Skip to main content
SpringerLink
Log in

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

  • Original Contribution
  • Published:
Rheologica Acta Aims and scope Submit manuscript

Abstract

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a≫1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  • Aidun CK (1995) Lattice Boltzmann simulation of solid particles suspended in fluid. J Stat Phys 81(1-2):49–61

    Article  Google Scholar 

  • Astarita G, Marrucci G (1974) Principles of non-Newtonian fluid mechanics (Vol 28). McGraw-Hill, New York

    Google Scholar 

  • Bagdassarov NS, Dingwell DB (1992) A rheological investigation of vesicular rhyolite. J Volcanol Geotherm Res 50(3):307–322

    Article  Google Scholar 

  • Batchelor GK (1967) An introduction to fluid dynamics. Cambridge university press

  • Batchelor GK, Green JT (1972) The determination of the bulk stress in a suspension of spherical particles to order c2. J Fluid Mech 56(03):401–427

    Article  Google Scholar 

  • Barnea E, Mizrahi J (1973) A generalized approach to the fluid dynamics of particulate systems: Part 1. General correlation for fluidization and sedimentation in solid multiparticle systems. The Chemical Engineering Journal 5(2):171–189

    Article  Google Scholar 

  • Barnea E, Mizrahi J (1975) A generalised approach to the fluid dynamics of particulate systems part 2: sedimentation and fluidisation of clouds of spherical liquid drops. The Canadian Journal of Chemical Engineering 53(5):461–468

    Article  Google Scholar 

  • Benito S, Bruneau CH, Colin T, Gay C, Molino F (2008) An elasto-visco-plastic model for immortal foams or emulsions. The European Physical Journal E 25(3):225–251

    Article  Google Scholar 

  • Boyer F, Guazzelli E., Pouliquen O (2011) Unifying suspension and granular rheology. Phys Rev Lett 107(18):188301

    Article  Google Scholar 

  • Boumonville B, Coussot P, Chateau X (2005) Modification du modle de Farris pour la prise en compte des interactions gomtriques d’un mlange polydisperse de particules. Rhologie 7:1–8

    Google Scholar 

  • Brady JF, Bossis G (1988) Stokesian dynamics. Ann Rev Fluid Mech 20:111–157

    Article  Google Scholar 

  • Brouwers HJH (2013) Random packing fraction of bimodal spheres: an analytical expression. Phys Rev E 87(3):032202

    Article  Google Scholar 

  • Brouwers HJH (2010) Viscosity of a concentrated suspension of rigid monosized particles. Phys Rev E 81(5):051402

    Article  Google Scholar 

  • Cantat I, Cohen-Addad S, Elias F, Graner F, Höhler R, Pitois O (2013) Foams: structure and dynamics. Oxford University Press

  • Chaffey CE, Brenner H (1967) A second-order theory for shear deformation of drops. J Colloid Interface Sci 24(2):258–269

    Article  Google Scholar 

  • Chan D, Powell RL (1984) Rheology of suspensions of spherical particles in a Newtonian and a non-Newtonian fluid. J Non-Newtonian Fluid Mech 15(2):165–179

    Article  Google Scholar 

  • Chang C, Powell RL (1994) Effect of particle size distributions on the rheology of concentrated bimodal suspensions. J Rheol (1978-present) 38(1):85–98

    Google Scholar 

  • Chang C, Powell RL (2002) Hydrodynamic transport properties of concentrated suspensions. AIChE j 48(11):2475–2480

    Article  Google Scholar 

  • Cheng Z, Zhu J, Chaikin PM, Phan SE, Russel WB (2002) Nature of the divergence in low shear viscosity of colloidal hard-sphere dispersions. Phys Rev E 65(4):041405

    Article  Google Scholar 

  • Choi SJ, Schowalter WR (1975) Rheological properties of nondilute suspensions of deformable particles. Phys Fluids 18:420

    Article  Google Scholar 

  • Chong JS, Christiansen EB, Baer AD (1971) Rheology of concentrated suspensions. J Appl Polym Sci 15(8):2007–2021

