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Temporal stability of free liquid threads with surface viscoelasticity

Published online by Cambridge University Press:  10 May 2018

A. Martínez-Calvo  and
A. Sevilla Open the ORCID record for A. Sevilla [Opens in a new window]
A. Martínez-Calvo*
Affiliation:
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Av. Universidad 30, 28911 Leganés (Madrid), Spain
A. Sevilla
Affiliation:
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos, Universidad Carlos III de Madrid, Av. Universidad 30, 28911 Leganés (Madrid), Spain
*
†Email address for correspondence: [email protected]
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Abstract

We analyse the effect of surface viscoelasticity on the temporal stability of a free cylindrical liquid jet coated with insoluble surfactant, extending the results of Timmermans & Lister (J. Fluid Mech., vol. 459, 2002, pp. 289–306). Our development requires, in particular, deriving the correct expressions for the normal and tangential stress boundary conditions at a general axisymmetric interface when surface viscosity is modelled with the Boussinesq–Scriven constitutive equation. These stress conditions are applied to obtain a new dispersion relation for the liquid thread, which is solved to describe its temporal stability as a function of four governing parameters, namely the capillary Reynolds number, the elasticity parameter, and the shear and dilatational Boussinesq numbers. It is shown that both surface viscosities have a stabilising influence for all values of the capillary Reynolds number and elasticity parameter, the effect being more pronounced at low capillary Reynolds numbers. The wavenumber of maximum amplification depends non-monotonically on the Boussinesq numbers, especially for very viscous threads at low values of the elasticity parameter. Finally, two different lubrication approximations of the equations of motion are derived. While the validity of the leading-order model is limited to small enough values of the elasticity parameter and of the Boussinesq numbers, a higher-order parabolic model is able to accurately capture the linearised behaviour for the whole range of values of the four control parameters.

JFM classification

Interfacial Flows (free surface): Capillary flows Instability: Instability Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 877 - 901
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Copyright
© 2018 Cambridge University Press 

