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General rheology of highly concentrated emulsions with insoluble surfactant

  • Alexander Z. Zinchenko (a1) and Robert H. Davis (a1)


A general constitutive model is constructed and validated for highly concentrated monodisperse emulsions of deformable drops with insoluble surfactant through long-time, large-scale and high-resolution multidrop simulations. There is the same amount of surfactant on each drop, and the linear model is assumed for the surface tension versus the surfactant concentration. The surfactant surface transport is coupled to multidrop hydrodynamics through the convective–diffusive equation and the interfacial stress balance. Only the limit of small surfactant diffusivities is addressed, when this parameter does not affect the rheology. An Oldroyd constitutive equation is postulated, with five variable coefficients depending on one instantaneous flow invariant (chosen as the drop-phase contribution to the dissipation rate). These coefficients are found by fitting the model to five precise rheological functions from two steady base flows at arbitrary deformation rates. One base flow is planar extension (PE) ( $\dot{\unicode[STIX]{x1D6E4}}x_{1},-\dot{\unicode[STIX]{x1D6E4}}x_{2},0$ ), the other one is planar mixed flow (PM) ( $\dot{\unicode[STIX]{x1D6FE}}x_{2}$ , $\dot{\unicode[STIX]{x1D6FE}}\unicode[STIX]{x1D712}x_{1}$ , 0) with $\unicode[STIX]{x1D712}=0.16$ . A small but finite $\unicode[STIX]{x1D712}$ (a precise choice in the range $\unicode[STIX]{x1D712}\sim 0.1$ is unimportant) provides a necessarily perturbation to exclude severe ergodic difficulties and abnormal, kinked behaviour inherent in simple shear for high drop volume fractions $c$ , especially at small capillary numbers $Ca$ and small drop-to-medium viscosity ratios $\unicode[STIX]{x1D706}$ . The database rheological functions are obtained for $c=0.45{-}0.6$ , $\unicode[STIX]{x1D706}=0.25{-}3$ and surfactant elasticities $\unicode[STIX]{x1D6FD}=0.05{-}0.2$ (based on the equilibrium surfactant concentration) from long-time simulations by a multipole-accelerated boundary-integral code with $N=100{-}200$ drops in a periodic cell and 2000–4000 boundary elements per drop. The code is an extension from Zinchenko & Davis (J. Fluid Mech., vol. 779, 2015, pp. 197–244) to account for surfactant transport and Marangoni stresses. Massive drop cusping or (sometimes) drop break-up limit the range of $Ca$ from above in the base flows, but there is no substantial lower limitation owing to the absence of phase transition difficulties. At small $\unicode[STIX]{x1D706}$ , even minimal surface contamination may have a strong effect on the rheology. The simulations remain accurate for quite strong drop interactions, when the PE emulsion viscosity is nine times that for the carrier fluid. The model validation against a steady PM flow with a different $\unicode[STIX]{x1D712}=0.5$ shows a very good agreement for various $Ca$ , $c$ and $\unicode[STIX]{x1D706}$ . In the three PE and PM time-dependent flow tests, the quasi-steady approximation is found to predict stresses poorly. In contrast, the combination of the steady-state results for PE and PM used in the present method to generate the Oldroyd parameters gives a model with much better predictions for these time-dependent flows.


