Rheology of granular flows: Role of the interstitial fluid

Transcription

1 Rheology of granular flows: Role of the interstitial fluid Colorado 2003, USGS Olivier Pouliquen, IUSTI, CNRS, Aix-Marseille University Marseille, France

2 Motivations : debris flows, landslides, avalanches, silo Not Fault!!! Low level of pressure: kpa in natural events kpa in the experiments => Rigid and non breakable particles

3 Particles of different sizes + liquid (non newtonian) + complex topography + unsteady flows????????????????? In this talk : 1) rheology of dry granular flows? 2) What happen when coupling with the interstitial fluid matters?

4 Dry granular material: Collection of grains No cohesion No brownian motion No fluid interaction Only contact interactions But not so easy

5 Dry granular flows Different flow regimes Gas Liquid Solid

6 Dry granular flows Different flow regimes Gas Liquid Solid

7 Quasi-static deformations : Soil mechanics and plasticity «!solid!» Focus on initial deformation What happens :! at large deformations?! for fast deformations?

8 Dry granular flows Different flow regimes Gas Liquid Solid

9 Kinetic theory for rapid granular gases «!gas!» Binary collisions+inelastic collisions! constitutive equations coupling Density, velocity and granular temperature But if not enough energy is injected:! finite duration contact,! multiple contact,

10 Dry granular flows Different flow regimes Gas Liquid Solid

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12 Different flow configurations studied both experimentally and numerically GDR Midi, Eur. Phys. J 04

13 P plane shear under controlled normal stress U Lois et al 2005 Da Cruz et al, PRE 05 GdR Midi, Eur. Phys. J 04 h P One imposes P and A single dimensionless number (inertial number) Shear stress "? Volume fraction #? (Savage 84, Ancey et al 99)

14 Inertial number * ratio between 2 times : : time scale of the mean shear : microscopic time for rearrangement

15 «!quasi-static!» «liquid» «!gas!» 0 1 I = " d P /#

16 P U Da Cruz et al, PRE 05 GdR Midi Eur. Phys. J 04 h P One imposes P and Shear stress "? Volume fraction #?

17 P " " = µ I ( )P " /P " = " I ( ) I = " d P /#

18 remark: No velocity weakening Peyneau & Roux PRE 08 Dacruz et al PRE 05

19 Forterre, Pouliquen ARFM 08 For spheres Data from Inclined plane exp. (Pouliquen 99) Inclined plane simulations (Baran et al 2006) Annular shear cell exp. (Sayed, Savage JFM, 84)

20 An empirical friction law: " = µ(i)p I = " d P /# µ(i) = µ s + µ 2 " µ s I 0 I +1

21 Constant pressure µ # I $ And shear at constant volume fraction?? Bagnold Proc. R. Soc 54 Lois et al PRE 07 Lemaitre PRE 05 Da Cruz et al PRE 05 f1 f2 # #

22 allows to describe (not perfectly) velocity profiles on inclined plane, Let s go further

23 Predicted velocity and volume fraction profiles Rheology µ(i) predicts - V % h (h-z) 1.5 and &=cte Gdr Midi et al, 2004, Da cruz et al 2002, Silbert et al Pb with thin flows and close to free surface Gdr Midi et al, 2004, Da cruz et al 2002, Rajchenbach 2003

24 allows to describe (not perfectly) velocity profiles on inclined plane, on pile, Let s go further

25 3D generalisation: a visco-plastic model (Jop et al Nature 06) assumptions : 1) P isotropic 2) and are colinear (Savage 83, Goddard 86, Schaeffer 87, ) Effective viscosity

26 3D generalisation of the friction law : granular flows as a viscoplastic fluid (Jop et al Nature 06) assumptions: 1) P isotropic 2) and are co-linear (Savage 83, Goddard 86, Schaeffer 87, ) Pressure dependent viscosity

27 flows on a heap : a full 3D problem (P. Jop et al Nature 06) Q W z/d y/d L = 1.5 m V(y,z)

