Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T11:48:59.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Impact of surfactants on the rise of deformable bubbles and interfacial gas–liquid mass transfer

Published online by Cambridge University Press:  29 August 2023

Kalyani Kentheswaran Open the ORCID record for Kalyani Kentheswaran [Opens in a new window] ,
William Antier ,
Nicolas Dietrich Open the ORCID record for Nicolas Dietrich [Opens in a new window]  and
Benjamin Lalanne Open the ORCID record for Benjamin Lalanne [Opens in a new window]
Kalyani Kentheswaran
Affiliation:
Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France Toulouse Biotechnology Institute, Université de Toulouse, CNRS, INRAE, INSA, Toulouse, France Fédération de recherche FERMaT, CNRS, Toulouse, France
William Antier
Affiliation:
Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France
Nicolas Dietrich
Affiliation:
Toulouse Biotechnology Institute, Université de Toulouse, CNRS, INRAE, INSA, Toulouse, France Fédération de recherche FERMaT, CNRS, Toulouse, France
Benjamin Lalanne*
Affiliation:
Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INPT, UPS, Toulouse, France Fédération de recherche FERMaT, CNRS, Toulouse, France
*
†Email address for correspondence: Benjamin.Lalanne@ensiacet.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Axisymmetric numerical simulations of the hydrodynamics around rising bubbles are performed in order to investigate the impact of surfactants on the bubble dynamics. Surfactants are assumed to be insoluble. The transport of the adsorbed surfactants is computed along the deforming surface at large surface Péclet number, and Marangoni stresses are taken into account. This simulation model leads to the stagnant-cap regime, with partially immobile interfaces. A parametric study is performed on cases at given Archimedes number, by varying the degree of contamination (Marangoni number) but maintaining a nearly constant Eötvös number. The presence of surfactants affects the rise velocity for oblate bubbles less than for spherical bubbles: the increase of the drag coefficient, due to interface contamination, is mitigated by a lower bubble deformation. When the cap angle $\theta _{cap}$ belongs to the southern hemisphere, the aspect ratio $\chi$ is found to decrease with contamination: the dynamic pressure responsible for the bubble distortion is lowered, related to the decline of kinetic energy. As soon as $\theta _{cap}$ lies in the northern hemisphere, the pressure stress causing distortion becomes independent on $\theta _{cap}: \chi$ no longer evolves with contamination, and already matches the prediction for fully immobile interfaces. Mass transfer of a passive scalar across the contaminated interface is also analysed. Surprisingly, the Sherwood number $Sh$ is found to follow the same law as for spherical shapes (Kentheswaran et al., Intl J. Heat Mass Transfer, vol. 198, 2022, 123325), allowing us to predict the decrease in $Sh$ due to contamination. These results reveal the couplings between interface immobilisation, bubble deformation, rise velocity and interfacial mass transfer.

JFM classification

Drops and Bubbles: Bubble dynamics Convection: Marangoni convection Mass Transport: Mass Transport
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 970 , 10 September 2023 , A5
Check for updates
Copyright
© The Author(s), 2023. Published by 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

Abadie, T., et al. 2022 Oxygen transfer of microbubble clouds in aqueous solutions – application to wastewater. Chem. Engng Sci. 257, 117693.CrossRefGoogle Scholar
Aoyama, S., Hayashi, K., Hosokawa, S. & Tomiyama, A. 2016 Shapes of ellipsoidal bubbles in infinite stagnant liquids. Intl J. Multiphase Flow 79, 23–30.CrossRefGoogle Scholar
Aoyama, S., Hayashi, K., Hosokawa, S. & Tomiyama, A. 2018 Shapes of single bubbles in infinite stagnant liquids contaminated with surfactant. Expl Therm. Fluid Sci. 96, 460–469.CrossRefGoogle Scholar
Aslam, T. 2003 A partial differential equation approach to multidimensional extrapolation. J. Comput. Phys. 193, 349–355.CrossRefGoogle Scholar
