Asymptotic modeling of viscous film flow inside a tube with time-dependent radius

The flow of a highly viscous film lining the interior of a tube with time-dependent radius is studied using a long-wave asymptotic model. The impact of tube expansion/contraction in time on net transport of the film (due to air flow and/or gravity) is explored first through prescribing the tube radius and air volume flow rate as oscillatory functions of time; specifically, the dependence of the film’s net transport on the magnitude of the phase lag between radius and airflow is quantified. First-order model corrections -- which incorporate the growth (and its saturation) of long-wave disturbances arising due to the Plateau-Rayleigh instability -- are then added to the model. Linear stability analysis of this periodically forced model show that tube contractions/expansions enhance instability growth compared to a rigid tube. Simulations of the full nonlinear model equation highlight the role of free-surface waves in enhancing transport. Parameter values used here are motivated by the human lung/airway system. If time permits, a compartment model for lungs and compliant airways will be coupled to the film flow model.