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SAND STIRRED BY
CHAOTIC ADVECTION
Work in collaboration with
Andrea Puglisi, Univ. di Roma
Introduction/Motivation
Model
-Numerical algorithm
-Some results on clustering
Continuum approach
(ongoing)
GRANULAR MATTER
L. Kadanoff, Rev. Mod. Phys. 71, 435 (1999).
H. Jaeger, S. Nagel, R.P. Behringer, Rev. Mod. Phys. 68, 1259 (1996).
A. Puglisi, V. Loreto, U. Marini, A. Petri, A. Vulpiani, PRL 81, 3848 (1998).
Y.Du, H. Li, L. Kadanoff, PRL 74, 1268 (1995).
CONTINUUM
D. Dean, J. Phys. A 29, L613 (1996).
U. Marini Bettolo Marconi and P. Tarazona, J. Chem. Phys. 110, 8032 (1999)
Papers de Kawasaki.
Particles in fluid
How does one deal with the extremely common situation of
suspensions, that is, fluids containing particles? Examples
include the transport of sand in the oceans, sand-forming dunes
in air, the motions of colloidal particles in fluids, and the
suspended particles that are used in catalytic reactors. Particles
moving in a fluid react to the background, but they also interact
with each other in complex ways. For these problems, it seems
that neither the particle nor the hydrodynamic (continuum)
approach adequately describes all observed phenomena. It may
be beneficial to combine the two approaches in creative ways.
Jerry Gollub – Physics Today January 2003
Transport of finite size particles by external flows
Turbulent community
Influence of collisions on
inertial non-reacting
particles advected by turb.
chaot. flows
Granular community
Influence of an external
turbulent or chaotic flow in
Inelastically colliding
particles
We study the granular gas
stirred by chaotic advection
Set of N particles colliding inelastically
and with low density
N identical particles with m=1 driven by an external flow u(x,t)
dvi ( t )
1

( v i ( t )  u ( x , t ))
dt

dxi ( t )
 vi (t )
dt
Stokes time
In addition, particles (i and j) mutually collide inelastically
(loosing energy in every collision)
1 r
v 'i ( j )  v i ( j ) 
(( v i ( j )  v j ( i ) )  nˆ )nˆ
2
Restitution coefficient [0,1]
Clarifications of the model at the light of the turbulent
community
In the absence of collisions are the equations of motion of an
spherical particle in a flow where the Faxen corrections, the
added mass term and the Bernoulli term are neglected (and
gravity is not considered). The term that remains is the Stokes
drag.
dv 3 du
1
 R
  (v  u( x , t ))
dt 2 dt

R
2 f
 f  2 p
For heavier particles than fluid the model is well-posed
Numerical algorithm
Direct Simulation MonteCarlo (DSMC) or Bird algorithm.
This scheme has been proved to converge to the Boltzmann
solution of the corresponding hard disk gas.
In every time step dt:
a) Free flow step: particles move according to the motion
equations without taking into account collisions.
b) Collision step: it is fixed a priori the mean collision time,
 c ,such that the probability that a particle collides is
p=dt/ c . For every particle i a random number is extracted rn:
If rn>p no collision,
Otherwise particle i collides with a particle j which is
close to it with probability proportional to their relative
velocity.
Velocities are updated after collisions.
A statistically steady state is reached when
the dissipation of energy due to collisions is
balanced with the continuos injection of energy coming from the
flow. The steady state is reached when the typical fluctuations of the
total energy of the system are small.
Four relevant time scales in the problem
c

T
f
Mean collision time
Stokes time
Typical time scale of the flow
The inverse of the Lyapunov exponent of
the chaotic flow. Gives a time scale for
separation of close fluid parcels
General anticipated results
 f  c
Chaoticity of the flow avoids
clusterization
c   f
Aggregation mechanism of inelastic
collisions resists dispersing due to the
chaotic flow
 dv i

 (v i  u)  Fcolisiones
T dt
c
Inertia irrelevant
Without
collisions
  T
c   f
r=1
 f  c
c   f
r=0.6
r=0.1
Elastic collisions produce clustering??

v  u  Fcolisiones
c
v  0
It is an inertia-like induced clustering
However, the effect of multiple elastic collisions is
equivalent to macroscopic diffusion,Dand
 1/  c
Increasing
the clustering for elastic collisions dissapear.
Inertia relevant
Without
collisions
  T
Elastic
c   f
r=0.75
 f  c
r=0.75
Continuum approach
Macroscopic equations for granular systems are not
free of controversy. Generally are obtained through
balance equations for mas, momentum and energy.
Here we try to obtain an equation for the density of
particles from the microscopic dynamics. We follow
the method of the Dynamic Density Functional widely
used for fluids, problems of solid-liquid transition,
glass transition,...
Consider N Brownian particles interacting vi an arbitrary pair potential
dri ( t )
   V ( ri  r j )   i ( t )
dt
j
Define the density of the system
 ( x , t )    i ( x , t )    ( ri ( t )  x )
i
i
Consider an arbitrary function f of the coordinates. Using
f ( ri ( t ))   dx i ( x , t ) f ( x )
and Ito Calculus, one derives
 ( x , t )
 D 2      ( x , t ) dy ( y , t )V ( x  y )  (1/ 2 )
t
Our model
dv i ( t )
1
1 r
  (v i  u) 
dt

2
  [( v
j
i
ˆ ]n
ˆ  ( t  t ij )
 v j ) n
t ij
dri ( t )
 vi (t )
dt
In the limit of small inertia
1 r
vi  u  

2 j
2
ˆ
ˆ
[(
u
(
r
)

u
(
r
))

n
]
n

(
t

t
)


Dn( ri )i ( ri , t )
 i
j
ij
t ij
Number of particles that collide with
Assuming mean collis. Time is the
smallest,u and density field smooth,
 ( x , t )
du  D
 ( u)  (  ) 
 (   )
t
dt
c
4
(1  r ) 
2

  [   dnˆ ( nˆ  u)nˆ ]
c
|nˆ | 1
2
Summary
The influence of collisions (inelastic) on
inertial particles transported by a chaotic
flow has been studied.
A continuum description has been proposed
which seems to work well. Numerical
simulations c oincide with the discrete ones.