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Influence of surface viscosity on droplets in shear flow

  • J. Gounley (a1), G. Boedec (a2), M. Jaeger (a1) and M. Leonetti (a2)
Abstract

The behaviour of a single droplet in an immiscible external fluid, submitted to shear flow is investigated using numerical simulations. The surface of the droplet is modelled by a Boussinesq–Scriven constitutive law involving the interfacial viscosities and a constant surface tension. A numerical method using Loop subdivision surfaces to represent droplet interface is introduced. This method couples boundary element method for fluid flows and finite element method to take into account the stresses due to the surface dilational and shear viscosities and surface tension. Validation of the numerical scheme with respect to previous analytic and computational work is provided, with particular attention to the viscosity contrast and the shear and dilational viscosities characterized both by a Boussinesq number $B_{q}$ . Then, influence of equal surface viscosities on steady-state characteristics of a droplet in shear flow are studied, considering both small and large deformations and for a large range of bulk viscosity contrast. We find that small deformation analysis is surprisingly predictive at moderate and high surface viscosities. Equal surface viscosities decrease the Taylor deformation parameter and tank-treading angle and also strongly modify the dynamics of the droplet: when the Boussinesq number (surface viscosity) is large relative to the capillary number (surface tension), the droplet displays damped oscillations prior to steady-state tank-treading, reminiscent from the behaviour at large viscosity contrast. In the limit of infinite capillary number $Ca$ , such oscillations are permanent. The influence of surface viscosities on breakup is also investigated, and results show that the critical capillary number is increased. A diagram $(B_{q};Ca)$ of breakup is established with the same inner and outer bulk viscosities. Additionally, the separate roles of shear and dilational surface viscosity are also elucidated, extending results from small deformation analysis. Indeed, shear (dilational) surface viscosity increases (decreases) the stability of drops to breakup under shear flow. The steady-state deformation (Taylor parameter) varies nonlinearly with each Boussinesq number or a linear combination of both Boussinesq numbers. Finally, the study shows that for certain combinations of shear and dilational viscosities, drop deformation for a given capillary number is the same as in the case of a clean surface while the inclination angle varies.

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Corresponding author
Email address for correspondence: leonetti@irphe.univ-mrs.fr
References
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Abreu, D., Levant, M., Steinberg, V. & Seifert, U. 2014 Fluid vesicles in flow. Adv. Colloid Interface Sci. 208, 129141.
Acrivos, A. 1983 The breakup of small drops and bubbles in shear flows. Ann. N.Y. Acad. Sci. 404, 111.
Barthès-Biesel, D. 2009 Capsule motion in flow: deformation and membrane buckling. C. R. Phys. 10 (8), 764774.
Barthès-Biesel, D. & Acrivos, A. 1973 Deformation and burst of a liquid droplet freely suspended in a linear shear field. J. Fluid Mech. 61 (01), 122.
Barthès-Biesel, D. & Sgaier, H. 1985 Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow. J. Fluid Mech. 160, 119135.
Boedec, G., Jaeger, M. & Leonetti, M. 2012 Settling of a vesicle in the limit of quasispherical shapes. J. Fluid Mech. 690, 227261.
Boedec, G., Jaeger, M. & Leonetti, M. 2014 Pearling instability of a cylindrical vesicle. J. Fluid Mech. 743, 262279.
Boedec, G., Leonetti, M. & Jaeger, M. 2011 3D vesicle dynamics simulations with a linearly triangulated surface. J. Comput. Phys. 230 (4), 10201034.
Boussinesq, M. 1913 Sur l’existence d’une viscosité superficielle, dans la mince couche de transition séparant un liquide d’un autre fluide contigue. Ann. Chim. Phys. 29, 349357.
Chaffey, C. E. & Brenner, H. 1967 A second-order theory for shear deformation of drops. J. Colloid Interface Sci. 24, 258269.
Chang, K. & Olbricht, W. 1993 Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow. J. Fluid Mech. 250 (1), 609633.
Cirak, F., Ortiz, M. & Schroder, P. 2000 Subdivision surfaces: a new paradigm for thin-shell finite-element analysis. Intl J. Numer. Meth. Engng 47 (12), 20392072.
