Finite Reynolds number effects in the propagation of an air finger into a liquid-filled flexible-walled channel

This paper investigates finite Reynolds number effects in the problem of the propagation of an air finger into a liquid-filled flexible-walled two-dimensional channel. The study is motivated by the physiological problem of pulmonary airway reopening. A fully consistent model of the fluid-structure interaction is formulated and solved numerically using coupled finite element discretizations of the free-surface Navier-Stokes equations and the Lagrangian wall equations. It is shown that for parameter values which are representative of the conditions in the lung and in typical laboratory experiments, fluid inertia plays a surprisingly important role: even for relatively modest ratios of Reynolds and capillary numbers (Re/Ca approximate to 5-10), the pressure required to drive the air finger at a given speed increases significantly compared to the zero Reynolds number case. Fluid inertia leads to significant changes in the velocity and pressure fields near the bubble tip and is responsible for a noticeable change in the wall deformation pattern ahead of the bubble. For some parameter variations (such as variations in the wall tension), finite Reynolds number effects are shown to lead to qualitative changes in the system's behaviour. Finally, the implications of the result for pulmonary airway reopening are discussed.