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)

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). D. Dean, J. Phys. A 29, L613 (1996). U. Marini Bettolo Marconi and P. Tarazona, J. Chem. Phys. 110, 8032 (1999) Papers de Kawasaki. CONTINUUM GRANULAR MATTER

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 Influence of collisions on inertial non-reacting particles advected by turb. chaot. flows Turbulent community Influence of an external turbulent or chaotic flow in Inelastically colliding particles Granular community

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) In addition, particles (i and j) mutually collide inelastically (loosing energy in every collision) Stokes time 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. 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,,such that the probability that a particle collides is p=dt/. 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 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 T

General anticipated results Chaoticity of the flow avoids clusterization Aggregation mechanism of inelastic collisions resists dispersing due to the chaotic flow

Inertia irrelevant Without collisions r=1 r=0.6 r=0.1

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.

Inertia relevant Without collisions Elastic 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 Define the density of the system Consider an arbitrary function f of the coordinates. Using and Ito Calculus, one derives

Our model In the limit of small inertia Number of particles that collide with i

Assuming mean collis. Time is the smallest,u and density field smooth,

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.