Atasi, O., Ravisankar, M., Legendre, D. & Zenit, R. 2023 Presence of surfactants controls the stability of bubble chains in carbonated drinks. Phys. Rev. Fluids 8 (5), 053601.CrossRefGoogle Scholar
Bel Fdhila, R. & Duineveld, P.C. 1996 The effect of surfactant on the rise of a spherical bubble at high Reynolds and Péclet numbers. Phys. Fluids 8, 310–321.CrossRefGoogle Scholar
Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 7 (6), 1265–1274.CrossRefGoogle Scholar
Bothe, D. 2022 Sharp-interface continuum thermodynamics of multicomponent fluid systems with interfacial mass. Intl J. Engng Sci. 179, 103731.CrossRefGoogle Scholar
Boussinesq, J. 1905 Calcul du pouvoir refroidissant des courants fluides. J. Math. Pures Appl. 6, 285–332.Google Scholar
Butler, C., Cid, E., Billet, A.-M. & Lalanne, B. 2021 Numerical simulation of mass transfer dynamics in Taylor flows. Intl J. Heat Mass Transfer 179, 121670.CrossRefGoogle Scholar
Chen, J., Hayashi, K., Hosokawa, S. & Tomiyama, A. 2019 Drag correlations of ellipsoidal bubbles in clean and fully contaminated systems. Multiphase Sci. Technol. 31, 215–234.CrossRefGoogle Scholar
Clift, R., Grace, J.R. & Weber, M.E. 1978 Bubbles, Drops and Particles. Academic Press.Google Scholar
Colombet, D., Legendre, D., Cockx, A. & Guiraud, P. 2013 Mass or heat transfer inside a spherical gas bubble at low to moderate Reynolds number. Intl J. Heat Mass Transfer 67, 1096–1105.CrossRefGoogle 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.CrossRefGoogle Scholar
Dalmon, A., Kentheswaran, K., Mialhe, G., Lalanne, B. & Tanguy, S. 2020 Fluids-membrane interaction with a full Eulerian approach based on the level set method. J. Comput. Phys. 406, 109171.CrossRefGoogle Scholar
Dani, A., Cockx, A., Legendre, D. & Guiraud, P. 2021 Effect of bubble interface contamination on gas–liquid mass transfer at intermediate Reynolds numbers: from DNS to Sherwood numbers. Chem. Engng Sci. 1, 116979.Google Scholar
Dani, A., Cockx, A., Legendre, D. & Guiraud, P. 2022 Effect of spheroid bubble interface contamination on gas–liquid mass transfer at intermediate Reynolds numbers: from DNS to Sherwood numbers. Chem. Engng Sci. 248, 116979.CrossRefGoogle Scholar
Dendy, J.E. 1982 Black box multigrid. J. Comput. Phys. 48, 366–386.CrossRefGoogle Scholar
Dijkhuizen, W., Roghair, I., Van Sint Annaland, M. & Kuipers, J.A.M. 2010 DNS of gas bubbles behaviour using an improved 3D front tracking model-drag force on isolated bubbles and comparison with experiments. Chem. Engng Sci. 65, 1415–1426.CrossRefGoogle Scholar
Dollet, B., Marmottant, P. & Garbin, V. 2019 Bubble dynamics in soft and biological matter. Annu. Rev. Fluid Mech. 51, 331–355.CrossRefGoogle Scholar
Duineveld, P.C. 1995 The rise velocity and shape of bubbles in pure water at high Reynolds number. J. Fluid Mech. 292, 325–332.CrossRefGoogle Scholar
Dukhin, S.S., Kovalchuk, V.I., Gochev, G.G., Lotfi, M., Krzan, M., Malysa, K. & Miller, R. 2015 Dynamics of rear stagnant cap formation at the surface of spherical bubbles rising in surfactant solutions at large Reynolds numbers under conditions of small Marangoni number and slow sorption kinetics. Adv. Colloid Interface Sci. 222, 260–274.CrossRefGoogle ScholarPubMed
Favelukis, M. & Hung Ly, C. 2005 Unsteady mass transfer around spheroidal drops in potential flow. Chem. Engng Sci. 60, 7011–7021.CrossRefGoogle Scholar
Fedkiw, R.P., Aslam, T., Merriman, B. & Osher, S. 1999 A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492.CrossRefGoogle Scholar
Figueroa-Espinoza, B. & Legendre, D. 2010 Mass or heat transfer from spheroidal gas bubbles rising through a stationary liquid. Chem. Engng Sci. 65, 6296–6309.CrossRefGoogle Scholar
Frössling, N. 1938 Uber die verdunstung fallender tropfen (the evaporation of falling drops). Gerlands Beitrage zur Geophysik 52, 170–216.Google Scholar
Garner, F.H. & Hale, A.R. 1953 The effect of surface active agents in liquid extraction processes. Chem. Engng Sci. 2, 157–163.CrossRefGoogle Scholar
Gibou, F., Fedkiw, R.P., Cheng, L. & Kang, M. 2002 A second-order-accurate symmetric discretization of the Poisson equation on irregular domains. J. Comput. Phys. 176, 205–227.CrossRefGoogle Scholar
