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The steady propagation of an air finger into a rectangular tube


The steady propagation of an air finger into a fluid-filled tube of uniform rectangular cross-section is investigated. This paper is primarily focused on the influence of the aspect ratio, α, on the flow properties, but the effects of a transverse gravitational field are also considered. The three-dimensional interfacial problem is solved numerically using the object-oriented multi-physics finite-element library oomph-lib and the results agree with our previous experimental results (de Lózar et al. Phys. Rev. Lett. vol. 99, 2007, article 234501) to within the ±1% experimental error.

At a fixed capillary number Ca (ratio of viscous to surface-tension forces) the pressure drops across the finger tip and relative finger widths decrease with increasing α. The dependence of the wet fraction m (the relative quantity of liquid that remains on the tube walls after the propagation of the finger) is more complicated: m decreases with increasing α for low Ca but it increases with α at high Ca. Our results also indicate that the system is approximately quasi-two-dimensional for α ≥ 8, when we obtain quantitative agreement with McLean & Saffman's two-dimensional model for the relative finger width as a function of the governing parameter 1/B = 12α2 Ca. The action of gravity causes an increase in the pressure drops, finger widths and wet fractions at fixed capillary number. In particular, when the Bond number (ratio of gravitational to surface-tension forces) is greater than one the finger lifts off the bottom wall of the tube leading to dramatic increases in the finger width and wet fraction at a given Ca.

For α ≥ 3 a previously unobserved flow regime has been identified in which a small recirculation flow is situated in front of the finger tip, shielding it from any contaminants in the flow. In addition, for α ≳ 2 the capillary number, Cac, above which global recirculation flows disappear has been observed to follow the simple empirical law: Cac2/3α = 1.21.

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Ajaev, V. S. & Homsy, G. M. 2006 Modeling shapes and dynamics of confined bubbles. Annu. Rev. Fluid Mech. 38, 277.
Baroud, C. N. & Willaime, H. 2004 Multiphase flows in microfluidics. Compt. Rendus Phys. 7, 547.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166.
Clanet, C., Héraud, P. & Searby, G. 2004 On the motion of bubbles in vertical tubes of arbitrary cross-sections: some complements to the Dumitrescu-Taylor problem. J. Fluid Mech. 519, 359.
Duff, I. S. & Scott, J. A. 1996 The design of a new frontal code for solving sparse, unsymmetric linear systems. ACM Trans. Math. Software 22, 3045.
Giavedoni, M. D. & Saita, F. A. 1997 The axisymmetric and plane cases of a gas phase steadily displacing a Newtonian liquid-A simultaneous solution of the governing equations. Phys. Fluids 9, 2420.
Gresho, P. M. & Sani, R. L. 2000 Incompressible Flow and the Finite Element Method. Volume Two: Isothermal Laminar Flow. John Wiley & Sons Ltd.
Hazel, A. L. & Heil, M. 2002 The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section. J. Fluid Mech. 470, 91.
Heil, M. & Hazel, A. L. 2006 oomph-lib–An Object-Oriented Multi-Physics Finite-Element Library. In Fluid-Structure Interaction (ed. Schafer, M. & Bungartz, H.-J.), p. 19. Springer.
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271.
Jensen, M. H., Libchaber, A., Pelcé, P. & Zocchi, G. 1987 Effect of gravity on Saffman-Taylor meniscus: Theory and experiment. Phys. Rev. A 35, 2221.
Kistler, S. F. & Scriven, L. E. 1983 Coating flows. In Computational Analysis of Polymer Processing (ed. Pearson, J. R. A. & Richardson, S. M.), p. 243. Applied Science Publishers.
Kolb, W. B. & Cerro, R. L. 1991 Coating the inside of a capillary of square cross section. Chem. Engng Sci. 46, 2181.
Landau, L. D. & Levich, V. G. 1942 Dragging of a liquid by a moving plate. Acta Physiocochimica URSS 17, 42.
Lister, J. R., Morrison, N. F. & Rallison, J. M. 2006 Sedimentation of a two-dimensional drop towards a rigid horizontal plane. J. Fluid Mech. 552, 345.
de Lózar, A., Hazel, A. L. & Juel, A. 2007 Scaling properties of coating flows in rectangular channels. Phys. Rev. Lett. 99, 234501.
McLean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455.
Moore, M. G., Juel, A., Burgess, J. M., McCormick, W. D. & Swinney, H. L. 2002 Fluctuations in viscous fingering. Phys. Rev. E 65, 030601(R).
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291.
Ratulowski, J. & Chang, H.-C. 1989 Transport of gas bubbles in capillaries. Phys. Fluids A 1, 16421655.
Reinelt, D. A. 1987 Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 183, 219.
Reinelt, D. A. & Saffmann, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6, 542.
Ruschak, K. J. 1980 A method for incorporating free boundaries with surface tension in finite element fluid-flow simulators. Intl J. Num. Meth. Engng 15, 639.
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.
Tabeling, P. & Libchaber, A. 1986 Film draining and the Saffman-Taylor problem. Phys. Rev. A 33, 794.
Tabeling, P., Zocchi, G. & Libchaber, A. 1987 An experimental study of the Saffman-Taylor instability. J. Fluid Mech. 177, 67.
Taylor, G. I. 1961 Deposition of viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161.
Whitesides, G. M. & Stroock, A. D. 2001 Flexible methods for microfluidics. Physics Today 54 (6), 42.
Wong, H., Radke, C. J. & Morris, S. 1995 a The motion of long bubbles in polygonal capillaries. part 1. thin films. J. Fluid Mech. 292, 71.
Wong, H., Radke, C. J. & Morris, S. 1995 b The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95.
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Journal of Fluid Mechanics
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