Oscillatory instability in a driven granular gas

Evgeniy KhainBaruch Meerson

Racah Institute of PhysicsHebrew University of Jerusalem

• Granular gas: a simple model of a fluidized granular medium

• Granular hydrodynamics

• Phase-separation instability

• Oscillatory instability

• Summary

• Granular Materials are ubiquitous:

sand, sugar, flour, …

• GMs are important:

powder metallurgy, pharmacology, …

• GMs are interesting

Surface Waves Avalanches

Brazil Nut Effect

Size separation

Motivation

The simplest model of granular gas:

Inelastic Hard Spheres

inelastic binary collisionscoefficient of normal

restitution:

elasticcollisions

The energy loss in each collision

4/)v)(vr-(1ΔE 22n1n

2 =

1r0

Hydrodynamics of gases with inelastic collisions

Continuous approach:coarse-grained variables

• Granular temperature T

• Granular density ρ

• Granular pressure P

Works well for nearly elastic collisions

Kinetic theory

Constitutive relations

1r-1

Eqs. of Granular Hydrodynamics

. Γv:Pqdt

dTρ

, fPdt

dvρ

, 0vρdt

dρ

These equations and constitutive relations can be derived

from kinetic theory(for nearly elastic collisions)

Jenkins and Richman (1985), …

)r-(1Γ 2

• P - stress tensor• q - heat flux• rate of energy losses

by collisions• f - external force

ρ grows, T decreases

1-D static cluster can becomeunstable!

1-D static cluster state

Simplest setting of driven granular gas

P = g(ρ)T =const

Grossman, Zhou and Ben-Naim (1997) – MD simulations + hydrodynamic model,

Kudrolli, Wolpert and Gollub (1997) - experiment

Thermalwall

Thermalwall

Thermalwall

Tobochnik (1999), Brey and Cubero (1999)

Khain and Meerson (2003)

Governing parameters

Governing equations

. RnGTεT)F(Tεpdt

dTn

,dt

dn

, 0ndt

dn

3/21

1/2

v

Pv

v

DIDP ˆTεF)]tr(εnGT-p[ 1/22

1/2 stress tensor

Relative heat loss parameter

Transport parameter 1L

d2ε

2ε

r)-(1

π

16R

Khain and Meerson (2003)

General scenario for instabilities: R exceeds a critical value

Area fractioncn

nf

A. Phase-separation instability

0 0.8 1.62.5

3

3.5

4 cR

yk

Livne et al. (2002), Khain and Meerson (2002)

R*c

Marginal stability:

unstable

stable

H

L

HΔ

Aspect ratio:

L

Two coexisting phases One phase

Meerson, Sasorov, Pöschel,  and Schwager

(2002)

MD simulations, hydro simulations:

Explanations and further exciting issues: wait for the lecture of Baruch Meerson

tomorrow

Let's consider a small aspect ratio.1-D static cluster can become

unstable even in this case !

cΔΔ cΔΔ

Linear stability analysis: instability threshold

B. Oscillatory instability

http://huji-phys.phys.huji.ac.il/staff/Khain/index.html

Khain and Meerson (2003)

Unstable region

1

2

Stable region

MD simulations:

Cluster oscillates back and forth

away from the thermal walls

MD simulations:

stable region unstable regionsmall-amplitude

noise

large-amplitudeoscillations

What happens for larger aspect ratios?

The two instabilities coexist

Small isolated cluster with broken symmetry

oscillates back and forth

Summary

• We found a novel oscillatory instability in a simple driven granular system

• Hydrodynamic linear stability analysis performed, instability threshold determined

• Predictions of linear theory verified in MD simulations. Next step should be nonlinear theory

• Hydrodynamics is instrumental in analysis of rapid granular flow.