Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T12:03:15.378Z Has data issue: false hasContentIssue false
  • English
  • Français

Flowing resistance and dilatancy of dense suspensions: lubrication and repulsion

Published online by Cambridge University Press:  07 November 2011

Pierre Grégoire Rognon ,
Itai Einav  and
Cyprien Gay
Pierre Grégoire Rognon*
Affiliation:
School of Civil Engineering, J05, The University of Sydney, Sydney, New South Wales 2006, Australia
Itai Einav
Affiliation:
School of Civil Engineering, J05, The University of Sydney, Sydney, New South Wales 2006, Australia
Cyprien Gay
Affiliation:
Matière et Systèmes Complexes, Université Paris Diderot – Paris 7, CNRS UMR 7057, Paris, France
*
†Email address for correspondence: pierre.rognon@sydney.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this study is to provide generic insights into the rheology of immersed particulate materials in dense configurations. We focus on a non-Brownian and Stokesian suspension of elastic spheres, as a way to identify the effect on the collective behaviour of two short-range interactions: lubrication and steric repulsion below a tunable range. The response of the material to shear under a prescribed shear rate and a prescribed normal stress is simulated using ‘soft dynamics’, a discrete element method which accounts for the dynamics of such a system. The material exhibits a visco-elastic behaviour, deforming elastically at high shear rates, and viscously at slower shear rates. When steady flow is established, the constitutive law can be expressed, as for dry grains, through a friction law and a dilatancy law, whose numerical constants depend in a non-trivial manner on the equilibrium gap. The analysis of the contribution of each interacting pair as a function of its distance and orientation sheds some light on this collective behaviour. Lubrication, which hinders grain approach, is responsible for a loss of contact and a low repulsion contribution to the stress. In contrast, lubrication viscously connects each grain to as many as 10 neighbours, which all contribute significantly to the shear stress. This forms a robust dynamic connected network controlling the collective resistance to flow. This study should be useful for modelling the rheological behaviour of real materials such as foams, emulsions and blood: beside the specific properties of their particles, these particulate fluids also involve lubrication and repulsion.

