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Effect of surfactant on motion and deformation of compound droplets in arbitrary unbounded Stokes flows

Published online by Cambridge University Press:  19 August 2016

Shubhadeep Mandal ,
Uddipta Ghosh  and
Suman Chakraborty
Shubhadeep Mandal
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India
Uddipta Ghosh
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India
Suman Chakraborty*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal 721302, India
*
†Email address for correspondence: [email protected]
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Abstract

This study deals with the motion and deformation of a compound drop system, subject to arbitrary but Stokesian far-field flow conditions, in the presence of bulk-insoluble surfactants. We derive solutions for fluid velocities and the resulting surfactant concentrations, assuming the capillary number and surface Péclet number to be small, as compared with unity. We first focus on a concentric drop configuration and apply Lamb’s general solution, assuming the far-field flow to be arbitrary in nature. As representative case studies, we consider two cases: (i) flow dynamics in linear flows and (ii) flow dynamics in a Poiseuille flow, although for the latter case, the concentric configuration does not remain valid in general. We further look into the effective viscosity of a dilute suspension of compound drops, subject to linear ambient flow, and compare our predictions with previously reported experiments. Subsequently, the eccentric drop configuration is addressed by using a bipolar coordinate system where the far-field flow is assumed to be axisymmetric but otherwise arbitrary in nature. As a specific example for eccentric drop dynamics, we focus on Poiseuille flow and study the drop migration velocities. Our analysis shows that the presence of surfactant generally opposes the imposed flows, thereby acting like an effective augmented viscosity. Our analysis reveals that maximizing the effects of surfactant makes the drops behave like solid particles suspended in a medium. However, in uniaxial extensional flow, the presence of surfactants on the inner drop, in conjunction with the drop radius ratio, leads to a host of interesting and non-monotonic behaviours for the interface deformation. For eccentric drops, the effect of eccentricity only becomes noticeable after it surpasses a certain critical value, and becomes most prominent when the two interfaces approach each other. We further depict that surfactant and eccentricity generally tend to suppress each other’s effects on the droplet migration velocities.

JFM classification

Drops and Bubbles: Drops Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 200 - 249
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© 2016 Cambridge University Press 

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References

Bandopadhyay, A., Mandal, S., Kishore, N. K. & Chakraborty, S. 2016 Uniform electric-field-induced lateral migration of a sedimenting drop. J. Fluid Mech. 792, 553–589.CrossRefGoogle Scholar
Barthès-Biesel, D. 2012 Microhydrodynamics and Complex Fluids. CRC Press.CrossRefGoogle Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (03), 545–570.CrossRefGoogle Scholar
Batchelor, G. K. & Green, J. T. 1972a The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56 (03), 401.CrossRefGoogle Scholar
Batchelor, G. K. & Green, J. T. 1972b The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (02), 375.Google Scholar
Behjatian, A. & Esmaeeli, A. 2013 Electrohydrodynamics of a compound drop. Phys. Rev. E 88 (3), 033012.Google ScholarPubMed