Cox, R. 1969 The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37 (03), 601623.
Cristini, V., Guido, S., Alfani, A., Bławzdziewicz, J. & Loewenberg, M. 2003 Drop breakup and fragment size distribution in shear flow. J.  Rheol. 47 (5), 12831298.
Danov, K. D. 2001 On the viscosity of dilute emulsions. J. Colloid Interface Sci. 235 (1), 144149.
Dickinson, E., Murray, B. S. & Stainsby, G. 1988 Coalescence stability of emulsion-sized droplets at a planar oil–water interface and the relationship to protein film surface rheology. J. Chem. Soc. Faraday Trans. 84 (3), 871883.
Erni, P. 2011 Deformation modes of complex fluid interfaces. Soft Matt. 7 (17), 75867600.
Erni, P., Fischer, P. & Windhab, E. J. 2005 Deformation of single emulsion drops covered with a viscoelastic adsorbed protein layer in simple shear flow. Appl. Phys. Lett. 87 (24), 244104.
Feigl, K., Megias-Alguacil, D., Fischer, P. & Windhab, E. J. 2007 Simulation and experiments of droplet deformation and orientation in simple shear flow with surfactants. Chem. Engng Sci. 62 (12), 32423258.
Fischer, P. & Erni, P. 2007 Emulsion drops in external flow fields – the role of liquid interfaces. Curr. Opin. Colloid Interface Sci. 12 (4), 196205.
Flumerfelt, R. W. 1980 Effects of dynamic interfacial properties on drop deformation and orientation in shear and extensional flow fields. J.  Colloid Interface Sci. 76 (2), 330349.
Georgieva, D., Schmitt, V., Leal-Calderon, F. & Langevin, D. 2009 On the possible role of surface elasticity in emulsion stability. Langmuir 25 (10), 55655573.
Guido, S., Greco, F. & Villone, M. 1999 Experimental determination of drop shape in slow steady shear flow. J. Colloid Interface Sci. 219, 298309.
Guido, S. & Villone, M. 1998 Three-dimensional shape of a drop under simple shear flow. J. Rheol. 42, 395415.
Harvey, P., Nguyen, A., Jameson, G. & Evans, G. 2005 Influence of sodium dodecyl sulphate and dowfroth frothers on froth stability. Miner. Engng 18 (3), 311315.
Huang, W.-X., Chang, C. B. & Sung, H. J. 2012 Three-dimensional simulation of elastic capsules in shear flow by the penalty immersed boundary method. J. Comput. Phys. 231, 33403364.
Kennedy, M., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comput. Fluids 23 (2), 251278.
Kwak, S. & Pozrikidis, C. 1998 Adaptive triangulation of evolving, closed, or open surfaces by the advancing-front method. J. Comput. Phys. 145 (1), 6188.
Langevin, D. 2000 Influence of interfacial rheology on foam and emulsion properties. Adv. Colloid Interface Sci. 88 (1), 209222.
Le, D. V. 2010 Effect of bending stiffness on the deformation of liquid capsules enclosed by thin shells in shear flow. Phys. Rev. E 82, 016318.
LeVan, M. D. 1981 Motion of a droplet with a Newtonian interface. J. Colloid Interface Sci. 83 (1), 1117.
Li, J., Renardy, Y. Y. & Renardy, M. 2000 Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids 12 (2), 269282.
Li, X. & Pozrikidis, C. 1997 The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341, 165194.
Loop, C. T.1987 Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah.
de Loubens, C., Deschamps, J., Boedec, G. & Leonetti, M. 2015a Stretching of capsules in an elongation flow, a route to constitutive law. J. Fluid Mech. 767, R3.
de Loubens, C., Deschamps, J., Edwards-Levy, F. & Leonetti, M. 2015b Tank-treading of microcapsules in shear flow. J. Fluid Mech., doi:10.1017/jfm.2015.758.
Manor, O., Lavrenteva, O. & Nir, A. 2008 Effect of non-homogeneous surface viscosity on the marangoni migration of a droplet in viscous fluid. J. Colloid Interface Sci. 321 (1), 142153.
Miller, R., Ferri, J. K., Javadi, A., Kragel, J., Mucic, N. & Wustneck, R. 2010 Rheology of interfacial layers. Colloid Polym. Sci. 288, 937950.
