Order Parameter Description of Shear Flows in Granular Media Igor Aronson (Argonne) Lev Tsimring, Dmitri Volfson (UCSD) Publications: Continuum Description of Avalanches in Granular Media, Phys. Rev. E 64, (2001); Theory of Partially Fluidized Granular Flows, Phys. Rev. E 65, (2002) Order Parameter Description of Stationary Gran Flows, Phys. Rev. Lett. 90, (2003) Partially Fluidized Granular Flows, Continuum theory and MD Simulations, Phys. Rev. E. 68, (2003) Stick-Slip Dynamics in a Granular Layer under Shear, Phys. Rev. E 69, (2004) Supported by US DOE, Office of Basic Energy Sciences SAMSI Workshop, NC 2004

Outline Introduction Experimental observations of partially fluidized granular flows Theoretical description: order parameter model Examples: –Near-surface shear flows –Stick-slips in shear flows –Avalanches in thin chute flows MD simulations and fitting the OP theory Conclusions

Onset of Motion & Fluidization: Quest for Constitutive Relation Various phenomena: avalanches, slides, surface flows, stick-slips are related to the transition from granular solid to granular liquid Theoretical descriptions of granular solid and granular liquid are very different: need for unification Universal description of partially fluidized flows requires a constitutive relation valid for both granular solid and granular liquid

Recent publications Experiments –Nasuno, Kudrolli, Bak and Gollub, PRE, 58, 2161 (1998). –Daerr and Douady, Nature, 399, 241 (1999) –Pouliquen, Phys. Fluids, 11, 542 (1999) –Veje, Howell, and Behringer, PRE, 59, 739 (1999) –Mueth, Debregeas, Karczmar, Eng, Nagel, Jaeger, Nature, 406, 385 (2000) –Komatsu, Inagaki, Nakagawa, Nasuno, PRL, 86, 1757 (2001) –Bocquet, Losert, Schalk, Lubensky, and Gollub, PRE 65, (2001) Molecular dynamics –Silbert et al, PRE 64, (2002);cond-mat/ –Aharonov and Sparks, PRE 65, (2002) Continuum theories –Bouchaud, Cates, Ravi Prakash, and Edwards, J.Phys.France, 4, 1383 (1994) –Boutreux, Raphael, and de Gennes, PRE, 58, 4692 (1998) –Bocquet, Losert, Schalk, Lubensky, and Gollub, PRE 65, (2001) –Rajchenbach, Phys. Rev. Lett. 88, (2002); 89, (2002) –Aranson and Tsimring, PRE, 64, (R) (2001);65, (2002)

Near-surface granular flows Komatsu, Inagaki, Nakagawa, Nasuno, PRL, 86, 1757 (2001) exponential velocity profile h

Driven shear flow under a heavy plate Light sheet Tsai, Voth, & Gollub, PRL 2003 U driving = 7.2mm/s =12 d/s Particle size d=0.6 mm Channel width 30d, circumference 750d, depth = 0~50d. load

Taylor-Couette granular flow (2D) Veje, Howell, and Behringer, Phys.Rev.E, 59, 739 (1999) Velocity profile:

Taylor-Couette granular flow (3D) Mueth, Debregeas, Karczmar, Eng, Nagel, Jaeger, Nature, 406, 385 (2000) 60mm

Taylor-Couette flow - 2 Bocquet et al, 2001

Chute flows Daerr & Douady, 1999 O.Pouliquen, 1999

Avalanches in chute flows Triangular (down-hill) Balloon (up-hill) Daerr & Douady, Nature, 399, 241 (1999)

Avalanches: phase diagram No flow spontaneous avalanching Bistability downhill avalanches uphill avalanches Daerr & Douady, Nature, 399, 241 (1999)

Stick-slip motion of grains Nasuno, Kudrolli, Bak and Gollub, PRE, 58, 2161 (1998). sliding speed V=11.33 mm/s sliding speed V=5.67 mm/s sliding speed V=5.67 m/s

Stick-slip motion - 2

Theoretical model Euler equation where - density of material ( =1) g - gravity acceleration v - hydrodynamic velocity D/Dt - material derivative - stress tensor div v=0 incompressibility condition

Stress-stain relation for partially fluidized granular flow Here - strain rate tensor - viscosity - quasistatic (contact) part fluid: solid: has non-zero off-diagonal elements

Stress-stain relation in partially fluidized granular matter the diagonal components (pressures) related to the components of the static stresses – may weakly depend on the order parameter shear stresses are strongly dependent on the order parameter viscosity may also weakly depend on the order parameter

Equation for the order parameter Ginzburg-Landau free energy for shear melting phase transition Two stable states: = f and =1 One unstable state u is a control parameter 0,l –characteristic time & length =0 – liquid; – solid 0 1 f 1 0 1

Near-surface granular flows Boundary condition Order parameter equation x z or by differentiating Control parameter no gravity

Near-surface granular flows - theory (cont.) Constitutive relationBalance of forces requires, In non-dimensional units,

Analogy: Maxwell stress relaxation condition In visco-elastic fluids, shear modulus, strain tensor relaxation time In our case, fluid solid

