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The motion of long bubbles in polygonal capillaries. Part 1. Thin films

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
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Abstract

Foam in porous media exhibits an unusually high apparent viscosity, making it useful in many industrial processes. The rheology of foam, however, is complex and not well understood. Previous pore-level models of foam are based primarily on studies of bubble flow in circular capillaries. A circular capillary, however, lacks the corners that characterize the geometry of the pores. We study the pressure–velocity relation of bubble flow in polygonal capillaries. A long bubble in a polygonal capillary acts as a leaky piston. The ‘piston’ is reluctant to move because of a large drag exerted by the capillary sidewalls. The liquid in the capillary therefore bypasses the bubble through the leaky corners at a speed an order higher than that of the bubble. Consequently, the pressure work is dissipated predominantly by the motion of the fluid and not by the motion of the bubble. This is opposite to the conclusion based on bubble flow in circular capillaries. The discovery of this new flow regime reconciles two groups of contradictory foam-flow experiments.

Part 1 of this work studies the fluid films deposited on capillary walls in the limit Ca → 0 (Ca ≡ μU/σ, where μ is the fluid viscosity, U the bubble velocity, and σ the surface tension). Part 2 (Wong et al. 1995) uses the film profile at the back end to calculate the drag of the bubble. Since the bubble length is arbitrary, the film profile is determined here as a general function of the dimensionless downstream distance x. For 1 [Lt ] x [Lt ] Ca−1, the film profile is frozen with a thickness of order Ca2/3 at the centre and order Ca at the sides. For x ∼ Ca−1, surface tension rearranges the film at the centre into a parabolic shape while the film at the sides thins to order Ca4/3. For x [Gt ] Ca−1, the film is still parabolic, but the height decreases as film fluid leaks through the side constrictions. For x ∼ Ca−5/3, the height of the parabola is order Ca2/3. Finally, for x [Gt ] Ca−5/3, the height decreases as Ca1/4x−1/4.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 292 , 10 June 1995 , pp. 71 - 94
Copyright
© 1995 Cambridge University Press

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References

Albrecht, R. A. & Marsden, S. S. 1970 Foams as blocking agents in porous media. Soc. Pet. Engng J. 10, 51.Google Scholar
Assar, G. R. & Burley, R. W. 1986 Hydrodynamics of foam flow in pipes, capillary tubes, and porous media. In Encyclopedia of Fluid Mechanics (ed. N. P. Cheremisinoff), vol. 3, chap. 2. Gulf Publishing Company, Houston, TX.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Burgess, D. & Foster, M. R. 1990 Analysis of the boundary conditions for a Hele-Shaw bubble. Phys. Fluids A 2, 1105–1117.Google Scholar
Chambers, K. T. & Radke, C. J. 1990 Capillary phenomena in foam flow through porous media. In Interfacial Phenomena in Oil Recovery (ed. N. R. Morrow). Marcel Dekker.
Courant, R. & Hilbert, D. 1953 Methods of Mathematical Physics, vol. I, Chap. 6. Interscience.
Falls, A. H., Musters, J. J. & Ratulowski, J. 1989 The apparent viscosity of foams in homogeneous beadpacks. SPE Reservoir Engng 4, 155–164.Google Scholar
Frankel, S. & Mysels, K. J. 1962 On the dimpling during the approach of two interfaces. J. Phys. Chem. 66, 190–191.Google Scholar
Hirasaki, G. J. 1989 The steam-foam process. J. Pet. Technol. 41, 449–456.Google Scholar
Holm, L. W. & Garrison, W. H. 1988 CO2 diversion with foam in an immiscible CO2 field project. SPE Reservoir Engng 3, 112–118.Google Scholar
Jones, A. F. & Wilson, S. D. R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263–288.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.
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochimica URSS 17, 42–54.Google Scholar
Mason, G. & Morrow, N. R. 1984 Meniscus curvatures in capillaries of uniform cross-section. J. Chem. Soc. Faraday Trans. 80, 2375–2393.Google Scholar
Mysels, K. J., Shinoda, K. & Frankel, S. 1959 Soap Films: Studies of their Thinning and a Bibliography. Pergamon.
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Reinelt, D. A. 1987 The effect of thin film variations and transverse curvature on the shape of fingers in a Hele-Shaw cell. Phys. Fluids 30, 2617–2623.Google Scholar
Reinelt, D. A. & Kraynik, A. M. 1990 On the shearing flow of foams and concentrated emulsions. J. Fluid Mech. 215, 431–455.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.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
Stebe, K. J., Lin, S. Y. & Maldarelli, C. 1991 Remobilizing surfactant retarded fluid particle interfaces. I. Stress-free conditions at the interfaces of micellar solutions of surfactants with fast sorption kinetics. Phys. Fluids A 3, 3–20.Google Scholar
Weatherburn, C. E. 1938 An Introduction to Riemannian Geometry and the Tensor Calculus. Cambridge University Press.
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. 1992a Two-dimensional menisci in non-axisymmetric capillaries. J. Colloid Interface Sci. 148, 284–287.Google Scholar
Wong, H., Morris, S. & Radke, C. J. 1992b Three-dimensional menisci in polygonal capillaries. J. Colloid Interface Sci. 148, 317–336.Google Scholar
Wong, H., Radke, C. J. & Morris, S. 1995 The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95–110.Google Scholar