Order Parameter Description of Shear Flows in Granular Media

1. 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, 020301 (2001); Theory of Partially Fluidized Granular Flows, Phys. Rev. E 65, 061303 (2002) Order Parameter Description of Stationary Gran Flows, Phys. Rev. Lett. 90, 254301 (2003) Partially Fluidized Granular Flows, Continuum theory and MD Simulations, Phys. Rev. E. 68, 021301 (2003) Stick-Slip Dynamics in a Granular Layer under Shear, Phys. Rev. E 69, 031302 (2004) Supported by US DOE, Office of Basic Energy Sciences SAMSI Workshop, NC 2004

2. 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

3. 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

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

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

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

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

8. Stick-slip motion of grains 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

9. 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

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

11. 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 r • viscosity h may also weakly depend on the order parameter

12. Equation for the order parameter f Ginzburg-Landau free energy for “shear melting” phase transition r 0 1 t0,l –characteristic time & length r 1 Two stable states: r = rf and r =1 One unstable state ru d is a control parameter d 1 0 r=0 – liquid;r=1– solid

13. What is the order parameter?

14. Molecular Dynamics Simulations We consider non-cohesive, dry, disk-like grains with 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 We use linear spring-dashpotmodel for normal impact, and Cundall-Strackmodel for oblique impact. Stress tensor Restitution coefficient e 2304 particles (48x48), e = 0.82; m = 0.3; Pext= 13.45,Vx=24 Friction coefficient m All quantities are normalized using particle size d, mass m, and gravity g

15. Order parameter for granular fluidization:static contacts vs. fluid contacts: Microscopic Point of View • Zstis the static coordination number: • the number of long-term ( >1.1tc) • contacts per particle. • Zis the total coordination number: • the total number of contacts per • particle. Stationary profiles of coordination numbers Z,Zst, and order parameter in a system of 4600 grains. e = 0.82; m = 0.3; Pext= 13.45,Vx=48

16. 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)

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

18. Couette flow in a thin granular layer (no gravity) 500 particles (50x10), e = 0.82; m = 0.3; Pext= 13.45 Adiabatic change in shear force:

19. Bifurcation diagram

20. Small initial perturbation in a bistable region

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

22. 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

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

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

25. Heavy plate under external forcing – no gravity

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

27. Slip event: MD simulations

28. Slip event: MD simulations

29. 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

30. Simplified theory: reduction to ODE • 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 y x g x

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

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

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

34. 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-r)a,  S =(1 -(1-r)a) , a 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.

35. 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