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 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 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 time-dependent 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.