28 Flow between rough lateral walls: Jop et al, Nature V surf gd h/d ,2 0 0,2 0,4 0,6 0,8 1 1,2 y/w ,2 0,4 0,6 0,8 1 y/w

29 Jop et al, Phys. Fluids 2007 Initiation of the flow?

30 Long wave instability in granular flows (Y. Forterre, JFM 06 )

31 Experimental Setup : forcing of the instability Power supply Function generator f, AGBF Stabilized alimentation Loudspeakers Slides projector Nozzle z y x h Photodiodes! Forterre and Pouliquen JFM 02

32 Instability threshold Forterre, JFM 06 Dispersion relation

33 Granular slumping (Lacaze and Kerswell 08) Lajeunesse et al Phys. Fluids 2004,2005 Lube et al JFM 2004, Larrieu et al JFM 2006, Staron & Hinch JFM 2005, Lacaze et al Phys. Fluids 2008 Lajeunesse et al 05

34 Lacaze and Kerswell (preprint 08)

35 Relative Success of the visco-plastic description. A starting point to adress other configurations (simulating the pressure dependent visco-plastic rheology is non trivial ) But there are problems when approaching the solid

36 Limits of the viscoplastic approach: 1) Quasistatic flows (shear band, finite size effect.) A need for non local approach 2) Transient flows when preparation plays a crucial role

37 Exponential tail Not predicted.. Velocity profile

38 Shear bands in quasi-static flow (Forterre & Pouliquen ARFM 08, Jop PRE 08) V/Vw y/d Howell et al PRL 99 Mueth et al Nature 00 Bocquet et al PRE 02 Not captured by the viscoplastic approach

39 Flow threshold hysteresis Finite size effects Not captured by the viscoplastic approach Limit of a local rheology?

40 to go further? Role of the fluctuations? Aranson and Tsimring PRE,01, Louge Phys. Fluids 03, Josserand et al 06 Lemaitre 02 Bazant 07 Nott 08 Behringer 08 Role of the correlations? Pouliquen et al 01, Ertas and Halsey 03, Mills et al 08 Jenkins and Chevoir 01, Jenkins Phys. Fluids 06, Radjai and Roux PRL Pouliquen PRL 04 Link with plasticity of other amorphous and glassy systems

41 F Evidence for non local effects: Microrheology experiments! Deflection angle (mrad) '=0 20 '! time (s) M. Van Hecke 2008 Pouliquen, Forterre, Nott

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43 Self activated process Pouliquen & Forterre, Phil. Trans, 2009

44 Limits of the viscoplastic approach: 1) Quasistatic flows (shear band, finite size effect.) A need for non local approach 2) Transient flows when preparation plays a crucial role

45 Influence of the initial Volume fraction on the Collapse of a pile. Daerr & Douady 99

46 Quasi-static case : critical state theory # # c ( " " c ( Coupling Friction- dilatancy

47 Reynolds Dilatancy

48 Simple critical state theory P Dilatancy angle ( dx dy # assumption: critical " volume fraction # c # c ( (Radjai and Roux 98)

49 Visco plastic theory : shear rate dependence but no dilatancy critical state theory : dilatancy but no shear rate dependence Shear rate dependent critical state theory

50 Shear rate dependent critical state theory : I = " d P /#

51 3D generalisation :......

52 Application to a dry flow: Initiation of flow on an inclined plane: z different )

53 z z Initially dense z z Initially loose Comparison with DEM simulations with N. Taberlet

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55 Changing time scales by putting the granular material in water (Cassar et al, Phys. Fluids 06) DV recorder d=112 µm glass beads Laser P.C!P

56 Dry Immersed

57 A naive idea : fluid only plays a role by changing the time scale of rearrangements " = P µ(i) with I = ( t micro viscous : dry :

58 µ=tg) Idry Ivisc Cassar et al. Phys. Fluids 05

59 Submarine flows on heap Doppler et al, JFM 07 Flow rate Velocity profile

60 And dilatancy??? And Pore pressure?? Cf In Faults Rice JGR 75, Rudnicki JGR 84,

61 How to explain the variety of landslides observed in nature? Iverson et al, (2000) Science Large scale experiments in the USGS facility

