APPENDIX A
1
Consider a granular material consisting of a binary mixture of differently colored smooth, spherical particles. For this system, light and dark colors are represented by subscripts l and d, respectively. The particles, which have the same masses m and radii

but different number densities l n and n d
, are sheared at a high shear rate. Due to the presence of gradients in the mean flow, random fluctuations in the local mean translational motion of the grains are generated. Since the particles in a rapid shear flow behave similar to molecules in a dense gas, methods from the kinetic theory of dense gases [58] might be used to describe the motion of particles in this system. For a nonequilibrium granular fluid, in the limit of two-body forces between the particles, the probability that at some instant of time t the particle of kind l will be at location r and have momentum p within the respective limits dr and dp may be governed by the following equation [51]:
 f
 t l
(1) c l
+ .
 f l
(1)
 r
F
 f l
(1)
 c
=-
  Χ .
 lm
( 2 ) f
 c d r m d p m
, (A1)
Here c l
is the instantaneous velocity of a particle of kind l , lm
X is the force on a particle of kind l due to all other particles and l
F is the force per unit mass acting on a particle of kind l due to external fields.
Eq. (A1) involves the pair distribution function f lm
(2)
, which is the probability that there is simultaneously a particle of kind l in the space element
 l r p l

about
 l l

and another particle of kind m , which can be either light or dark, in the space element
 d r m d p m

about

 m r , p m

. Thus, it is needed to find an expression for the pair distribution function, in terms of
2 single particle distribution functions. If the particle contacts can be regarded as nearly instantaneous collisions, the distribution functions do not change appreciably in a time interval comparable to the duration of a collision, the gradients of inhomogeneities are small, and the assumption of particle chaos remains valid, Eq. (A1) reduces to the generalized Boltzmann equation which governs the temporal change of single particle distribution function of the particle of kind l . Since the particle diffusion processes involved in a granular fluid are of interest in the present study, the new variable C l c l
V , which is the particle peculiar velocity, is introduced in place of c l
into Eq. (A1). As a result the following kinetic equation is obtained
[58] d f l
(1) r C t d     2
=



C l

 r
.
+
 
 u r
 
: C l


C
 
+ d u

.

C

F l

 
C
 f l
(1) r C t g lm
 r r
 k n
  r , C l ´ , t
 f m
(1)
 r +
 k , C m ´ , t

 g lm

,
  k f l
(1)
 r , C l , t
 f m
(1)
 r
  k , C m , t
  c lm  k
  c lm  k
 d d m , n


(A2)
Here d/d = / + . / r is the substantial time derivative,
   l n n d

are solid component densities, V = d m m c m / d   m m is the mean mass velocity of the mixture, lm m c c c is the relative velocity of the two particles of kind l and kind m , k is the unit vector directed along the line from the center of the particle of kind m to the center of the particle of kind l at contact, and

is the
Heaviside step function. Also, factors lm g are introduced to take into account the difference in position of the colliding particles in binary collision dynamics, and the resulting increase in the frequency of collisions. Following van Beijeren & Ernst [59], g lm are chosen to be the mixture

radial distribution functions at contact, which are evaluated as nonlocal functionals of density
3 fields of the two components in the granular fluid mixture. Thus, g lm

,
  k d  r q lmq d H
 n s r
 g lm c
 
,
  k r n s r
 n s r
  r q
  n q
 r

O(
 2 ).
(A3)
In the above equation, g c lm is the equilibrium value of the radial distribution function at contact for particles of kind l and m : g lm c


|
 n s r
  q d 

1 n q

V lmp

,
  k r q
 d r q 
(A4)
Here, H lmq

,
  k r q
, n

V lmq

,
  k r q

+ m  q
 n q ´
  
V lmqq ´

,
  q q ´ k r r
 d r q ´ 
...
, which is symmetric under the interchange of superscript l and m . Also, V

