Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T07:19:58.284Z Has data issue: false hasContentIssue false
  • English
  • Français

Level-set simulations of a 2D topological rearrangement in a bubble assembly: effects of surfactant properties

Published online by Cambridge University Press:  12 January 2018

A. Titta ,
M. Le Merrer Open the ORCID record for M. Le Merrer [Opens in a new window] ,
F. Detcheverry ,
P. D. M. Spelt  and
A.-L. Biance
A. Titta
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, 69622 Villeurbanne, France Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509, Université de Lyon, 69134 Écully, France
M. Le Merrer*
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, 69622 Villeurbanne, France
F. Detcheverry
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, 69622 Villeurbanne, France
P. D. M. Spelt
Affiliation:
Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509, Université de Lyon, 69134 Écully, France
A.-L. Biance
Affiliation:
Université de Lyon, Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, 69622 Villeurbanne, France
*
†Email address for correspondence: marie.le-merrer@univ-lyon1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A liquid foam is a dispersion of gas bubbles in a liquid matrix containing surface-active agents. Its flow involves the relative motion of bubbles, which switch neighbours during a so-called topological rearrangement of type 1 (T1). The dynamics of T1 events, as well as foam rheology, have been extensively studied, and experimental results point to the key role played by surfactants in these processes. However, the complex and multiscale nature of the system has so far impeded a complete understanding of the mechanisms involved. In this work, we investigate numerically the effect of surfactants on the rheological response of a 2D sheared bubble cluster. To do so, a level-set method previously employed for simulation of two-phase flow has been extended to include the effects of surfactants. The dynamical processes of the surfactants – diffusion in the liquid and along the interface, adsorption/desorption at the interface – and their coupling with the flow – surfactant advection and Laplace and Marangoni stresses at the interface – are all taken into account explicitly. Through a systematic study of the Biot, capillary and Péclet numbers that characterise the surfactant properties in the simulation, we find that the presence of surfactants can affect the liquid/gas hydrodynamic boundary condition (from a rigid-like situation to a mobile one), which modifies the nature of the flow in the volume from a purely extensional situation to a shear. Furthermore, the work done by surface tension (the 2D analogue of the work by pressure forces), resulting from surfactant and interface dynamics, can be interpreted as an effective dissipation, which reaches a maximum for a Péclet number of order unity. Our results, obtained at high liquid fraction, should provide a reference point, with which experiments and models of T1 dynamics and foam rheology can be compared.

JFM classification

Drops and Bubbles: Bubble dynamics Complex Fluids: Foams Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 222 - 247
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aussillous, P. & Quéré, D. 2002 Bubbles creeping in a viscous liquid along a slightly inclined plane. Europhys. Lett. 59, 370–376.Google Scholar
Bel Fdhila, R. & Duineveld, P. C. 1996 The effect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet numbers. Phys. Fluids 8 (2), 310–321.Google Scholar
Besson, S., Debregeas, G., Cohen-Addad, S. & Höhler, R. 2008 Dissipation in a sheared foam: from bubble adhesion to foam rheology. Phys. Rev. Lett. 101 (21), 214504.CrossRefGoogle Scholar