    Article  Google Scholar 

  • Cichocki B, Felderhof BU (1991) Linear viscoelasticity of semidilute hard-sphere suspensions. Phys Rev A 43(10):5405

    Article  Google Scholar 

  • Costa A, Caricchi L, Bagdassarov N (2009) A model for the rheology of particlebearing suspensions and partially molten rocks. Geochem Geophys Geosyst 10(3)

  • Dai SC, Bertevas E, Qi F, Tanner RI (2013) Viscometric functions for noncolloidal sphere suspensions with Newtonian matrices. J Rheol (1978-present) 57(2):493–510

    Google Scholar 

  • D’Avino G, Greco F, Hulsen MA, Maffettone PL (2013) Rheology of viscoelastic suspensions of spheres under small and large amplitude oscillatory shear by numerical simulations. J Rheol (1978-present) 57(3):813–839

    Google Scholar 

  • Ducamp VC, Raj R (1989) Shear and densification of glass powder compacts. J Am Ceram Soc 72(5):798–804

    Article  Google Scholar 

  • Einstein A (1911) Berichtigung zu meiner arbeit: eine neue bestimmung der moleköul-dimensionen. Annln Phys 339:591–592

    Article  Google Scholar 

  • Eilers VH (1943) Die viskositöat-konzentrationsabhöangigkeit kolloider systeme in organischen losungsmitteln. Kolloid-Z 102: 154–169

    Article  Google Scholar 

  • Faroughi SA, Huber C (2014) Crowding-based rheological model for suspensions of rigid bimodal-sized particles with interfering size ratios. Phys Rev E 90(052303)

  • Faroughi SA, Parmigiani A, Huber C (2013) Volatile dynamics in crystal-rich magma bodies, perspectives from laboratory experiments and theory, AGU Fall Meeting Abstracts, V31B2689F, 2689

  • Farris RJ (1968) Prediction of the viscosity of multimodal suspensions from unimodal viscosity data. Transactions of The Society of Rheology (1957-1977) 12(2):281–301

    Article  Google Scholar 

  • Frankel NA, Acrivos A (1967) On the viscosity of a concentrated suspension of solid spheres. Chem Eng Sci 22(6):847–853

    Article  Google Scholar 

  • Frankel NA, Acrivos A (1970) The constitutive equation for a dilute emulsion. J Fluid Mech 44(01):65–78

    Article  Google Scholar 

  • Goddard JD, Miller C (1967) Nonlinear effects in the rheology of dilute suspensions. J Fluid Mech 28(part 4):657–673. Chicago

    Article  Google Scholar 

  • Greco F (2002) Second-order theory for the deformation of a Newtonian drop in a stationary flow field. Phys Fluids (1994-present) 14(3):946–954

    Article  Google Scholar 

  • Happel J, Brenner H (eds) (1983) Low Reynolds number hydrodynamics: with special applications to particulate media (Vol 1). Springer

  • Happel J (1958) Viscous flow in multiparticle systems: slow motion of fluids relative to beds of spherical particles. AIChE J 4(2):197–201

    Article  Google Scholar 

  • Hatschek E (1913) The general theory of viscosity of two-phase systems. Trans Faraday Soc 9:80–92

    Article  Google Scholar 

  • Kim S, Russel WB (1985) Modelling of porous media by renormalization of the Stokes equations. J Fluid Mech 154:269–286

    Article  Google Scholar 

  • Koelman J MVA, Hoogerbrugge PJ (1993) Dynamic simulations of hard-sphere suspensions under steady shear. EPL (Europhys Lett) 21(3):363

    Article  Google Scholar 

  • Kramer TA, Clark MM (1999) Incorporation of aggregate breakup in the simulation of orthokinetic coagulation. J Colloid Interface Sci 216(1):116–126

    Article  Google Scholar 

  • Krieger IM, Dougherty TJ (1959) A mechanism for non-Newtonian flow in suspensions of rigid spheres. J Rheol 3:137

