Viscous Airflow Through a Rigid Tube With a Compliant Lining: A Simple Model for the Air-Mucus Interaction in Pulmonary Airways

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Abstract
The respiratory tract of mammals is lined with a layer of mucus, described as viscoelastic semi-solid, above a layer of watery serous fluid. The interaction of these compliant layers with pulmonary airflow plays a major role in lung clearance by two-phase gas-liquid flow and in increased flow resistance in patients with obstructive airway diseases such as cystic fibrosis, chronic bronchitis and asthma. Experiments have shown that such coupled systems of flow-compliant-layers are quite susceptible to sudden shear instabilities, leading to formation of relatively large amplitude waves at the interface. Although these waves enhance the lung clearance by mobilizing the secretions, they increase the flow resistance in airways. The objective of this paper is to understand the basic interaction mechanism between the two media better by studying airflow through a rigid pipe that is lined by a compliant layer. The mathematical model that has been developed for this purpose is capable of explaining some of the published experimental observations. Wave instability theory is applied to the coupled air-mucus system to explore the stability of the interface. The results show that the onset flow speed for the initiation of unstable surface waves, and the resulting wavelength, are both very sensitive to mucus thickness. The model predicts that the instabilities initiate in the form of propagating waves for the elastic mucus where the wave speed is about 40 percent of the flow speed. The wavelength and phase speed to air velocity ratio are shown to increase with increasing mucus thickness. Also, results show that the mucus viscosity causes the onset air velocity to increase and the wave speed to decrease. The predictions of the model for the viscoelastic case are in good qualitative and quantitative agreement with some of the published experimental observations.

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