Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-19T11:44:56.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Marangoni effects of trace impurities on the motion of long gas bubbles in capillaries

Published online by Cambridge University Press:  26 April 2006

J. Ratulowski  and
H.-C. Chang
J. Ratulowski
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
H.-C. Chang
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

When a viscous liquid is displaced by a long air bubble in a capillary, it leaves behind a wetting liquid film. A lubrication analysis by Bretherton (1961), which assumes a mobile surface, underpredicts the film thickness at low bubble speeds. In this investigation, the Marangoni effect of small amounts of impurities is shown to be capable of explaining this discrepancy. We carry out an asymptotic analysis for different convective, diffusive and kinetic timescales and show that, if transport in the film is mass-transfer limited such that a bulk concentration gradient exists in the film, the film thickness increases by a maximum factor of 4 2/3; over Bretherton's mobile result at low bubble speeds. Moreover, at large bubble speeds, Bretherton's mobile prediction is approached for all ranges of timescales. For intermediate bubble speeds, the film thickness varies with respect to the bubble speed with an exponent smaller than 2/3 of the mobile theory. These results are favourably compared to literature data on film thickness.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 210 , January 1990 , pp. 303 - 328
Copyright
© 1990 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

Bretherton, F. P. 1961 The motion of long drops and bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Chang, H.-C. & Ratulowski, J. 1987 Bubble transport in capillaries. AIChE Annual Meeting, November 15-20, New York, paper 681.Google Scholar
Chen, J. D. 1986 Measuring the film thickness surrounding a bubble inside a capillary. J. Colloid Interface Sci. 109, 34–39.Google Scholar
Davis, R. E. & Acrivos, A. 1966 The influence of surfactants on the creeping motion of bubbles. Chem. Engng Sci. 21, 681–685.Google Scholar
Fairbrother, F. & Stubbs, A. E. 1935 Studies in electroendosmosis. Part VI. The bubble-tube methods of measurement. J. Chem. Soc. 1, 527–529.Google Scholar
Ginley, G. M. & Radke, C. J. 1989 The influence of soluble surfactants on the flow of the long bubbles through a cylindrical capillary. ACS Symposium Series 396, pp. 480–501.Google Scholar
Goldshtik, M. A., Khanin, V. M. & Ligai, V. G. 1986 A liquid drop on an air cushion as an analogue of Leidenfrost boiling. J. Fluid Mech. 166, 1–20.Google Scholar
Goldsmith, H. L. & Mason, S. G. 1963 The flow of suspensions through tubes II. Single large bubbles. J. Colloid Sci. 18, 237–261.Google Scholar
Hedge, M. G. & Slattery, J. C. 1971 Capillary waves at a gas—liquid interface. J. Colloid Interface Sci. 35, 183–199.Google Scholar
Herbolzheimer, E. 1987 The effect of surfactant on the motion of a bubble in a capillary. AIChE Annual Meeting, November 15-20, New York, paper 68j.Google Scholar
Hirasaki, G. J. & Lawson, J. B. 1985 Mechanisms of foam flow in porous media: Apparent viscosity in smooth capillaries. Soc. Pet. Engrs J. 25, 176–190.Google Scholar
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physio-chim. USSR 17, 42–45.Google Scholar
Levich, B. 1962 Physiochemical Hydrodynamics, pp. 415–429. Prentice Hall.
Lu, W. Q. & Chang, H.-C. 1988 An extension of the biharmonic boundary integral method to free surface flow in channels. J. Comp. Phys. 77, 340–360.Google Scholar
Marchessault, R. N. & Mason, S. G. 1960 Ind. Chem. Engng 52, 79–81.
Park, C. W. & Homsy, G. W. 1984 Two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 139, 291–308.Google Scholar
Ratulowski, J. 1988 Mathematical modeling of mechanisms in bubble transport: A lubrication approach. PhD thesis, University of Houston.
Ratulowski, J. & Chang, H.-V. 1989 Transport of air bubbles in capillaries. Phys. Fluids (in press).Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and a tube. SIAM J. Sci. Stat. Comput. 6, 542–561.Google 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. 126, 237–250.Google Scholar
Schwartz, L. W., Princen, H. M. & Kiss, A. D. 1986 On the motion of bubbles in capillary tubes. J. Fluid Mech. 172, 259–275.Google Scholar
Secomb, T. W. & Skalak, R. 1982 Two-dimensional model for capillary flow of an Asymmetric Cell. Microvascular Res. 24, 194–203.Google Scholar
Shen, E. I. & Udell, K. S. 1985 A finite element study of low Reynolds number two-phase flow in cylindrical tubes. J. Appl. Mech. 107, 253–256.Google Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165.Google Scholar
Teletzke, G. F. 1983 Thin liquid films: Molecular theory and hydrodynamic implications. PhD thesis, University of Minnesota.
Whitaker, S. & Pierson, F. W. 1976 Studies of the drop-weight method for surfactant solutions I. Mathematical analysis of the adsorption of surfactants at the surface of a growing drop. J. Colloid Interface Sci. 51, 203–218.Google Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16, 209–221.Google Scholar