Biance, A.-L., Cohen-Addad, S. & Höhler, R. 2009 Topological transition dynamics in a strained bubble cluster. Soft Matt. 5 (23), 4672–4679.CrossRefGoogle Scholar
Biance, A.-L., Delbos, A. & Pitois, O. 2011 How topological rearrangements and liquid fraction control liquid foam stability. Phys. Rev. Lett. 106, 068301.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10 (2), 166–188.Google Scholar
Breward, C. J. W. & Howell, P. D. 2002 The drainage of a foam lamella. J. Fluid Mech. 458, 379–406.Google Scholar
Buzza, D. M. A., Lu, C.-Y. D. & Cates, M. E. 1995 Linear shear rheology of incompressible foams. J. Physique II 5 (1), 37–52.Google Scholar
Cantat, I. 2011 Gibbs elasticity effect in foam shear flows: a non quasi-static 2D numerical simulation. Soft Matt. 7, 448–455.CrossRefGoogle Scholar
Cantat, I. 2013 Liquid meniscus friction on a wet plate: bubbles, lamellae, and foams. Phys. Fluids 25 (3), 1–21.CrossRefGoogle Scholar
Cantat, I., Cohen-Addad, S., Elias, F., Graner, F., Höhler, R., Pitois, O., Rouyer, F. & Saint-Jalmes, A. 2013 Foams: Structure and Dynamics. Oxford University Press.Google 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
Cohen-Addad, S., Höhler, R. & Pitois, O. 2013 Flow in foams and flowing foams. Annu. Rev. Fluid Mech. 45 (1), 241–267.Google Scholar
Costa, S., Höhler, R. & Cohen-Addad, S. 2013 The coupling between foam viscoelasticity and interfacial rheology. Soft Matt. 9 (4), 1100–1112.Google Scholar
Cuenot, B., Magnaudet, J. & Spennato, B. 1997 The effects of slightly soluble surfactants on the flow around a spherical bubble. J. Fluid Mech. 339, 25–53.Google Scholar
Dangla, R.2012 2D droplet microfluidics driven by confinement gradients. PhD thesis, Ecole Polytechnique.Google Scholar
Denkov, N. D., Subramanian, V., Gurovich, D. & Lips, A. 2005 Wall slip and viscous dissipation in sheared foams: effect of surface mobility. Colloids Surf. A 263, 129–145.Google Scholar
Denkov, N. D., Tcholakova, S., Golemanov, K., Ananthapadmanabhan, K. P. & Lips, A. 2008 Viscous friction in foams and concentrated emulsions under steady shear. Phys. Rev. Lett. 100 (13), 138301.Google Scholar
Denkov, N. D., Tcholakova, S., Golemanov, K., Subramanian, V. & Lips, A. 2006 Foam wall friction: effect of air volume fraction for tangentially immobile bubble surface. Colloids Surf. A 282, 329–347.CrossRefGoogle Scholar
Dieter-Kissling, K., Marschall, H. & Bothe, D. 2015 Direct numerical simulation of droplet formation processes under the influence of multiple surfactants. Comput. Fluids 113, 93–105.Google Scholar
Durand, M., Martinoty, G. & Langevin, D. 1999 Liquid flow through aqueous foams: from the Plateau border-dominated regime to the node-dominated regime. Phys. Rev. E 60 (6), R6307–R6308.Google Scholar
Durand, M. & Stone, H. A. 2006 Relaxation time of the topological T1 process in a two-dimensional foam. Phys. Rev. Lett. 97 (22), 226101.CrossRefGoogle Scholar
Durian, D. J. 1995 Foam mechanics at the bubble scale. Phys. Rev. Lett. 75, 4780.Google Scholar
Durian, D. J. 1997 Bubble-scale model of foam mechanics: melting, nonlinear behavior, and avalanches. Phys. Rev. E 55, 1739–1751.Google Scholar
Edwards, D., Brenner, H. & Wasan, D. T. 1991 Interfacial Transport Processes and Rheology. Butterworth-Heinemann.Google Scholar
Eggleton, C. D., Tsai, T. M. & Stebe, K. J. 2001 Tip streaming from a drop in the presence of surfactants. Phys. Rev. Lett. 87 (4), 048302.CrossRefGoogle ScholarPubMed
Erni, P. 2011 Deformation modes of complex fluid interfaces. Soft Matt. 7 (17), 7586–7600.Google Scholar
Golemanov, K., Denkov, N. D., Tcholakova, S., Vethamuthu, M. & Lips, A. 2008 Surfactant mixtures for control of bubble surface mobility in foam studies. Langmuir 24, 9956–9961.Google Scholar
