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

Convection of grains under vertical vibrations

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

Steady state configurations

initially flat layer. L=45 and =10

N=225

N=675

=0.1

=0.1

=0.05

=0.3

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:-

Steady Heaps hs(x)

B.C. :

Solution:

ho given by:

Assume only freq. dependent length is 1/k, then

= dimensionless dissipation strength

Steady state heaping profiles:

Initial flat layer downward heap upward heap

As k increases, steady heap changes from

downward (h(0)/H <1) to upward (h(0)/H >1)

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)

aspect ratio offlat layer:

Heaping angle

Comparison with Experimentalmeasurement on Heaping angles

Identifying:

Hisau et al.,Adv. Powder Tech. 7, 173 (96)

Thicker layer can beexcited to steeper heap

Effect of layer thickness:Effect of layer thickness:

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.

Dynamics of heap formation

Heap formation time

Layered bidispersed Granular Bed: oscillations

Du et. al, PRE 84, 041307 (2010)

Oscillating layer video

Cu

Ala

Anti-symmetric profile h(x,t)

Steady state

Stability:

Flat profile remains stable

=ko/ho

Another Layer on top

Steady state profile

Heaping angle

=ko/ho

Flat interface becomes

=ko/ho

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,

Oscillating layer instability

=ko/ho

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