Hostname: page-component-848d4c4894-pftt2 Total loading time: 0 Render date: 2024-06-04T09:38:34.427Z Has data issue: false hasContentIssue false

Marangoni effects caused by contaminants adsorbed on bubble surfaces

Published online by Cambridge University Press:  18 March 2010

PETER LAKSHMANAN
Affiliation:
Fluid Mechanics, Biochemical & Chemical Engineering, Technische Universität Dortmund, Emil-Figge-Strasse 68, D-44221 Dortmund, Germany
PETER EHRHARD*
Affiliation:
Fluid Mechanics, Biochemical & Chemical Engineering, Technische Universität Dortmund, Emil-Figge-Strasse 68, D-44221 Dortmund, Germany
*
Email address for correspondence: p.ehrhard@bci.tu-dortmund.de

Abstract

The influence of Marangoni stresses, caused by contaminants adsorbed on the surface of small air bubbles, rising in water, is examined by numerical simulations. A modified level set method is used to represent the deformable bubble interface, extended by a model for the contaminant transport on the bubble surface. We show that surface tension variations of less than 2% are sufficient to generate Marangoni stresses that are strong enough to change the rising characteristics of a bubble to that of a corresponding solid particle. In such situations, we find that the bubble surface is fully covered with contaminant and the shear stress profile resembles the shear stress profile around a solid sphere.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Anderson, D. M. & McFadden, G. B. 1998 Diffusive-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139168.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modelling surface tension. J. Comput. Phys. 100, 335354.CrossRefGoogle Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops, and Particles. Academic Press.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
Drumright-Clark, M. A. & Renardy, Y. 2004 The effect of insoluble surfactant at dilute concentration on drop breakup under shear with inertia. Phys. Fluids 16, 1421.CrossRefGoogle Scholar
Enright, D., Fedkiw, R., Ferizger, J. & Mitchel, I. 2002 A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83116.CrossRefGoogle Scholar
Frumkin, A. & Levich, V. G. 1947 On surfactants and interfacial motion. Zhurnal Fizicheskoi Khimii 21, 11831208.Google Scholar
Harper, J. F. 1974 On spherical bubbles rising steadily in dilute surfactant solutions. Q. J. Mech. Appl. Math. 17, 87100.CrossRefGoogle Scholar
Hirt, C. W & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201225.CrossRefGoogle Scholar
James, A. J. & Lowengrub, J. 2004 A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 201, 685722.CrossRefGoogle Scholar
Li, J. 2006 The effect of an insoluble surfactant on the skin friction of a bubble. Eur. J. Mech. B/Fluids 25, 5973.CrossRefGoogle Scholar
McLaughlin, J. B. 1996 Numerical simulation of bubble motion in water. J. Colloid Interface Sci. 184, 614625.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 6, 418420.CrossRefGoogle Scholar
Olsson, E. & Kreiss, G. 2005 A conservative level set method for two phase flow. J. Comput. Phys. 210, 225246.CrossRefGoogle Scholar
Palaparthi, R., Papageorgiou, D. T. & Maldarelli, C. 2006 Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles. J. Fluid Mech. 559, 144.CrossRefGoogle Scholar
Peskin, C. S. 2002 The immersed boundary method. Acta Numer. 11, 479517.CrossRefGoogle Scholar
Renardy, Y. Y., Renardy, M. & Cristini, V. 2002 A new volume-of-fluid formulation for surfactants and simulations of drop deformation under shear at low viscosity ratio. Eur. J. Mech. B/Fluids 21, 4959.CrossRefGoogle Scholar
Sadhal, S. S. & Johnson, R. E. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film — exact solution. J. Fluid Mech. 129, 237.CrossRefGoogle Scholar
Savic, P. 1953 Circulation and distortion of liquid drops falling through a viscous medium. Natl Res. Counc. Can. Div. Mech. Engng Rep. MT–22, 135.Google Scholar
Schiller, L. & Naumann, A. Z. 1933 Über die grundlegenden Berechnungen bei der Schwerkraftaufbereitung. Zeitung VDI 77, 318320.Google 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, 111112.CrossRefGoogle Scholar
Sussman, M. 1994 A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 149159.CrossRefGoogle Scholar
Sussman, M. & Puckett, E. G. 2000 A coupled level set and volume of fluid method for computing 3D and axissymmetric incompressible two-phase flows. J. Comput. Phys. 162, 301337.CrossRefGoogle Scholar
Tomiyama, A., Kataoka, I., Zun, I. & Sakaguchi, T. 1998 Drag coefficients of single bubbles under normal and micro gravity conditions. Japan Soc. Mech. Engnng Intl J. Ser. B 41, 472479.Google Scholar
Wang, Y., Papageorgiou, D. T. & Maldarelli, C. 1999 Increased mobility of a surfactant–retarded bubble at high bulk concentrations. J. Fluid Mech. 390, 251270.CrossRefGoogle Scholar
Xu, J.-J., Li, Z., Lowengrub, J. & Zhao, H. 2006 A level-set method for interfacial flows with surfactant. J. Comput. Phys. 212, 590616.CrossRefGoogle Scholar