JFM classification

Granular media: Granular media Non-Newtonian Flows: Rheology Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 689 , 25 December 2011 , pp. 75 - 96
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Batchelor, G. K. & Green, J. T. 1972a The determination of the bulk stress in a suspension of spherical particles to order . J. Fluid Mech. 56 (3), 401–427.CrossRefGoogle Scholar
2. Batchelor, G. K. & Green, J. T. 1972b The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (2), 375–400.CrossRefGoogle Scholar
3. Bonn, D., Rodts, S., Groenink, M., Rafaï, S., Shahidzadeh-Bonn, N. & Coussot, P. 2008 Some applications of magnetic resonance imaging in fluid mechanics: complex flows and complex fluids. Annu. Rev. Fluid Mech. 40 (1), 209–233.CrossRefGoogle Scholar
4. Bonnoit, C., Darnige, T., Clement, E. & Lindner, A. 2010 Inclined plane rheometry of a dense granular suspension. J. Rheol. 54, 65–79.CrossRefGoogle Scholar
5. Bossis, G. & Brady, J. F. 1984 Dynamic simulation of sheared suspensions. Part I. General method. J. Chem. Phys. 80, 5141–5154.CrossRefGoogle Scholar
6. Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155, 105–129.CrossRefGoogle Scholar
7. Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20 (1), 111–157.CrossRefGoogle Scholar
8. Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.CrossRefGoogle Scholar
9. Cantat, I., Cohen-Addad, S., Elias, F., Graner, F., Höhler, R., Pitois, O., Rouyer, F. & Saint-Jalmes, A. 2010 Les Mousses: Structure et Dynamique. Belin.Google Scholar
10. Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.CrossRefGoogle Scholar
11. Coussot, P. 2005 Rheometry of Pastes, Suspensions, and Granular Materials. Wiley-Interscience.CrossRefGoogle Scholar
12. Coussot, P. 2007 Rheophysics of pastes: a review of microscopic modelling approaches. Soft Matt. 3 (5), 528–540.CrossRefGoogle ScholarPubMed
13. Coussot, P., Nguyen, Q. D., Huynh, H. T. & Bonn, D. 2002a Viscosity bifurcation in thixotropic, yielding fluids. J. Rheol. 43 (3), 1–17.Google Scholar
14. Coussot, P., Raynaud, J. S., Moucheront, P, Guilbaud, J. P. & Huynh, H. T. 2002b Coexistence of liquid and solid phases in flowing soft-glassy materials. Phys. Rev. Lett. 88 (21), 218301.CrossRefGoogle ScholarPubMed
15. Coussot, P., Tabuteau, H., Chateau, X., Tocquer, L. & Ovarlez, G. 2006 Aging and solid or liquid behaviour in pastes. J. Rheol. 50, 975–994.CrossRefGoogle Scholar
16. da Cruz, F., Emam, S., Prochnow, M., Roux, J.-N. & Chevoir, F. 2005 Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72, 021309.CrossRefGoogle ScholarPubMed
17. Cundall, P. A. & Strack, O. D. L. 1979 A discrete numerical model for granular assemblies. Géotech. 29, 47–65.CrossRefGoogle Scholar
18. Davis, R. H., Zhao, Y., Galvin, K. P. & Wilson, H. J. 2003 Solid–solid contacts due to surface roughness and their effects on suspension behaviour. Phil. Trans. R. Soc. Lond. A 361 (1806), 871–894.CrossRefGoogle ScholarPubMed
19. Deboeuf, A., Gauthier, G., Martin, J., Yurkovetsky, Y. & Morris, J. F. 2009 Particle pressure in a sheared suspension: a bridge from osmosis to granular dilatancy. Phys. Rev. Lett. 102 (10), 108301.CrossRefGoogle Scholar
20. Divoux, T. & Geminard, J. C. 2007 friction and dilatancy in immersed granular matter. Phys. Rev. Lett. 99 (25), 258301.CrossRefGoogle ScholarPubMed
21. Dratler, D. I. & Schowalter, W. R. 1996 Dynamic simulation of suspensions of non-Brownian hard spheres. J. Fluid Mech. 325, 53–77.CrossRefGoogle Scholar
22. Einstein, A. 1906 Eine neue Bestimmung der Molekuldimensionen. Ann. Phys. 19, 289 and 34, 591.CrossRefGoogle Scholar
23. Fall, A., Bertrand, F., Ovarlez, G. & Bonn, D. 2009 Yield stress and shear banding in granular suspensions. Phys. Rev. Lett. 103 (17), 178301.CrossRefGoogle ScholarPubMed
24. Forterre, Y. & Pouliquen, O. 2008 Flows of dense granular media. Annu. Rev. Fluid Mech. 40, 1–24.CrossRefGoogle Scholar
25. GDR MiDi, 2004 On dense granular flows. Euro. Phys. J. E 14, 341–365.CrossRefGoogle Scholar
26. Huang, N. & Bonn, D. 2007 Viscosity of a dense suspension in Couette flow. J. Fluid Mech. 590, 497–507.CrossRefGoogle Scholar
27. Huang, N., Ovarlez, G., Bertrand, F., Rodts, S., Coussot, P. & Bonn, D. 2005 Flow of wet granular materials. Phys. Rev. Lett. 94 (2), 28301.CrossRefGoogle ScholarPubMed
28. Iverson, R. M. 2005 Regulation of landslide motion by dilatancy and pore pressure feedback. J. Geophys. Res. 110 (F2), F02015.Google Scholar
29. Jenkins, J. T. & Koenders, M. A. 2005 Hydrodynamic interaction of rough spheres. Geophys. Mag. 7 (1), 13–18.Google Scholar
