Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-19T18:29:44.807Z Has data issue: false hasContentIssue false
  • English
  • Français

The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow

Published online by Cambridge University Press:  26 April 2006

Harris Wong ,
C. J. Radke  and
S. Morris
Harris Wong
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Present address: The Levich Institute, The City College of CUNY, New York, NY 10031, USA
C. J. Radke
Affiliation:
Earth Sciences Division of Lawrence Berkeley Laboratory and Department of Chemical Engineering, University of California, Berkeley, CA 94720, USA
S. Morris
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca2/3 in the limit Ca → 0 (Ca μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary.

The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa2/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc [Lt ] 10−6. For Qc [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q2/3. For Q [Lt ] Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 292 , 10 June 1995 , pp. 95 - 110
Copyright
© 1995 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

Ali, J., Burley, R. W. & Nutt, C. W. 1985 Foam enhanced oil recovery from sand packs. Chem. Engng Res. Des. 63, 101–111.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Dussan, V. E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Ann. Rev. Fluid Mech. 11, 371–400.Google Scholar
Kovscek, A. R. & Radke, C. J. 1994 Fundamentals of foam transport in porous media. In Foams: Fundamentals and Applications (ed. L. L. Schramm), vol. 242, pp. 115–163. Washington, DC. American Chemical Society.
Mayer, R. P. & Stowe, R. A. 1965 Mercury porosimetry-breakthrough pressure for penetration between packed spheres. J. Colloid Sci. 20, 893–911.Google Scholar
Ransohoff, T. C. & Radke, C. J. 1988 Laminar flow of a wetting liquid along the corners of a predominantly gas-occupied noncircular pore. J. Colloid Interface Sci. 121, 392–401.Google Scholar
Ratulowski, J. & Chang, H. C. 1989 Transport of gas bubbles in capillaries. Phys. Fluids A 1, 1642–1655.Google Scholar
Ruschak, K. J. & Scriven, L. E. 1977 Developing flow on a vertical wall. J. Fluid Mech. 81, 305–316.Google Scholar
Wong, H. 1992 The motion of a long bubble in polygonal capillaries at low capillary numbers. PhD thesis, University of California, Berkeley.
Wong, H., Morris, S. & Radke, C. J. 1992 Two-dimensional menisci in non-axisymmetric capillaries. J. Colloid Interface Sci. 148, 284–287.Google Scholar
Wong, H., Radke, C. J. & Morris, S. 1995 The motion of long bubbles in polygonal capillaries. Part 1. Thin films. J. Fluid Mech. 292, 71–94.Google Scholar