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The rheology of non-dilute dispersions of highly deformable viscoelastic particles in Newtonian fluids

Published online by Cambridge University Press:  17 December 2014

Reza Avazmohammadi  and
Pedro Ponte Castañeda
Reza Avazmohammadi
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA
Pedro Ponte Castañeda*
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USA
*
†Email address for correspondence: ponte@seas.upenn.edu
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Abstract

We present a model for the rheological behaviour of non-dilute suspensions of initially spherical viscoelastic particles in viscous fluids under uniform Stokes flow conditions. The particles are assumed to be neutrally buoyant Kelvin–Voigt solids undergoing time-dependent finite deformations and exhibiting generalized neo-Hookean behaviour in their purely elastic limit. We investigate the effects of the shape dynamics and constitutive properties of the viscoelastic particles on the macroscopic rheological behaviour of the suspensions. The proposed model makes use of known homogenization estimates for composite material systems consisting of random distributions of aligned ellipsoidal particles with prescribed two-point correlation functions to generate corresponding estimates for the instantaneous (incremental) response of the suspensions, together with appropriate evolution laws for the relevant microstructural variables. To illustrate the essential features of the model, we consider two special cases: (i) extensional flow and (ii) simple shear flow. For each case, we provide the time-dependent response and, when available, the steady-state solution for the average particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspensions. The results exhibit shear thickening for extensional flows and shear thinning for simple shear flows, and it is found that the volume fraction and constitutive properties of the particles significantly influence the rheology of the suspensions under both types of flows. In particular, for extensional flows, suspensions of particles with finite extensibility constraints are always found to reach a steady state, while this is only the case at sufficiently low strain rates for suspensions of (less realistic) neo-Hookean particles, as originally reported by Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) and Gao et al. (J. Fluid Mech., vol. 687, 2011, pp. 209–237). For shear flows, viscoelastic particles with high viscosities can experience a damped oscillatory motion of decreasing amplitude before reaching the steady state.

JFM classification

Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Non-Newtonian Flows: Non-Newtonian Flows Multiphase and Particle-laden Flows: Particle/fluid flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 763 , 25 January 2015 , pp. 386 - 432
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© 2014 Cambridge University Press 

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Footnotes

‡

Present address: Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, Austin, TX 78712, USA.

