oscillatory instability in a driven granular gas evgeniy khain baruch meerson racah institute of...
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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.