Lecture IV: Dense Granular Flows

1. Lecture IV: Dense Granular Flows Igor Aronson Materials Science Division Argonne National Laboratory

2. Outline • Stationary Flow Stability • Phase Diagram • Avalanches in Shallow Layers • Deep Layers/Connection to BCRE • Comparison with MD simulations • Granular Stick-Slips Friction

3. Summary of Lecture III Order parameter equation Constitutive relation for shear stress Mass conservation + Momentum conservation

4. Shear Temperature: Simplified Version • j1, j2 – dynamic/static repose angles • for j=j2 d=1 – granular solid is unstable • for j=j1 d=1/2 – solid/liquid equilibrium • (note slightly different definition of j1)

5. z y g h x j Chute: stability of solid state r=1 Boundary conditions: r= 1 for z = -h(rough bottom) rz = 0 for z = 0 (free surface) OPE: Perturbation: Eigenvalue: Stability limit:

6. z y g h x j Stationary Flow Stationary OPE: 1st integral: Velocity profile – from stress constitutive relations: Boundary conditions:

7. Stationary Flow Existence Limit Solution exists only for Solving the first integral:

8. Phase Diagram solid & liquid flow liquid only hmin no flow solid only

9. Single mode approximation:depth averaging Close to the stability boundary A(x,y,t) – slowly varying amplitude Orthogonality/Solvability condition Order parameter equation

10. z Correction to d j j0 x j0- chute angle, j –local slope - local slope contribution

11. Mass Conservation Law down-hill flux of grains “dimensionless” mobility Transverse flux Jy is neglected since Jx>> Jy

12. Model • Order parameter equation • Local “shear temperature” • Evolution of layer depth

13. Boundary conditions • Inlet x=0 • No-flux condition: Jx=0 • Fixed flux condition Jx=aAh3=J0 (grains supplied from hopper with the fixed rate)

14. Numerical Methods • Implicit Crank-Nicholson code for A • Number of mesh point in x – 600-2400 • Number of mesh points in y – 600 • Time of integration op to 2000 units • h – 2-16 a=0.025, b=3, Lx=400, Ly=200 • Unit of length is about grain diameter

15. Fixed Flux at the Inlet of the Chute t 1000 0 Space-time plot of the height h Red – max, blue - min • Large flux – steady flow, • h is adjusted according to J0 • Small flux – periodic sequence • of avalanches, Period T ~1/J0 500 x continuous flow periodic avalanches

16. Two types of avalanches (theory) Downhill Uphill

17. Two types of avalanches (experiment) Daerr & Douady, Nature, 399, 241 (1999) Triangular (downhill) Balloon (uphill)

18. 0.6 0.5 0.4 V 0.3 0.2 0.1 0.0 1.05 1.06 1.07 1.08 1.09 1.10 d Transition from down-hill to up-hill:1D analysis of avalanche cross-sections uphill 20 Uphill front speed Secondary avalanche h 10 0 downhill 20 h 10 discontinuous transition! 0 0 200 400 600 x

19. Quantitative comparison with experiment Model parameters t, characteristic time l, characteristic length j1,j2, static/dynamic repose angles h, viscosity coefficient Daerr & Douady: (particle diameter)

20. Phase diagram (theory & experiment)

21. Infinite Layers: Exact front solution for d=1/2 New variable z0=const - position of the front • does not satisfy boundary conditions • non-stationary for d≠1/2

22. Avalanches in deep chute Universal approximation for r (exact for small and large z) Bi-stable function F New variable z0 : depth of fluidized layer d=0.9 d=0.7 Evolution of z0 d=0.5

23. Deep chute (cont) Expression for flux - x Symmetry x No triangular avalanches in sandpiles!

24. Connection with BCRE theory • BCRE (Bouchaud, Cates, Rave Prakash & Edwards, 1995) operates with: • H-thickness of immobile fraction, R-thickness of rolling fraction Boutreux, Raphael & de Gennes modified instability term • Our theory: • reproduces BCRE for small R (or z0) • reproduces Boutreux et al for large R • has hysteresis missing in both theories

