SAND STIRRED BY CHAOTIC ADVECTION

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SAND STIRRED BY CHAOTIC ADVECTION
Work in collaboration with Andrea Puglisi, Univ.
di Roma
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Introduction/Motivation Model -Numerical
algorithm -Some results on clustering Continuum
approach (ongoing)
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GRANULAR MATTER

• L. Kadanoff, Rev. Mod. Phys. 71, 435 (1999).
• H. Jaeger, S. Nagel, R.P. Behringer, Rev. Mod.
  Phys. 68, 1259 (1996).
• 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.
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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
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Transport of finite size particles by external
flows
Turbulent community
Granular community
Influence of collisions on inertial non-reacting
particles advected by turb. chaot. flows
Influence of an external turbulent or chaotic
flow in Inelastically colliding particles
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We study the granular gas stirred by chaotic
advection
Set of N particles colliding inelastically and
with low density
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N identical particles with m1 driven by an
external flow u(x,t)
Stokes time
In addition, particles (i and j) mutually collide
inelastically (loosing energy in every collision)
Restitution coefficient 0,1
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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.
For heavier particles than fluid the model is
well-posed
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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
• 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,
• ,such that the probability that a
  particle collides is
• pdt/ . For every particle i a random number
  is extracted rn
• If rngtp 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.

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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
Mean collision time
Stokes time
Typical time scale of the flow
T
The inverse of the Lyapunov exponent of
the chaotic flow. Gives a time scale for
separation of close fluid parcels
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General anticipated results
Chaoticity of the flow avoids clusterization Aggre
gation mechanism of inelastic collisions resists
dispersing due to the chaotic flow
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Inertia irrelevant
Without collisions
r1
r0.1
r0.6
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Elastic collisions produce clustering??
It is an inertia-like induced clustering However,
the effect of multiple elastic collisions is
equivalent to macroscopic diffusion, and
Increasing the clustering for
elastic collisions dissapear.
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Inertia relevant
Without collisions
Elastic
r0.75
r0.75
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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,...
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Consider N Brownian particles interacting vi an
arbitrary pair potential
Define the density of the system
Consider an arbitrary function f of the
coordinates. Using
and Ito Calculus, one derives
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Our model
In the limit of small inertia
Number of particles that collide with i
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(No Transcript)
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Assuming mean collis. Time is the smallest,u and
density field smooth,
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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.