Models of Heaping Pik-Yin Lai ( 黎璧賢 ) Dept. of Physics and Center for Complex Systems,...
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Transcript of Models of Heaping Pik-Yin Lai ( 黎璧賢 ) Dept. of Physics and Center for Complex Systems,...
Models of HeapingModels of Heaping
Pik-Yin Lai (黎璧賢 )Dept. of Physics and Center for Complex Systems, National Central University, Taiwan
Symmetric heap formation
Anti-symmetric heaps & oscillations in bi-layer granular bed
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
Heap formation of granular materials in a vertical vibrating bed: amplitude A, freq.
No vibration
Steady heap formedfor > 1.2
Steady Downward Heap (mountain) at low vibrations:(downward convection current next to the walls)
Glass beads with d=0.61mm in a 100mm x 43mm
container. =1.9; f=50Hz
Upward Heap (valley) at strong vibrations:
Glass beads with d=0.61mm in a 100mm x 37mm container.
=5.9; f=50Hz
Experimental Data from K.M. Aoki et al.
Density fluctuations due to vibration & convection can be induced
Surface flow is needed to complete a convection cycle
Density fluctuations realized by creation of empty sites/voids in the bulk
Surface flow taken care by sandpile rules
Empty site sandpile model
Dynamic rules for Empty site & grains
•empty sites are created randomly and uniformly with a probability
•empty sites exchange their positions to regions of lower pressure.
•pressure at an empty site ~ the number of grains on top of that site.
•empty site gets to the top of the pile, it disappears
•grains topple above critical slope with rate
N= 675 and L=45
is similar to in experiments—enhance fluctuations
: relaxation of height— suppress fluctuations
Competition between & produces different steady state heaps
Phase diagram of steady heaps
a simple analytic model to predict the structures of steady state upward and downward heaps
Height profile h(x,t) as the only dynamical variable
Three basic factors:
(1) energy pumped into the medium by vibration that causes density fluctuations & layer expansion
(2) grains roll down the slope by surface flow and cause the profile to flatten
(3) dissipation due to grain collisions --- nonlinear suppression of height
Phenomenological Model :Phenomenological Model :
(2) (1) (3)Grain rolling layer expansion dissipation
Boundary Conditions: (i) Symmetric profile (identical left & right walls) (ii) Total Volume under h(x,t) is constant (vibrations not too violent)
N grain of size a in a H x 2 l bed
Initial Flat Profile:
Model
steady-state profile:
approx. correct for small vibrations(low k):
Non-dissipative (linear) solution:-
Assume only freq. dependent length is 1/k, then
= dimensionless dissipation strength
Steady state heaping profiles:
Downward Heap Profile
Glass beads with d=3mm in a 190mm x 30mm container.
=1.5; f=50Hz
Hisau et al.,Adv. Powder Tech. 7, 173 (96)
Upward Heap Profile
Glass beads with d=0.61mm in a 100mm x 37mm container.
=5.9; f=50Hz
Aoki et al., PRE 54, 874 (1996)
Comparison with Experimentalmeasurement on Heaping angles
Identifying:
Hisau et al.,Adv. Powder Tech. 7, 173 (96)
Heap Equation:
Continuity Equation:
Conservation Law:
Surface flow bulk flow under the profile
Effective Current :
Effective Current agrees with convective pattern
Downward heap formation:Surface current >0 for x>0 but total j<0, so bulk current <0 deep in layer.
Upward heap formation:Surface current <0 for x>0 but total j>0, so bulk current >0 deep in layer.
Layered bidispersed Granular Bed: oscillations
Du et. al, PRE 84, 041307 (2010)
Oscillating layer video
Cu
Ala
Oscillating Layer for c
the heap is so large that it either(i) hits the bottom of the container, i.e.
or (ii) pinches off the total height of the layers,
must become unstable first for heaping to occur before the second oscillation instabilitycan take place: c
c= Maxc,
Summary
Phenomenological model for heap formation using h(x,t) Energy input to the system by the increase in height Dissipation is represented by the nonlinear terms Upward and Downward heaps can be modeled.
Strong enough vibration leads to anti-symmetric interface in a bi-layer pile.
Oscillating layers can occur. Model with cubic non-linearity can model the interface
profile, heaping angle and threshold vibration strengths.
CollaboratorsCollaborators
C.K. Chan Institute of Physics, Academia SinicaL.C. Jia Dept. of Physics, Nat’l Central Univ.
Phys. Rev. Lett. 83, 3832 (1999); Phys. Rev. E 61, 5593 (2000)Chin. J. Phys. 38, 814 (2000); J. Phys. A 33, 8241 (2000)
Ning Zheng Dept. of Physics , Beijing Institute of TechnologyEurophy. Lett. 100, 44002 (2012).
Thank you