Hodges, S. R., Jensen, O. E. & Rallison, J. M. 2004 Sliding, slipping and rolling: the sedimentation of a viscous drop down a gently inclined plane. J. Fluid Mech. 512, 95–131.Google Scholar
Höhler, R. & Cohen-Addad, S. 2005 Rheology of liquid foam. J. Phys.: Condens. Matter 17 (41), R1041–R1069.Google Scholar
Hutzler, S., Saadatfar, M., van der Net, A., Weaire, D. & Cox, S. J. 2008 The dynamics of a topological change in a system of soap films. Colloids Surf. A 323 (1–3), 123–131.Google Scholar
Israelachvili, J. 2010 Intermolecular and Surface Forces, 3rd edn. Academic Press.Google Scholar
Kern, N., Weaire, D, Martin, A., Hutzler, S. & Cox, S. J. 2004 Two-dimensional viscous froth model for foam dynamics. Phys. Rev. E 70 (4), 041411.Google Scholar
Kraynik, A. M., Reinelt, D. A. & Princen, H. M. 1991 The nonlinear elastic behavior of polydisperse hexagonal foams and concentrated emulsions. J. Rheol. 35 (6), 1235–1253.Google Scholar
Krishan, K., Helal, A., Höhler, R. & Cohen-Addad, S. 2010 Fast relaxations in foam. Phys. Rev. E 82 (1), 011405.Google Scholar
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42.Google Scholar
Langevin, D. 2014 Rheology of adsorbed surfactant monolayers at fluid surfaces. Annu. Rev. Fluid Mech. 46, 47–65.Google Scholar
Le Merrer, M., Cohen-Addad, S. & Höhler, R. 2012 Bubble rearrangement duration in foams near the jamming point. Phys. Rev. Lett. 108 (18), 188301.Google Scholar
Le Merrer, M., Cohen-Addad, S. & Höhler, R. 2013 Duration of bubble rearrangements in a coarsening foam probed by time-resolved diffusing-wave spectroscopy: impact of interfacial rigidity. Phys. Rev. E 88 (2), 022303.Google Scholar
Lorenceau, E., Louvet, N., Rouyer, F. & Pitois, O. 2009 Permeability of aqueous foams. Eur. Phys. J. E 28, 293–304.Google ScholarPubMed
Lucassen, J. & van den Tempel, M. 1972 Dynamic measurements of dilational properties of a liquid interface. Chem. Engng Sci. 27 (6), 1283–1291.Google Scholar
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37 (1), 239–261.Google Scholar
Ó Náraigh, L., Valluri, P., Scott, D. M., Bethune, I. & Spelt, P. D. M. 2014 Linear instability, nonlinear instability and ligament dynamics in three-dimensional laminar two-layer liquid–liquid flows. J. Fluid Mech. 750, 464–506.Google Scholar
Osher, S. & Fedkiw, R. 2003 Level Set Methods and Dynamic Implicit Surfaces. Springer.Google Scholar
Ou Ramdane, O. & Quéré, D. 1997 Thickening factor in Marangoni coating. Langmuir 13 (11), 2911–2916.Google Scholar
Park, C.-W. 1991 Effects of insoluble surfactants on dip coating. J. Colloid Interface Sci. 146 (2), 382–394.Google Scholar
Pereira, A., Trevelyan, P. M. J., Thiele, U. & Kalliadasis, S. 2007 Dynamics of a horizontal thin liquid film in the presence of reactive surfactants. Phys. Fluids 19, 112102.CrossRefGoogle Scholar
Petit, P., Seiwert, J., Cantat, I. & Biance, A.-L. 2015 On the generation of a foam film during a topological rearrangement. J. Fluid Mech. 763, 286–301.Google Scholar
Pozrikidis, C. 2001 Interfacial dynamics for Stokes flow. J. Comput. Phys. 169 (2), 250–301.CrossRefGoogle Scholar
Pozrikidis, C. 2011 Introduction to Theoretical and Computational Fluid Dynamics, 2nd edn. Oxford University Press.Google Scholar
Princen, H. M. 1983 Rheology of foams and highly concentrated emulsions. Part 1. Elastic properties and yield stress of a cylindrical model system. J. Colloid Interface Sci. 91 (1), 160–175.Google Scholar
Princen, H. M. 1985 Rheology of foams and highly concentrated emulsions. Part 2. Experimental study of the yield stress and wall effects for concentrated oil-in-water emulsions. J. Colloid Interface Sci. 105 (1), 150–171.Google Scholar
