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Computational analysis of single rising bubbles influenced by soluble surfactant

Published online by Cambridge University Press:  09 October 2018

Chiara Pesci
Mathematical Modelling and Analysis, Technische Universität Darmstadt, Darmstadt, 64287, Germany
Andre Weiner
Mathematical Modelling and Analysis, Technische Universität Darmstadt, Darmstadt, 64287, Germany
Holger Marschall
Mathematical Modelling and Analysis, Technische Universität Darmstadt, Darmstadt, 64287, Germany
Dieter Bothe*
Mathematical Modelling and Analysis, Technische Universität Darmstadt, Darmstadt, 64287, Germany
Email address for correspondence:


This paper presents novel insights into the influence of soluble surfactants on bubble flows obtained by direct numerical simulation (DNS). Surfactants are amphiphilic compounds which accumulate at fluid interfaces and significantly modify the respective interfacial properties, influencing also the overall dynamics of the flow. With the aid of DNS, local quantities like the surfactant distribution on the bubble surface can be accessed for a better understanding of the physical phenomena occurring close to the interface. The core part of the physical model consists of the description of the surfactant transport in the bulk and on the deformable interface. The solution procedure is based on an arbitrary Lagrangian–Eulerian (ALE) interface-tracking method. The existing methodology was enhanced to describe a wider range of physical phenomena. A subgrid-scale (SGS) model is employed in the cases where a fully resolved DNS for the species transport is not feasible due to high mesh resolution requirements and, therefore, high computational costs. After an exhaustive validation of the latest numerical developments, the DNS of single rising bubbles in contaminated solutions is compared to experimental results. The full velocity transients of the rising bubbles, especially the contaminated ones, are correctly reproduced by the DNS. The simulation results are then studied to gain a better understanding of the local bubble dynamics under the effect of soluble surfactant. One of the main insights is that the quasi-steady state of the rise velocity is reached without ad- and desorption being necessarily in equilibrium.

