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