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Natural break-up and satellite formation regimes of surfactant-laden liquid threads

Published online by Cambridge University Press:  26 November 2019

A. Martínez-Calvo Open the ORCID record for A. Martínez-Calvo [Opens in a new window] ,
J. Rivero-Rodríguez Open the ORCID record for J. Rivero-Rodríguez [Opens in a new window] ,
B. Scheid Open the ORCID record for B. Scheid [Opens in a new window]  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
J. Rivero-Rodríguez
Affiliation:
TIPs, Université Libre de Bruxelles, CP 165/67, Avenue F. D. Roosevelt 50, 1050Bruxelles, Belgium
B. Scheid
Affiliation:
TIPs, Université Libre de Bruxelles, CP 165/67, Avenue F. D. Roosevelt 50, 1050Bruxelles, Belgium
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: amcalvo@ing.uc3m.es
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Abstract

We report a numerical analysis of the unforced break-up of free cylindrical threads of viscous Newtonian liquid whose interface is coated with insoluble surfactants, focusing on the formation of satellite droplets. The initial conditions are harmonic disturbances of the cylindrical shape with a small amplitude $\unicode[STIX]{x1D716}$, and whose wavelength is the most unstable one deduced from linear stability theory. We demonstrate that, in the limit $\unicode[STIX]{x1D716}\rightarrow 0$, the problem depends on two dimensionless parameters, namely the Laplace number, $La=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}_{0}\bar{R}/\unicode[STIX]{x1D707}^{2}$, and the elasticity parameter, $\unicode[STIX]{x1D6FD}=E/\unicode[STIX]{x1D70E}_{0}$, where $\unicode[STIX]{x1D70C}$, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70E}_{0}$ are the liquid density, viscosity and initial surface tension, respectively, $E$ is the Gibbs elasticity and $\bar{R}$ is the unperturbed thread radius. A parametric study is presented to quantify the influence of $La$ and $\unicode[STIX]{x1D6FD}$ on two key quantities: the satellite droplet volume and the mass of surfactant trapped at the satellite’s surface just prior to pinch-off, $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$, respectively. We identify a weak-elasticity regime, $\unicode[STIX]{x1D6FD}\lesssim 0.05$, in which the satellite volume and the associated mass of surfactant obey the scaling law $V_{sat}=\unicode[STIX]{x1D6F4}_{sat}=0.0042La^{1.64}$ for $La\lesssim 2$. For $La\gtrsim 10$, $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$ reach a plateau of about $3\,\%$ and $2.9\,\%$, respectively, $V_{sat}$ being in close agreement with previous experiments of low-viscosity threads with clean interfaces. For $La<7.5$, we reveal the existence of a discontinuous transition in $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$ at a critical elasticity, $\unicode[STIX]{x1D6FD}_{c}(La)$, with $\unicode[STIX]{x1D6FD}_{c}\rightarrow 0.98$ for $La\lesssim 0.2$, such that $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$ abruptly increase at $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{c}$ for increasing $\unicode[STIX]{x1D6FD}$. The jumps experienced by both quantities reach a plateau when $La\lesssim 0.2$, while they decrease monotonically as $La$ increases up to $La=7.5$, where both become zero.

JFM classification

Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Breakup/coalescence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A35
Copyright
© 2019 Cambridge University Press

