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Extensional and shear flows, and general rheology of concentrated emulsions of deformable drops

Published online by Cambridge University Press:  14 August 2015

Alexander Z. Zinchenko*
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424, USA
Robert H. Davis
Affiliation:
Department of Chemical and Biological Engineering, University of Colorado, Boulder, CO 80309-0424, USA
*
Email address for correspondence: alexander.zinchenko@colorado.edu

Abstract

The rheology of highly concentrated monodisperse emulsions is studied by rigorous multidrop numerical simulations for three types of steady macroscopic flow, (i) simple shear ($\dot{{\it\gamma}}x_{2}$, 0 0), (ii) planar extension (PE) ($\dot{{\it\Gamma}}x_{1},-\dot{{\it\Gamma}}x_{2},0$) and (iii) mixed ($\dot{{\it\gamma}}x_{2}$, $\dot{{\it\gamma}}{\it\chi}x_{1}$, 0), where $\dot{{\it\gamma}}$ and $\dot{{\it\Gamma}}$ are the deformation rates, and ${\it\chi}\in (-1,1)$ is the flow parameter, in order to construct and validate a general constitutive model for emulsion flows with arbitrary kinematics. The algorithm is a development of the multipole-accelerated boundary-integral (BI) code of Zinchenko & Davis (J. Fluid Mech., vol. 455, 2002, pp. 21–62). It additionally incorporates periodic boundary conditions for (ii) and (iii) (based on the reproducible lattice dynamics of Kraynik–Reinelt for PE), control of surface overlapping, much more robust controllable surface triangulations for long-time simulations, and more efficient acceleration. The emulsion steady-state viscometric functions (shear viscosity and normal stress differences) for (i) and extensiometric functions (extensional viscosity and stress cross-difference) for (ii) are studied in the range of drop volume fractions $c=0.45{-}0.55$, drop-to-medium viscosity ratios ${\it\lambda}=0.25{-}10$ and various capillary numbers $\mathit{Ca}$, with 100–400 drops in a periodic cell and 2000–4000 boundary elements per drop. High surface resolution is important for all three flows at small $\mathit{Ca}$. Large system size and strains $\dot{{\it\gamma}}t$ of up to several thousand are imperative in some shear-flow simulations to identify the onset of phase transition to a partially ordered state, and evaluate (although still not precisely) the viscometric functions in this state. Below the phase transition point, the shear viscosity versus $\mathit{Ca}$ shows a kinked behaviour, with the local minimum most pronounced at ${\it\lambda}=1$ and $c=0.55$. The ${\it\lambda}=0.25$ emulsions flow in a partially ordered manner in a wide range of $\mathit{Ca}$ even when $c=0.45$. Increase of ${\it\lambda}$ to 3–10 shifts the onset of ordering to much smaller $\mathit{Ca}$, often outside the simulation range. In contrast to simple shear, phase transition is never observed in PE or mixed flow. A generalized five-parameter Oldroyd model with variable coefficients is fitted to our extensiometric and viscometric functions at arbitrary flow intensities (but outside the phase transition range). The model predictions compare very well with precise simulation results for strong mixed flows, ${\it\chi}=0.25$. Time-dependent PE flow is also considered. Ways to overcome the phase transition and drop breakup limitations on constitutive modelling are discussed.

