m m n
j1 = ¡
D 11 r ln n 1 ¡ 1 2 2 D 12 r ln n 2 ¡ ½D T r ln T
½
½
µ
P° ¯ = p±° ¯ ¡ ´
¶
2
r ° U¯ + r ¯ U° ¡ r ¢U ¡ · r ¢U
d
q = ¡ T 2 D q;1 r ln n 1 ¡ T 2 D q;2 r ln n 2 ¡ ¸ r T
n
Eight transport coefficients:
o
D 11 ; D 12 ; D T ; ´ ; · ; D q;1 ; D q;2 ; ¸
Parameter space: f m 1 =m 2 ; ¾1 =¾2 ; x 1 ; Á; ®11 ; ®22 ; ®12 g
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Transport coefficients given in terms of solutions of coupled linear
integral equations. Complex mathematical problem
Sonine polynomial approximation. Only leading terms are
usually considered
To test the accuracy of the Sonine solution: comparison with
numerical solutions of the RET (DSMC)
VG, J. Dufty, C. Hrenya, PRE 76, 031303 (2007); 031304 76(2007);
J.A. Murray, VG, C. Hrenya, Powder Tech. 220, 24 (2012)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Some limiting cases
 Mechanically equivalent particles [Garzó&Dufty PRE 59,
5895 (1999)+Lutsko, PRE 72 021306 (2005)]
 Binary mixtures at low-density [Garzó&Dufty, PF 14, 1476
(2002)+Garzó&Montanero, JSP 129, 27(2007)]
 Elastic hard spheres
[López de Haro, Cohen&Kincaid, JCP 78, 2746 (1983)]
Self-consistency of our results
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Shear viscosity coefficient of a (heated) granular fluid
VG&J.M. Montanero, PRE 68, 041302 (2003)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Tracer diffusion coefficient: Diffusion of
impurities in granular gas under HCS
When the excess gas is in HCS, the diffusion equation is
@t x 1 ( r ; t ) = D 11 ( t ) r 2 x 1 ;
q
D 11 ( t ) /
T ( t)
Mean square position of impurity after a time interval t
@
2dD 11 ( t )
2
hj r ( t ) ¡ r ( 0) j i =
@t
n2
Einstein form is used to measure D11 in DSMC simulations
[Brey et al. PF 12, 876 (2000)]
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
First two Sonine approximations are considered !!
D11()/D11(1)
1.5
1.4
First Sonine
1.3
Second Sonine
1.2
DSMC
1.1
1.0
0.5
0.6
0.7

0.8
0.9
1.0
m 1 =m 2 = 1=4; ¾1 =¾2 = 1=2; Á = 0:2
®22 = ®12 ´ ®; x 1 !
0
VG&Montanero, PRE 68, 021301 (2004)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
VG&F.Vega Reyes, PRE 79, 041303 (2009); JFM 623, 387 (2009)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
INSTABILITIES IN FREELY COOLING
DENSE GRANULAR BINARY MIXTURES
In contrast to ordinary fluids, instabilities (such as dynamic particle
clusters) occur in freely cooling (HCS) granular gases
Pionnering work of Goldhirsch&Zanetti
(PRL 70, 1619 (1993))
Typical configuration of particles
exhibing clustering
® = 0:6; Á = 0:05
(Peter Mitrano, CU)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
General trends: (i) instabilities are more likely in larger domains;
and (ii) velocity vortices manifest more readily than
particle clusters
This is also a very good problem to assess Navier-Stokes
hydrodynamics derived from Kinetic Theory
For given values of the mechanical parameters of the system, there
exists a critical system lenght demarcates (stable) homogeneous
flow from one with velocity-vortex instabilities or one exhibiting
the clustering instability
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
LINEAR STABILITY ANALYSIS
HCS is unstable with respect to long enough
wave-length perturbations. Stability analysis of the
nonlinear hydrodynamic equations with respect to HCS
for small initial excitations
HCS solution
r x 1H = r n H = r TH = 0; U H = 0;
@t ln TH = ¡ ³ 0H
This basic solution is unstable to linear perturbations
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
We linearize the Navier-Stokes equations
with respect to the HCS solution. Deviations of the hydrodynamic
fields from their values in HCS are small
x 1 ( r ; t ) = x 1H + ±x 1 ( r ; t ) ; n( r ; t ) = n H + ±n( r ; t ) ;
U ( r ; t ) = ±U ( r ; t ) ; T ( r ; t ) = TH + ±T ( r ; t )
Equation for the velocity field :
µ
½H @t ±U + r ` pH = ´ H r 2 ±U +
d¡ 2
2
´H ¡ · H
d
d
¶
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
r ( r ¢±U )
Linearization about HCS yields a set of partial
differential eqs. with coefficients that are independent of space
