Motivated by discrepancies between recent bench-top experiments A.
Juel and A. Heap, J. Fluid Mech.
572, 287 (2007) and numerical simulations A. L.
Hazel and M. Heil, ASME J. Biomech. Eng.
128, 573 (2006) we employ computational methods to
examine the effects of transverse gravity on the steady propagation of
a semi-infinite, inviscid air finger into a two-dimensional elastic
channel filled with a Newtonian fluid. The special case of
propagation in a rigid channel is also discussed in Appendix B. The
coupled free-surface, fluid-structure-interaction problem is solved
numerically using the object-oriented multiphysics finite-element
library oomph-lib. In the absence of gravity the relationship
between the applied pressure and the propagation speed of the finger
is nonmonotonic, with a turning point at small values of the
propagation speed. We demonstrate that the turning point disappears
when a modest gravitational force is applied and conjecture that it is
this effect of gravity rather than any instability of the zero-gravity
solution, as postulated in previous studies, that explains why the
turning point has never been observed in experiments. At large
propagation speeds, the presence of transverse gravity is shown to
increase the pressure required to drive the air finger at a given
speed, which is consistent with the observed discrepancies between
previous zero-gravity simulations and the experimental results.
Finally, we briefly discuss the possible implications of our results
for the physiological problem of pulmonary airway reopening.