25. Connection with hydrodynamics & kinetic theory • For flow with finite granular temperature Control parameter • T-granular temperature, d0-shear temperature • T1,2-critical temperatures for instability of overheated solid/overcooled liquid Resulting equations • momentum conservation • order parameter • granular temperature evolution Transition to conventional granular hydrodynamics for T>>d0

26. Validation of Theory by MD simulations • non-cohesive, dry, disk-like grains • three degrees of freedom. • A grain pis specified by radius Rp, position rp, translational and angular velocities vpand wp. • Grains pand qinteract whenever they overlap,Rp+ Rq-| rp –rp| > 0 Restitution coefficient e Friction coefficient m • linear spring-dashpotmodel for normal impact • Cundall-Strackmodel for oblique impact. • Detail: Silbert et al, Phys Rev E, 64, 051302 (2001) All quantities are normalized using particle size d, mass m, and gravity g 2304 particles (48x48), e = 0.82; m = 0.3; Pext= 13.45,Vx=24 Simulations: IBM SP2 at NERSC, fastest unclassified computer in the world

27. Testbed system:Couette flow in a thin granular layer without gravity 500 particles (50x10), e= 0.82; m = 0.3; P= 13.45, no gravity Adiabatic change in shear force:

28. Fitting free energy: fixed points of the order parameter OP equation MD simulations • 500 particles (50x10), • = 0.82; m = 0.3

29. Fitting the constitutive relation Fit: qy(r) = (1-r1.2)1.9 Fit: q(r) = (1-r)2.5 Phenomen. theory: q(r)=1-r qx(r) = (1-r)1.9

30. Newtonian Fluid + Contact Part Kinematic viscosity in slow dense flows: h≈12

31. Relation to Bagnold Scaling Bagnold relation (1954): Silbert, Ertas, Grest, Halsey, Levine, and Plimpton, Phys. Rev. E 64, 051302 (2001) Fitting Bagnold scaling relation

32. Shear granular flows and stick-slips Nasuno, Kudrolli, Bak and Gollub, PRE, 58, 2161 (1998). sliding speed V=5.67 mm/s sliding speed V=5.67 mm/s sliding speed V=11.33 mm/s

33. Slip event: MD simulations

34. Example: stick-slips thick surface driven granular flow with gravity m k V0 y Set of equations for sand g Ly x Equations for heavy plate 5000 particles (50x100), e = 0.82; m = 0.3; Pext= 10,50,Vtop=5,50

35. Simplified theory: reduction to ODE y x • Stationary OP profile: • x –width of fluidized layer (depends on shear stress), r1=(4r*-1)/3 • Stationary solutionexists only for specific value ofd(y) (symmetry between the roots of OP equations)which fixes position of the front g x

36. Perturbation theory • Substituting r into OP equation and performing orthogonality one obtains • Regularization for x<<1 (l –is the growth rate of small perturbations)

37. Resulting 3 ODE • 2 Equations for Plate • 1 Equation for width of fluidized layer

38. Comparison: Spring deflection vs time theory: PDE theory: ODE MD simulations

39. Conclusions • We introduced a theoretical description for partially fluidized granular flows based on theorder parameterwhich is defined as a fraction of persistent contacts among the grains. • Stress tensorin granular flows is separated into a “fluid” part and a “solid” part. The ratio of the fluid and solid parts is controlled the order parameter • The dynamics of the order parameter is described by the Ginzburg-Landau equation with a bistable free energy functional. • Thefree energycontrolling the dynamics of the order parameter, can be extracted from molecular dynamics simulations. • The viscosity coefficient calculated as a ratio of thefluid shear stress to the strain ratedoes notdiverge at small strain rates. • The model successfully describes dynamics of various shear granular flow: from avalanches to stick slips.