62 Dense preparation Courtesy of Dick Iverson

63 Loose preparation Courtesy of Dick Iverson

64 Pore Pressure feedback argument (Iverson Rev. Geo. 97, JGR 05) Loose case Dense case &!! Fluid expelled! P fluid!! P eff "! Friction "! Less friction between grains &"! Fluid sucked! P fluid "! P eff!! Friction!! higher friction between grains In faults: rice JGR 75, Rudnicki JGR 84,

65 A simple experiment: Dense sample Loose sample

66 Experimental setup (Pailha et al 08) Glass beads : 160µm Liquid: mixture of water and Ucon oil: *= kg/m.s *= kg/m.s 20cm 7cm 1m

67 Experimental procedure 0.60 Compaction by taps 0.59! Result of sedimentation # tap 15 20

68 Velocity of the Free surface Typical results Surface velocity (mm/s) ,&i=0.568 &i=0.571,&i=0.562 &i=0.578 &i=0.584 &i=0.588,&i= =25º h=5mm *= Pa.s Pressure under The avalanche Pore pressure (Pa) Time (s) (Pailha et al Pof 08)

69 Velocity of the Free surface Typical results Surface velocity (mm/s) ,&i=0.568 &i=0.571,&i=0.562 &i=0.578 Loose &i<&c &i=0.584 &i=0.588,&i=0.592 Dense &i>&c +=25º h=5mm *= Pa.s Pressure under The avalanche Pore pressure (Pa) Time (s) (Pailha et al Pof 08)

70 Triggering time in the dense case

71 Triggering time + =25 + = = 28 + = 30 h=3.7 mm h=6.1 mm Time (s) *= kg/m.s Phi

72 Deformation Surface velocity (mm.s -1 ) Time (s) Surface velocity (mm.s -1 ) !i=0.588!i=0.584!i= Deformation For initial preparation erased

73 Two phase flow model 1 Coupling with the liquid: Two phase equations Fluids mechanics 2 Rheology of the granular phase 3 Dilatancy Granular matter Soil mechanics

74 submarine avalanches : Depth averaged approach (Pitman and Le 05) : z h )

75 Submarine granular avalanches: Shear rate critical state theory (Cassar et al 05, Doppler et al 07 ) Particle -fluid coupling Relative weight Viscous drag due to the Vertical displacement

76 µ Calibration of the model looking at the steady state I µ ( I ) v x10-3! " ( eq I ) v 2 I 3 4x10-3! start K A single free Parameter K "

77 Velocity Predictions : Pressure u (mm.s-1) Pore pressure (Pa) t (s) t (s) u (mm.s-1) Pore pressure (Pa) t (s) t (s)

78 Scaling of the triggering time t/t0 Different h Different viscosities

79 Pore Pressure Maximum acceleration

80 Conclusions for submarine avalanches A simple critical state approach+ a viscoplastic rheology + two phase flow equations!semi quantitative predictions in the complex dynamics of the flow initiation of submarine avalanches Beyond the depth averaged approach? Question of the numerical implementation of such models?

81 Index matching method Mickael Pailha unpublished

82 Conclusions for constitutive modeling of granular flows Visco-plastic approach gives the order zero of viscous behavior of granular flows It can serve as a base for further developments: -irregular particles? -cohesive particles? -polydispersed materials? -breakable particles? (dilatancy, underwater granular flows, cohesive flows ) -link with the microscopic physics? -how to capture non local effects (role of fluctuations, link with glassy systems,.)?

83 Towards more complex granular media: Polydispersed : Felix et Thomas PRE 04 Rognon et al 06 Cohesive granular matter: Rognon et al 08 Halsey et al 06 Richefeu et al 06 granular matter with fluid interactions

84 Merci à Mickael Pailha Pierre Jop, Cyril Cassar Yoël Forterre Pascale Aussillous Maxime Nicolas Prabhu Nott Jeff Morris Neil Balmforth Bruno Andreotti Olivier Dauchot