1. John Drozd Colin Denniston Simulations of Collision Times and Stress In Gravity Driven Granular Flow bottom sieve particles at bottom go to top reflecting left and right walls periodic or reflecting front and back walls  Snapshot of 2d simulation from paper: “Dynamics and stress in gravity-driven granular flow” Phys. Rev. E. Vol. 59, No. 3, March 1999 Colin Denniston and Hao Li 3d simulation 

2. Outline Profiles Mechanics Stress Momentum Impulse Collision times

3. Velocity q Collision rules for dry granular media as modelled by inelastic hard spheres

4. 300 (free fall region) 250 (fluid region) 200 (glass region) 150

5. 300 (free fall region) 250 (fluid region) 200 (glass region) 150 Y Velocity Distribution Plug flow kink fracture Poiseuille flow

6. 300 (free fall region) 250 (fluid region) 200 (glass region) 150 Granular Temperature 235 (At Equilibrium Temperature) fluid equilibrium glass

7. Nehal Al Tarhuni  Experiment by N. Menon and D. J. Durian, Science, 275, 1997.  Simulation results by summer student Fluctuating and Flow Velocity v2v2 vfvf

8. Normal Stresses Along Height Weight not supported by a pressure gradient.  k  ik +  g i =  t (  v i )+  k (  v i v k )  k  ik +  g i = 0,  v  v  Momentum Conservation

10. Shear Stress

11. Experimental data from the book: “Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials” By Jacques Duran.

12. Simulation  Experiment  Quasi-1d Theory  (Coppersmith, et al) (Longhi, Easwar) Impulse defined: Magnitude of velocity after collision minus velocity before collision. Most frequent collisions contributing to smallest impulses Related to Forces: Impulse Distribution

13. Distribution of Impulses from "spheres in 2d" simulation fluid region glassy region Upturn in glassy region Downturn in fluid region

14. Collision time distribution P(t) = probability that a molecule survives a time t without collision P(t) t 0 1 ? w dt = the probability that a molecule suffers a collision between time t and t+dt dt w is independent of past history (assumption of molecular chaos)

15. Hence P(t+dt) = P(t) [probability that molecule avoids collision in interval dt] = P(t) (1- w dt) Thus or So that The “collision time” or “relaxation time” is

16. Power Laws for Collision Times Similar power laws for 2d and 3d simulations! 2d disks 3d spheres Collision time = time between collisions 1) 2d disks 2) spheres in 2d 3) 3d spheres

17. Comparison With Experiment  Figure from experimental paper: “Large Force Fluctuations in a Flowing Granular Medium” Phys. Rev. Lett. 89, 045501 (2002) E. Longhi, N. Easwar, N. Menon  : experiment 1.5 vs. simulation 2.7 Discrepancy as a result of Experimental response time and sensitivity of detector.  Experiment “Spheres in 2d”: 3d Simulation with front and back reflecting walls separated one diameter apart  Pressure Transducer

18. 1) grain grain 2) grain wall Wall collisions only  = 1.7  0.2  1.5  = 2.9  0.2  2.7

19.  random packing at early stage  = 2.75 crystallization  at later stage  = 4.3 Is there any difference between this glass and a solid? Answer: Look at Monodisperse grains Disorder has a universal effect on Collsion Time power law.

20. Radius Polydispersity 2d disksSpheres in 2d 3d spheres 0 % (monodisperse) 4 4.3 3.4 7.5 % (polydisperse) 2.75 2.85 3.4 15 % (polydisperse) 2.75 2.85 3.4 Summary of Power Laws

21. Conclusions Power laws dependent on disorder. Using power laws can differentiate between granular fluid and glass. Can differentiate between granular glass and solid! Is this universal? Not done before! Dynamics (collision directions) lead to static stresses.

22. Future Work Plot impulse distributions along height for 3d simulation to fully compare to 2D results Power law for distances between collisions Viscosity vs. Temperature power law (Lubensky) Constitutive stress relations (Cates) Chaos and diffusion (3D Tetrahedral Voronoi cells) Heat flux analysis for 3D simulation

23. THE END