The Bretherton Problem In Elastic-Walled Channels: Finite Reynolds Number Effects

Abstract

This paper investigates the effect of fluid inertia on the propagation of an air finger into a channel with elastic walls, a problem which can be regarded as a generalisation of the classical Bretherton problem. The study is motivated by the physiological problem of pulmonary airway reopening. Numerical results show that fluid inert ia plays a surprisingly important role in thi s problem: Even for relatively modest ratios of Reynolds and Capillary numbers (Re/Ca ≈ 5 – 10), the pressure required to drive the air finger at a given speed increases significantly compared to the zero Reynolds number case. Inertial effects are also shown to be responsible for a notice able change in the wall deformation ahead of the bubble tip. This is analysed by a Karman-Pohlhausen approximation which yields a linear ODE, the eigenvalues of which determine the wavelength and decay rate of the oscillatory wall displacement field in this region.