Unsteady bubble propagation in a flexible channel: predictions of a viscous stick-slip instability

Abstract

We investigate the unsteady motion of a long bubble advancing under either prescribed pressure p b or prescribed volume flux q b into a fluid-filled flexible-walled channel at zero Reynolds number, an idealized model for the reopening of a liquidlined lung airway. The channel walls are held under longitudinal tension and are supported by external springs; the bubble moves with speed U. Provided p b exceeds a critical pressure p crit , the system exhibits two types of steady motion. At low speeds, the bubble acts like a piston, slowly pushing a column of fluid ahead of itself, and U decreases with increasing p b. At high speeds, the bubble rapidly peels the channel walls apart and U increases with increasing p b. Using two independent timedependent simulation techniques (a two-dimensional boundary-element method and a one-dimensional asymptotic approximation), we show that with prescribed p b > p crit , peeling motion is stable and the steady pushing solution is unstable; for p b < p crit the system ultimately exhibits unsteady pushing behaviour for which U continually diminishes with time. When q b is prescribed, peeling motion (with large q b) is again stable, but pushing motion (with small q b) loses stability at long times to a novel instability leading to spontaneous relaxation oscillations of increasing amplitude and period, for which the bubble switches abruptly between slow unsteady pushing and rapid quasi-steady peeling. This stick-slip motion is characterized using a third-order lumped-parameter model which in turn is reduced to a nonlinear map. Implications for the inflation of occluded lung airways are discussed.

Acoustically detectable cellular-level lung injury induced by fluid mechanical stresses in microfluidic airway systems

“…Considering that the length of a single cell (Ϸ40 m) is greater than that of the capillary wave, the maximum pressure drop and shear stress change within a cell can be estimated to be the largest change in the magnitude of the wall pressure and shear stress in the capillary wave region, which are 6,454.9 dynes/cm 2 and 97.58 dynes/cm 2 , respectively. It should be noted, however, that these values may change because of the factors that have not been taken into account in our study, such as pulmonary surfactant (42,43,44,56,57), airway compliance (35)(36)(37), and non-planar topography of the airway wall resulting from the protrusion of airway epithelial cells (38).…”

“…In addition, transient pressure waves generated by plug rupture are believed to produce abnormal breath sounds known as respiratory crackles that are routinely used as an indicator of a wide range of respiratory disorders in clinics (30)(31)(32)(33). Although clinically considered more as a symptom of respiratory diseases than as a cause, several theoretical investigations have suggested that the progression of liquid plugs or air bubbles during airway reopening can potentially generate deleterious fluid mechanical stresses characterized by large wall shear and normal stresses (34)(35)(36)(37)(38)(39). There have also been experimental studies based on excised lungs or in vivo animal models that corroborate the theoretical predictions and demonstrate severe tissue damage in surfactantdeficient lungs as a result of repetitive airway reopening (40,41).…”

Transient elastohydrodynamic drag on a particle moving near a deformable wall

“…Theory (23) predicts that the left-hand steady "pushing" solutions are unstable (at sufficiently long times) and therefore unrealizable experimentally when either the bubble pressure or the bubble volume flux is prescribed, although slow unsteady pushing may be an accessible mode of reopening transiently, particularly at low bubble pressures. Under low fixed volume flux, an oscillatory instability is predicted to develop at large times (23) for which the tube switches abruptly between slow pushing motion (when the bubble enlarges predominantly by transverse inflation without advancing appreciably) and rapid peeling motion (when the bubble enlarges by elongating rapidly while also shrinking in width); the longer the bubble, the greater the propensity to exhibit this unsteady oscillatory behavior. In conjunction with unpredictable avalanche effects operating throughout the airway network (2), such an instability would make airway recruitment under external ventilation difficult to control in the sense that imposing a steady volume flux can lead to strongly timedependent motion.…”

“…This low pressure provides the adhesive force that holds the tube walls together; the associated pressure gradient drives viscous flows within the tube.) For a bubble advancing with P b prescribed, provided Pb Ͼ Pbcrit, the peeling solution has been shown to be stable to time-dependent disturbances both theoretically (23,44) and experimentally (16,47,48); steady pushing motion, in contrast, is unstable to perturbations at long times if either the bubble pressure or the bubble volume flux is prescribed (23) and is therefore not accessible experimentally, although unsteady pushing motion may occur transiently at low Pb and low U. In summary, with pb prescribed, steady reopening is predicted to occur only for sufficiently large pb and it occurs preferentially at high bubble speeds in peeling mode; unsteady pushing motion may occur at low pressures and speeds (23), but this falls outside the framework of the present (steady) model.…”

“…However, instead of solving the full problem computationally, which is feasible but challenging, we reduce the problem to a more tractable form by exploiting some systematic approximations developed previously (15,23,33,44). One necessary condition allowing this approach is that the wall tension T should be large compared with ␥, which ensures that wall slopes are uniformly small.…”