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Granular Clock & Temperature Oscillations in a bidisperse Granular Gas Pik-Yin Lai ( 黎璧賢 ) Dept. of Physics & Center for Complex Systems, National Central.

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Presentation on theme: "Granular Clock & Temperature Oscillations in a bidisperse Granular Gas Pik-Yin Lai ( 黎璧賢 ) Dept. of Physics & Center for Complex Systems, National Central."— Presentation transcript:

1 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

2 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

3 Grains 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

4 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)

5 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

6 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

7 Molecular gases

8 monodisperse granular gas in compartments: Maxwell’s Demon Eggers, PRL, 83 5322 (1999) v

9 Clustering Granular gas in Compartmentalized chamber under vertical vibration D. Lohse’s group

10 Maxwell’s Demon is possible in granular system Steady state: input energy rate = kinetic energy loss rate due to inelastic collisions N v kinetic temp Evaporation-condensation Unstable ! Bottom plate velocity (input) Dissipation (output) u Evaporation condensation characteristic

11 Flux model n h 1-n large V stable; as V decrease  bifurcation  ! uniform  cluster to 1 side is always a fixed point Eggers, PRL, 83 5322 (1999)

12 What happens for a binary mixture?

13 Granular Oscillations in compartmentalized bidisperse granular gas NA grain A NB grain B co =NA/NB

14 Phase Diagram

15 Objectives Quantitative description A model to understand the quantitative data

16 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

17 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 Change of K.E. of A grain due to A-B inelastic collision: Dissipation rate of A grain due to A-B inelastic collision:

18 Binary mixture in a single compartment 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 Balancing input energy rate from vibrating plate with total dissipation due to collision:

19 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 T BR gets higher, then B evaporates to L Enough B jumped to L to heat up As, T AL increases  A evaporates from L to R A oscillates ! (B heats up A & A slows down B)

20 Flux Model for binary mixture of A & B grains in 2 compartments L R PRL, 100, 068001 (2008) J. Phys. Soc. Jpn. 78, 041001 (2009)

21 Theoretical result for p & q Balancing input energy and dissipation due to inelastic collisions:

22 p(c) & q(c) can be calculated theoretically

23 is always a fixed point, stable for V>Vc For V<Vc, Hopf bifurcation  oscillation L R

24 V>Vc V<Vc V<V f Numerical solution

25 Model Results V>Vc, A & B evenly distributed in 2 chambers Supercritical Hopf bifurcation near V c V<Vc, limit cycle. Granular clock for A & B. Amplitude  (v-v c ) 0.5 [Hopf] Period  ~ (v- v f ) -  (numerical solution of Flux model) V < V f, clustering into one chamber Saddle-node bifurcation at V f (to be proved rigorously)

26 Vc-V (cm/s) Oscillation amplitude: exptal data Numerical soln. of Flux model

27 Oscillation period

28 Phase diagram

29 Analytic results Fix point (0,0) loses stability at v c

30 Supercritical Hopf bifurcation at vc Theorem: supercritical Hopf if verified Expanding near (0,0):

31 Analytic result for phase boundary Fix point (N A /2,N B /2) loses stability at v c Vc calculated from

32 Analytic result for emergent frequency at vc Hopf bifurcation at v c : Larger c (more A), longer time to heat up for evaporation  smaller freq.

33 Saddle-node bifurcation at vf Phase boundary of vf: New stable fixed point emerges: V < Vf, clustering

34 Other interesting cases: Tri-dispersed grains : A, B,C 3-dim nonlinear dynamical system  complex dynamics, Chaos…

35 Other interesting cases: Bi-dispersed grains in M-compartments: 2(M-1)-dim nonlinear dynamical system  complex dynamics,…… 3 12

36 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


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