Stationary shear flow profile at large W z V a b

Relaxation oscillations at small W V V max Bifurcation diagram

Asymptotic velocity profiles (2D vs 3D) Why? Experiments: Komatsu et al, PRL,86, 1757 (2001) Lemieux&Durian, PRL, 85, 4273 (2000) Nasuno et al, PRE, 58, 2161 (1998) Losert et al, PRL, 58, 1428 (2000) Mueth et al, Nature, 406, 385 (2000) Theory:

3D shear granular flow y x z g V0V0 2L2L In deep granular layers, Control parameter are major and minor principal stresses and Force balance:

3D shear granular flow (cont.) Explicit z -dependence in the control parameter! From OP equation near and the same scaling for v

Chute flow Equilibrium conditions Stresses: 1, 2 - static/dynamic repose angles) Control parameter h y x g z

Chute flow Boundary conditions: = 1 for z h (sticky bottom) z = 0 for z free surface Stability of the uniform solid state = 1 OPE: Perturbation: Eigenvalue: Stability limit: h y x g z

Flow existence limit Solution exists only for Stationary OPE: 1st integral:

Single mode approximation Close to the stability boundary Here

Constant feeding flux at the top J/ h 0 80 J/ x t Stationary flow solutions continuous flow periodic avalanches

Two types of avalanches (theory) Downhill Uphill

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

Avalanche cross-sections h h x uphill downhill Secondary avalanche V Uphill front speed 1st order transition!

Deep chute (sandpile) For h>>1, Symmetry x x No triangular avalanches in sanpiles!

Quantitative comparison with experiment Model parameters, characteristic time l, characteristic length 1, 2, static/dynamic repose angles viscisity coefficient Daerr & Douady: (particle diameter) Ertas et al (MD simulations of chute flow):

Phase diagram (theory and experiment)

What is the order parameter?

Molecular Dynamics Simulations We consider non-cohesive, dry, disk-like grains with three degrees of freedom. A grain p is specified by radius R p, position r p, translational and angular velocities v p and p. Grains p and q interact whenever they overlap, R p + R q r p –r p | > 0 We use linear spring-dashpot model for normal impact, and Cundall-Strack model for oblique impact. Stress tensor All quantities are normalized using particle size d, mass m, and gravity g Restitution coefficient Friction coefficient 2304 particles (48x48), = 0.82; = 0.3; P ext = 13.45,V x =24

Order parameter for granular fluidization: static contacts vs. fluid contacts: Microscopic Point of View Z st is the static coordination number: the number of long-term ( >1.1t c ) contacts per particle. Z is the total coordination number: the total number of contacts per particle. Stationary profiles of coordination numbers Z, Z st, and order parameter in a system of 4600 grains. = 0.82; = 0.3; P ext = 13.45, V x =48

Order parameter for granular fluidization: Elastic vs Kinetic Energy: Macroscopic Point of View U – elastic energy stored in grains T – fluctuational kinetic energy (granular temperature)

Stick-slip granular friction

Stress tensor Reynolds stress tensor part of the stress tensor due to short-term collisional contacts ( < 1.1t c ). part of the stress tensor due to force chains between particles ( > 1.1t c ). Static stress tensor Fluid stress tensor

Stationary near-surface flow: shear stress

Couette flow in a thin granular layer (no gravity) 500 particles (50x10), = 0.82; = 0.3; P ext = Adiabatic change in shear force:

Bifurcation diagram

Small initial perturbation in a bistable region

Order parameter fixed points MD simulations OP equation 500 particles (50x10), = 0.82; = 0.3

Fitting the constitutive relation Fit: q( ) = (1 ) 2.5 Phenomen. theory: q( )=1- Fit: q y ( ) = (1 ) 1.9 q x ( ) = (1 ) 1.9

Newtonian Fluid + Contact Part Kinematic viscosity in slow dense flows:

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

Fit: f( ) =1 (1 ) 3 Fitting the constitutive relation

Fluid stress vs. shear strain rate 2300 grains; = 0.82; = 0.3; P ext = Bagnold scaling?

Heavy plate under external forcing – no gravity Equation of motion for the plate Constitutive relation Order parameter equation

Heavy plate under external forcing – no gravity

Shear flow of grains with gravity MD simulations, box 48x48 upper plate is dragged with a constant velocity

Shear flow with gravity: continuum model

Shear flow under gravity: continuum model

Slip event: MD simulations

Shear flow of grains with gravity MD simulations; box 96x48

Example: stick-slips thick surface driven granular flow with gravity 5000 particles (50x100), = 0.82; = 0.3; P ext = 10,50,V top =5,50 x y g Set of equations for sand LyLy V0V0 m Equations for heavy plate

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

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

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

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

Constitutive relations for stress components & indeterminacy Static stress tensor Force balance: Linear relation between shear and normal stresses, force chains and stress inhomogeneity (only six relations, need 3 more!) Symmetry: Veje, Howell, and Behringer, Phys.Rev.E, 59, 739 (1999) Linear elasticity & Hooks law do not apply

Conclusions Stress tensor in granular flows is separated into a fluid part and a solid part. The ratio of the fluid and solid parts is fitted by the function of the order parameter: F = (1 ), S =(1 (1 ) ), 2.5. The dynamics of the order parameter is descibed by the Ginzburg-Landau equation with a bistable free energy functional. The free energy controlling the dynamics of the order parameter, can be extracted from molecular dynamics simulations.

Future directions Self-consistent description of stress evolution Elaboration of statistical features of fluidization transition, effect of fluctuations. Extraction of order parameter from experimental data