temperature oscillations in a compartmetalized bidisperse granular gas c. k. chan 陳志強...
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Temperature Oscillations in a Compartmetalized Bidisperse
Granular Gas
C. K. Chan陳志強
Institute of Physics, Academia Sinica, Dept of Physics,National Central University,
Taiwan
Collaborators
• May Hou, Institute of Physics, CAS• 厚美英
• P. Y. Lai, National Central University• 黎璧賢
Content
• What is a clock?
• What is special about a granular clock?
• Unstable Evaporation/Condensation
• Two temperature in a bi-disperse system
• Model for bidisperse oscillation
• Summary
What is a clock ?
Periodic motion
sun, moon, pendulum etc …
Periodic Reaction
BZ reaction, enzyme circadian rhythm
Periodic Collective behavior
suprachiasmatic nuclei, sinoatrial node, comparmentalized granular gases, etc…
BZ reaction
From S. Mueller
Granular Oscillation
Second Law no clock?
• Belousov-Zhabotinsky reaction
A B A B; Why not: A B
• Two-compartment granular Clock
Molecular gases
Properties of Granular Gases
• Particles in “random” motion and collisions• “similar” to molecular gases
But …
• Inelastic Collisions / Highly dissipative• Energy input from vibration table
• Far from thermal equilibrium Brazil Nut Effect, Clustering, Maxwell’s demon
monodisperse granular gas in compartments: Maxwell’s Demon
Eggers, PRL, 83 5322 (1999)
v
Clustering
• Granular gas in Compartmentalized chamber under vertical vibration
D. Lohse’s group
Maxwell’s Demon is possible in granular systemSteady state: input energy rate = kinetic energy loss rate due to inelastic collisions
N
v
kinetic temp
Evaporation-condensationUnstable !
Bottom plate velocity (input)
Dissipation (output)
Tu
N
VT
grain ~
~2
uRL TT
Evaporation condensation
characteristic
Heaping
Flux model
kT
mgz
ekT
mgNzn
)(
22 )1(22 )1( naan enendt
dn
n h 1-n
large V stable; as V decrease bifurcation !
uniform cluster to 1 side
2
1n
2
1n
2
1n is always a fixed point
Eggers, PRL, 83 5322 (1999)
)(hnuareadt
dN
What happens for a binary mixture?
What are the steady state?
How many granular temperatures ?
Oscillation of millet (小米 , N=4000) and
mung beans (绿豆 , N=400)
F = 20Hz. Amp = 2mm
soda lime glass138 small spheres diameter : 2 mm27 large spheres diameter 4 mmbox height:7.7 cmx0.73cmx5 cm
Effects of compartments + bidispersity: Granular Clock
Markus et al, Phys. Rev. E, 74, 04301 (2006)
Big and small grains. Explained by Reverse Brazil Nuts effects
a=6 mm, f =20 Hz. Times: a=0, b=3.1, c=58.3, d=66.2, e=103.2 s.
Granular Oscillationsin compartmentalized bidisperse granular gas
2.6cmx5.4cmx13.3cm
barrier at1.5 cm
Steel glass balls Radius = 0.5 mm
N = 960
f = 60 Hz
Phase Diagram
B
Ao N
N
Model of two temperatures
• Very large V, A & B are uniform in L & R,
• As V is lowered, at some point only
A is free to exchange:
clustering instability of A• TBR gets higher, then B evaporates to L
• Enough B jumped to L to heat up As,
TAL increases A evaporates from L to R
A oscillates !
ABBRBLARAL TTTTTT ;;
(B heats up A & A slows down B)
Model Objectives
• Quantitative description
• A model to understand the quantitative data
Binary mixture in a single compartment
A B inelastic collision is asymmetric:
A can get K.E. from B (B heats up A & A slows down B)TB is lowered by the presence of A grains ABAB mme
Change of K.E. of A grain due to A-B inelastic collision:BuAu
Dissipation rate of A grain due to A-B inelastic collision:
Binary mixture in a single compartment
)()(
~
)()(
~
2
2
2
2
BB
AA
vq
VT
vp
VT
A B inelastic collision is asymmetric: suppose A gets K.E. from B (B heats up A & A slows down B)TB is lowered by the presence of A grains
ABAB mme
0;0
AB N
q
N
p
AB TT B
A
N
N
Balancing input energy rate from vibrating plate with total dissipation due to collision:
Flux Model for binary mixture of A & B grains in 2 compartments
L RBL
ALL N
N
BR
ARR N
N
PRL, 100, 068001 (2008)J. Phys. Soc. Jpn. 78, 041001 (2009)
)()(
~
)()(
~
2
2
2
2
BB
AA
vq
VT
vp
VT
• is always a fixed point, • stable for V>Vc
• For V<Vc, Hopf bifurcation oscillation
2;
2B
BLA
AL
NN
NN
L R
BL
ALL N
N
BR
ARR N
N
V>Vc
V<Vc
V<Vc
V<Vf
Numerical solution
Model Results• V>Vc, A & B evenly distributed in 2 chambers
• Supercritical Hopf bifurcation near Vc
• V<Vc, limit cycle. Granular clock for A & B.
• Amplitude(v-vc)0.5 [Hopf]
• Period ~ (v- vf)- (numerical solution of Flux model)
• V < Vf , clustering into one chamber
• Saddle-node bifurcation at Vf (??? to be proved rigorously???)
Vc-V (cm/s)
Oscillation amplitude: exptal data
Numerical soln. ofFlux model
Oscillation period
Phase diagram
Other interesting cases:• Tri-dispersed grains : A, B ,C
3-dim nonlinear dynamical system complex dynamics, Chaos…
Other interesting cases:• Bi-dispersed grains in M-compartments:
2(M-1)-dim nonlinear dynamical system complex dynamics,……
3
1 2
Summary
• Dissipation is density dependent “Maxwell demon”
• Different collision dissipations in binary system existence of two “granular temperatures”
• Non-homogeneous temperature with homogenous energy input both spatially and temporally
• Granular steady state + compartment oscillations
Thermophoresis or Janus ?
A worm in a temperature bath
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