    Article  Google Scholar 

  • Ladd AJC, Verberg R (2001) Lattice-Boltzmann simulations of particle-fluid suspensions. J Stat Phys 104(5-6):1191–1251

    Article  Google Scholar 

  • Landau LD, Lifshitz EM (1987) Fluid Mechanics: Volume 6 (Course Of Theoretical Physics). Bu

  • Lejeune AM, Bottinga Y, Trull TW, Richet P (1999) Rheology of bubble-bearing magmas. Earth Planet Sci Lett 166(1):71–84

    Article  Google Scholar 

  • Leighton D, Acrivos A (1986) Viscous resuspension. Chem Eng Sci 41(6):1377–1384

    Article  Google Scholar 

  • Lewis TB, Nielsen LE (1968) Viscosity of dispersed and aggregated suspensions of spheres. Transactions of The Society of Rheology (1957–1977) 12(3):421–443

    Article  Google Scholar 

  • Lim YM, Seo D, Youn JR (2004) Rheological behavior of dilute bubble suspensions in polyol. Korea-Australia Rheology Journal 16(1):47–54

    Google Scholar 

  • Llewellin EW, Mader HM, Wilson SDR (2002) The rheology of a bubbly liquid. Proceedings of the Royal Society of London. Series A: Mathematical. Phys Eng Sci 458(2020):987–1016

    Article  Google Scholar 

  • Llewellin EW, Manga M (2005) Bubble suspension rheology and implications for conduit flow. J Volcanol Geotherm Res 143(1):205–217

    Article  Google Scholar 

  • Mackenzie JK (1950) The elastic constants of a solid containing spherical holes. Proc. Phys. Soc. B 63(1):2

    Article  Google Scholar 

  • Manga M, Loewenberg M (2001) Viscosity of magmas containing highly deformable bubbles. J Volcanol Geotherm Res 105(1):19–24

    Article  Google Scholar 

  • Marmottant P, Raufaste C, Graner F (2008) Discrete rearranging disordered patterns, part II: 2D plasticity, elasticity and flow of a foam. The European Physical Journal E 25(4):371–384

    Article  Google Scholar 

  • Maron SH, Pierce PE (1956) Application of Ree-Eyring generalized flow theory to suspensions of spherical particles. J Colloid Sci 11(1):80–95

    Article  Google Scholar 

  • Mendoza CI (2011) Effective static and high-frequency viscosities of concentrated suspensions of soft particles. The Journal of chemical physics 135(054904)

  • Mooney ME (1951) The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 6(2):162–170

    Article  Google Scholar 

  • Morris JF, Boulay F (1213) Curvilinear flows of noncolloidal suspensions: The role of normal stresses. J rheol:43

  • Mueller S, Llewellin EW, Mader HM (2009) The rheology of suspensions of solid particles. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, rspa20090445

  • Norris AN, Callegari AJ, Sheng P (1985) A generalized differential effective medium theory. Journal of the Mechanics and Physics of Solids 33(6):525–543

    Article  Google Scholar 

  • Oldroyd JG (1953) The elastic and viscous properties of emulsions and suspensions. Proceedings of the Royal Society of London. Series A. Math Phys Sci 218(1132):122–132

    Article  Google Scholar 

  • Oldroyd JG (1959). In: Mill C C (ed) Complicated rheological properties. In Rheology of disperse systems. Pergamon Press, London, pp 1–15

  • Oliver DR, Ward SG (1953) Relationship between relative viscosity and volume concentration of stable suspensions of spherical particles. Nature 171:396–397

    Article  Google Scholar 

  • Pal R (2004) Rheological constitutive equation for bubbly suspensions. Ind Eng Chem Res 43(17):5372–5379

    Article  Google Scholar 

  • Pal R (2003) Rheological behavior of bubble-bearing magmas. Earth Planet Sci Lett 207(1):165–179

    Article  Google Scholar 

  • Pal R (a 2003) Rheology of concentrated suspensions of deformable elastic particles such as human erythrocytes. J Biomech 36(7):981–989