,
  k r q

and
V

,
  k r r q ´

represent Husimi functions. As discussed earlier, the above approximation for the radial distribution function at contact is not quite appropriate for the present study of a sheared granular flow where a spatial preference for collisions is imposed by the velocity field.
However, there is presently no better approximation available.
To derive the hydrodynamic equations of a granular fluid mixture, which are valid outside the time regime for which Eq. (A2) is exact. This equation is first multiplied by m
 l
, where
 l is any property of particles of kind l . The resulting equation is then integrated over the instantaneous velocity c l
to give the equation of change for the assembly of particles of kind l in terms of the mass-weighted mean values:

d d t

 l m
 l
 l

+
 l m
 l
 l

 r
 l m
 l d
 l dt


 r
.

 l m
 l
C l
 l

  l m
 l d V d t
.
 

C l

 l m
 l
 

V r

: C l
 

C l
+
 l m
 l
F l



C l
.
+
 l m
 l
C l


 r l
.
+ d  m=l


 r
.
 lm

1
2
 m

3
 


 k


 l ´   l


C l


:




V r




 c lp  k

Η
 c lp  k
 f m
(1)
 r C m
, , t
  r C l
, , t
 g lp c



1+


2 k


 r
 ln f m
(1) f l
(1) r C m t r C l t

 
 l k c c m
+... d d d ,

(A5)
4
Here
 l m
and
 l are the material density and solid volume fraction of the particles of kind l .
Expressions for the particle-particle collisional flux
 lm and the source-like term
 lm
are given as follows: f
 lm =

1
2
 m

 
 l k

3
 c lm  k
  c lm  k
 lm g f c m
(1)
 r C m
, , t

 l
(1)
 r C l , t



1+ 1
2
 k

 r

 ln f f m
(1) l
(1)

 r C m
, , r C l
, , t t

 
  
 k c c m , f
 lm
= 1
2
 m

 l ´ 

2
 c lm  k
  c lm  k
 lm g f c m
(1)
 r C m , t

 l
(1)
 r C l
, t



1+ 1
2
 k


 r

 l n f f m
(1) l
(1)

 r C m
, , r C l
, , t t

 
 

 l k c c m
for m
 l ,
(A6) f
 lm
=
 m

 l ´   l

 2
 c lm  k
  c lm  k
 lm g f c m
(1)
 r C m
, , t

 l
(1)
 r C l
, t



1+ 1
2
 k

 r

 ln f f m
(1) l
(1)

 r C m
, , r C l , t t

 
  
 l k c c m
for m
 l .
The equations of conservation of mass, momentum and energy for granular assembly of kind l may be derived by taking
  in Eq. (A5) to be C l
1, , and l
2
C , respectively:

5
I) Balance of mass d
 l d t
+ where
 l

 r

V

+
 r


 l u l

=0,
 l
=
 l m
 l
is the apparent density and u l
(A7)
is the diffusion velocity of particles of kind l .
II) Balance of linear momentum
 l d V l d t

V l

 r where V l
, and


 l l
V

 

 r
 Π l
+
 l F l   l d V
  l d t

V
 r
 V l
+ d  m
 l
 lm
, (A8)
Π l
are the velocity vector and the macroscopic pressure tensor of particles of kind l , respectively.
III) Balance of kinetic energy (pseudo-thermal energy)
3
 n l d T
2 d t

T

 r
  

 r q
 l
:

V
 r
  l l  l   l l
C
 d V d t
, (A9) where T

1 3 C l
2
is the mixture pseudo-thermal energy, l q is the energy flux vector of particles of kind l and
 l
is the rate of energy dissipation of particles of kind l per unit volume of the mixture due to the inelastic nature of the collisions. The effect of the thermal energy on the particle elasticity is not considered in the present treatment.
Assuming that the rapid shearing motion of a binary mixture of differently colored granular fluid can be adequately described by consideration of the 13-moment approximation [60], expressions for the macroscopic pressure tensor
Π l
, energy flux vector l q and source like terms
 lm
 
and
 l
are given as follows [25]:

