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

We investigate the unsteady motion of a long bubble advancing under either prescribed pressure pb or prescribed volume flux qb into a fluid-filled flexible-walled channel at zero Reynolds number, an idealized model for the reopening of a liquid-lined lung airway. The channel walls are held under longitudinal tension and are supported by external springs; the bubble moves with speed U . Provided pb exceeds a critical pressure pcrit , 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 pb . At high speeds, the bubble rapidly peels the channel walls apart and U increases with increasing pb . Using two independent time-dependent simulation techniques (a two-dimensional boundary-element method and a one-dimensional asymptotic approximation), we show that with prescribed pb\,>\,pcrit , peeling motion is stable and the steady pushing solution is unstable; for pb\,<\,pcrit the system ultimately exhibits unsteady pushing behaviour for which U continually diminishes with time. When qb is prescribed, peeling motion (with large qb ) is again stable, but pushing motion (with small qb ) 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.