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Astarita, G. & Marrucci, G. 1974 Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill.
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.
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.
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545570.
Bazhlekov, I. B., Anderson, P. D. & Meijer, H. E. H. 2006 Numerical investigation of the effects of insoluble surfactants on drop deformation and breakup in simple shear flow. J. Colloid Interface Sci. 298, 369394.
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.
Eggers, J. & du Pont, S. C. 2009 Numerical analysis of tips in viscous flow. Phys. Rev. E 79, 066311.
Eggleton, C. D., Pawar, Y. P. & Stebe, K. J. 1999 Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces. J. Fluid Mech. 385, 7999.
Eggleton, C. D. & Stebe, K. J. 1998 An adsorption-desorption-controlled surfactant on a deforming droplet. J. Colloid Interface Sci. 208, 6880.
Eggleton, C. D., Tsai, T.-M. & Stebe, K. J. 2001 Tip streaming from a drop in the presence of surfactants. Phys. Rev. Lett. 87, 048302.
Farutin, A., Biben, T. & Misbah, C. 2014 3D numerical simulations of vesicle and inextensible capsule dynamics. J. Comput. Phys. 275, 539568.
Frankel, N. A. & Acrivos, A. 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44, 6578.
Hand, G. L. 1962 A theory of anisotropic fluids. J. Fluid Mech. 13, 3346.
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.
He, Y., Yazhgur, P., Salonen, A. & Langevin, D. 2015 Adsorbtion-desorption kinetics of surfactants at liquid surfaces. Adv. Colloid Interface Sci. 222, 377384.
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.
Jeong, J.-T. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 122.
Johnson, R. A. & Borhan, A. 1999 Effect of insoluble surfactants on the pressure-driven motion of a drop in a tube in the limit of high surface coverage. J. Colloid Interface Sci. 218, 184200.
Klaseboer, E., Sun, Q. & Chan, D. Y. C. 2012 Non-singular boundary integral methods for fluid mechanics applications. J. Fluid Mech. 696, 468478.
Kraynik, A. M. & Reinelt, D. A. 1992 Extensional motions of spatially periodic lattices. Intl J. Multiphase Flow 18, 10451059.
Li, X. & Pozrikidis, C. 1997 The effects of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341, 165194.
Lin, Y., Mckeigue, K. & Maldarelli, C. 1991 Diffusion-limited interpretation of the induction period in the relaxation in surface tension due to the adsorption of strain chain, small polar group surfactants: theory and experiment. Langmuir 7, 10551066.
Loewenberg, M. & Hinch, E. J. 1996 Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech. 321, 395419.
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, 759.
Milliken, W. J., Stone, H. A. & Leal, L. G. 1993 The effect of surfactant on the transient motion of Newtonian drops. Phys. Fluids A 5, 6979.
Oldroyd, J. G. 1958 Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids. Proc. R. Soc. Lond. A 245, 278297.
Pawar, Y. & Stebe, K. J. 1996 Marangoni effects on drop deformation in an extensional flow: the role of surfactant physical chemistry. I. Insoluble surfactants. Phys. Fluids 8, 17381751.
Princen, H. M. & Kiss, A. D. 1989 Rheology of foams and highly concentrated emulsions. IV. An experimental study of the shear viscosity and yield stress of concentrated emulsions. J. Colloid Interface Sci. 128, 176187.
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.
Rother, M. A., Zinchenko, A. Z. & Davis, R. H. 2006 Surfactant effects on buoyancy-driven viscous interactions of deformable drops. Colloids Surf. A 282–283, 5060.
Schowalter, W. R., Chaffey, C. E. & Brenner, H. 1968 Rheological behavior of a dilute emulsion. J. Colloid Interface Sci. 26, 152160.
Stone, H. A. 1990 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2, 111112.
Stone, H. A. & Leal, L. G. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161186.
Vlahovska, P. M., Bławzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293337.
Vlahovska, P. M., Loewenberg, M. & Bławzdziewicz, J. 2005 Deformation of a surfactant-covered drop in a linear flow. Phys. Fluids 17, 103103.
Yiantsios, S. G. & Davis, R. H. 1990 On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface. J. Fluid Mech. 217, 547573.
Yiantsios, S. G. & Davis, R. H. 1991 Close approach and deformation of two viscous drops due to gravity and van der Waals forces. J. Colloid Interface Sci. 144, 412433.
Yon, S. & Pozrikidis, C. 1998 A finite-volume/boundary-element method for flow past interfaces in the presence of surfactants, with application to shear flow past a viscous drop. J. Comput. Fluids 27, 879902.
Zinchenko, A. Z. 1994 An efficient algorithm for calculating multiparticle thermal interaction in a concentrated dispersion of spheres. J. Comput. Phys. 111, 120135.
Zinchenko, A. Z. 1998 Effective conductivity of loaded granular materials by numerical simulation. Phil. Trans. R. Soc. Lond. A 356, 29532998.
Zinchenko, A. Z. & Davis, R. H. 2000 An efficient algorithm for hydrodynamical interaction of many deformable drops. J. Comput. Phys. 157, 539587.
Zinchenko, A. Z. & Davis, R. H. 2002 Shear flow of highly concentrated emulsions of deformable drops by numerical simulations. J. Fluid Mech. 455, 2162.
Zinchenko, A. Z. & Davis, R. H. 2003 Large-scale simulations of concentrated emulsion flows. Phil. Trans. R. Soc. Lond. A 361, 813845.
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.
Zinchenko, A. Z. & Davis, R. H. 2008 Algorithm for direct numerical simulation of emulsion flow through a granular material. J. Comput. Phys. 227, 78417888.
Zinchenko, A. Z. & Davis, R. H. 2013 Emulsion flow through a packed bed with multiple drop breakup. J. Fluid Mech. 725, 611663.
Zinchenko, A. Z. & Davis, R. H. 2015 Extensional and shear flows, and general rheology of concentrated emulsions of deformable drops. J. Fluid Mech. 779, 197244.
Zinchenko, A. Z., Rother, M. A. & Davis, R. H. 1999 Cusping, capture and breakup of interacting drops by a curvatureless boundary-integral algorithm. J. Fluid Mech. 391, 249292.
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