Mun, S. & McClements, D. J. 2006 Influence of interfacial characteristics on Ostwald ripening in hydrocarbon oil-in-water emulsions. Langmuir 22 (4), 15511554.
Narsimhan, V., Spann, A. P. & Shaqfeh, E. S. G. 2015 Pearling, wrinkling buckling of vesicles in elongation flows. J. Fluid Mech. 777, 126.
Oldroyd, J. 1955 The effect of interfacial stabilizing films on the elastic and viscous properties of emulsions. Proc. R. Soc. Lond. A 232 (1191), 567577.
Pawar, Y. & Stebe, K. J. 1996 Marangoni effects on drop deformation in an extensional flow: the role of surfactnt physical chemistry. i. Insoluble surfactants. Phys. Fluids 8 (7), 17381751.
Phillips, W. J., Graves, R. W. & Flumerfelt, R. W. 1980 Experimental studies of drop dynamics in shear fields: role of dynamic interfacial effects. J. Colloid Interface Sci. 76 (2), 350370.
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Pozrikidis, C. 1994 Effects of surface viscosity on the finite deformation of a liquid drop and the rheology of dilute emulsions in simple shearing flow. J. Non-Newtonian Fluid Mech. 51 (2), 161178.
Rallison, J. 1980 Note on the time-dependent deformation of a viscous drop which is almost spherical. J. Fluid Mech. 98 (03), 625633.
Rallison, J. 1981 A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 109, 465482.
Rallison, J. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89 (01), 191200.
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 4566.
Reusken, A. & Zhang, Y. 2013 Numerical simulation of incompressible two-phase flows with a Boussinesq–Scriven interface stress tensor. Intl J. Numer. Meth. Fluids 73 (12), 10421058.
Rodrigues, D. S., Ausas, R. F., Mut, F. & Buscaglia, G. C. 2015 A semi-implicit finite element method for viscous lipid membranes. J. Comput. Phys. 298, 565584.
Rumscheidt, F.-D. & Mason, S. 1961 Particle motions in sheared suspensions XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. Colloid Sci. 16 (3), 238261.
Schwalbe, J. T., Phelan, F. R. Jr, Vlahovska, P. M. & Hudson, S. D. 2011 Interfacial effects on droplet dynamics in Poiseuille flow. Soft Matt. 7 (17), 77977804.
Scriven, L. 1960 Dynamics of a fluid interface equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98108.
Secomb, T. & Skalak, R. 1982 Surface flow of viscoelastic membranes in viscous fluids. Q. J. Mech. Appl. Maths 35 (2), 233247.
Spann, A. P., Zhao, H. & Shaqfeh, E. S. G. 2014 Loop subdivision surface boundary integral method simulations of vesicles at low reduced volume ratio in shear and extensional flow. Phys. Fluids 26, 031902.
Stone, H. & Leal, L. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161186.
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26 (1), 65102.
Taylor, G. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501523.
Valkovska, D., Danov, K. & Ivanov, I. 1999 Surfactants role on the deformation of colliding small bubbles. Colloids Surf. A 156 (1), 547566.
Vlahovska, P. M., Blawzdziewicz, J. & Loewenberg, M. 2005 Deformation of a surfactant-covered drop in a linear flow. Phys. Fluids 17, 103103.
Vlahovska, P. M., Blawzdziewicz, J. & Loewenberg, M. 2009a Small-deformation theory for a surfactant drop in linear flows. J. Fluid Mech. 624, 293337.
Vlahovska, P. M., Podgorski, T. & Misbah, C. 2009b Vesicles and red blood cells in flow: from individual dynamics to rheology. C. R. Phys. 10 (8), 775789.
Yazdani, A. & Bagchi, P. 2013 Influence of membrane viscosity on capsule dynamics in shear flow. J. Fluid Mech. 718, 569595.
Zell, Z. A., Nowbahar, A., Mansard, V., Leal, L. G., Deshmukh, S. S., Mecca, J. M., Tucker, C. J. & Squires, T. M. 2014 Surface shear inviscidity of soluble surfactants. Proc. Natl Acad. Sci. USA 111 (10), 36773682.
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