  One mechanism for airway closure in the lung is the surface-tension-driven
instability of the mucus layer which lines the airway wall. We study the
instability of an axisymmetric layer of viscoplastic Bingham liquid coating the
interior of a rigid tube, which is a simple model for an airway that takes into
account the yield stress of mucus. An evolution equation for the thickness of
the liquid layer is derived using long-wave theory, from which we also derive a
simpler thin-film evolution equation. In the thin-film case, we show that two
branches of marginally-yielded static solutions of the evolution equation can
be used to both predict the size of initial perturbation required to trigger
instability and quantify how increasing the capillary Bingham number (a
parameter measuring yield stress relative to surface tension) reduces the final
deformation of the layer. Using numerical solutions of the long-wave evolution
equation, we quantify how the critical layer thickness required to form a
liquid plug in the tube increases as the capillary Bingham number is increased.
We discuss the significance of these findings for modelling airway closure in
obstructive conditions such as cystic fibrosis, where the mucus layer is often
thicker and has a higher yield stress.
