# Airway Closure: The Model Problem

We model the airway wall as a long thin elastic shell which deforms in response to the external pressure and to the additional loads due to the surface tension of the liquid bridge. The liquid bridge is enclosed between two air-liquid interfaces (menisci) which meet the tube wall at a constant contact angle along a contact line whose position has to be determined as part of the solution. The pressure jump over the menisci implies that the tube wall is compressed more strongly in the region which is wetted by the liquid bridge. For zero gravity (or small Bond number) the menisci are surfaces of constant mean curvature. Here's a sketch of the model in an axisymmetric state:

Since we want to model non-axisymmetrically buckled tubes, the real model is slightly more complicated than indicated in the above sketch: We need two sets of surface coordinates to parametrise the tube wall and the meniscus. We also have to parametrise the position of the contact line on the tube wall. The following picture shows how this is done (only one half of the tube and one meniscus need to be modelled since the configuration is symmetric in the axial direction -- we cut the tube in the plane of symmetry at z=0).

To solve the problem mathematically/computationally, the wall deformation was described by geometrically non-linear shell theory, coupled to a variational description of the meniscus shape. A spine method was used to implement the parametric description of the meniscus shape. The resulting highly non-linear equations were discretised with Finite Element Methods and solved with a global Newton-Raphson method, embedded in an adaptive continuation technique.

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