Behjatian, A. & Esmaeeli, A. 2015 Transient electrohydrodynamics of compound drops. Acta Mechanica 226 (8), 2581–2606.CrossRefGoogle Scholar
Blawzdziewicz, J., Vlahovska, P. & Loewenberg, M. 2000 Rheology of a dilute emulsion of surfactant-covered spherical drops. Phys. A 276 (1), 50–85.Google Scholar
Borhan, A., Haj-Hariri, H. & Nadim, A. 1992 Effect of surfactants on the thermocapillary migration of a concentric compound drop. J. Colloid Interface Sci. 149 (2), 553–560.CrossRefGoogle Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3–4), 242–251.CrossRefGoogle Scholar
Brenner, H. 1964 The Stokes resistance of a slightly deformed sphere. Chem. Engng Sci. 19 (8), 519–539.CrossRefGoogle Scholar
Chan, D. & Powell, R. L. 1984 Rheology of suspensions of spherical particles in a Newtonian and a non-Newtonian fluid. J. Non-Newtonian Fluid Mech. 15 (2), 165–179.CrossRefGoogle Scholar
Chan, P. C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92 (01), 131–170.CrossRefGoogle Scholar
Chen, Y., Liu, X. & Shi, M. 2013 Hydrodynamics of double emulsion droplet in shear flow. Appl. Phys. Lett. 102 (5), 051609.Google Scholar
Chen, Y., Liu, X., Zhang, C. & Zhao, Y. 2015a Enhancing and suppressing effects of an inner droplet on deformation of a double emulsion droplet under shear. Lab on a Chip 15 (5), 1255–1261.CrossRefGoogle ScholarPubMed
Chen, Y., Liu, X. & Zhao, Y. 2015b Deformation dynamics of double emulsion droplet under shear. Appl. Phys. Lett. 106 (14), 141601.CrossRefGoogle Scholar
Cox, R. G. 1969 The deformation of a drop in a general time-dependent fluid flow. J. Fluid Mech. 37 (03), 601–623.CrossRefGoogle Scholar
Davis, A. M. J. & Brenner, H. 1981 Emulsions containing a third solid internal phase. J. Engng Mech. Div. 107 (3), 609–621.CrossRefGoogle Scholar
Dickinson, E. 1999 Adsorbed protein layers at fluid interfaces: interactions, structure and surface rheology. Colloids Surf. B 15 (2), 161–176.CrossRefGoogle Scholar
Draxler, J. & Marr, R. 1986 Emulsion liquid membranes part I: phenomenon and industrial application. Chem. Engng Process. 20 (6), 319–329.Google Scholar
Elfring, G. J., Leal, L. G. & Squires, T. M. 2016 Surface viscosity and Marangoni stresses at surfactant laden interfaces. J. Fluid Mech. 792, 712–739.CrossRefGoogle Scholar
Fabiilli, M. L., Lee, J. A., Kripfgans, O. D., Carson, P. L. & Fowlkes, J. B. 2010 Delivery of water-soluble drugs using acoustically triggered perfluorocarbon double emulsions. Pharmaceut. Res. 27 (12), 2753–2765.CrossRefGoogle ScholarPubMed
Faroughi, S. A. & Huber, C. 2015 A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number. Rheol. Acta 54 (2), 85–108.CrossRefGoogle Scholar
Ficheux, M., Bonakdar, L. & Bibette, J. 1998 Some stability criteria for double emulsions. Langmuir 14 (11), 2702–2706.CrossRefGoogle Scholar
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), 330–349.Google Scholar
Frankel, N. A. & Acrivos, A. 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44 (01), 65.CrossRefGoogle Scholar
Frith, W. J., Mewis, J. & Strivens, T. A. 1987 Rheology of concentrated suspensions: experimental investigations. Powder Technol. 51 (1), 27–34.Google Scholar
Garti, N. & Bisperink, C. 1998 Double emulsions: progress and applications. Curr. Opin. Colloid Interface Sci. 3 (6), 657–667.Google Scholar
Gounley, J., Boedec, G., Jaeger, M. & Leonetti, M. 2016 Influence of surface viscosity on droplets in shear flow. J. Fluid Mech. 791, 464–494.Google Scholar
Gouz, H. N. & Sadhal, S. S. 1989 Fluid dynamics and stability analysis of a compound droplet in an electric field. Q. J. Mech. Appl. Maths 42 (1), 65–83.CrossRefGoogle Scholar