Princen, H. M. & Kiss, A. D. 1986 Rheology of foams and highly concentrated emulsions. Part 3. Static shear modulus. J. Colloid Interface Sci. 112 (2), 427–437.Google Scholar
Princen, H. M. & Kiss, A. D. 1989 Rheology of foams and highly concentrated emulsions. Part 4. An experimental study of the shear viscosity and yield stress of concentrated emulsions. J. Colloid Interface Sci. 128 (1), 176–187.Google Scholar
Ratulowski, J. & Chang, H. C. 1990 Marangoni effects of trace impurities on the motion of long gas bubbles in capillaries. J. Fluid Mech. 210 (1), 303–328.Google Scholar
Rio, E. & Biance, A.-L. 2014 Thermodynamic and mechanical timescales involved in foam film rupture and liquid foam coalescence. Chem. Phys. Chem. 15 (17), 3692–3707.Google Scholar
Rognon, P., Einav, I. & Gay, C. 2010 Internal relaxation time in immersed particulate materials. Phys. Rev. E 81, 061304.Google Scholar
Sagis, L. M. C. 2011 Dynamic properties of interfaces in soft matter: experiments and theory. Rev. Mod. Phys. 83 (4), 1367–1403.Google Scholar
Satomi, R., Grassia, P. & Oguey, C. 2013 Modelling relaxation following T1 transformations of foams incorporating surfactant mass transfer by the Marangoni effect. Colloids Surf. A 438, 77–84.Google Scholar
Saye, R. I. & Sethian, J. A. 2013 Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams. Science 340 (6133), 720–724.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 in liquid film formation. Europhys. Lett. 90 (2), 24002.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), 7797–7804.CrossRefGoogle Scholar
Seiwert, J., Monloubou, M., Dollet, B. & Cantat, I. 2013 Extension of a suspended soap film: a two-step process. Phys. Rev. Lett. 111, 094501.Google Scholar
Seth, J. R., Mohan, L., Locatelli-Champagne, C., Cloitre, M. & Bonnecaze, R. T. 2011 A micromechanical model to predict the flow of soft particle glasses. Nat. Mater. 10, 838–843.Google Scholar
Sethian, J. A. 1999 Level Set Methods and Fast Marching Methods. Cambridge University Press.Google Scholar
Sexton, M. B., Möbius, M. E. & Hutzler, S. 2011 Bubble dynamics and rheology in sheared two-dimensional foams. Soft Matt. 7 (23), 11252–11258.Google Scholar
Solomenko, Z., Spelt, P. D. M., Ó Náraigh, L. & Alix, P. 2017 Mass conservation and reduction of parasitic interfacial waves in level-set methods for the numerical simulation of two-phase flows: a comparative study. Intl J. Multiphase Flow 95, 235–256.Google Scholar
Sonin, A. A., Bonfillon, A. & Langevin, D. 1993 Role of surface elasticity in the drainage of soap films. Phys. Rev. Lett. 71 (14), 2342–2345.CrossRefGoogle ScholarPubMed
Stone, H. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26 (1), 65–102.Google Scholar
Sussman, M. & Fatemi, E. 1999 An efficient, interface-preserving level set redistancing application to interfacial incompressible fluid flow. SIAM J. Sci. Comput. 20, 1165–1191.Google Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114 (1), 146–159.Google Scholar
Takagi, S. & Matsumoto, Y. 2011 Surfactant effects on bubble motion and bubbly flows. Annu. Rev. Fluid Mech. 43, 615–636.Google Scholar
Tcholakova, S., Denkov, N. D., Golemanov, K., Ananthapadmanabhan, K. P. & Lips, A. 2008 Theoretical model of viscous friction inside steadily sheared foams and concentrated emulsions. Phys. Rev. E 78, 011405.Google Scholar
Teigen, E. K., Li, X., Lowengrub, J., Wang, F. & Voigt, A. 2009 A diffuse-interface approach for modeling transport, diffusion and adsorption/desorption of material quantities on a deformable interface. Commun. Math. Sci. 7 (4), 1009–1037.Google Scholar
Teigen, E. K., Song, P., Lowengrub, J. & Voigt, A. 2011 A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230 (2), 375–393.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 (2), 708–759.Google 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