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Rheology of a suspension of elastic particles in a viscous shear flow

Published online by Cambridge University Press:  14 October 2011

Tong Gao ,
Howard H. Hu  and
Pedro Ponte Castañeda
Tong Gao
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA
Howard H. Hu*
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA
Pedro Ponte Castañeda
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA
*
†Email address for correspondence: hhu@seas.upenn.edu
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Abstract

In this paper we consider a suspension of elastic solid particles in a viscous liquid. The particles are assumed to be neo-Hookean and can undergo finite elastic deformations. A polarization technique, originally developed for analogous problems in linear elasticity, is used to establish a theory for describing the finite-strain, time-dependent response of an ellipsoidal elastic particle in a viscous fluid flow under Stokes flow conditions. A set of coupled, nonlinear, first-order ODEs is obtained for the evolution of the uniform stress fields in the particle, as well as for the shape and orientation of the particle, which can in turn be used to characterize the rheology of a dilute suspension of elastic particles in a shear flow. When applied to a suspension of cylindrical particles with initially circular cross-section, the theory confirms the existence of steady-state solutions, which can be given simple analytical expressions. The two-dimensional, steady-state solutions for the particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspension, are in excellent agreement with direct numerical simulations of multiple-particle dispersions in a shear flow obtained by using an arbitrary Lagrangian–Eulerian (ALE) finite element method (FEM) solver. The corresponding solutions for the evolution of the microstructure and the rheological properties of suspensions of initially spherical (three-dimensional) particles in a simple shear flow are also obtained, and compared with the results of Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) in the steady-state regime. Interestingly, the results show that sufficiently soft elastic particles can be used to reduce the effective viscosity of the suspension (relative to that of the pure fluid).

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Non-Newtonian Flows: Rheology Non-Newtonian Flows: Viscoelasticity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 209 - 237
Copyright
Copyright © Cambridge University Press 2011

