** Abstract: **

###
Hazel, A. L. & Heil, M. (2003) Three-dimensional airway reopening: The steady
propagation of a semi-infinite bubble into a buckled
elastic tube.
*Journal of Fluid Mechanics* (in press)

####
We consider the steady propagation of an air finger into a buckled elastic
tube initially filled with viscous fluid. This study is
motivated by the physiological
problem of pulmonary airway reopening. The system is modelled using
geometrically non-linear, Kirchhoff-Love shell theory coupled to the
free-surface Stokes equations.
The resulting three-dimensional fluid-structure-interaction
problem is solved numerically by a fully-coupled finite element
method.

The system is governed by three dimensionless parameters:
(i) the Capillary number,
Ca = mu U /sigma^{*}, represents the ratio of viscous
to surface-tension forces, where mu is the fluid viscosity,
U is the finger's propagation speed and sigma^{*} is the
surface tension at the air-liquid interface; (ii) sigma = sigma^{*}/(RK)
represents the ratio of surface tension to elastic forces, where R
is the undeformed radius of the tube and K its bending modulus;
and (iii) A_{infinity} =
A^{*}_{infinity}/(4R^{2}), characterises the initial
degree of tube collapse, where A^{*}_{infinity} is the
cross-sectional area of the tube far ahead of the bubble.

The generic behaviour of the system is found to be very similar to
that observed
in previous two-dimensional models (Gaver et al. 1996, Heil 2000).
In particular, we find a two-branch behaviour in the
relationship between dimensionless
propagation speed, Ca, and dimensionless bubble pressure, p_{b} =
p^{*}_{b}/(sigma^{*}/R).
At low Ca, a decrease in p_{b} is required
to increase the propagation speed. We present a simple model which
explains this behaviour and why it occurs in both two and
three dimensions.
At high Ca, p_{b} increases monotonically with propagation speed and
p_{b} is proportional to Ca for sufficiently large values of
sigma and
Ca. In a frame of reference moving with the finger velocity, an open
vortex develops ahead of the bubble tip at low Ca. As Ca increases,
the flow topology changes and the vortex disappears.

An increase in dimensional surface tension sigma^{*} causes an
increase in the bubble pressure required to drive the air finger at a
given speed.
p_{b} also increases with A^{*}_{infinity}
and higher bubble
pressures are required to open less strongly buckled tubes. This
unexpected finding could have important physiological
ramifications. Furthermore, we find that the maximum wall shear
stresses exerted
on the airways during reopening may be large enough to damage the lung tissue.

Page last modified: February 28, 2003

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