temperature oscillations in a compartmetalized bidisperse granular gas
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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