White, J.P. & M. Heil (2004) Three-dimensional instabilities of liquid-lined elastic tubes -- a lubrication theory model. Physics of Fluids 17

We develop a theoretical model of surface-tension-driven, three-dimensional instabilities of liquid-lined elastic tubes --- a model for pulmonary airway closure. The model is based on large-displacement shell theory, coupled to the equations of lubrication theory, modified to ensure the exact representation of the system's equilibrium configurations. The liquid film that lines the initially uniform, axisymmetric tube can become unstable to a surface-tension-driven instability. We show that, if the surface tension of the liquid lining is sufficiently large (relative to the tube's bending stiffness), the axisymmetric redistribution of fluid by this instability can increase the wall compression to such an extent that the system becomes unstable to a secondary, non-axisymmetric instability which causes the tube wall to buckle. We establish the conditions for the occurrence of the non-axisymmetric instability by a linear stability analysis and use finite element simulations to explore the system's subsequent evolution in the large-displacement regime. The simulations show that non-axisymmetric instabilities allow the formation of occluding liquid bridges in situations in which the volume of fluid is insufficient to occlude the tube in its axisymmetric state. Finally, we discuss the implications of our results for the physiological problem of pulmonary airway closure.

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