Models of Heaping

1. 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

2. 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

3. Heap formation of granular materials in a vertical vibrating bed: amplitude A, freq. w No vibration Steady heap formed forG> 1.2

4. Convection of grains under vertical vibrations

5. 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

6. 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.

7. Empty site sandpile model 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

8. Dynamic rules for Empty site & grains • empty sites are created randomly and uniformly with a probability a • 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 g

9. Steady state configurations L=45 and g=10 initially flat layer. a=0.05 N=225 a=0.1 N=675 a=0.1 a=0.3

10. Phase diagram of steady heaps • is similar to Gin experiments—enhance fluctuations • : relaxation of height— suppress fluctuations Competition between g & aproduces different steady state heaps N=675 and L=45

11. Phenomenological Model : 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

12. 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:

13. Non-dissipative (linear) solution:- steady-state profile: approx. correct for small vibrations(low k):

14. Steady Heaps hs(x) B.C. : Solution: ho given by:

15. Steady state heaping profiles: Assume only freq. dependent length is 1/k, then m = dimensionless dissipation strength

16. Initial flat layer downward heap upward heap

17. As k increases, steady heap changes from downward (h(0)/H <1) to upward (h(0)/H >1)

18. Downward Heap Profile Hisau et al.,Adv. Powder Tech. 7, 173 (96) Glass beads with d=3mm in a 190mm x 30mm container. G=1.5; f=50Hz

19. Upward Heap Profile Aoki et al., PRE 54, 874 (1996) Glass beads with d=0.61mm in a 100mm x 37mm container. G=5.9; f=50Hz

20. Heaping angle aspect ratio of flat layer:

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

22. Effect of layer thickness:Effect of layer thickness: Thicker layer can be excited to steeper heap

23. Effective Current : Surface flow bulk flow under the profile Heap Equation: Continuity Equation: Conservation Law:

24. 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.

25. Dynamics of heap formation

26. Heap formation time

27. Layered bidispersed Granular Bed: oscillations Cu Ala Oscillating layer video Du et. al, PRE 84, 041307 (2010)

28. Anti-symmetric profile h(x,t)

29. Steady state Stability: Flat profile remains stable

30. Another Layer on top c=ko/ho

31. Steady state profile q

32. Heaping angle c=ko/ho

33. Flat interface becomes G* c=ko/ho

34. Oscillating Layer for G > Gc 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,

35. must become unstable first for heaping to occur before the second oscillation instability can take place: Gc > G* Gc= Max(Gc, G*)

36. Oscillating layer instability c=ko/ho

37. 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.

38. 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, 44002 (2012). Thank you