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 $p_{\rm b}$ or prescribed volume flux $q_{\rm b}$ 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 $p_{\rm b}$ exceeds a critical pressure $p_{\rm crit}$, 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 $p_{\rm b}$. At high speeds, the bubble rapidly peels the channel walls apart and $U$ increases with increasing $p_{\rm b}$. Using two independent time-dependent simulation techniques (a two-dimensional boundary-element method and a one-dimensional asymptotic approximation), we show that with prescribed $p_{\rm b}\,{>}\,p_{\rm crit}$, peeling motion is stable and the steady pushing solution is unstable; for $p_{\rm b}\,{<}\,p_{\rm crit}$ the system ultimately exhibits unsteady pushing behaviour for which $U$ continually diminishes with time. When $q_{\rm b}$ is prescribed, peeling motion (with large $q_{\rm b}$) is again stable, but pushing motion (with small $q_{\rm b}$) 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.