The system is governed by three dimensionless parameters: (i) the Capillary number, Ca = mu U /sigma*, represents the ratio of viscous to surface-tension forces, where mu is the fluid viscosity, U is the finger's propagation speed and sigma* is the surface tension at the air-liquid interface; (ii) sigma = sigma*/(RK) represents the ratio of surface tension to elastic forces, where R is the undeformed radius of the tube and K its bending modulus; and (iii) Ainfinity = A*infinity/(4R2), characterises the initial degree of tube collapse, where A*infinity is the cross-sectional area of the tube far ahead of the bubble.
The generic behaviour of the system is found to be very similar to that observed in previous two-dimensional models (Gaver et al. 1996, Heil 2000). In particular, we find a two-branch behaviour in the relationship between dimensionless propagation speed, Ca, and dimensionless bubble pressure, pb = p*b/(sigma*/R). At low Ca, a decrease in pb is required to increase the propagation speed. We present a simple model which explains this behaviour and why it occurs in both two and three dimensions. At high Ca, pb increases monotonically with propagation speed and pb is proportional to Ca for sufficiently large values of sigma and Ca. In a frame of reference moving with the finger velocity, an open vortex develops ahead of the bubble tip at low Ca. As Ca increases, the flow topology changes and the vortex disappears.
An increase in dimensional surface tension sigma* causes an increase in the bubble pressure required to drive the air finger at a given speed. pb also increases with A*infinity and higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications. Furthermore, we find that the maximum wall shear stresses exerted on the airways during reopening may be large enough to damage the lung tissue.