Models 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. w No vibration Steady heap formed for G > 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. G =1.9; f=50Hz

Upward Heap (valley) at strong vibrations: Glass beads with d=0.61mm in a 100mm x 37mm container. G =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  =0.05  =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 :

(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 h s(x) B.C. : Solution: h o given by:

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

Downward Heap Profile Glass beads with d=3mm in a 190mm x 30mm container. G =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. G =5.9; f=50Hz Aoki et al., PRE 54, 874 (1996)

aspect ratio of flat layer: Heaping angle

Comparison with Experimental measurement on Heaping angles Identifying: Hisau et al.,Adv. Powder Tech. 7, 173 (96)

Thicker layer can be excited 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 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, (2010) Oscillating layer video Cu Ala

Anti-symmetric profile h(x,t)

Steady state Stability: Flat profile remains stable

c =ko/ho Another Layer on top

Steady state profile q

Heaping angle c =ko/ho

Flat interface becomes G* c =ko/ho

Oscillating Layer for G > G 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 instability can take place: G c > G* G c = Max (G c, G*)

Oscillating layer instability c =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.

Collaborators Collaborators C.K. Chan Institute of Physics, Academia Sinica L.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 Technology Europhy. Lett. 100, (2012). Thank you