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References

Ambravaneswaran, Bala & Basaran, O. A. 1999 Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching liquid bridges. Phys. Fluids 11 (5), 997–1015.CrossRefGoogle Scholar
Aris, R. 1962 Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice Hall.Google Scholar
Boussinesq, J. V. 1913 Sur l’existence d’une viscosité superficielle, dans la mince couche de transition séparant un liquide d’un autre fluide contigu. J. Ann. Chim. Phys. 29, 349–357.Google Scholar
Campana, D. M. & Saita, F. A. 2006 Numerical analysis of the Rayleigh instability in capillary tubes: the influence of surfactant solubility. Phys. Fluids 18, 022104.CrossRefGoogle Scholar
Campana, D. M., Ubal, S., Giavedoni, M. D. & Saita, F. A. 2011 A deeper insight into the dip coating process in the presence of insoluble surfactants: a numerical analysis. Phys. Fluids 23 (5), 052102.CrossRefGoogle Scholar
Carroll, B. J. & Lucassen, J. 1974 Effect of surface dynamics on the process of droplet formation from supported and free liquid cylinders. J. Chem. Soc. Faraday Trans. 70, 1228–1239.CrossRefGoogle Scholar
Champougny, L., Scheid, B., Restagno, F., Vermant, J. & Rio, E. 2015 Surfactant-induced rigidity of interfaces: a unified approach to free and dip-coated films. Soft Matt. 11 (14), 2758–2770.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.Google Scholar
Craster, R. V., Matar, O. K. & Papageorgiou, D. T. 2002 Pinchoff and satellite formation in surfactant covered viscous threads. Phys. Fluids 14 (4), 1364–1376.CrossRefGoogle Scholar
Craster, R. V., Matar, O. K. & Papageorgiou, D. T. 2009 Breakup of surfactant-laden jets above the critical micelle concentration. J. Fluid Mech. 629, 195–219.Google Scholar
Dravid, V., Songsermpong, S., Xue, Z., Corvalan, C. M. & Sojka, P. E. 2006 Two-dimensional modeling of the effects of insoluble surfactant on the breakup of a liquid filament. Chem. Engng Sci. 61, 3577–3585.Google Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458.CrossRefGoogle ScholarPubMed
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–222.CrossRefGoogle Scholar
García, F. J. & Castellanos, A. 1994 One-dimensional models for slender axisymmetric viscous liquid jets. Phys. Fluids 6 (8), 2676–2689.Google Scholar
Hajiloo, A., Ramamohan, T. R. & Slattery, J. C. 1987 Effect of interfacial viscosities on the stability of a liquid thread. J. Colloid Interface Sci. 117 (2), 384–393.CrossRefGoogle Scholar
Hameed, M., Siegel, M., Young, Y.-N., Li, J., Booty, M. R. & Papageorgiou, D. T. 2008 Influence of insoluble surfactant on the deformation and breakup of a bubble or thread in a viscous fluid. J. Fluid Mech. 594, 307–340.Google Scholar
Hansen, S., Peters, G. W. M. & Meijer, H. E. H. 1999 The effect of surfactant on the stability of a fluid filament embedded in a viscous fluid. J. Fluid Mech. 382, 331–349.Google Scholar
Joye, J.-L., Hirasaki, G. J. & Miller, C. A. 1994 Asymmetric drainage in foam films. Langmuir 10, 3174–3179.CrossRefGoogle Scholar
Kwak, S. & Pozrikidis, C. 2001 Effect of surfactants on the stability of instability of a liquid thread or annular layer. Part 1. Quiescent fluids. Intl J. Multiphase Flow 27, 1–37.CrossRefGoogle Scholar
Langevin, D. 2014 Rheology of adsorbed surfactant monolayers at fluid surfaces. Annu. Rev. Fluid Mech. 46, 47–65.CrossRefGoogle Scholar
Liao, Y. C., Franses, E. I. & Basaran, O. A. 2006 Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer. Phys. Fluids 18 (2), 022101.CrossRefGoogle Scholar
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 (1), 69–79.CrossRefGoogle Scholar
Pereira, A. & Kalliadasis, S. 2008 On the transport equation for an interfacial quantity. Eur. Phys. J. Appl. Phys. 44 (2), 211–214.CrossRefGoogle Scholar
Plateau, J. 1873 Statique Expérimentale et Théorique des Liquides. Gauthier-Villars.Google Scholar
Ponce-Torres, A., Herrada, M. A., Montanero, J. M. & Vega, J. M. 2016a Linear and nonlinear dynamics of an insoluble surfactant-laden liquid bridge. Phys. Fluids 28, 112103.CrossRefGoogle Scholar
Ponce-Torres, A., Montanero, J. M., Herrada, M. A., Vega, E. J. & Vega, J. M. 2017 Influence of the surface viscosity on the breakup of a surfactant-laden drop. Phys. Rev. Lett. 118, 024501.CrossRefGoogle ScholarPubMed
Ponce-Torres, A., Vega, E. J. & Montanero, J. M. 2016b Effect of surface-active impurities on the liquid bridge dynamics. Exp. Fluids 57, 67.Google Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. R. Soc. Lond. 10, 4–13.Google Scholar
Rayleigh, Lord 1892 On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. J. Sci. 34, 145–154.CrossRefGoogle Scholar
Rivero-Rodríguez, J. & Scheid, B. 2018 Bubble dynamics in microchannels: inertial and capillary migration forces. J. Fluid Mech. 842, 215–247.Google Scholar
Roché, M., Aytouna, M., Bonn, D. & Kellay, H. 2009 Effect of surface tension variations on the pinch-off behavior of small fluid drops in the presence of surfactants. Phys. Rev. Lett. 103 (26), 264501.Google Scholar
Rodríguez-Rodríguez, J., Sevilla, A., Martínez-Bazán, C. & Gordillo, J. M. 2015 Generation of microbubbles with applications to industry and medicine. Annu. Rev. Fluid Mech. 47, 405–429.CrossRefGoogle Scholar
Scheid, B., Delacotte, J., Dollet, B., Rio, E., Restagno, F., van Nierop, E. A., Cantat, I., Langevin, D. & Stone, H. A. 2010 The role of surface rheology on liquid film formation. Europhys. Lett. 90, 24002.Google Scholar
Scriven, L. E. 1960 Dynamics of a fluid interface. Equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12 (2), 98–108.CrossRefGoogle Scholar
Scriven, L. E. & Sternling, C. V. 1960 The Marangoni effects. Nature 187, 186–188.Google Scholar
Seiwert, J., Dollet, B. & Cantat, I. 2014 Theoretical study of the generation of soap films: role of interfacial visco-elasticity. J. Fluid Mech. 739, 124–142.CrossRefGoogle Scholar
Slattery, J. C., Sagis, L. & Oh, E. S. 2007 Interfacial Transport Phenomena. Springer.Google Scholar
Stone, HA 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. & Leal, L. G. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186.Google Scholar
Timmermans, M.-L. & Lister, J. R. 2002 The effect of surfactant on the stability of a liquid thread. J. Fluid Mech. 459, 289–306.CrossRefGoogle Scholar
Ting, L., Wasan, D. T., Miyano, K. & Xu, S. Q. 1984 Longitudinal surface waves for the study of dynamic properties of surfactant systems. Part II. Air–solution interface. J. Colloid Interface Sci. 102 (1), 248–259.Google Scholar
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. 150, 322–337.Google Scholar
Tricot, Y. M. 1997 Surfactants: static and dynamic surface tension. In Liquid Film Coating: Scientific Principles and their Technological Implications (ed. Schweizer, P. M. & Kistler, S. F.), pp. 99–136. Springer.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.CrossRefGoogle ScholarPubMed
Van Golde, L. M., Batenburg, J. J. & Robertson, B. 1988 The pulmonary surfactant system: biochemical aspects and functional significance. Phys. Rev. 68 (2), 374–455.Google Scholar
Whitaker, S. 1976 Studies of the drop-weight method for surfactant solutions. Part III. Drop stability, the effect of surfactants on the stability of a column of liquid. J. Colloid Interface Sci. 54 (2), 231–248.CrossRefGoogle Scholar
Wong, H., Rumschitzki, D. & Maldarelli, C. 1996 On the surfactant mass balance at a deforming fluid interface. Phys. Fluids 8 (11), 3203–3204.Google Scholar