Π l =

 l
T I
  l † l a

+ d  
1
    3 e n g
0 l m lm c
4
15
1
2 


 
V
 r

5
6

 r


 





3
T I


15
 a
† l  a
† m


(A10)
6 q l
=
5
2
 l

T v l 
1
5 a l
+ d 
 3 g lm c

1+ e
0

-
 
1
2


 m
6
T

1e
0


 n m
 n l
 r
 n l
 n m
 r



1
3 n m
 l

T
 r



  l
T


1
6
 u m  u l


1
4

1e
0
  u m  u l
 



20 n m
 l


 a m  a l


1
2

1e
0
  a m  a l
 

,
(A11)
 lm
 
=

2

1
 e
 g lm l m c
 n



3





T
1
15
1
2  a l  a m



, n l n m

 r

 n n l m






4
3
1
2
 u l  u m


(A12)
 l
= d 
2

2

1
 e
2
0
 lm g n c m
 l
T


  1
2


4


 r
 u


.
(A13)
In the above, a
† m is the pressure deviator, a m is the transport pseudo-thermal energy flux vector, e is the coefficient of restitution, and
0

V
 r
is non-divergent symmetric part of the mean mass mixture velocity-gradient tensor.
For steady shear flow, the energy supplied to the grains by continuous shearing is balanced by the energy dissipation due to inelastic collisions, and Eq. (A9) reduces to the following form:

0
 d 



 m † m a xy
 d  
1
 e
0
  m p
 3 n g c mp d d  m l p

1
2



15
 a
† m xy
 a
† xy p


2
15
 
T
1
2
 z





 z
2

1
 2
 mp p
  1
2 ,

(A14)
7
In the above equation,
  
is the local rate of shear.
Expressions for the pressure deviators a
† l
and xy a
† d
are given as follows: xy d  p=l
 2

1
  pn p n e g n
0 c
  1
2
2

1
3 5
2

1 e
0

  n
T 1
 d  
3

1
 e
0
 g c np n p a
† n xy

1
5
2

1 e
0
 
2
3 a
† xy p

1
5

1

1 e
0
 
1
3
 n = l, d

(A15)
Eq. (A15) can be inserted into Eq. (A14) to obtain Eq. (17) for the mixture. It is worth mentioning that Eq. (A15) may be derived from the balance law for the deviant part of the mean of the second moment of velocity fluctuation by assuming that the spatial gradients of the mean fields are small and that the dimensionless quantities

/L ,

/t T
0 0
1
2 , u l /T
0
1
2 , u d /T
0
1
2 , a
† l T
0
,
† d a T
0
, a l /T
0
3
2 and a d /T
0
3
2 are all of the same order of magnitude and small. Here, L , t
0
, and T
0 are the characteristic length, time, and the mixture granular temperature, respectively.
Assuming that u l z
 u l x
 u l x
 u d x

, and that
 u l x
 u d x

T
1
2 and
 a l x
 a d x

T
 1
2 are of the same order of magnitude, in the presence of non-uniform shear stress in the direction of shear gradient, namely the z direction, the balances of momentum (A7) in the direction of shear plane for particles of kind l , for which l n n d
 n , may be reduced to
 l l u z


 z



 l a
† l xz
 d  
1
 e
0
   3 g c



15
 a
† l a xz
 † xz p


2
15
   





.
(A16)

Substituting for a
† l
and xz a
† d
from Eq. (A15), assuming that the parameter xz
T
  
2
does
8 not change significantly in the direction of shear gradient [61], gives a reduced form of this drift
 l l u z

  
15
 1
2

3


T e
1
2
0
  
2  z
, (A17) where the term in the brackets, B
 
T
1
2 15
 1
2

3
 e
0
  
2
, is the particle mobility, and
U

   e
0
 g
 c s
18 3 e
0

2
    
5
  
2
is a potential field, which is a function of position. The term on right side of Eq. (A17) represents the contribution of inhomogeneous shear to the drift as well as the extra diffusive flux arising from viscosity gradients.