The steady propagation of an air finger into a fluid-filled tube of uniform rectangularcross-section is investigated. This paper is primarily focused on the influence of theaspect ratio, α, on the flow properties, but the effects of a transverse gravitational fieldare also considered. The three-dimensional interfacial problem is solved numericallyusing the object-oriented multi-physics finite-element library oomph-lib and theresults agree with our previous experimental results (de Lo´ zar 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 pressuredrops 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 onthe tube walls after the propagation of the finger) is more complicated: m decreaseswith increasing α for low Ca but it increases with α at high Ca. Our results alsoindicate that the system is approximately quasi-two-dimensional for α 8, when weobtain quantitative agreement with McLean & Saffman’s two-dimensional model forthe relative finger width as a function of the governing parameter 1/B =12α2Ca.The action of gravity causes an increase in the pressure drops, finger widths and wetfractions at fixed capillary number. In particular, when the Bond number (ratio ofgravitational to surface-tension forces) is greater than one the finger lifts off the bottomwall of the tube leading to dramatic increases in the finger width and wet fraction at agiven Ca.For α 3 a previously unobserved flow regime has been identified in which asmall recirculation flow is situated in front of the finger tip, shielding it from anycontaminants in the flow. In addition, for α 2 the capillary number, Cac, abovewhich global recirculation flows disappear has been observed to follow the simpleempirical law: Ca2/3c α =1.21.
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