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References

Ambravaneswaran, B. & 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
Ambravaneswaran, B., Subramani, H. J., Phillips, S. D. & Basaran, O. A. 2004 Dripping-jetting transitions in a dripping faucet. Phys. Rev. Lett. 93, 034501.CrossRefGoogle Scholar
Ambravaneswaran, B., Wilkes, E. D. & Basaran, O. A. 2002 Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. Phys. Fluids 14 (8), 2606–2621.CrossRefGoogle Scholar
Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48, 285–309.CrossRefGoogle Scholar
Anthony, C. R., Kamat, P. M., Harris, M. T. & Basaran, O. A. 2019 Dynamics of contracting filaments. Phys. Rev. Fluids 4 (9), 093601.CrossRefGoogle Scholar
Ashgriz, N. & Mashayek, F. 1995 Temporal analysis of capillary jet breakup. J. Fluid Mech. 291, 163–190.CrossRefGoogle Scholar
Bogy, D. B. 1979 Drop formation in a circular liquid jet. Annu. Rev. Fluid Mech. 11, 207–228.CrossRefGoogle Scholar
Boussinesq, J. V. 1913 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
Castrejon-Pita, A. A., Castrejon-Pita, J. R. & Hutchings, I. M. 2012 Breakup of liquid filaments. Phys. Rev. Lett. 108 (7), 074506.CrossRefGoogle ScholarPubMed
Castrejón-Pita, J. R., Castrejón-Pita, A. A., Thete, S. S., Sambath, K., Hutchings, I. M., Hinch, J., Lister, J. R. & Basaran, O. A. 2015 Plethora of transitions during breakup of liquid filaments. Proc. Natl Acad. Sci. USA 112 (15), 4582–4587.CrossRefGoogle ScholarPubMed
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, ed and Transl. E MacCurdy. George Brazillier.Google Scholar
Chaudhary, K. C. & Maxworthy, T. 1980 The nonlinear capillary instability of a liquid jet. Part 3. Experiments on satellite drop formation and control. J. Fluid. Mech. 96 (2), 287–297.CrossRefGoogle Scholar
Christopher, G. F. & Anna, S. L. 2007 Microfluidic methods for generating continuous droplet streams. J. Phys. D: Appl. Phys. 40, R319–R336.CrossRefGoogle 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.CrossRefGoogle Scholar
Day, R. F., Hinch, E. J. & Lister, J. R. 1998 Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80 (4), 704.CrossRefGoogle Scholar
Delacotte, J., Montel, L., Restagno, F., Scheid, B., Dollet, B., Stone, H. A., Langevin, D. & Rio, E. 2012 Plate coating: influence of concentrated surfactants on the film thickness. Langmuir 28 (8), 3821–3830.CrossRefGoogle ScholarPubMed
Derby, B. 2010 Inkjet printing of functional and structural materials: Fluid property requirements, feature stability, and resolution. Annu. Rev. Mater. Res. 40, 395–414.CrossRefGoogle Scholar
Donnelly, R. J. & Glaberson, W. I. 1966 Experiments on the capillary instability of a liquid jet. Proc. R. Soc. Lond. A 290, 547–566.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.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3d axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458.CrossRefGoogle ScholarPubMed
Eggers, J 1997 Nonlinear dynamics and breakup of free surface flows. Rev. Mod. Phys. 69, 865–929.CrossRefGoogle Scholar
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
Eggers, J. & Fontelos, M. A. 2005 Isolated inertialess drops cannot break up. J. Fluid Mech. 530, 177–180.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation, vol. 53. Cambridge University Press.CrossRefGoogle Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
Fuller, G. G. & Vermant, J. 2012 Complex fluid-fluid interfaces: rheology and structure. Ann. Rev. Chem. Biol. Engng 3, 519–543.CrossRefGoogle ScholarPubMed
García, F. J. & Castellanos, A. 1994 One-dimensional models for slender axisymmetric viscous liquid jets. Phys. Fluids 6 (8), 2676–2689.CrossRefGoogle Scholar
Goedde, E. F. & Yuen, M. C. 1970 Experiments on liquid jet instability. J. Fluid Mech. 40 (3), 495–511.CrossRefGoogle Scholar
González, H. & García, F. J. 2009 The measurement of growth rates in capillary jets. J. Fluid Mech. 619, 179–212.CrossRefGoogle 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.CrossRefGoogle Scholar
Kalaaji, A., Lopez, B., Attane, P. & Soucemarianadin, A. 2003 Breakup length of forced liquid jets. Phys. Fluids 15, 2469–2479.CrossRefGoogle Scholar
Kamat, P. M., Wagoner, B. W., Thete, S. S. & Basaran, O. A. 2018 Role of marangoni stress during breakup of surfactant-covered liquid threads: Reduced rates of thinning and microthread cascades. Phys. Rev. Fluids 3 (4), 043602.CrossRefGoogle Scholar
Karapetsas, G. & Bontozoglou, V. 2013 The primary instability of falling films in the presence of soluble surfactants. J. Fluid Mech. 729, 123–150.CrossRefGoogle Scholar
Keller, J. B., Rubinow, S. I. & Tu, Y. O. 1973 Spatial instability of a jet. Phys. Fluids 16, 2052–2055.CrossRefGoogle Scholar
Keller, J. B. & Miksis, M. J 1983 Surface tension driven flows. SIAM J. Appl. Maths 43 (2), 268–277.CrossRefGoogle Scholar
Kovalchuk, N. M., Jenkinson, H., Miller, R. & Simmons, M. J. H. 2018 Effect of soluble surfactants on pinch-off of moderately viscous drops and satellite size. J. Colloid Interface Sci. 516, 182–191.CrossRefGoogle ScholarPubMed
