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Influence of pressure-dependent surface viscosity on dynamics of surfactant-laden drops in shear flow

Published online by Cambridge University Press:  02 November 2018

Zheng Yuan Luo Open the ORCID record for Zheng Yuan Luo [Opens in a new window] ,
Xing Long Shang  and
Bo Feng Bai
Zheng Yuan Luo
Affiliation:
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Xing Long Shang
Affiliation:
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Bo Feng Bai*
Affiliation:
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China
*
†Email address for correspondence: [email protected]
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Abstract

We study numerically the dynamics of an insoluble surfactant-laden droplet in a simple shear flow taking surface viscosity into account. The rheology of drop surface is modelled via a Boussinesq–Scriven constitutive law with both surface tension and surface viscosity depending strongly on the surface concentration of the surfactant. Our results show that the surface viscosity exhibits non-trivial effects on the surfactant transport on the deforming drop surface. Specifically, both dilatational and shear surface viscosity tend to eliminate the non-uniformity of surfactant concentration over the drop surface. However, their underlying mechanisms are entirely different; that is, the shear surface viscosity inhibits local convection due to its suppression on drop surface motion, while the dilatational surface viscosity inhibits local dilution due to its suppression on local surface dilatation. By comparing with previous studies of droplets with surface viscosity but with no surfactant transport, we find that the coupling between surface viscosity and surfactant transport induces non-negligible deviations in the dynamics of the whole droplet. More particularly, we demonstrate that the dependence of surface viscosity on local surfactant concentration has remarkable influences on the drop deformation. Besides, we analyse the full three-dimensional shape of surfactant-laden droplets in simple shear flow and observe that the drop shape can be approximated as an ellipsoid. More importantly, this ellipsoidal shape can be described by a standard ellipsoidal equation with only one unknown owing to the finding of an unexpected relationship among the drop’s three principal axes. Moreover, this relationship remains the same for both clean and surfactant-laden droplets with or without surface viscosity.

JFM classification

Complex Fluids: Complex Fluids Drops and Bubbles: Drops Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 858 , 10 January 2019 , pp. 91 - 121
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Copyright
© 2018 Cambridge University Press 

