** Abstract: **

###
Hazel, A. L. & Heil, M. (2002) The steady propagation of a semi-infinite
bubble into a tube of elliptical or rectangular cross-section.
*Journal of Fluid Mechanics* **470**, 91-114.

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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 alpha. 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, Delta p, 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
Delta p. In both elliptical and rectangular tubes, the ratio
Delta p (alpha)/Delta p(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.

Page last modified: November 11, 2002

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