Ha, J.-W. & Yang, S.-M. 1995 Effects of surfactant on the deformation and stability of a drop in a viscous fluid in an electric field. J. Colloid Interface Sci. 175 (2), 369–385.CrossRefGoogle Scholar
Ha, J.-W. & Yang, S.-M. 1999 Fluid dynamics of a double emulsion droplet in an electric field. Phys. Fluids 11 (5), 1029.CrossRefGoogle Scholar
Haber, S. & Hetsroni, G. 1971 The dynamics of a deformable drop suspended in an unbounded Stokes flow. J. Fluid Mech. 49 (02), 257–277.CrossRefGoogle Scholar
Haber, S. & Hetsroni, G. 1972 Hydrodynamics of a drop submerged in an unbounded arbitrary velocity field in the presence of surfactants. Appl. Sci. Res. 25 (1), 215–233.CrossRefGoogle Scholar
Haber, S., Hetsroni, G. & Solan, A. 1973 On the low Reynolds number motion of two droplets. Intl J. Multiphase Flow 1 (1), 57–71.Google Scholar
Haj-Hariri, H., Nadim, A. & Borhan, A. 1993 Recriprocal theorem for concentric compound drops in arbitrary Stokes flows. J. Fluid Mech. 252, 265.Google Scholar
Hanna, J. A. & Vlahovska, P. M. 2010 Surfactant-induced migration of a spherical drop in Stokes flow. Phys. Fluids 22 (1), 013102.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1981 Low Reynolds Number Hydrodynamics. Springer.Google Scholar
Hetsroni, G. & Haber, S. 1970 The flow in and around a droplet or bubble submerged in an unbound arbitrary velocity field. Rheol. Acta 9 (4), 488–496.CrossRefGoogle Scholar
Hua, H., Shin, J. & Kim, J. 2014 Dynamics of a compound droplet in shear flow. Intl J. Heat Fluid Flow 50, 63–71.CrossRefGoogle Scholar
Johnson, R. E. & Sadhal, S. S. 1985 Fluid mechanics of compound multiphase drops and bubbles. Annu. Rev. Fluid Mech. 17 (1), 289–320.Google Scholar
Kan, H.-C., Udaykumar, H. S., Shyy, W. & Tran-Son-Tay, R. 1998 Hydrodynamics of a compound drop with application to leukocyte modeling. Phys. Fluids 10 (4), 760.CrossRefGoogle Scholar
Kim, S. & Karrila, S. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.Google Scholar
Kim, S.-H., Kim, J. W., Cho, J.-C. & Weitz, D. A. 2011 Double-emulsion drops with ultra-thin shells for capsule templates. Lab on a Chip 11 (18), 3162–3166.CrossRefGoogle ScholarPubMed
Kita, Y., Matsumoto, S. & Yonezawa, D. 1977 Viscometric method for estimating the stability of W/O/W-type multiple-phase emulsions. J. Colloid Interface Sci. 62 (1), 87–94.Google Scholar
Krieger, I. M. & Dougherty, T. J. 1959 A mechanism for non-Newtonian flow in suspensions of rigid spheres. J. Rheol. 3 (1), 137.Google Scholar
Lamb, H. 1975 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Langevin, D. 2000 Influence of interfacial rheology on foam and emulsion properties. Adv. Colloid Interface Sci. 88 (1–2), 209–222.CrossRefGoogle ScholarPubMed
Lavrenteva, O. M., Rosenfeld, L. & Nir, A. 2011 Shape change, engulfment, and breakup of partially engulfed compound drops undergoing thermocapillary migration. Phys. Rev. E 84 (5), 056323.Google ScholarPubMed
Leal, L. G. 2007 Advanced Transport Phenomena. Cambridge University Press.CrossRefGoogle Scholar
Lee, Y. 2001 Synthesis and electrorheological characteristics of microencapsulated polyaniline particles with melamine–formaldehyde resins. Polymer (Guildf) 42 (19), 8277–8283.CrossRefGoogle Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2015 Effect of interfacial slip on the cross-stream migration of a drop in an unbounded Poiseuille flow. Phys. Rev. E 92 (2), 023002.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2016 Dielectrophoresis of a surfactant-laden viscous drop. Phys. Fluids 28 (6), 062006.CrossRefGoogle Scholar
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), 142–153.CrossRefGoogle ScholarPubMed
