The steady propagation of a bubble in a flexible-walled channel: Asymptotic and computational models

The steady motion at zero Reynolds number of a semi-infinite bubble through a fluid-filled, flexible-walled channel, a model for the reopening of a collapsed lung airway, is described using asymptotic and numerical methods. The channel walls are membranes that are supported by external springs and are held under large longitudinal tension. An asymptotic analysis is presented under the assumption that membrane slopes are uniformly small. Near the bubble tip, the flow is equivalent to that of a semi-infinite bubble in a weakly tapered channel. Key features of this two-dimensional flow are matched to long-wavelength approximations describing the remainder of the solution domain. The analysis is valid for a wide range of bubble speeds, and it takes a particularly simple form when the bubble peels apart the channel walls as it advances. Predictions of bubble pressure as a function of bubble speed are validated by comparison with existing computations, new boundary-element simulations describing bubble motion in a channel with one rigid and one deformable wall, and experiments.