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References

Aggarwal, N. & Sarkar, K. 2007 Deformation and breakup of a viscoelastic drop in a newtonian matrix under steady shear. J. Fluid Mech. 584, 1–21.Google Scholar
Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48, 285–309.Google Scholar
Bai, B. F., Luo, Z. Y., Wang, S. Q., He, L., Lu, T. J. & Xu, F. 2013 Inertia effect on deformation of viscoelastic capsules in microscale flows. Microfluid. Nanofluid. 14, 817–829.Google Scholar
Carroll, R. M. & Gupta, N. R. 2014 Inertial and surfactant effects on the steady droplet flow in cylindrical channels. Phys. Fluids 26, 122102.Google Scholar
Cristini, V., Guido, S., Alfani, A., Blawzdziewicz, J. & Loewenberg, M. 2003 Drop breakup and fragment size distribution in shear flow. J. Rheol. 47, 1283–1298.Google Scholar
Derkach, S. R. 2009 Rheology of emulsions. Adv. Colloid Interface Sci. 151, 1–23.Google Scholar
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, 3242–3258.Google Scholar
Fischer, P. & Erni, P. 2007 Emulsion drops in external flow fields - the role of liquid interfaces. Curr. Opin. Colloid Interface Sci. 12, 196–205.Google 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, 330–349.Google Scholar
Frijters, S., Gunther, F. & Harting, J. 2012 Effects of nanoparticles and surfactant on droplets in shear flow. Soft Matt. 8, 6542–6556.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
Guido, S. 2011 Shear-induced droplet deformation: Effects of confined geometry and viscoelasticity. Curr. Opin. Colloid Interface Sci. 16, 61–70.Google Scholar
Guido, S. & Villone, M. 1998 Three-dimensional shape of a drop under simple shear flow. J. Rheol. 42, 395–415.Google Scholar
Hermans, E. & Vermant, J. 2014 Interfacial shear rheology of dppc under physiologically relevant conditions. Soft Matt. 10, 175–186.Google Scholar
Jesus, W. C. d., Roma, A. M., Pivello, M. R., Villar, M. M. & da Silveira-Neto, A. 2015 A 3d front-tracking approach for simulation of a two-phase fluid with insoluble surfactant. J. Comput. Phys. 281, 403–420.Google Scholar
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, 184–200.Google Scholar
Johnson, R. A. & Borhan, A. 2003 Pressure-driven motion of surfactant-laden drops through cylindrical capillaries: Effect of surfactant solubility. J. Colloid Interface Sci. 261, 529–541.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, 251–278.Google Scholar
Kim, K., Choi, S. Q., Zasadzinski, J. A. & Squires, T. M. 2011 Interfacial microrheology of dppc monolayers at the air–water interface. Soft Matt. 7, 7782–7789.Google Scholar
Kim, K., Choi, S. Q., Zell, Z. A., Squires, T. M. & Zasadzinski, J. A. 2013 Effect of cholesterol nanodomains on monolayer morphology and dynamics. Proc. Natl Acad. Sci. USA 110, E3054–E3060.Google Scholar
Komrakova, A. E., Shardt, O., Eskin, D. & Derksen, J. J. 2014 Lattice boltzmann simulations of drop deformation and breakup in shear flow. Intl J. Multiphase Flow 59, 24–43.Google Scholar
Kwak, S. & Pozrikidis, C. 1998 Adaptive triangulation of evolving, closed, or open surfaces by the advancing-front method. J. Comput. Phys. 145, 61–88.Google Scholar
Langevin, D. 2014 Rheology of adsorbed surfactant monolayers at fluid surfaces. Annu. Rev. Fluid Mech. 46, 47–65.Google Scholar
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, 269–282.Google Scholar
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, 165–194.Google Scholar
Li, X. Y. & Sarkar, K. 2008 Front tracking simulation of deformation and buckling instability of a liquid capsule enclosed by an elastic membrane. J. Comput. Phys. 227, 4998–5018.Google Scholar
Luo, Z. Y. & Bai, B. F. 2016 Dynamics of nonspherical compound capsules in simple shear flow. Phys. Fluids 28, 101901.Google Scholar
Luo, Z. Y. & Bai, B. F. 2018 Dynamics of capsules enclosing viscoelastic fluid in simple shear flow. J. Fluid Mech. 840, 656–687.Google Scholar
Luo, Z. Y., He, L. & Bai, B. F. 2015 Deformation of spherical compound capsules in simple shear flow. J. Fluid Mech. 775, 77–104.Google Scholar
Luo, Z. Y., Shang, X. L. & Bai, B. F. 2018 Marangoni effect on the motion of a droplet covered with insoluble surfactant in a square microchannel. Phys. Fluids 30, 077101.Google Scholar
Luo, Z. Y., Wang, S. Q., He, L., Xu, F. & Bai, B. F. 2013 Inertia-dependent dynamics of three-dimensional vesicles and red blood cells in shear flow. Soft Matt. 9, 9651–9660.Google Scholar
Maffettone, P. & Minale, M. 1998 Equation of change for ellipsoidal drops in viscous flow. J. Non-Newtonian Fluid Mech. 78, 227–241.Google Scholar
Mandal, S., Das, S. & Chakraborty, S. 2017 Effect of marangoni stress on the bulk rheology of a dilute emulsion of surfactant-laden deformable droplets in linear flows. Phys. Rev. Fluids 2, 113604.Google Scholar
Mandal, S., Ghosh, U. & Chakraborty, S. 2016 Effect of surfactant on motion and deformation of compound droplets in arbitrary unbounded stokes flows. J. Fluid Mech. 803, 200–249.Google Scholar
Manikantan, H. & Squires, T. M. 2017 Pressure-dependent surface viscosity and its surprising consequences in interfacial lubrication flows. Phys. Rev. Fluids 2, 023301.Google Scholar
Minale, M. 2010 Models for the deformation of a single ellipsoidal drop: A review. Rheol. Acta 49, 789–806.Google Scholar
Muradoglu, M. & Tryggvason, G. 2014 Simulations of soluble surfactants in 3d multiphase flow. J. Comput. Phys. 274, 737–757.Google Scholar
Ni, M. J., Komori, S. & Morley, N. 2003 Projection methods for the calculation of incompressible unsteady flows. Numer. Heat Transfer B-Fund. 44, 533–551.Google Scholar
Olgac, U. & Muradoglu, M. 2013 Effects of surfactant on liquid film thickness in the bretherton problem. Intl J. Multiphase Flow 48, 58–70.Google Scholar
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, 350–370.Google Scholar
Ponce-Torres, A., Montanero, J., Herrada, M., Vega, E. & Vega, J. 2017 Influence of the surface viscosity on the breakup of a surfactant-laden drop. Phys. Rev. Lett. 118, 024501.Google Scholar
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, 161–178.Google Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 45–66.Google Scholar
Sagis, L. M. 2011 Dynamic properties of interfaces in soft matter: Experiments and theory. Rev. Mod. Phys. 83, 1367–1403.Google Scholar
Samaniuk, J. R. & Vermant, J. 2014 Micro and macrorheology at fluid–fluid interfaces. Soft Matt. 10, 7023–7033.Google Scholar
Scriven, L. E. 1960 Dynamics of a fluid interface equation of motion for newtonian surface fluids. Chem. Engng Sci. 12, 98–108.Google Scholar
Stone, H. & Leal, L. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186.Google Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26, 65–102.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146, 0501–0523.Google Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y. J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.Google Scholar
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–liquid Multiphase Flows. Cambridge University Press.Google Scholar
Tucker III, C. L. & Moldenaers, P. 2002 Microstructural evolution in polymer blends. Annu. Rev. Fluid Mech. 34, 177–210.Google Scholar
Underhill, P. T., Hirsa, A. H. & Lopez, J. M. 2017 Modelling steady shear flows of newtonian liquids with non-newtonian interfaces. J. Fluid Mech. 814, 5–23.Google Scholar
Vananroye, A., Janssen, P. J. A., Anderson, P. D., Van Puyvelde, P. & Moldenaers, P. 2008 Microconfined equiviscous droplet deformation: Comparison of experimental and numerical results. Phys. Fluids 20, 013101.Google Scholar
Vananroye, A., Van Puyvelde, P. & Moldenaers, P. 2011 Deformation and orientation of single droplets during shear flow: combined effects of confinement and compatibilization. Rheol. Acta 50, 231–242.Google Scholar
Vlahovska, P. M., Blawzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293–337.Google Scholar
Vlahovska, P. M., Loewenberg, M. & Blawzdziewicz, J. 2005 Deformation of a surfactant-covered drop in a linear flow. Phys. Fluids 17, 103103.Google Scholar
Yazdani, A. & Bagchi, P. 2013 Influence of membrane viscosity on capsule dynamics in shear flow. J. Fluid Mech. 718, 569–595.Google Scholar
Yu, W. & Zhou, C. 2011 Dynamics of droplet with viscoelastic interface. Soft Matt. 7, 6337–6346.Google Scholar
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, 3677–3682.Google Scholar