References

Abreu, D., Levant, M., Steinberg, V. & Seifert, U. 2014 Fluid vesicles in flow. Adv. Colloid Interface Sci. 208, 129–141.CrossRefGoogle Scholar
Almusallam, A. S., Larson, R. G. & Solomon, M. J. 2000 A constitutive model for the prediction of ellipsoidal droplet shapes and stresses in immiscible blends. J. Rheol. 44 (5), 1055–1083.CrossRefGoogle Scholar
Aravas, N. & Ponte Castañeda, P. 2004 Numerical methods for porous metals with deformation-induced anisotropy. Comput. Meth. Appl. Mech. Engng 193, 3767–3805.Google Scholar
Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100, 831–853.CrossRefGoogle Scholar
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.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570.Google Scholar
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order $c^{2}$ . J. Fluid Mech. 56, 401–427.Google Scholar
Bernstein, B. 1960 Hypo-elasticity and elasticity. Arch. Rat. Mech. Anal. 6 (1), 89–104.Google Scholar
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
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
Brady, J. F. & Bossis, G. 1985 The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech. 155 (June), 105–129.CrossRefGoogle Scholar
Brady, J. F., Khair, A. S. & Swaroop, M. 2006 On the bulk viscosity of suspensions. J. Fluid Mech. 554, 109–123.Google Scholar
Brooks, D. E., Goodwin, J. W. & Seaman, G. V. 1970 Interactions among erythrocytes under shear. J. Appl. Physiol. 28 (2), 172–177.Google Scholar
Cerf, R. 1952 On the frequency dependence of the viscosity of high polymer solutions. J. Chem. Phys. 20, 395–402.Google Scholar
Chen, H.-S. & Acrivos, A. 1978 The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Intl J. Solids Struct. 14 (5), 349–364.Google Scholar
Clausen, J. R. & Aidun, C. K. 2010 Capsule dynamics and rheology in shear flow: particle pressure and normal stress. Phys. Fluids 22, 123302.Google Scholar
Clausen, J. R., Reasor, D. A. & Aidun, C. K. 2011 The rheology and microstructure of concentrated non-colloidal suspensions of deformable capsules. J. Fluid Mech. 685, 202–234.CrossRefGoogle Scholar
Danker, G. & Misbah, C. 2007 Rheology of a dilute suspension of vesicles. Phys. Rev. Lett. 98, 088104.Google Scholar
Einstein, A. 1906 A new determination of molecular dimensions. Ann. Phys. 19, 289–306.Google Scholar
Ekeland, I. & Témam, R. 1999 Convex Analysis and Variational Problems. SIAM.Google Scholar
Eshelby, J. D. 1957 The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A 241, 376–396.Google Scholar
Frankel, N. A. & Acrivos, A. 1967 On the viscosity of a concentrated suspension of solid spheres. Chem. Engng Sci. 22, 847–853.Google Scholar
Fröhlich, H. & Sack, R. 1946 Theory of the rheological properties of dispersions. Proc. R. Soc. Lond. A 185, 415–430.Google Scholar
Gadala-Maria, F. & Acrivos, A. 1980 Shear-induced structure in a concentrated suspension of solid spheres. J. Rheol. 24 (6), 799–814.CrossRefGoogle Scholar
Gao, T., Hu, H. H. & Ponte Castañeda, P. 2011 Rheology of a suspension of elastic particles in a viscous shear flow. J. Fluid Mech. 687, 209–237.Google Scholar
Gao, T., Hu, H. H. & Ponte Castañeda, P. 2012 Shape dynamics and rheology of soft elastic particles in a shear flow. Phys. Rev. Lett. 108, 058302.Google Scholar
Gao, T., Hu, H. H. & Ponte Castañeda, P. 2013 Dynamics and rheology of elastic particles in an extensional flow. J. Fluid Mech. 715, 573–596.Google Scholar
Gent, A. N. 1996 A new constitutive relation for rubber. Rubber Chem. Tech. 69, 59–61.Google Scholar
Ghigliotti, G., Biben, T. & Misbah, C. 2010 Rheology of a dilute two-dimensional suspension of vesicles. J. Fluid Mech. 653, 489–518.Google Scholar
Goddard, J. D. 1977 An elastohydrodynamic theory for the rheology of concentrated suspensions of deformable particles. J. Non-Newtonian Fluid Mech. 2 (2), 169–189.Google Scholar
Goddard, J. D. & Miller, C. 1967 Nonlinear effects in the rheology of dilute suspensions. J. Fluid Mech. 28, 657–673.Google Scholar
Hashin, Z. & Shtrikman, S. 1963 A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11, 127–140.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102, 161–179.Google Scholar
Jeffrey, D. J. & Acrivos, A. 1976 The rheological properties of suspensions of rigid particles. AIChE J. 22, 417–432.Google Scholar
Joseph, D. D. 1990 Fluid Dynamics of Viscoelastic Liquids. Springer.Google Scholar
Kailasam, M.1998 A general constitutive theory for particulate composites and porous materials with evolving microstructures. PhD thesis, University of Pennsylvania.Google Scholar
