Interplay of surfactant and viscoplasticity in the Rayleigh-Plateau instability of a film coating a tube

Airway closure in the lungs can be caused by a capillary instability of the mucus layer that lines the airway wall. To investigate the combined effects of surfactant and mucus yield stress in a simple airway closure model, we study the Rayleigh-Plateau instability of a film of Bingham liquid coating a rigid tube with insoluble surfactant at the air-liquid interface. We use thin-film and long-wave approximations to derive equations describing the evolution of the liquid layer. Numerical solutions and asymptotic analyses are used to quantify how surfactant amplifies the stabilising effect of the yield stress. For thin films, increasing the surfactant strength increases the size of perturbation to the free surface required to trigger instability, slows growth of the instability and reduces the final deformation of the layer. The dynamics of a thin film with very strong surfactant coincide with those of a surfactant-free film but with time slowed by a factor of four and the capillary Bingham number, which measures yield stress relative to capillary stresses, doubled. The critical layer thickness required for a liquid plug to form in the tube is shown to be increased by strong surfactant, in contrast to the Newtonian case where surfactant only acts to delay plug formation.