BUT depend on time. Time dependence can be eliminated by
¿=
Zt
0
dt 0º
H
( t 0) ; `
d¡ 1
º H / n H ¾12
vH ; vH =
º H ( t)
=
r
vH ( t )
q
2TH =( m 1 + m 2 )
Set of coupled linear differential equations with
constant coefficients
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Set of Fourier transformed dimensionless variables:
±x 1 k ( ¿)
½1;k ( ¿) =
;
x 1H
±U k ( ¿)
w k ( ¿) =
;
vH ( ¿)
Z
±yk ¯ ( ¿) =
±n k ( ¿)
½k ( ¿) =
nH
±Tk ( ¿)
µk ( ¿) =
TH ( ¿)
d` e¡ i k ¢` ±y¯ ( ` ; ¿)
n
±yk ¯ ´
o
½1;k ; ½k ; w k ; µk
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Transversal component of the velocity field is decoupled
from the other modes. This identifies “d-1” shear (transversal) modes
µ
¶
@
1 ¤ 2
¤
¡ ³ 0H + ´ H k w k ? = 0
@¿
2
s? ¿
w k ? ( ¿) = w k ? ( 0) e
; s? ( k) =
There exists a critical wave number:
k < k ?c
¤
³ 0H
1 ¤ 2
¡ ´Hk
2
v
u
¤
u
2³
0H
k ?c = t
¤
´H
Shear modes grow exponentially!!
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
The remaining 4 longitudinal modes are coupled and are
the eigenvalues of a 4X4 matrix
There exists two critical wave numbers:
v
u
¤
u
2³
c
0H ;
k? = t
¤
´H
c
kk
Solution of a quartic equation
For wave numbers smaller than these critical values,
the system becomes unstable
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Periodic boundary conditions, the smallest k is
2¼=L
If the system length L>Lc, then the system becomes unstable
n
o
2¼
c
c
=
m
ax
k
;
k
?
k
¤
Lc
In most of the studied cases, the linear stability
analysis predicts that the HCS is unstable to velocity vortices
and linearly stable to particle clusters.
c
c
k? > kk
Stringent assessment of kinetic theory calculations!!!
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
MONOCOMPONENT GRANULAR FLUIDS
VG, PRE 72, 021106 (2005)
P. Mitrano et al. PF 23, 093303 (2011)
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Standard Sonine approximation
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
HIGHLY DISSIPATIVE GRANULAR FLUIDS
Modified Sonine approximation (VG et al., Physica A 376 94 (2007))
Mitrano, VG, Hilger, Ewasko,
Hrenya, PRE 85, 041303 (2012)
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
BINARY GRANULAR DENSE MIXTURES
P. Mitrano, VG and C.M. Hrenya, 2013
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Quantitative agreement, even for e=0.7. Agreement could be
improved for disparate mass/size binary mixtures by considering
the second Sonine solution
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
d1 + d2
d=
2
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
Á = 0:1
33
MD e=0.7
MD e=0.8
28
Lvortex /d
Theory e=0.7
Theory
23
18
13
8
0
2
4
6
8
10
12
 = m1/m2
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
CONCLUSIONS
Hydrodynamic description (derived from kinetic theory) appears
to be a powerful tool for analysis and predictions of rapid
flow gas dynamics of granular mixtures at moderate densities.
New and interesting result: partial temperatures (which measure
the mean kinetic energy of each species) are different (breakdown of
energy equipartition theorem). Energy nonequipartition has important
and new quantitative effects on macroscopic properties
Instability of HCS: Good agreement with MD even for
finite dissipation and moderate densities. Stringent test of
kinetic theory results
IMA Conference on Dense Granular Flows, July 1-4, Cambridge
SOME OPEN PROBLEMS
Extension to inelastic rough spheres
Influence of interstitial fluid on grains (suspensions).
Previous results for monodisperse gas-solid flows
(VG, Tenneti, Subramaniam, Hrenya, JFM 712, 129 (2012))
Granular hydrodynamics for far from equilibrium steady states
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UEx (Spain)
•José María Montanero
•Francisco Vega Reyes
External collaborations
•James W. Dufty (University of Florida, USA)
•Christine Hrenya, Peter Mitrano (University of Colorado, USA)
Ministerio de Educación y Ciencia (Spain) Grant no. FIS2010-16587
Junta de Extremadura Grant No. GRU10158
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