Kowalewski, T. A. 1996 On the separation of droplets from a liquid jet. Fluid Dyn. Res. 17, 121–145.CrossRefGoogle Scholar
Lafrance, P. 1975 Nonlinear breakup of a laminar liquid jet. Phys. Fluids 18 (4), 428–432.CrossRefGoogle Scholar
Langevin, D. 2014 Rheology of adsorbed surfactant monolayers at fluid surfaces. Annu. Rev. Fluid Mech. 46, 47–65.CrossRefGoogle Scholar
Lee, H. C. 1974 Drop formation in a liquid jet. IBM J. Res. Dev. 18 (4), 364–369.CrossRefGoogle Scholar
Leib, S. J. & Goldstein, M. E. 1986a Convective and absolute instability of a viscous liquid jet. Phys. Fluids 29 (4), 952–954.CrossRefGoogle Scholar
Leib, S. J. & Goldstein, M. E. 1986b The generation of capillary instabilities on a liquid jet. J. Fluid Mech. 168, 479–500.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
Magnus, G. 1859 Hydraulische undersuchungen. Ann. Phys. Chem. 106, 1.CrossRefGoogle Scholar
Mansour, N. N. & Lundgren, T. S. 1990 Satellite formation in capillary jet breakup. Phys. Fluids A: Fluid Dyn. 2 (7), 1141–1144.CrossRefGoogle Scholar
Martínez-Calvo, A., Rubio-Rubio, M. & Sevilla, A. 2018 The nonlinear states of viscous capillary jets confined in the axial direction. J. Fluid Mech. 834, 335–358.CrossRefGoogle Scholar
Martínez-Calvo, A. & Sevilla, A. 2018 Temporal stability of free liquid threads with surface viscoelasticity. J. Fluid Mech. 846, 877–901.CrossRefGoogle Scholar
Mashayek, F. & Ashgriz, N. 1995 Nonlinear instability of liquid jets with thermocapillarity. J. Fluid Mech. 283, 97–123.CrossRefGoogle Scholar
McGough, P. T. & Basaran, O. A. 2006 Repeated formation of fluid threads in breakup of a surfactant-covered jet. Phys. Rev. Lett. 96 (5), 054502.CrossRefGoogle ScholarPubMed
Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid filament. J. Fluid Mech. 512, 223–256.CrossRefGoogle Scholar
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7 (7), 1529–1544.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 et Cie.Google 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
Rayleigh, W. S. 1878 On the instability of jets. Proc. R. Soc. Lond. 10, 4–13.Google Scholar
Rayleigh, W. S. 1882 Further observations upon liquid jets, in continuation of those recorded in the royal society’s ‘proceedings’ for march and may. Proc. R. Soc. Lond. 130–145.Google Scholar
Lord Rayleigh Sec., R. S. 1892 XVI. On the instability of a cylinder of viscous liquid under capillary force. Lond. Edinb. Dublin Phil. Mag. J. Sci. 34 (207), 145–154.Google Scholar
Rivero-Rodríguez, J. & Scheid, B. 2018a Bubble dynamics in microchannels: inertial and capillary migration forces. J. Fluid Mech. 842, 215–247.CrossRefGoogle Scholar
Rivero-Rodríguez, J. & Scheid, B. 2018b Bubble dynamics in microchannels: inertial and capillary migration forces – CORRIGENDUM. J. Fluid Mech. 855, 1242–1245.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Rubio-Rubio, M., Sevilla, A. & Gordillo, J. M. 2013 On the thinnest steady threads obtained by gravitational stretching of capillary jets. J. Fluid Mech. 729, 471–483.CrossRefGoogle Scholar
Rutland, D. F. & Jameson, G. J. 1970 Theoretical prediction of the sizes of drops formed in the breakup of capillary jets. Chem. Engng Sci. 25 (11), 1689–1698.CrossRefGoogle Scholar
Rutland, D. F. & Jameson, G. J. 1971 A non-linear effect in the capillary instability of liquid jets. J. Fluid Mech. 46 (2), 267–271.CrossRefGoogle Scholar
Savart, F. 1833 Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi. Ann. Chim. 53, 337–386.Google 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. EPL 90, 24002.CrossRefGoogle 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
Siderius, A., Kehl, S. K. & Leaist, D. G. 2002 Surfactant diffusion near critical micelle concentrations. J. Sol. Chem. 31 (8), 607–625.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
Subramani, H. J., Yeoh, H. K., Suryo, R., Xu, Q., Ambravaneswaran, B. & Basaran, O. A. 2006 Simplicity and complexity in a dripping faucet. Phys. Fluids 18 (3), 032106.CrossRefGoogle 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
Wang, F., Contò, F. P., Naz, N., Castrejón-Pita, J. R., Castrejón-Pita, A. A., Bailey, C. G., Wang, W., Feng, J. J. & Sui, Y. 2019 A fate-alternating transitional regime in contracting liquid filaments. J. Fluid Mech. 860, 640–653.CrossRefGoogle Scholar
Whitaker, S. 1976 Studies of the drop-weight method for surfactant solutions III. Drop stability, the effect of surfactants on the stability of a column of liquid. J. Colloid Interf. 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.CrossRefGoogle Scholar
Xu, Q., Liao, Y.-C. & Basaran, O. A. 2007 Can surfactant be present at pinch-off of a liquid filament? Phys. Rev. Lett. 98 (5), 054503.CrossRefGoogle ScholarPubMed
Yildirim, O. E., Xu, Q. & Basaran, O. A. 2005 Analysis of the drop weight method. Phys. Fluids 17, 062107.CrossRefGoogle Scholar
Yuen, M.-C. 1968 Non-linear capillary instability of a liquid jet. J. Fluid Mech. 33 (1), 151–163.CrossRefGoogle Scholar