Air-flow-induced instability of yield-stress fluid films

Excess mucus in the lungs may be cleared by coughing. This air-driven mucus clearance can be critically impacted in obstructive lung diseases such as cystic fibrosis, in which the mucus yield stress can be significantly enhanced. To explore the effect of a yield stress on air-driven mucus transport, we study the dynamics of viscoplastic liquid films driven by an overlying turbulent air flow. We develop a model using long-wave theory to approximate transport in the liquid layer. Numerical solutions to the long-wave model demonstrate the emergence of travelling waves when the air drag is sufficient to yield the liquid layer. Periodic travelling-wave solutions are constructed explicitly to quantify how increasing the yield stress, relative to air-induced stresses, can reduce rates of liquid transport. When waves are not spatially periodic, wave growth can instead be enhanced by viscoplasticity if the liquid ahead of an oncoming wave is unyielded or minimally yielded. We compare modelling results with measurements of free-surface shape evolution in laboratory experiments with viscous and yield-stress fluids exposed to air flow in a rectangular duct. For yield-stress fluid layers, isolated waves grow rapidly while the fluid ahead of them remains static, in accordance with the behaviour observed in model simulations.