Granular Clock Temperature Oscillations in a bidisperse Granular
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 Everywhere 1 trillion US$/year 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
Physical aspects of Grains • Discrete & Macroscopic • Hardcore interactions & Dissipative 12 • Zero temperature: mgd/k. T ~ 10 • Breakdown of hydrodynamics • Friction is important • Dynamically Driven • Inhomogeneous static stresses • Complex Many-body systems • Mixing & Segregation • Pattern formation • Complex Flow Rev. Mod. Phys. 68, 1259 (1996) 71, S 374 (1999) 71, 435 (1999) 物理雙月刊 23, 503 (2001)
Grains under excitations: vibrating bed • Far from equilibrium • Dissipation: inelastic collisions • Energy input: vibrating bed (bottom collision) • Dissipation rate ~ input rate steady state • Heap formation • Granular gas
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 v Eggers, PRL, 83 5322 (1999)
Clustering • Granular gas in Compartmentalized chamber under vertical vibration D. Lohse’s group
Maxwell’s Demon is possible in granular system Steady state: input energy rate = kinetic energy loss rate due to inelastic collisions Bottom plate velocity (input) Dissipation (output) characteristic kinetic temp u Evaporation-condensation Evaporation N Unstable ! v condensation
Flux model n h Eggers, PRL, 83 5322 (1999) 1 -n is always a fixed point large V stable; as V decrease bifurcation uniform ! cluster to 1 side
What happens for a binary mixture?
Granular Oscillations in compartmentalized bidisperse granular gas NA grain A NB grain B co=NA/NB
Phase Diagram
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 Change of K. E. of A grain due to A-B inelastic collision: Dissipation rate of A grain due to A-B inelastic collision: 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
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:
binary mixture of A & B grains in 2 compartments (B heats up A & A slows down B) • 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 !
Flux Model for binary mixture of A & B grains in 2 compartments L PRL, 100, 068001 (2008) J. Phys. Soc. Jpn. 78, 041001 (2009) R
Theoretical result for p & q • Balancing input energy and dissipation due to inelastic collisions:
p(c) & q(c) can be calculated theoretically
• • is always a fixed point, stable for V>Vc • For V<Vc, Hopf bifurcation oscillation L R
Numerical solution V>Vc V<Vc V<Vf
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 D ~ (v-vc)0. 5 [Hopf] Period t ~ (v- vf)-a (numerical solution of Flux model) V < Vf , clustering into one chamber Saddle-node bifurcation at Vf (to be proved rigorously)
Oscillation amplitude: exptal data Numerical soln. of Flux model Vc-V (cm/s)
Oscillation period
Phase diagram
Analytic results • Fix point (0, 0) loses stability at vc
Supercritical Hopf bifurcation at vc Expanding near (0, 0): • Theorem: supercritical Hopf if verified
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 c (more A), longer time to heat up for evaporation smaller freq.
Saddle-node bifurcation at vf New stable fixed point emerges: V < Vf , clustering • Phase boundary of vf:
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, …… 1 2 3
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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