THE STEADY MOTION OF A SEMIINFINITE BUBBLE THROUGH A FLEXIBLE-WALLED CHANNEL

We performed a theoretical investigation of the progression of a finge r of air through a liquid-filled flexible-walled channel-an initial mo del of pulmonary airway reopening. Positive pressure, P-b, drives the bubble forward, and separates flexible walls that are modelled as mem branes under tension, T, supported by linearly elastic springs with el asticity K. The gap width between the walls under stress-free conditio ns is 2H, and the liquid has constant surface tension, gamma, and visc osity, mu. Three parameters define the state of the system: Ca = mu U/ mu is a dimensionless velocity that represents the ratio of viscous to capillary stresses; eta = T/gamma is the wall tension to surface tens ion ratio, and Gamma = KH2/gamma is the wall elastance parameter. We e xamined steady-state solutions as a function of these parameters using lubrication analysis and the boundary element method. These studies s howed multiple-branch behaviour in the P-b-Ca relationship, where P-b = P-b/(gamma/H) is the dimensionless bubble pressure. Low Ca hows (Ca much less than min (1, (Gamma(3)/eta)(1/2))) are dominated by the cou pling of surface tension and elastic stresses. In this regime, P-b dec reases as Ca increases owing to a reduction in the downstream resistan ce to flow, caused by the shortening of the section connecting the ope n end of the channel to the fully collapsed region. High Ca behaviour (max(1, (Gamma(3)/eta)(1/2)) much less than Ca much less than eta) is dominated by the balance between fluid viscous and longitudinal wall t ension forces, resulting in a monotonically increasing P-b-Ca relation ship. Increasing eta or decreasing Gamma reduces the Ca associated wit h the transition from one branch to the other. Low Ca streamlines show closed vortices at the bubble tip, which disappear with increasing Ca . Start-up yield pressures are predicted to range from 1 less than or equal to P-yield/(gamma/L*) less than or equal to 2, which is less th an the minimum pressure for steady-state reopening, P-min(gamma/L*), where L is the upstream channel width. Since P-yield* < P-min*, the t heory implies that low Ca reopening may be unsteady, a behaviour that has been observed experimentally. Our results are consistent with expe rimental observations showing that P-b in highly compliant channels s cales with gamma/L. In contrast, we find that wall shear stress scale s with gamma/H. These results imply that wall shear and normal stresse s during reopening are potentially very large and may be physiological ly significant.