Matsumoto, S. & Kohda, M. 1980 The viscosity of W/O/W emulsions: an attempt to estimate the water permeation coefficient of the oil layer from the viscosity changes in diluted systems on aging under osmotic pressure gradients. J. Colloid Interface Sci. 73 (1), 13–20.CrossRefGoogle Scholar
Mooney, M. 1951 The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 6 (2), 162–170.CrossRefGoogle Scholar
Morton, D. S., Subramanian, R. S. & Balasubramaniam, R. 1990 The migration of a compound drop due to thermocapillarity. Phys. Fluids A 2 (12), 2119.CrossRefGoogle Scholar
Nakano, M. 2000 Places of emulsions in drug delivery. Adv. Drug Deliv. Rev. 45 (1), 1–4.CrossRefGoogle ScholarPubMed
Oguz, H. N. & Sadhal, S. S. 1987 Growth and collapse of translating compound multiphase drops: analysis of fluid mechanics and heat transfer. J. Fluid Mech. 179, 105.Google Scholar
Oliveira, T. F. & Cunha, F. R. 2011 A theoretical description of a dilute emulsion of very viscous drops undergoing unsteady simple shear. Trans. ASME J. Fluids Engng 133 (10), 101208.Google Scholar
Pak, O. S., Feng, J. & Stone, H. A. 2014 Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers. J. Fluid Mech. 753, 535–552.CrossRefGoogle Scholar
Pal, R. 1996a Effect of droplet size on the rheology of emulsions. AIChE J. 42 (11), 3181–3190.Google Scholar
Pal, R. 1996b Multiple O/W/O emulsion rheology. Langmuir 7463 (16), 2220–2225.CrossRefGoogle Scholar
Pal, R. 1999 Rheology of blends of suspensions and emulsions. Ind. Engng Chem. Res. 38 (12), 5005–5010.Google Scholar
Pal, R. 2007 Rheology of double emulsions. J. Colloid Interface Sci. 307 (2), 509–515.CrossRefGoogle ScholarPubMed
Pal, R. 2008 Viscosity models for multiple emulsions. Food Hydrocoll. 22 (3), 428–438.CrossRefGoogle Scholar
Pal, R. 2011 Rheology of simple and multiple emulsions. Curr. Opin. Colloid Interface Sci. 16 (1), 41–60.Google Scholar
Palaniappan, D. & Daripa, P. 2000 Compound droplet in extensional and paraboloidal flows. Phys. Fluids 12 (10), 2377.CrossRefGoogle Scholar
Pozrikidis, C. 2001 Interfacial dynamics for Stokes flow. J. Comput. Phys. 169 (2), 250–301.CrossRefGoogle Scholar
Princen, H. 1983 Rheology of foams and highly concentrated emulsions. J. Colloid Interface Sci. 91 (1), 160–175.CrossRefGoogle Scholar
Qu, X. & Wang, Y. 2012 Dynamics of concentric and eccentric compound droplets suspended in extensional flows. Phys. Fluids 24 (12), 123302.Google Scholar
Raja, R. V., Subramanian, G. & Koch, D. L. 2010 Inertial effects on the rheology of a dilute emulsion. J. Fluid Mech. 646, 255.Google Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16 (1), 45–66.CrossRefGoogle Scholar
Ramachandran, A. & Leal, L. G. 2012 The effect of interfacial slip on the rheology of a dilute emulsion of drops for small capillary numbers. J. Rheol. 56 (6), 1555.CrossRefGoogle Scholar
Ramachandran, A., Tsigklifis, K., Roy, A. & Leal, G. 2012 The effect of interfacial slip on the dynamics of a drop in flow: part I. Stretching, relaxation, and breakup. J. Rheol. 56 (1), 45.Google Scholar
Rosenfeld, L., Lavrenteva, O. M. & Nir, A. 2009 On the thermocapillary motion of partially engulfed compound drops. J. Fluid Mech. 626, 263.Google Scholar
Rushton, E. & Davies, G. A. 1973 The slow unsteady settling of two fluid spheres along their line of centres. Appl. Sci. Res. 28 (1), 37–61.Google Scholar
Rushton, E. & Davies, G. A. 1983 Settling of encapsulated droplets at low Reynolds numbers. Intl J. Multiphase Flow 9 (3), 337–342.Google Scholar
Rust, A. C. & Manga, M. 2002 Effects of bubble deformation on the viscosity of dilute suspensions. J. Non-Newtonian Fluid Mech. 104 (1), 53–63.CrossRefGoogle Scholar
Sadhal, S. S., Ayyaswamy, P. S. & Chung, J. N. 1996 Transport Phenomena with Drops and Bubbles. Springer.Google Scholar
