We consider the steady propagation of an air finger into a buckled elastic tube initially filled with viscous fluid. This study is motivated by the physiological problem of pulmonary airway reopening. The system is modelled using geometrically nonlinear Kirchhoff–Love shell theory coupled to the free-surface Stokes equations. The resulting three-dimensional fluid–structure-interaction problem is solved numerically by a fully coupled finite element method.
The system is governed by three dimensionless parameters: (i) the capillary number, Ca=μU/σ*, represents the ratio of viscous to surface-tension forces, where μ is the fluid viscosity, U is the finger's propagation speed and σ* is the surface tension at the air–liquid interface; (ii) σ=σ*/(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) A∞=A*∞/(4R2), characterizes the initial degree of tube collapse, where A*∞ 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, p*b/(σ*/R). At low Ca, a decrease in p*b is required to increase the propagation speed. We present a simple model that explains this behaviour and why it occurs in both two and three dimensions. At high Ca, p*b increases monotonically with propagation speed and p*b/(σ*/R) ∝ Ca for sufficiently large values of σ and Ca. In a frame of reference moving with the finger velocity, an open vortex develops ahead of the bubble tip at low Ca, but as Ca increases, the flow topology changes and the vortex disappears.
An increase in dimensional surface tension, σ*, causes an increase in the bubble pressure required to drive the air finger at a given speed; p*b also increases with A*∞ and higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications. If σ* is sufficiently small, steady airway reopening can occur when the bubble pressure is lower than the external (pleural) pressure, in which case the airway remains buckled (non-axisymmetric) after the passage of the air finger. Furthermore, we find that the maximum wall shear stresses exerted on the airways during reopening may be large enough to damage the lung tissue.
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