    Article  Google Scholar 

  • Pal R (b 2003) Viscous behavior of concentrated emulsions of two immiscible Newtonian fluids with interfacial tension. J Colloid Interface Sci 263(1):296–305

    Article  Google Scholar 

  • Pal R (2001) Evaluation of theoretical viscosity models for concentrated emulsions at low capillary numbers. Chem Eng J 81(1):15–21

    Article  Google Scholar 

  • Pal R (2000) Relative viscosity of non-Newtonian concentrated emulsions of noncolloidal droplets. Ind Eng Chem Res 39(12):4933–4943

    Article  Google Scholar 

  • Pal R (1996) Viscoelastic properties of polymer-thickened oil-in-water emulsions. Chem Eng Sci 51(12):3299–3305

    Article  Google Scholar 

  • Pal R (1245) Rheology of polymer-thickened emulsions. J Rheol:36

  • Pal R, Rhodes E (1021) Viscosity concentration relationships for emulsions. J Rheol:33

  • Pasquino R, Grizzuti N, Maffettone PL, Greco F (2008) Rheology of dilute and semidilute noncolloidal hard sphere suspensions. J Rheol (1978-present) 52(6):1369–1384

    Google Scholar 

  • Qi F, Tanner RI (2011) Relative viscosity of bimodal suspensions. Korea-Australia Rheology Journal 23(2):105–111

    Article  Google Scholar 

  • Quemada D (1977) Rheology of concentrated disperse systems and minimum energy dissipation principle. Rheol Acta 16(1):82–94

    Article  Google Scholar 

  • Rexha G, Minale M (2011) Numerical predictions of the viscosity of non-Brownian suspensions in the semidilute regime. J Rheol (1978-present) 55(6):1319–1340

    Google Scholar 

  • Robinson JV (1949) The Viscosity of Suspensions of Spheres. The Journal of Physical Chemistry 53(7):1042–1056

    Article  Google Scholar 

  • Rodriguez BE, Kaler EW, Wolfe MS (1992) Binary mixtures of mono-disperse latex dispersions. 2. Viscosity. Langmuir 8(10):2382–2389

    Article  Google Scholar 

  • Roscoe R (1952) The viscosity of suspensions of rigid spheres. Br J Appl Phys 3(8):267

    Article  Google Scholar 

  • Rust AC, Manga M (2002) Effects of bubble deformation on the viscosity of dilute suspensions. J Non-Newtonian Fluid Mech 104(1):53–63

    Article  Google Scholar 

  • Rutgers IR (1962) Relative viscosity and concentration. Rheol Acta 2(4):305–348

    Article  Google Scholar 

  • Saito N (1950) Concentration dependence of the viscosity of high polymer solutions. J Phys Soc Jpn 5(1):4–8

    Article  Google Scholar 

  • Santiso E, Muller EA (2002) Dense packing of binary and polydisperse hard spheres. Mol Phys 100(15):2461–2469

    Article  Google Scholar 

  • Schaink HM, Slot JJM, Jongschaap RJJ, Mellema J (2000) The rheology of systems containing rigid spheres suspended in both viscous and viscoelastic media, studied by Stokesian dynamics simulations. J Rheol (1978-present) 44(3):473–498

    Google Scholar 

  • Schowalter WR, Chaffey C, Brenner H (1968) Rheological behavior of a dilute emulsion. J Colloid Interface Sci 26(2):152–160

    Article  Google Scholar 

  • Scherer GW (1979) Sintering inhomogeneous glasses: application to optical waveguides. J Non-Cryst Solids 34(2):239–256

    Article  Google Scholar 

  • Schramm LL (2006) Emulsions, foams, and suspensions: fundamentals and applications. John Wiley

  • Scott GD, Kilgour DM (1969) The density of random close packing of spheres. J Phys D Appl Phys 2(6):863