Hessenkemper, H., Ziegenhein, T., Lucas, D. & Tomiyama, A. 2021 Influence of surfactant contaminations on the lift force of ellipsoidal bubbles in water. Intl J. Multiphase Flow 145, 103833.CrossRefGoogle Scholar
Jimenez, M., Dietrich, N., Grace, J.R. & Hebrard, G. 2014 Oxygen mass transfer and hydrodynamic behaviour in wastewater: determination of local impact of surfactants by visualization techniques. Water Res. 58, 111–121.CrossRefGoogle ScholarPubMed
Johnson, T.A. & Patel, V.C. 1999 Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378, 19–70.CrossRefGoogle Scholar
Kang, M., Fedkiw, R. & Liu, X.-D. 2000 A boundary condition capturing method for multiphase incompressible flow. J. Sci. Comput. 15, 323–360.CrossRefGoogle Scholar
Kentheswaran, K., Dietrich, N., Tanguy, S. & Lalanne, B. 2022 Direct numerical simulation of gas–liquid mass transfer around a spherical contaminated bubble in the stagnant-cap regime. Intl J. Heat Mass Transfer 198, 123325.CrossRefGoogle Scholar
Kishore, N. & Gu, S. 2011 Momentum and heat transfer phenomena of spheroid particles at moderate Reynolds and Prandtl numbers. Intl J. Heat Mass Transfer 54, 2595–2601.CrossRefGoogle Scholar
Lakshmanan, P. & Ehrhard, P. 2010 Marangoni effects caused by contaminants adsorbed on bubble surfaces. J. Fluid Mech. 647, 143–161.CrossRefGoogle Scholar
Lalanne, B., Abi Chebel, N., Vejražka, J., Tanguy, S., Masbernat, O. & Risso, F. 2015 a Non-linear shape oscillations of rising drops and bubbles: experiments and simulations. Phys. Fluids 27, 123305.CrossRefGoogle Scholar
Lalanne, B., Masbernat, O. & Risso, F. 2020 Determination of interfacial concentration of a contaminated droplet from shape oscillation damping. Phys. Rev. Lett. 124, 194501.CrossRefGoogle ScholarPubMed
Lalanne, B., Rueda Villegas, L., Tanguy, S. & Risso, F. 2015 b On the computation of viscous terms for incompressible two-phase flows with Level Set/Ghost Fluid Method. J. Comput. Phys. 301, 289–307.CrossRefGoogle Scholar
Lebrun, G., Xu, F., Le Men, C., Hébrard, G. & Dietrich, N. 2021 Gas–liquid mass transfer around a rising bubble : combined effect of rheology and surfactant. Fluids 6, 84.CrossRefGoogle Scholar
Legendre, D. 2007 On the relation between the drag and the vorticity produced on a clean bubble. Phys. Fluids 19, 018102.CrossRefGoogle Scholar
Legendre, D., Zenit, R. & Rodrigo Velez-Cordero, J. 2012 On the deformation of gas bubbles in liquids. Phys. Fluids 24, 043303.CrossRefGoogle Scholar
Lepilliez, M., Popescu, E.-R., Gibou, F. & Tanguy, S. 2016 On two-phase flow solvers in irregular domains with contact line. J. Comput. Phys. 321, 1217–1251.CrossRefGoogle Scholar
Levich, V.G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
Lochiel, A.C. & Calderbank, P.H. 1964 Mass transfer in the continuous phase around axisymmetric bodies of revolution. Chem. Engng Sci. 19, 471–484.CrossRefGoogle Scholar
Loth, E. 2008 Quasi-steady shape and drag deformable bubbles and drops. Intl J. Multiphase Flow 34, 523–546.CrossRefGoogle Scholar
Magnaudet, J. & Mougin, G. 2007 Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311–337.CrossRefGoogle Scholar
Manikantan, H. & Squires, T.M. 2020 Surfactant dynamics: hidden variables controlling fluid flows. J. Fluid Mech. 892, P1.CrossRefGoogle ScholarPubMed
Mei, R., Klausner, J.F. & Lawrence, C.J. 1994 A note on the history force on a spherical bubble at finite Reynolds number. Phys. Fluids 418, 418–420.CrossRefGoogle Scholar
Moore, D.W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.CrossRefGoogle Scholar
Moore, D.W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002 The generalized Kirchoff equations and their application to the interaction between a rigid body and an arbitrary time-dependent viscous flow. Intl J. Multiphase Flow 28, 1837–1851.CrossRefGoogle Scholar
Palaparthi, R., Demetrios, T.P. & Maldarelli, C. 2006 Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles. J. Fluid Mech. 559, 1–44.CrossRefGoogle Scholar
Pereira, A. & Kalliadasis, S. 2008 On the transport equation for an interfacial quantity. Eur. Phys. J. Appl. Phys. 44, 211–214.CrossRefGoogle Scholar