JFM Papers
© 2018 Cambridge University Press 

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Ahn, H. T. & Shashkov, M. 2008 Geometric algorithms for 3D interface reconstruction. In Proceedings of the 16th International Meshing Roundtable (ed. Brewer, M. L. & Marcum, D.), pp. 405422. Springer.CrossRefGoogle Scholar
Aksenenko, E. V., Makievski, A. V., Miller, R. & Fainerman, V. B. 1998 Dynamic surface tension of aqueous alkyl dimethyl phosphine oxide solutions. Effect of the alkyl chain length. Colloids Surf. A 143, 311321.CrossRefGoogle Scholar
Albert, C., Kromer, J., Robertson, A. M. & Bothe, D. 2015 Dynamic behaviour of buoyant high viscosity droplets rising in a quiescent liquid. J. Fluid Mech. 778, 485533.CrossRefGoogle Scholar
Alke, A. & Bothe, D. 2009 3D numerical modelling of soluble surfactant at fluid interfaces based on the Volume-of-Fluid method. Fluid Dyn. Mater. Process. 5 (4), 345372.Google Scholar
Bothe, D. & Fleckenstein, S. 2013 A Volume-of-Fluid-based method for mass transfer processes at fluid particles. Chem. Engng Sci. 101, 283302.CrossRefGoogle Scholar
Bothe, D., Prüss, J. & Simonett, G. 2005 Well-posedness of a two-phase flow with soluble surfactant. In Nonlinear Elliptic and Parabolic Problems (ed. Escher, J. & Chipot, M.), pp. 3761. Birkhäuser.CrossRefGoogle Scholar
Cano-Lozano, J. C., Martínez-Bazán, C., Magnaudet, J. & Tchoufag, J. 2016 Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability. Phys. Rev. Fluids 1 (5), 053604.CrossRefGoogle Scholar
Chang, C. H. & Franses, E. I. 1995 Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms. Colloids Surf. A 100, 145.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles, 2nd edn. Dover Publications.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, 2553.CrossRefGoogle Scholar
Davis, R. E. & Acrivos, A. 1966 The influence of surfactants on the creeping motion of bubbles. Chem. Engng Sci. 21, 681685.CrossRefGoogle Scholar
Dieter-Kissling, K., Marschall, H. & Bothe, D. 2015a Direct Numerical Simulation of droplet formation processes under the influence of soluble surfactant mixtures. Comput. Fluids 113, 93105.CrossRefGoogle Scholar
Dieter-Kissling, K., Marschall, H. & Bothe, D. 2015b Numerical method for coupled interfacial surfactant transport on dynamic surface meshes of general topology. Comput. Fluids 109, 168184.CrossRefGoogle Scholar
Duineveld, P. C. 1995 The rise velocity and shape of bubbles in pure water at high Reynolds number. J. Fluid Mech. 292, 325332.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, 260274.CrossRefGoogle ScholarPubMed
Dukhin, S. S., Kretzschmar, G. & Miller, R. 1995 Dynamics of Adsorption at Liquid Interfaces. Elsevier.Google Scholar
Dukhin, S. S., Lotfi, M., Kovalchuck, V. I., Bastani, D. & Miller, R. 2016 Dynamics of rear stagnant cap formation at the surface of rising bubbles in surfactant solutions at large Reynolds and Marangoni numbers and for slow sorption kinetics. Colloids Surf. A 492, 127137.CrossRefGoogle Scholar
Ellingsen, K. & Risso, F. 2001 On the rise of an ellipsoidal bubble in water: oscillatory paths and liquid-induced velocity. J. Fluid Mech. 440, 235268.CrossRefGoogle Scholar
Fdhila, R. B. & Duineveld, P. C. 1996 The effect of surfactant on the rise of a spherical bubble at high Reynolds and Peclet numbers. Phys. Fluids 8, 310321.CrossRefGoogle Scholar
Ferziger, J. H. & Perić, M. 1996 Computational Methods for Fluid Dynamics. Springer.CrossRefGoogle Scholar
He, Z., Maldarelli, C. & Dagan, Z. 1991 The size of stagnant caps of bulk soluble surfactant on the interface of translating fluid droplets. J. Colloid Interface Sci. 146, 442451.CrossRefGoogle Scholar
Hirt, C. W., Amsden, A. A. & Cook, J. L. 1974 An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14, 227253.CrossRefGoogle Scholar
Huang, J. & Saito, T. 2017a Discussion about the differences in mass transfer, bubble motion and surrounding liquid motion between a contaminated system and a clean system based on consideration of three-dimensional wake structure obtained from LIF visualization. Chem. Engng Sci. 170, 105115.CrossRefGoogle Scholar
Huang, J. & Saito, T. 2017b Influence of gas–liuid interface contamination on bubble motions, bubble wakes, and instantaneous mass transfer. Chem. Engng Sci. 157, 182199.CrossRefGoogle Scholar
Issa, R. 1986 Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62 (1), 4065.CrossRefGoogle Scholar
Jasak, H. & Tuković, Z. 2006 Automatic mesh motion for the unstructured finite volume method. Trans. FAMENA 30 (2), 120.Google Scholar
Kim, I. & Pearlstein, A. 1990 Stability of the flow past a sphere. J. Fluid Mech. 211, 7393.CrossRefGoogle Scholar
Kovalchuk, V. I., Krägel, J., Makievski, A. V., Ravera, F., Liggieri, L., Loglio, G., Fainerman, V. B. & Miller, R. 2004 Rheological surface properties of C12DMPO solution as obtained from amplitude- and phase-frequency characteristics of an oscillating bubble system. J. Colloid Interface Sci. 280 (2), 498505.CrossRefGoogle ScholarPubMed