Type
Papers
Copyright
© 2015 Cambridge University Press 

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References

Ahamadi, M. & Harlen, O. G. 2008 A Lagrangian finite element method for simulation of a suspension under planar extensional flow. J. Comput. Phys. 227, 75437560.CrossRefGoogle Scholar
Astarita, G. & Marucci, G. 1974 Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill.Google Scholar
Barnes, H. A., Hutton, J. F. & Walters, K. 1989 An Introduction to Rheology. Elsevier.Google Scholar
Barthés-Biesel, D. & Acrivos, A. 1973a Deformation and burst of a liquid droplet freely suspended in a linear shear field. J. Fluid Mech. 61, 122.Google Scholar
Barthés-Biesel, D. & Acrivos, A. 1973b Rheology of suspensions and its relation to phenomenological theories for non-Newtonian fluids. Intl J. Multiphase Flow 1, 124.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545570.Google Scholar
Bentley, B. J. & Leal, L. G. 1986 A computer-controlled four-roll mill for investigations of particle and drop dynamics in two-dimensional linear shear flows. J. Fluid Mech. 167, 219240.Google Scholar
Cristini, V., Bławzdziewicz, J. & Loewenberg, M. 2001 An adaptive mesh algorithm for evolving surfaces: simulations of drop breakup and coalescence. J. Comput. Phys. 168, 445463.Google Scholar
Cristini, V., Guido, S., Alfani, A., Bławzdziewicz, J. & Loewenberg, M. 2003 Drop breakup and fragment size distribution in shear flow. J. Rheol. 47, 12831298.CrossRefGoogle Scholar
Derkach, S. R. 2009 Rheology of emulsions. Adv. Colloid Interface Sci. 151, 123.CrossRefGoogle ScholarPubMed
Frankel, N. A. & Acrivos, A. 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44, 6578.Google Scholar
Golemanov, K., Tcholakova, S., Denkov, N. D., Ananthapadmanabhan, K. P. & Lips, A. 2008 Breakup of bubbles and drops in steadily sheared foams and concentrated emulsions. Phys. Rev. E 78, 051405.CrossRefGoogle ScholarPubMed
Hand, G. L. 1962 A theory of anisotropic fluids. J. Fluid Mech. 13, 3346.Google Scholar
Hansen, J. P. & McDonald, I. R. 1976 Theory of Simple Liquids. Academic.Google Scholar
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5, 317328.CrossRefGoogle Scholar
Hinch, E. G. & Leal, L. G. 1975 Constitutive equations in suspension mechanics. Part 1. General formulation. J. Fluid Mech. 71, 481495.CrossRefGoogle Scholar
Hinch, E. G. & Leal, L. G. 1976 Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J. Fluid Mech. 76, 187208.Google Scholar
Hoover, W. G. & Ree, F. H. 1968 Melting transition and communal entropy for hard spheres. J. Chem. Phys. 49, 36093617.Google Scholar
Hunt, A., Bernardi, S. & Todd, B. D. 2010 A new algorithm for extended nonequilibrium molecular dynamics simulations of mixed flow. J. Chem. Phys. 133, 154116.Google Scholar
Jansen, K. M. B., Agterof, G. M. & Mellema, J. 2001 Droplet breakup in concentrated emulsions. J. Rheol. 45, 227236.Google Scholar
Kennedy, M. R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput. Fluids 23, 251278.Google Scholar
Kraynik, A. M. & Reinelt, D. A. 1992 Extensional motions of spatially periodic lattices. Intl J. Multiphase Flow 18, 10451059.Google Scholar
Loewenberg, M. 1998 Numerical simulation of concentrated emulsion flows. Trans. ASME: J. Fluids Engng 120, 824832.Google Scholar
Loewenberg, M. & Hinch, E. J. 1996 Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech. 321, 395419.CrossRefGoogle Scholar
Martin, R., Zinchenko, A. & Davis, R. 2014 A generalized Oldroyd’s model for non-Newtonian liquids with applications to a dilute emulsion of deformable drops. J. Rheol. 58, 759777.Google Scholar
Oldroyd, J. G. 1958 Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. R. Soc. Lond. A 245, 278297.Google Scholar
Proudman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237262.CrossRefGoogle Scholar
Rallison, J. M. 1981 A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 109, 465482.Google Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 4566.Google Scholar
Rivlin, R. S. & Ericksen, J. L. 1955 Stress-deformation relations for isotropic materials. Arch. Rat. Mech. Anal. 4, 323425.Google Scholar
Rózanska, S., Rózanski, J., Ochowiak, M. & Mitkowski, P. T. 2014 Extensional viscosity measurements of concentrated emulsions with the use of the opposed nozzles device. Braz. J. Chem. Engng 31, 4755.Google Scholar
Schowalter, W. R., Chaffey, C. E. & Brenner, H. 1968 Rheological behavior of a dilute emulsion. J. Colloid Interface Sci. 26, 152160.Google Scholar
Van Dyke, M. 1967 Perturbation Methods in Fluid Mechanics. Academic.Google Scholar
Vlahovska, P. M., Loewenberg, M. & Bławzdziewicz, J. 2005 Deformation of a surfactant-covered drop in a linear flow. Phys. Fluids 17, 103103.Google Scholar
Vlahovska, P. M., Bławzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293337.CrossRefGoogle Scholar
Zinchenko, A. Z. & Davis, R. H. 2000 An efficient algorithm for hydrodynamical interaction of many deformable drops. J. Comput. Phys. 157, 539587.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2002 Shear flow of highly concentrated emulsions of deformable drops by numerical simulations. J. Fluid Mech. 455, 2162.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2003 Large-scale simulations of concentrated emulsion flows. Phil. Trans. R. Soc. Lond. A 361, 813845.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2004 Hydrodynamical interaction of deformable drops. In Emulsions: Structure Stability and Interactions (ed. Petsev, D. N.), pp. 391447. Elsevier.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2005 A multipole-accelerated algorithm for close interaction of slightly deformable drops. J. Comput. Phys. 207, 695735.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2006 A boundary-integral study of a drop squeezing through interparticle constrictions. J. Fluid Mech. 564, 227266.Google Scholar
Zinchenko, A. Z. & Davis, R. H. 2008 Algorithm for direct numerical simulation of emulsion flow through a granular material. J. Comput. Phys. 227, 78417888.CrossRefGoogle Scholar
Zinchenko, A. Z. & Davis, R. H. 2013 Emulsion flow through a packed bed with multiple drop breakup. J. Fluid Mech. 725, 611663.Google Scholar
Zinchenko, A. Z., Rother, M. A. & Davis, R. H. 1997 A novel boundary-integral algorithm for viscous interaction of deformable drops. Phys. Fluids 9, 14931511.CrossRefGoogle Scholar