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References

1. Aravas, N. & Ponte Castañeda, P. 2004 Numerical methods for porous metals with deformation-induced anisotropy. Comput. Meth. Appl. Mech. Engng 193, 3767–3805.CrossRefGoogle Scholar
2. Barnes, H. A. 1994 Rheology of emulsions: a review. Colloids Surf. A 91, 89–95.CrossRefGoogle Scholar
3. Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.CrossRefGoogle Scholar
4. Barthès-Biesel, D. & Rallison, J. M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefGoogle Scholar
5. Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.CrossRefGoogle Scholar
6. Batchelor, G. K. 1971 The stress generated in a non-dilute suspension of elongated particles by pure straining motion. J. Fluid Mech. 46, 813–829.CrossRefGoogle Scholar
7. Bilby, B. A., Eshelby, J. D. & Kundu, A. K. 1975 The change of shape of a viscous ellipsoidal region embedded in a slowly deforming matrix having a different viscosity. Tectonophysics 28, 265–274.CrossRefGoogle Scholar
8. Bilby, B. A. & Kolbuszewski, M. L. 1977 The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow. Proc. R. Soc. Lond. A 355, 335–353.Google Scholar
9. Bird, R. B., Armstrong, R. C., Hassager, O. & Curtiss, C. F. 1987 Dynamics of Polymeric Liquids, Kinetic Theory, vol. 2. Wiley.Google Scholar
10. Cerf, R. 1952 On the frequency dependence of the viscosity of high polymer solutions. J. Chem. Phys. 20, 395–402.CrossRefGoogle Scholar
11. Chien, S., Usami, S., Dellenback, R. J. & Gregersen, M. I. 1967a Blood viscosity: influence of erythrocyte aggregation. Science 157, 829–831.CrossRefGoogle ScholarPubMed
12. Chien, S., Usami, S., Dellenback, R. J. & Gregersen, M. I. 1967b Blood viscosity: influence of erythrocyte deformation. Science 157, 827–829.CrossRefGoogle ScholarPubMed
13. Einstein, A. 1906 Eine neue Bestimmung der Molekül-dimensionen. Ann. Phys. 324, 289–306. Corrections, ibid, 339, 591–592.CrossRefGoogle Scholar
14. Eshelby, J. D. 1957 The determination of the elastic eld of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond. A 241, 376–396.Google Scholar
15. Eshelby, J. D. 1959 The elastic eld outside an ellipsoidal inclusion. Proc. R. Soc. Lond. A 252, 561–569.Google Scholar
16. Fröhlich, H. & Sack, R. 1946 Theory of the rheological properties of dispersions. Proc. R. Soc. Lond. A 185, 415–430.Google ScholarPubMed
17. Gao, T. & Hu, H. H. 2009 Deformation of elastic particles in viscous shear flow. J. Comput. Phys. 228, 2132–2151.CrossRefGoogle Scholar
18. Gel’fand, I. M. & Shilov, G. E. 1964 Generalized Functions, vol. 1: Properties and Operations. Academic.Google Scholar
19. Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.CrossRefGoogle Scholar
20. Goddard, J. D. & Miller, C. 1967 Nonlinear effects in a rheology of dilute suspensions. J. Fluid Mech. 28, 657–673.CrossRefGoogle Scholar
21. Hirt, C. W., Amsden, A. A. & Cook, J. L. 1974 An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227–253.CrossRefGoogle Scholar
22. Hu, H. H., Zhu, M. Y. & Patankar, N. 2001 Direct numerical simulations of fluid–solid systems using the arbitrary Lagrangian–Eulerian technique. J. Comput. Phys. 169, 427–462.CrossRefGoogle Scholar
23. Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
24. Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids, Applied Mathematical Sciences , vol. 84. Springer.CrossRefGoogle Scholar
25. Kailasam, M. & Ponte Castañeda, P. 1998 A general constitutive theory for linear and nonlinear particulate media with microstructure evolution. J. Mech. Phys. Solids 46, 427–465.CrossRefGoogle Scholar
26. Keller, S. R. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47.CrossRefGoogle Scholar
27. Lac, E., Barthès-Biesel, D., Pelekasis, N. A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
28. Le Tallec, P. & Mouro, J. 2001 Fluid structure interaction with large structural displacements. Comput. Meth. Appl. Mech. Engng 190, 3039–3067.CrossRefGoogle Scholar
29. Macosko, C. W. 1994 Rheology: Principles, Measurements and Applications. VCH.Google Scholar
30. Masud, A. & Hughes, T. J. R. 1997 A space–time Galerkin/least-squares finite element formulation of the Navier–Stokes equations for moving domain problems. Comput. Meth. Appl. Mech. Engng 146, 91–126.CrossRefGoogle Scholar
31. Mattsson, J., Wyss, H. M., Fernandez-Nieves, A., Miyazaki, K., Hu, Z., Reichman, D. R. & Weitz, D. A. 2009 Soft colloids make strong glasses. Nature 462, 83–86.CrossRefGoogle ScholarPubMed
32. Murata, T. 1981 Deformation of an elastic particle suspended in an arbitrary flow field. J. Phys. Soc. Japan 50, 1009–1016.CrossRefGoogle Scholar
33. Ogden, R. W. 1984 Nonlinear Elastic Deformations. Dover.Google Scholar
34. Oldroyd, J. G. 1953 The elastic and viscous properties of emulsions and suspensions. Proc. R. Soc. Lond. A 218, 122–132.Google Scholar
35. Ponte Castañeda, P. 2005 Heterogeneous materials. Lecture Notes , Department of Mechanics, Ecole Polytechnique.Google Scholar
36. Ponte Castañeda, P. & Willis, J. R. 1995 The effect of spatial distribution on the effective behaviour of composite materials and cracked media. J. Mech. Phys. Solids 43, 1919–1951.CrossRefGoogle Scholar
37. Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of capsule viscosity. J. Fluid Mech. 361, 117–143.CrossRefGoogle Scholar
38. Rivlin, R. S. 1948 Large elastic deformation of isotropic materials. I. fundamental concepts. Phil. Trans. R. Soc. A 240, 459–490.Google Scholar
39. Roscoe, R. 1967 On the rheology of a suspension of viscoelastic spheres in a viscous liquid. J. Fluid Mech. 28, 273–293.CrossRefGoogle Scholar
40. Rumscheidt, F. D. & Mason, S. G. 1961 Particle motions in sheared suspensions. Part XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. Colloid Sci. 16, 238–261.CrossRefGoogle Scholar
41. Schowalter, W. R., Chaffey, C. E. & Brenner, H. 1968 Rheological behaviour of a dilute emulsion. J. Colloid Interface Sci. 26, 152–160.CrossRefGoogle ScholarPubMed
42. Snabre, P. & Mills, P. 1999 Rheology of concentrated suspensions of viscoelastic particles. Colloids Surf. A Physicochem Engng Asp. 152, 79–88.CrossRefGoogle Scholar
43. Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous flows. Annu. Rev. Fluid Mech. 26, 65–102.CrossRefGoogle Scholar
44. Subramaniam, A. B., Abkarian, M., Mahadevan, L. & Stone, H. A. 2005 Non-spherical gas bubbles. Nature 438, 930.CrossRefGoogle Scholar
45. Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. Roy. Soc. Lond. A 146, 501–523.Google Scholar
46. Wetzel, E. D. & Tucker, C. L. 2001 Droplet deformation in dispersions with unequal viscosities and zero interfacial tension. J. Fluid Mech. 426, 199–228.CrossRefGoogle Scholar
47. Willis, J. R. 1981 Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78.CrossRefGoogle Scholar