Computer Simulation of Gravity-Driven Granular Flow University of Western Ontario Department of Applied Mathematics John Drozd and Dr. Colin Denniston Western Canadian Conference for Young Researchers in Mathematics
Granular Matter Granular matter definition –Small discrete particles vs. continuum Granular matter interest –Biology, engineering, geology, material science, physics. –Mathematics and computer science. Granular motion –Energy input and dissipation. Granular experiments –Vibration
Small Amplitude Surface Waves 3-Node Arching Large Amplitude Surface Waves C. Wassgren et al. 1996
Simulation
Other Phenomena in Granular Materials Shear flow Vertical shaking Horizontal shaking Conical hopper Rotating drum Cylindrical pan
Cylindrical Pan Oscillations Oleh Baran et al. 2001
Raw Feed Grinding Media, Rods Vibrator Motor Ground Product Vibratory Drum Grinder Isolation Spring Reactor Springs Vibratory Drum Grinder
Goal Find optimum oscillation that results in a force between the rods which achieves the ultimate stress of a particular medium that is to be crushed between the rods. Minimize the total energy required to grind the medium. Mixing is also important.
Event-driven Simulation Without Gravity
Event-driven Simulation With Gravity
Typical Simulation
Harry Swinney et al. 1997
Harry Swinney et al. 1997
Granular Materials: A window to studying the Transition from a non- Newtonian Granular Fluid To A "Glassy" system
Velocity q Collision rules for dry granular media as modelled by inelastic hard spheres As collisions become weaker (relative velocity v n small), they become more elastic. C. Bizon et. al., PRL 80, 57, Savage and Jeffrey J. Fluid Mech. 130, 187, 1983.
300 (free fall region) 250 (fluid region) 200 (glass region) 150 polydisperse monodisperse Polydispersity means Normal distribution of particle radii y xz y vyvy dv y /dt P Donev et al PRE 71 Donev et al PRL96 "Do Binary Hard Disks Exhibit an Ideal Glass Transition?"
0 = 0.9 0 = 0.95 0 = 0.99 The density in the glassy region is a constant. In the interface between the fluid and the glass does the density approach the glass density exponentially? Interface width seems to increase as 0 1 y vyvy How does depend on (1 0 ) ?
Density vs Height in Fluid-Glass Transition
Length Scale in Transition "interface width diverges" Slope = 0.42 poly Slope = 0.46 mono
300 (free fall region) 250 (fluid region) 200 (glass region) 150 Y Velocity Distribution Plug flow snapshot Mono kink fracture Poiseuille flow y z x Mono-disperse (crystallized) only
300 (free fall region) 250 (fluid region) 200 (glass region) 150 Granular Temperature 235 (At Equilibrium Temperature) fluid equilibrium glass y xz
Experiment by N. Menon and D. J. Durian, Science, 275, Simulation results Fluctuating and Flow Velocity vv vfvf In Glassy Region ! Europhysics Letters, 76 (3), 360, 2006 J.J. Drozd and C. Denniston 16 x x 32 "questionable" averaging over nonuniform regions gives 2/3
1 in fluid glass transition For 0 = 0.9,0.95,0.96,0.97,0.98,0.99 Subtracting of T g and v c and not averaging over regions of different v x 2 Slope = 1.0 Down centre
Experiment: (W. Losert, L. Bocquet, T.C. Lubensky and J.P. Gollub) Physical Review Letters 85, Number 7, p (2000) "Particle Dynamics in Sheared Granular Matter"
Velocity Fluctuations vs. Shear Rate Physical Review Letters 85, Number 7, (2000) From simulation Experiment Must Subtract T g ! Slope = Slope = 0.4 U
Shear Stress q x y
Viscosity vs Temperature …can do slightly better… …"anomalous" viscosity. Is a fluid with "infinite" viscosity a useful description of the interior phase? Slope ~ 2 Transformation from a liquid to a glass takes place in a continuous manner. Relaxation times of a liquid and its Shear Viscosity increase very rapidly as Temperature is lowered.
Experimental data from the book: “Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials” By Jacques Duran.
Simulation Experiment Quasi-1d Theory (Coppersmith, et al) (Longhi, Easwar) Impulse defined: Magnitude of momentum after collision minus momentum before collision. Related to Forces: Impulse Distribution Most frequent collisions contributing to smallest impulses
Power Laws for Collision Times Similar power laws for 2d and 3d simulations! Collision time = time between collisions 1) spheres in 2d 2) 2d disks 3) 3d spheres
Comparison With Experiment Figure from experimental paper: “Large Force Fluctuations in a Flowing Granular Medium” Phys. Rev. Lett. 89, (2002) E. Longhi, N. Easwar, N. Menon : experiment 1.5 vs. simulation 2.8 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
= 2.75 = 1.50 Probability Distribution for Impulses vs. Collision Times (log scale)
random packing at early stage = 2.75 crystallization at later stage = 4.3 Is there any difference between this glass and a crystal? Answer: Look at Monodisperse grains Disorder has a universal effect on Collision Time power law.
Radius Polydispersity 2d disksSpheres in 2d 3d spheres 0 % (monodisperse) % (polydisperse) Summary of Power Laws
Conclusions A gravity-driven hard sphere simulation was used to study the glass transition from a granular hard sphere fluid to a jammed glass. We get the same 2/3 power law for velocity fluctuations vs. flow velocity as found in experiment, when each data point is averaged over a nonuniform region. When we look at data points averaged from a uniform region we find a power law of 1 as expected. We found a diverging length scale at this jamming (glass) to unjamming (granular fluid) transition. We compared our simulation to experiment on the connection between local velocity fluctuations and shear rate and found quantitative agreement. We resolved a discrepancy with experiment on the collision time power law which we found depends on the level of disorder (glass) or order (crystal).
Normal Stresses Along Height Weight not supported by a pressure gradient. Momentum Conservation k ik + g i = 0
Momentum Conservation