    Article  Google Scholar 

  • Segre PN, Meeker SP, Pusey PN, Poon WCK (1995) Viscosity and structural relaxation in suspensions of hard-sphere colloids. Phys rev lett 75(5):958

    Article  Google Scholar 

  • Sierou A, Brady JF (2002) Rheology and microstructure in concentrated noncolloidal suspensions. J Rheol 46:1031

    Article  Google Scholar 

  • Simha R (1952) A treatment of the viscosity of concentrated suspensions. J Appl Phys 23(9):1020–1024

    Article  Google Scholar 

  • Song C, Wang P, Makse HA (2008) A phase diagram for jammed matter. Nature 453(7195):629–632

    Article  Google Scholar 

  • Stein DJ, Spera FJ (1992) Rheology and microstructure of magmatic emulsions: theory and experiments. J Volcanol Geotherm Res 49(1):157–174

    Article  Google Scholar 

  • Stein DJ, Spera FJ (2002) Shear viscosity of rhyolite-vapor emulsions at magmatic temperatures by concentric cylinder rheometry. J Volcanol Geotherm Res 113(1): 243–258

    Article  Google Scholar 

  • Stickel JJ, Powell RL (2005) Fluid mechanics and rheology of dense suspensions. Annu Rev Fluid Mech 37:129–149

    Article  Google Scholar 

  • Strating P (1999) Brownian dynamics simulation of a hard-sphere suspension. Phys Rev E 59(2):2175

    Article  Google Scholar 

  • Sutherland W (1905) A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. Philos Mag 9:781–785

    Article  Google Scholar 

  • Taylor GI (1932) The viscosity of a fluid containing small drops of another fluid. Proc R Soc A 138:41–48

    Article  Google Scholar 

  • Thomas DG (1965) Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles. J Colloid Sci 20(3):267–277

    Article  Google Scholar 

  • Torquato S, Truskett TM, Debenedetti PG (2000) Is random close packing of spheres well defined? Phys Rev Lett 84(10):2064

    Article  Google Scholar 

  • Tran-Duc T, Phan-Thien N, Khoo BC (2013) Rheology of bubble suspensions using dissipative particle dynamics. Part I: a hard-core DPD particle model for gas bubbles. J Rheol 57:1715

    Article  Google Scholar 

  • Vand V (1948) Viscosity of solutions and suspensions. I. Theory. The Journal of Physical Chemistry 52(2):277–299

    Article  Google Scholar 

  • Verberg R, De Schepper IM, Cohen EGD (1997) Viscosity of colloidal suspensions. Phys Rev E 55(3):3143

    Article  Google Scholar 

  • Villone MM, D’Avino G, Hulsen MA, Greco F, Maffettone PL (2014) Numerical simulations of linear viscoelasticity of monodisperse emulsions of Newtonian drops in a Newtonian fluid from dilute to concentrated regime. Rheol Acta 53(5-6): 401–416

    Article  Google Scholar 

  • Winterwerp JC (1998) A simple model for turbulence induced flocculation of cohesive sediment. J Hydraul Res 36(3):309–326

    Article  Google Scholar 

  • Zarraga IE, Hill DA, Leighton D T Jr (2000) The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J Rheol 44:185

    Article  Google Scholar 

  • Zinchenko AZ, Davis RH (2003) Largescale simulations of concentrated emulsion flows. Philosophical Transactions of the Royal Society of London. Series A: Mathematical. Phys. Eng. Sci 361(1806):813–845

    Article  Google Scholar 

Download references

Acknowledgments

S.A.F and C.H. thank on anonymous reviewer and editor Jan Vermant for helpful suggestions that improved the quality of the manuscript. The work presented here was funded by NSF grant EAR 1144957.

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Salah Aldin Faroughi

  2. Department of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Christian Huber

Authors
  1. Salah Aldin Faroughi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Huber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Salah Aldin Faroughi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Faroughi, S.A., Huber, C. A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number. Rheol Acta 54, 85–108 (2015). https://doi.org/10.1007/s00397-014-0825-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00397-014-0825-8

Keywords

Search

Navigation