Kailasam, M. & Ponte Castañeda, P. 1998 A general constitutive theory for linear and nonlinear particulate media with microstructure evolution. J. Mech. Phys. Solids 46 (3), 427–465.CrossRefGoogle Scholar
Kailasam, M., Ponte Castañeda, P. & Willis, J. R. 1997 The effect of particle size, shape, distribution and their evolution on the constitutive response of nonlinearly viscous composites. I. Theory. Phil. Trans. R. Soc. Lond. A 355, 1835–1852.Google Scholar
Keller, J. B., Rubenfeld, L. A. & Molyneux, J. E. 1967 Extremum principles for slow viscous flows with applications to suspensions. J. Fluid Mech. 30 (01), 97–125.Google Scholar
Keller, S. R. & Skalak, R. 1982 Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120, 27–47.Google Scholar
Krieger, I. M. & Dougherty, T. J. 1959 A mechanism for non-Newtonian flow in suspensions of rigid spheres. J. Rheol. 3, 137–152.Google Scholar
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.Google Scholar
Lahellec, N., Ponte Castañeda, P. & Suquet, P. 2011 Variational estimates for the effective response and field statistics in thermoelastic composites with intra-phase property fluctuations. Proc. R. Soc. Lond. A 467 (2132), 2224–2246.Google Scholar
Lowenberg, M. & Hinch, E. J. 1996 Numerical simulation of concentrated emulsion in shear flow. J. Fluid Mech. 321, 395–419.Google Scholar
Milton, G. W. 2002 The Theory of Composites. Cambridge University Press.Google Scholar
Ogden, R. W., Saccomandi, G. & Sgura, I. 2004 Fitting hyperelastic models to experimental data. Comput. Mech. 34, 484–502.Google Scholar
Oldroyd, J. G. 1953 The elastic and viscous properties of emulsions and suspensions. Proc. R. Soc. Lond. A 218, 122–132.Google Scholar
Pal, R. 2003 Rheology of concentrated suspensions of deformable elastic particles such as human erythrocytes. J. Biomech. 36, 981–989.Google Scholar
Pal, R. 2010 Rheology of Particulate Dispersions and Composites. CRC Press.Google Scholar
Palierne, J. F. 1990 Linear rheology of viscoelastic emulsions with interfacial-tension. Rheol. Acta 29 (3), 204–214.Google Scholar
Phung, T. N., Brady, J. F. & Bossis, G. 1996 Stokesian dynamics simulations of Brownian suspensions. J. Fluid Mech. 313, 181–207.CrossRefGoogle Scholar
Ponte Castañeda, P. 2005 Heterogeneous Materials, Lecture Notes, Department of Mechanics, Ecole Polytechnique, ISBN: 2-7302-1267-1.Google Scholar
Ponte Castañeda, P. & Suquet, P. 1998 Nonlinear composites. Adv. Appl. Mech. 34, 171–302.Google Scholar
Ponte Castañeda, P. & Willis, J. R. 1995 The effect of spatial distribution on the effective behavior of composite materials and cracked media. J. Mech. Phys. Solids 43 (12), 1919–1951.Google Scholar
Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities. J. Fluid Mech. 361, 117–143.Google Scholar
Roscoe, R. 1967 On the rheology of a suspension of viscoelastic spheres in a viscous liquid. J. Fluid Mech. 28, 273–293.Google Scholar
Saito, N. 1950 Concentration dependence of the viscosity of high polymer solutions. I. J. Phys. Soc. Japan 5 (1), 4–8.Google Scholar
Skalak, R., Ozkaya, N. & Skalak, T. C. 1989 Biofluid mechanics. Annu. Rev. Fluid Mech. 21, 167–200.Google Scholar
Snabre, P. & Mills, P. 1999 Rheology of concentrated suspensions of viscoelastic particles. Colloids Surf. A 152, 79–88.Google Scholar
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.Google Scholar
Talbot, D. R. S. & Willis, J. R. 1985 Variational principles for inhomogeneous nonlinear media. IMA J. Appl. Maths 35, 39–54.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Tucker, C. L. & Moldenaers, P. 2002 Microstructural evolution in polymer blends. Annu. Rev. Fluid Mech. 34 (1), 177–210.CrossRefGoogle Scholar
Villone, M. M., Hulsen, M. A., Anderson, P. D. & Maffettone, P. L. 2014 Simulations of deformable systems in fluids under shear flow using an arbitrary Lagrangian Eulerian technique. Comput. Fluids 90, 88–100.Google Scholar
Vlahovska, P. M., Blawzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293–337.Google Scholar
Wetzel, E. D. & Tucker, C. L. 2001 Droplet deformation in dispersions with unequal viscosities and zero interfacial tension. J. Fluid Mech. 426, 199–228.Google Scholar
Willis, J. R. 1977 Bounds and self-consistent estimates for the overall moduli of anisotropic composites. J. Mech. Phys. Solids 25, 185–202.Google Scholar
Willis, J. R. 1981 Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78.Google Scholar
Zhao, H. & Shaqfeh, E. S. G. 2013 The dynamics of a non-dilute vesicle suspension in a simple shear flow. J. Fluid Mech. 725, 709–731.CrossRefGoogle Scholar