We performed a theoretical investigation of the progression of a finger of air through a liquid-filled flexible-walled channel - an initial model of pulmonary airway reopening. Positive pressure, Pb* drives the bubble forward, and separates flexible walls that are modelled as membranes under tension, T, supported by linearly elastic springs with elasticity K. The gap width between the walls under stress-free conditions is 2H, and the liquid has constant surface tension, γ, and viscosity, μ. Three parameters define the state of the system: Ca = μU/γ is a dimensionless velocity that represents the ratio of viscous to capillary stresses; η = T/γ is the wall tension to surface tension ratio, and γ = KH2/γ is the wall elastance parameter. We examined steady-state solutions as a function of these parameters using lubrication analysis and the boundary element method.These studies showed multiple-branch behaviour in the Pb-Ca relationship, where Pb = Pb*/(γ/H) is the dimensionless bubble pressure. Low Ca flows (Ca [Lt ] min (1, (Γ3/η)1/2)) are dominated by the coupling of surface tension and elastic stresses. In this regime, Pb decreases as Ca increases owing to a reduction in the downstream resistance to flow, caused by the shortening of the section connecting the open end of the channel to the fully collapsed region. High Ca behaviour (max (1, (γ3/η)1/2) [Lt ] Ca [Lt ] η) is dominated by the balance between fluid viscous and longitudinal wall tension forces, resulting in a monotonically increasing Pb–Ca relationship. Increasing η or decreasing Γ reduces the Ca associated with 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 [les ] Pyield*/(γ/L*) [les ] 2, which is less than the minimum pressure for steady-state reopening, Pmin/(γ/L*), where L* is the upstream channel width. Since Pyield* < Pmin*, the theory implies that low Ca reopening may be unsteady, a behaviour that has been observed experimentally. Our results are consistent with experimental observations showing that Pb* in highly compliant channels scales with γ/L*. In contrast, we find that wall shear stress scales with γ/H. These results imply that wall shear and normal stresses during reopening are potentially very large and may be physiologically significant.

The dynamics of a thin film of Newtonian fluid coating the inner surface of an elastic circular tube is analysed. This problem is motivated by an interest in the closure of small airways of the lungs either by formation of a liquid bridge, the collapse of the airway wall or a combination of both processes. Liquid bridge formation is due to the destabilization of the liquid film that coats the inner surface of airways, while wall collapse can be due to either the high surface tension of the air–liquid interface or the flexibility of the wall.Nonlinear evolution equations for the film thickness and wall position are derived using lubrication theory, but an accurate representation of the curvatures of both the liquid and wall interfaces is employed which is valid for thick films. These approximations allow closure to be predicted. In addition, these approximations are justified by comparison with rigid-wall results obtained by solving the full Navier–Stokes equations and because fluid inertia only becomes important in the very late stages of closure. The linear stability of these equations is examined using normal-mode analysis for infinitesimal disturbances and the nonlinear stability is investigated by solving the governing equations numerically using the method of lines. Solutions show that there is a critical film thickness, strongly dependent on fluid and wall properties, above which unstable waves grow to form liquid bridges. The critical film thickness decreases with increasing surface tension or wall compliance since waves grow faster. Even for relatively stiff airways, the volume of fluid in the liquid lining required for closure can be approximately 70% of the volume for the rigid-tube case. Wall damping is an important effect only when the airway is sufficiently compliant. Airway closure occurs more rapidly with increasing unperturbed film thickness, surface tension and wall flexibility and decreasing wall damping.

A theoretical analysis is presented predicting the closure of small airways in the region of the terminal and respiratory bronchioles. The airways are modelled as thin elastic tubes, coated on the inside with a thin viscous liquid lining. This model produces closure by a coupled capillary-elastic instability leading to liquid bridge formation, wall collapse or a combination of both. Nonlinear evolution equations for the film thickness, wall position and surfactant concentration are derived using an extended version of lubrication theory for thin liquid films. The positions of the air-liquid and wall-liquid interfaces and the surfactant concentration are perturbed about uniform states and the stability of these perturbations is examined by solving the governing equations numerically. Solutions show that there is a critical film thickness, dependent on fluid, wall and surfactant properties above which liquid bridges form. The critical film thickness, epsilon c, decreases with increasing mean surface-tension or wall compliance. Surfactant increases epsilon c by as much as 60 percent for physiological conditions, consistent with physiological observations. Airway closure occurs more rapidly with increasing film thickness and wall flexibility. The closure time for a surfactant rich interface can be approximately five times greater than an interface free of surfactant.

Articles you may be interested inEffect of surfactants on the instability of a two-layer film flow down an inclined plane Phys. Fluids 26, 094105 (2014); 10.1063/1.4896144 Shear banding in time-dependent flows of polymers and wormlike micellesThe linear stability of a two-fluid shear flow with an insoluble surfactant on the flat interface is investigated in the Stokes approximation. Gravity is neglected in order to isolate the Marangoni effect of the surfactant. In contrast to all earlier studies of related fluid systems, we encounter ͑i͒ the destabilization ͑here, of a shear flow͒ caused solely by the introduction of an interfacial surfactant and ͑ii͒ the destabilization ͑here, of a system with a surfactant͒ caused solely by the imposition of a Stokes flow. Asymptotic long-wave expressions for the growth rates are obtained.

A computational study is presented for the transport of liquids and insoluble surfactant through the lung airways, delivered from a source at the distal end of the trachea. Four distinct transport regimes are considered: 1) the instilled bolus may create a liquid plug that occludes the large airways but is forced peripherally during mechanical ventilation; 2) the bolus creates a deposited film on the airway walls, either from the liquid plug transport or from direct coating, that drains under the influence of gravity through the first few airway generations; 3) in smaller airways, surfactant species form a surface layer that spreads due to surface-tension gradients, i.e., Marangoni flows; and 4) the surfactant finally reaches the alveolar compartment where it is cleared according to first-order kinetics. The time required for a quasi-steady-state transport process to evolve and for the subsequent delivery of the dose is predicted. Following fairly rapid transients, on the order of seconds, steady-state transport develops and is governed by the interaction of Marangoni flow and alveolar kinetics. Total delivery time is approximately 24 h for a typical first dose. Numerical solutions show that both transit and delivery times are strongly influenced by the strength of the preexisting surfactant and the geometric properties of the airway network. Delivery times for follow-up doses can increase significantly as the level of preexisting surfactant rises.