Sadhal, S. S. & Oguz, H. N. 1985 Stokes flow past compound multiphase drops: the case of completely engulfed drops/bubbles. J. Fluid Mech. 160, 511–529.Google Scholar
Sadhal, S. S. & Oguz, H. N. 1989 Fluid dynamics and stability analysis of a compound droplet in an electric field. Q. J. Mech. Appl. Maths 42 (4), 65.Google Scholar
Sapei, L., Naqvi, M. A. & Rousseau, D. 2012 Stability and release properties of double emulsions for food applications. Food Hydrocoll. 27 (2), 316–323.Google Scholar
Schowalter, W. R., Chaffey, C. E. & Brenner, H. 1968 Rheological behavior of a dilute emulsion. J. Colloid Interface Sci. 26 (2), 152–160.Google Scholar
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), 7797.CrossRefGoogle Scholar
Smith, K., Ottino, J. & Olvera de la Cruz, M. 2004 Encapsulated drop breakup in shear flow. Phys. Rev. Lett. 93 (20), 204501.CrossRefGoogle ScholarPubMed
Song, Y., Xu, J. & Yang, Y. 2010 Stokes flow past a compound drop in a circular tube. Phys. Fluids 22 (7), 072003.CrossRefGoogle Scholar
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 (1), 111–112.CrossRefGoogle Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26 (1), 65–102.CrossRefGoogle Scholar
Stone, H. A. & Leal, L. G. 1990a Breakup of concentric double emulsion droplets in linear flows. J. Fluid Mech. 211, 123.CrossRefGoogle Scholar
Stone, H. A. & Leal, L. G. 1990b The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161.CrossRefGoogle Scholar
Subramanian, G., Koch, D. L., Zhang, J. & Yang, C. 2011 The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles. J. Fluid Mech. 674, 307–358.CrossRefGoogle Scholar
Subramanian, R. S. & Balasubramaniam, R. 2005 The Motion of Bubbles and Drops in Reduced Gravity. Cambridge University Press.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138 (834), 41–48.Google Scholar
Torza, S. & Mason, S.. 1970 Three-phase interactions in shear and electrical fields. J. Colloid Interface Sci. 33 (1), 67–83.Google Scholar
Tsemakh, D., Lavrenteva, O. M. & Nir, A. 2004 On the locomotion of a drop, induced by the internal secretion of surfactant. Intl J. Multiphase Flow 30 (11), 1337–1367.CrossRefGoogle Scholar
Tsukada, T., Mayama, J., Sato, M. & Hozawa, M. 1997 Theoretical and experimental studies on the behavior of a compound drop under a uniform DC electric field. J. Chem. Engng Japan 30 (2), 215–222.CrossRefGoogle Scholar
Utada, A. S., Lorenceau, E., Link, D. R., Kaplan, P. D., Stone, H. A. & Weitz, D. A. 2005 Monodisperse double emulsions generated from a microcapillary device. Science 308 (5721), 537–541.Google Scholar
Valkovska, D. S. & Danov, K. D. 2000 Determination of bulk and surface diffusion coefficients from experimental data for thin liquid film drainage. J. Colloid Interface Sci. 223 (2), 314–316.Google Scholar
Vlahovska, P., Blawzdziewicz, J. & Loewenberg, M. 2002 Nonlinear rheology of a dilute emulsion of surfactant-covered spherical drops in time-dependent flows. J. Fluid Mech. 463, 1–24.CrossRefGoogle Scholar
Vlahovska, P. M. 2011 On the rheology of a dilute emulsion in a uniform electric field. J. Fluid Mech. 670, 481–503.CrossRefGoogle Scholar
Vlahovska, P. M., Bławzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293.CrossRefGoogle Scholar
Vlahovska, P. M., Loewenberg, M. & Blawzdziewicz, J. 2005 Deformation of a surfactant-covered drop in a linear flow. Phys. Fluids 17 (10), 103103.CrossRefGoogle Scholar
Wilde, P. J. 2000 Interfaces: their role in foam and emulsion behaviour. Curr. Opin. Colloid Interface Sci. 5 (3–4), 176–181.Google Scholar
Yu, W. & Zhou, C. 2011 Dynamics of droplet with viscoelastic interface. Soft Matt. 7 (13), 6337.Google Scholar
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