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Lecture IV: Dense Granular Flows

Explore stationary flow stability, phase diagrams, avalanches in layers, connection to BCRE, and comparisons with MD simulations in dense granular flows. Dive into order parameter equations, constitutive relations, and mass conservation. Analyze deep chute dynamics and granular stick-slips friction. Validate theories through numerical methods and MD simulations.

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Lecture IV: Dense Granular Flows

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  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.

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