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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 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
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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
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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
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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)
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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
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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
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Molecular gases
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monodisperse granular gas in compartments: Maxwell’s Demon Eggers, PRL, 83 5322 (1999) v
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Clustering Granular gas in Compartmentalized chamber under vertical vibration D. Lohse’s group
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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
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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)
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What happens for a binary mixture?
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Granular Oscillations in compartmentalized bidisperse granular gas NA grain A NB grain B co =NA/NB
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Phase Diagram
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Objectives Quantitative description A model to understand the quantitative data
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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
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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:
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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:
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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)
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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)
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Theoretical result for p & q Balancing input energy and dissipation due to inelastic collisions:
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p(c) & q(c) can be calculated theoretically
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is always a fixed point, stable for V>Vc For V<Vc, Hopf bifurcation oscillation L R
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V>Vc V<Vc V<V f Numerical solution
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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)
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Vc-V (cm/s) Oscillation amplitude: exptal data Numerical soln. of Flux model
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Oscillation period
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Phase diagram
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Analytic results Fix point (0,0) loses stability at v c
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Supercritical Hopf bifurcation at vc Theorem: supercritical Hopf if verified Expanding near (0,0):
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Analytic result for phase boundary Fix point (N A /2,N B /2) loses stability at v c Vc calculated from
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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.
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Saddle-node bifurcation at vf Phase boundary of vf: New stable fixed point emerges: V < Vf, clustering
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Other interesting cases: Tri-dispersed grains : A, B,C 3-dim nonlinear dynamical system complex dynamics, Chaos…
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Other interesting cases: Bi-dispersed grains in M-compartments: 2(M-1)-dim nonlinear dynamical system complex dynamics,…… 3 12
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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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