Krzan, M. & Malysa, K. 2002 Profiles of local velocities of bubbles in n-butanol, n-hexanol and n-nonanol solutions. Colloids Surf. A 207, 279291.CrossRefGoogle Scholar
Krzan, M., Zawala, J. & Malysa, K. 2007 Development of steady state adsorption distribution over interface of a bubble rising in solutions of n-alkanols (C5, C8) and n-alkyl trimethyl ammonium bromides (C8, C12, C16). Colloids Surf. A 298, 4251.CrossRefGoogle Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics, 2nd edn. Prentice-Hall.Google Scholar
Liao, Y. & McLaughlin, J. B. 2000 Bubble motion in aqueous surfactant solutions. J. Colloid Interface Sci. 224, 297310.CrossRefGoogle ScholarPubMed
Lochiel, A. C. & Calderbank, P. H. 1964 Mass Transfer in the continuous phase around axisymmetric bodies of revolution. Chem. Engng Sci. 19, 471484.CrossRefGoogle Scholar
Małysa, K., Zawala, J., Krzan, M. & Krasowska, M. 2011 Bubbles rising in solutions; local and terminal velocities, shape variations and collisions with free surface. In Bubble and Drop Interfaces (ed. Miller, R. & Liggieri, L.), Progress in Colloid and Interface Science, vol. 2. CRC Press, Taylor & Francis Group.Google Scholar
Miller, R., Fainerman, V. B., Pradines, V., Kovalchuk, V. I., Kovalchuk, N. M., Aksenenko, E. V., Liggieri, L., Ravera, F., Loglio, G., Sharipova, A., Vysotsky, Y., Vollhardt, D., Mucic, N., Wüstneck, R., Krägel, J. & Javadi, A. 2014 Surfactant adsorption layers at liquid interfaces. In Surfactant Science and Technology. Retrospects and Prospects (ed. Romsted, L. S.). CRC Press.Google Scholar
Mougin, G. & Magnaudet, J. 2002 Path instability of a rising bubble. Phys. Rev. Lett. 88, 14502.CrossRefGoogle ScholarPubMed
Mougin, G. & Magnaudet, J. 2006 Wake-induced forces and torques on a zigzagging/spiralling bubble. J. Fluid Mech. 567, 185194.CrossRefGoogle Scholar
Muzaferija, S. & Perić, M. 1997 Computation of free-surface flows using the finite-volume method and moving grids. Numer. Heat Transfer B 32 (4), 369384.CrossRefGoogle Scholar
Pesci, C., Dieter-Kissling, K., Marschall, H. & Bothe, D. 2015 Finite volume/finite area interface tracking method for two-phase flows with fluid interfaces influenced by surfactant. In Progress in Colloid and Interface Science (ed. Rahni, M. T., Karbaschi, M. & Miller, R.). CRC Press, Taylor & Francis Group.Google Scholar
Pesci, C., Marschall, H., Ulaganathan, V., Kairaliyeva, T., Miller, R. & Bothe, D. 2017 Experimental and computational analysis of fluid interfaces influenced by soluble surfactant. In Transport Processes at Fluidic Interfaces (ed. Bothe, D. & Reusken, A.), chap. 15. Springer International Publishing, AG.Google Scholar
Sam, A., Gomez, C. O. & Finch, J. A. 1996 Axial velocity profiles of single bubbles in water/frother solutions. Intl J. Miner. Process. 47, 177196.CrossRefGoogle Scholar
Satapathy, R. & Smith, W. 1960 The motion of single immiscible drops through a liquid. J. Fluid Mech. 10, 561570.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), 111112.CrossRefGoogle Scholar
Tagawa, Y., Takagi, S. & Matsumoto, Y. 2014 Surfactant effects on path instability of a rising bubble. J. Fluid Mech. 378, 124142.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
Tasoglu, S., Demirci, U. & Muradoglu, M. 2008 The effect of soluble surfactant on the transient motion of a buoyancy-driven bubble. Phys. Fluids 20 (4), 040805.CrossRefGoogle Scholar
Tomiyama, A., Kataoka, I., Zun, I. & Sakaguchi, T. 1998 Drag coefficients of single bubbles under normal and micro gravity condition. JSME Intl J. 41 (2), 472479.CrossRefGoogle Scholar
Tsuge, H. & Hibino, S. 1971 The motion of single gas bubbles rising in various liquids. Chem. Engng 35 (1), 6571.CrossRefGoogle Scholar
Tuković, Z. & Jasak, H. 2008 Simulation of free-rising bubble with soluble surfactant using moving mesh finite volume/area method. In 6th International Conference on CFD in Oil & Gas, Metallurgical and Process Industries SINTEF/NTNU, Trondheim, Norway.Google Scholar
Tuković, Z. & Jasak, H. 2012 A moving mesh finite volume interface tracking method for surface tension dominated interfacial fluid flow. Comput. Fluids 55, 7084.CrossRefGoogle Scholar
Ulaganathan, V.2016 Molecular Fundamentals of foam fractionation. PhD thesis, Universität Potsdam, Potsdam.Google Scholar
Versteeg, H. K. & Malalasekera, W. 1995-2007 An Introduction to Computational Fluid Dynamics. Pearson Education Limited.Google Scholar
de Vries, A. W. G., Biesheuvel, A. & van Wijngaarden, L. 2002 Notes on the path and wake of a gas bubble rising in pure water. Intl J. Multiphase Flow 28, 18231835.CrossRefGoogle Scholar
Weber, P. S.2016 Modeling and numerical simulation of multi-component single- and two-phase fluid systems. PhD thesis, Technische Universität Darmstadt.Google Scholar
Weiner, A. & Bothe, D. 2017 Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles. J. Comput. Phys. 347, 261289.CrossRefGoogle Scholar
Zhang, Y. & Finch, J. A. 2001 A note on single bubble motion in surfactant solutions. J. Fluid Mech. 429, 6366.CrossRefGoogle Scholar
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