The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio a. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations. In rectangular tubes, our results for the pressure drop across the bubble tip, Deltap, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate Deltap. In both elliptical and rectangular tubes, the ratio Deltap(alpha)/Deltap(alpha = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures. In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with alpha = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r(infinity). In this regime, we find that r(infinity) in elliptical and rectangular tubes is related to r(infinity) in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio. For smaller Ca (Ca < CA), the air finger is always non-axisymmetric and the persisting draining flows in the thin film regions far behind the bubble tip ultimately lead to dry regions on the tube wall. Ca increases with increasing alpha and for alpha > alpha dry spots will develop on the tube walls at all values of Ca.