30. Lemaître, A., Roux, J. N. & Chevoir, F. 2009 What do dry granular flows tell us about dense non-Brownian suspension rheology? Rheol. Acta 48 (8), 925–942.CrossRefGoogle Scholar
31. Mewis, J. & Wagner, N. J. 2009 Current trends in suspension rheology. J. Non-Newtonian Fluid Mech. 157 (3), 147–150.CrossRefGoogle Scholar
32. Mills, P. & Snabre, P. 2009 Apparent viscosity and particle pressure of a concentrated suspension of non-Brownian hard spheres near the jamming transition. Eur. Phys. J. E 30 (3), 309–316.CrossRefGoogle ScholarPubMed
33. Morris, J. F. 2009 A review of microstructure in concentrated suspensions and its implications for rheology and bulk flow. Rheol. Acta 48 (8), 909–923.CrossRefGoogle Scholar
34. Morris, J. F. & Boulay, F. 1999 Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J. Rheol. 43, 1213–1237.CrossRefGoogle Scholar
35. Ovarlez, G., Bertrand, F. & Rodts, S. 2006 Local determination of the constitutive law of a dense suspension of noncolloidal particles through magnetic resonance imaging. J. Rheol. 50, 259–292.CrossRefGoogle Scholar
36. Pailha, M., Nicolas, M. & Pouliquen, O. 2008 Initiation of underwater granular avalanches: influence of the initial volume fraction. Phys. Fluids 20, 111701.CrossRefGoogle Scholar
37. Rognon, P., Einav, I., Bonivin, J. & Miller, T. 2010a A scaling law for heat conductivity in sheared granular materials. Europhys. Lett. 89, 58006.CrossRefGoogle Scholar
38. Rognon, P., Einav, I. & Gay, C. 2010b Internal relaxation time in immersed particulate materials. Phys. Rev. E 81, 061304.CrossRefGoogle ScholarPubMed
39. Rognon, P. & Gay, C. 2008 Soft dynamics simulation. Part 1. Normal approach of two deformable particles in a viscous fluid and optimal-approach strategy. Eur. Phys. J. E 27, 253–260.CrossRefGoogle Scholar
40. Rognon, P. & Gay, C. 2009 Soft dynamics simulation. Part 2. Elastic spheres undergoing a T1 process in a viscous fluid. Eur. Phys. J. E 30, 291–301.CrossRefGoogle Scholar
41. Rognon, P. G., Roux, J. N., Naaim, M. & Chevoir, F. 2008 Dense flows of cohesive granular materials. J. Fluid Mech. 596, 21–47.CrossRefGoogle Scholar
42. Rognon, P. G., Roux, J.-N., Wolf, D., Naaim, M. & Chevoir, F. 2006 Rheophysics of cohesive granular materials. Europhys. Lett. 74, 644–650.CrossRefGoogle Scholar
43. Roussel, N., Lemaı¯tre, A., Flatt, R. J. & Coussot, P. 2010 Steady state flow of cement suspensions: a micromechanical state of the art. Cem. Concr. Res. 40 (1), 77–84.CrossRefGoogle Scholar
44. Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115–146.CrossRefGoogle Scholar
45. Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol. 46, 1031–1056.CrossRefGoogle Scholar
46. Singh, A. & Nott, P. R. 2000 Normal stresses and microstructure in bounded sheared suspensions via Stokesian dynamics simulations. J. Fluid Mech. 412, 279–301.CrossRefGoogle Scholar
47. Singh, A. & Nott, P. R. 2003 Experimental measurements of the normal stresses in sheared Stokesian suspensions. J. Fluid Mech. 490, 293–320.CrossRefGoogle Scholar
48. Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37 (1), 129–149.CrossRefGoogle Scholar
49. Tabakova, S. S. & Danov, K. D. 2009 Effect of disjoining pressure on the drainage and relaxation dynamics of liquid films with mobile interfaces. J. Colloid Interface Sci. 336, 273–284.CrossRefGoogle ScholarPubMed
50. Vinogradova, O. I. & Yakubov, G. E. 2006 Surface roughness and hydrodynamic boundary conditions. Phys. Rev. E 73 (4), 45302.CrossRefGoogle ScholarPubMed
51. Weaire, D. & Hutzler, S. 1999 The Physics of Foams. Oxford University Press.Google Scholar
52. Wilson, H. J. 2005 An analytic form for the pair distribution function and rheology of a dilute suspension of rough spheres in plane strain flow. J. Fluid Mech. 534, 97–114.CrossRefGoogle Scholar
53. Yang, L., Seddon, J. R. T., Mullin, T., Del Pino, C. & Ashmore, J. 2006 The motion of a rough particle in a Stokes flow adjacent to a boundary. J. Fluid Mech. 557, 337–346.CrossRefGoogle Scholar
54. Yılmaz, F. & Gündoğdu, M. Y. 2008 A critical review on blood flow in large arteries: relevance to blood rheology, viscosity models, and physiologic conditions. Korea–Aust. Rheol. J. 20 (4), 197–211.Google Scholar
55. Zarraga, I. E., Hill, D. A. & Leighton, D. T. 2000 The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J. Rheol. 44, 185–220.CrossRefGoogle Scholar
56. Zarraga, I. E. & Leighton, D. T. 2001 Normal stress and diffusion in a dilute suspension of hard spheres undergoing simple shear. Phys. Fluids 13, 565–577.CrossRefGoogle Scholar