Pesci, C., Weiner, A., Marschall, H. & Bothe, D. 2018 Computational analysis of single rising bubbles influenced by soluble surfactant. J. Fluid Mech. 856, 709–763.CrossRefGoogle Scholar
Piedfert, A., Lalanne, B., Masbernat, O. & Risso, F. 2018 Numerical simulations of a rising drop with shape oscillations in the presence of surfactants. Phys. Rev. Fluids 3, 103605.CrossRefGoogle Scholar
Ranz, W.E. & Marshall, W.R. 1952 Evaporation from drops. Chem. Engng Prog. 48, 141–146.Google Scholar
Rastello, M., Marié, J.-L. & Lance, M. 2011 Drag and lift forces on clean spherical and ellipsoidal bubbles in a solid-body rotating flow. J. Fluid Mech. 682, 434–459.CrossRefGoogle Scholar
Rueda Villegas, L., Tanguy, S., Castanet, G., Caballina, O. & Lemoine, F. 2017 Direct numerical simulation of the impact of a droplet onto a hot surface above the leidenfrost temperature. Intl J. Heat Mass Transfer 104, 1090–1109.CrossRefGoogle Scholar
Ryskin, G. & Leal, L.G. 1984 a Numerical solution of free-boundary problems in fluid mechanics. Part 1. The finite-difference technique. J. Fluid Mech. 148, 1–17.CrossRefGoogle Scholar
Ryskin, G. & Leal, L.G. 1984 b Numerical solution of free-boundary problems in fluid mechanics. Part 2. J. Fluid Mech. 148, 19–35.CrossRefGoogle Scholar
Sadhal, S. & Johnson, S.E. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant filmexact solution. J. Fluid Mech. 126, 237–250.CrossRefGoogle Scholar
Schiller, L. & Naumann, A. 1935 A drag coefficient correlation. Z. Verein. Deutsch. Ing. 77, 318–320.Google Scholar
Sharaf, D.M., Premlata, A.R., Tripathi, M.K., Karri, B. & Sahu, K.C. 2017 Shapes and paths of an air bubble rising in quiescent liquids. Phys. Fluids 29, 122104.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 2, 111.CrossRefGoogle Scholar
Sussman, M., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159.CrossRefGoogle Scholar
Tagawa, Y., Takagi, S. & Matsumoto, Y. 2014 Surfactant effect on path instability of a rising bubble. J. Fluid Mech. 738, 124–142.CrossRefGoogle Scholar
Takagi, S. & Matsumoto, Y. 2011 Surfactant effects on bubble motion and bubbly flows. Annu. Rev. Fluid Mech. 43, 615–636.CrossRefGoogle Scholar
Takemura, F. 2005 Adsorption of surfactants onto the surface of a spherical rising bubble and its effect on the terminal velocity of the bubble. Phys. Fluids 17, 048104.CrossRefGoogle Scholar
Takemura, F. & Yabe, A. 1998 Gas dissolution process of spherical rising bubbles. Chem. Engng Sci. 53, 2691–2699.CrossRefGoogle Scholar
Takemura, F. & Yabe, A. 1999 Rising speed and dissolution rate of a carbon dioxide bubble in slightly contaminated water. J. Fluid Mech. 334, 319–334.CrossRefGoogle Scholar
Tanguy, S., Menard, T. & Berlemont, A. 2007 A level set method for vaporizing two-phase flows. J. Comput. Phys. 221, 837–853.CrossRefGoogle Scholar
Tomiyama, A., Kataoka, I., Zun, I. & Sakaguchi, T. 1998 Drag coefficients of single bubbles under normal and micro gravity conditions. JSME Intl J. 41, 472–479.CrossRefGoogle 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, 314–316.CrossRefGoogle ScholarPubMed
Vasconcelos, J.M.T., Orvalho, S.P. & Alves, S.S. 2002 Gas–liquid mass transfer to single bubbles : effect of surface contamination. AIChE J. 48, 1145–1154.CrossRefGoogle Scholar
Veldhuis, C. 2007 Leonardos paradox: path and shape instabilities of particles and bubbles. PhD thesis, University of Twente, The Netherlands.Google Scholar
Verwijlen, T., Imperiali, L. & Vermant, J. 2014 Separating viscoelastic and compressibility contributions in pressure-area isotherm measurements. Adv. Colloid Interface Sci. 206, 428–436.CrossRefGoogle ScholarPubMed
Wang, Y., Papageorgiou, D.T. & Maldarelli, C. 2002 Using surfactants to control the formation and size of wakes behind moving bubbles at order-one Reynolds numbers. J. Fluid Mech. 453, 1–19.CrossRefGoogle Scholar
Weiner, A., Timmermann, J., Pesci, C., Grewe, J., Hoffmann, M., Schlüter, M. & Bothe, D. 2019 Experimental and numerical investigation of reactive species transport around a small rising bubble. Chem. Engng Sci. 1, 100007.Google Scholar
Zhang, Y. & Finch, J.A. 2001 A note on single bubble motion in surfactant solutions. J. Fluid Mech. 429, 63–66.CrossRefGoogle Scholar