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80<br />
The influence of gravity on the propagation of an air finger<br />
into a fluid-filled, elastic-walled channel<br />
A. L. Hazel ∗ ,M.Heil ∗<br />
Airway closure occurs when the gas-conveying vessels of the lung collapse and<br />
become occluded by the liquid normally present in a thin lining film. Closure of the<br />
larger airways occurs only under certain pathological conditions; and if the vessels<br />
remain closed for the majority of the breathing cycle, lung function is severely compromised.<br />
The aim of any clinical treatment is to reopen the closed airways as rapidly<br />
as possible, but without damaging the lung tissue.<br />
Previous two- 1 and three-dimensional 2 theoretical studies of the system have neglected<br />
gravitational effects, an assumption justified by the relatively small size of<br />
the terminal airways (diameter ≈ 0.25mm) and, hence, small values of the Bond<br />
number (ratio of gravitational forces to surface-tension forces). Recent bench-top experiments<br />
3 have demonstrated, however, that gravity may yet play an important role<br />
in the reopening process. In particular, the internal pressure of the air finger measured<br />
experimentally was consistently greater than that predicted by a three-dimensional,<br />
theoretical model in which gravitational forces were absent. 2<br />
We model the reopening of an isolated, collapsed airway by considering the steady<br />
propagation of an inviscid air finger into a flexible-walled, two-dimensional channel<br />
initially filled by an incompressible viscous liquid. This fluid-structure-interaction<br />
problem is solved numerically using a fully-coupled, finite-element method. The motion<br />
of the fluid is described by the Navier–Stokes equations and the channel walls<br />
are modelled as two pre-stressed elastic beams supported on elastic foundations. We<br />
present results from simulations that include the effects of gravity and fluid inertia.<br />
The results indicate that modest values of the Bond and Reynolds numbers, the latter<br />
being the ratio of inertial to viscous forces in the fluid, lead to macroscopic changes<br />
in the geometrical configuration of the system near the finger tip and to significant<br />
increases in the pressure required to drive the air finger at a given speed.<br />
∗ School of Mathematics, University of Manchester, U.K.<br />
1 Gaver et al. J. Fluid. Mech. 319, 25 (1996)<br />
2 Hazel & Heil ASME J. Biomech. Eng. accepted<br />
3 Heap & Juel J. Fluid Mech. submitted
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