Granular Clock & Temperature Oscillations in a bidisperse Granular Gas Pik-Yin Lai ( 黎璧賢 )...
-
Upload
phyllis-day -
Category
Documents
-
view
213 -
download
0
Transcript of Granular Clock & Temperature Oscillations in a bidisperse Granular Gas Pik-Yin Lai ( 黎璧賢 )...
Granular Clock & Temperature Oscillations in a bidisperse Granular Gas
Pik-Yin Lai ( 黎璧賢 )Dept. of Physics & Center for Complex Systems,
National Central University, Chung-Li,Taiwan
Collaborators:C. K. Chan( 陳志強 ), Institute of Physics Academia Sinica
May Hou ( 厚美英 ), IOP, Chinese Academy of Sciences
Granular materials(顆粒體 )
refer to collections of a large number of discrete solid components.日常生活中所易見的穀物、土石、砂、乃至公路上的車流、輸送帶上的物流等
Granular materials have properties betwixt-and -between solids and fluids (flow).
Basic physics is NOT understood
Complex and non-linear medium
Grains EverywhereGrains Everywhere
Food: almost everything we eat,: rice, cereal, peas...
Engineering: Powder mechanics, soil mechanics
Construction: Rocks, bricks, sand..
Agriculture: transport, storage & manipulation of seeds, grains & foodstuffs
Pharmaceutical: pills & powder processing
Transportation: shock absorption packing
Industries: Mixing & segregation of grains & powder by shaking or rotation
Geological: desert, landslides, earthquake dynamics
1 trillion US$/year
•Discrete & Macroscopic•Hardcore interactions & Dissipative•Zero temperature: mgd/kT ~ 10•Breakdown of hydrodynamics•Friction is important•Dynamically Driven•Inhomogeneous static stresses•Complex Many-body systems
•Mixing & Segregation•Pattern formation•Complex Flow
Physical aspects of Grains
12
Rev. Mod. Phys. 68, 1259 (1996) 71, S374 (1999) 71, 435 (1999)物理雙月刊 23, 503 (2001)
•Far from equilibrium•Dissipation: inelastic collisions•Energy input: vibrating bed (bottom collision)•Dissipation rate ~ input rate steady state
•Heap formation•Granular gas
Grains under excitations: vibrating bed
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
Molecular gases
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
Flux model
kT
mgz
ekT
mgNzn
)(
22 )1(22 )1( naan enendt
dn TV
ha
1~
2
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?
Granular Oscillationsin compartmentalized bidisperse granular gas
NA grain ANB grain B
=NA/NB
Phase Diagram
B
Ao N
N
Objectives
• Quantitative description
• A model to understand the quantitative data
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
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:
binary mixture of A & B grains in 2 compartments
• 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)
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
Theoretical result for p & q• Balancing input energy and dissipation
due to inelastic collisions:
p() & q() can be calculated theoretically
• 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
Analytic results
• Fix point (0,0) loses stability at vc
Supercritical Hopf bifurcation at vc
• Theorem: supercritical Hopf if
verified
Expanding near (0,0):
Analytic result for phase boundary• Fix point (NA/2,NB/2) loses stability at vc
• Vc calculated from
Analytic result for emergent frequency at vc
• Hopf bifurcation at vc :
Larger (more A), longer time to heat up for evaporation smaller freq.
Saddle-node bifurcation at vf
• Phase boundary of vf:
New stable fixed point emerges:
V < Vf , clustering
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• Evaporation /Condensation in granular compartmentalized gas is
unstable when dissipations become important “Maxwell demon”
• Temperature difference is generated spontaneously.
• Each grain type has difference temperature in a bi-disperse vibrating grain mixture because of asymmetric properties of collisions (mass, size,…) [even for single compartment]
• Binary mixtures can generate oscillatory temperature differences in the two compartments
• Oscillations: Hopf bifurcation at vc• Clustering: saddle-node bifurcation at vf• Our model is confirmed by experiments.• Systems with rich and complex dynamics