The effect of viscoelasticity on the stability of a pulmonary airway liquid layer

Abstract

The lungs consist of a network of bifurcating airways that are lined with a thin liquid film. This film is a bilayer consisting of a mucus layer on top of a periciliary fluid layer. Mucus is a non-Newtonian fluid possessing viscoelastic characteristics. Surface tension induces flows within the layer, which may cause the lung's airways to close due to liquid plug formation if the liquid film is sufficiently thick. The stability of the liquid layer is also influenced by the viscoelastic nature of the liquid, which is modeled using the Oldroyd-B constitutive equation or as a Jeffreys fluid. To examine the role of mucus alone, a single layer of a viscoelastic fluid is considered. A system of nonlinear evolution equations is derived using lubrication theory for the film thickness and the film flow rate. A uniform film is initially perturbed and a normal mode analysis is carried out that shows that the growth rate g for a viscoelastic layer is larger than for a Newtonian fluid with the same viscosity. Closure occurs if the minimum core radius, R(min)(t), reaches zero within one breath. Solutions of the nonlinear evolution equations reveal that R(min) normally decreases to zero faster with increasing relaxation time parameter, the Weissenberg number We. For small values of the dimensionless film thickness parameter epsilon, the closure time, t(c), increases slightly with We, while for moderate values of epsilon, ranging from 14% to 18% of the tube radius, t(c) decreases rapidly with We provided the solvent viscosity is sufficiently small. Viscoelasticity was found to have little effect for epsilon>0.18, indicating the strong influence of surface tension. The film thickness parameter epsilon and the Weissenberg number We also have a significant effect on the maximum shear stress on tube wall, max(tau(w)), and thus, potentially, an impact on cell damage. Max(tau(w)) increases with epsilon for fixed We, and it decreases with increasing We for small We provided the solvent viscosity parameter is sufficiently small. For large epsilon approximately 0.2, there is no significant difference between the Newtonian flow case and the large We cases.

Figure 1

Variables describing geometry of the air-liquid interface: r^*=a is the tube radius, r^*=b is the unperturbed location of the air-liquid interface, and r^*=R^*(z^*,t^*) denotes the perturbed position of the interface at time t^*.

Figure 2

Results from normal mode analysis using the Jeffreys model with μ_s=0.5 showing how the growth rate g varies with wave number k for different values of We.

Figure 3

The effect of the solvent viscosity μ_s=μ₂∕We on the growth rate using the Jeffreys model, with We=15. Note that in the limit as μ_s→0 (λ₂→0), the linear Maxwell model is recovered, and for this choice of We, the growth rate is singular at k=0.526 and k=0.851.

Figure 4

The effect of the Weissenberg number We and the ratio of solvent viscosity to total viscosity μ_s on the maximum growth rate, g_max. Note that the linear Maxwell is obtained in the limit μ_s→0, and in this case g_max is singular at We=12. For μ_s>0 and any We the maximum growth rate remains finite.

Figure 5

Minimum core radius vs time for a Newtonian liquid layer with ε=0.13. Comparison between lubrication theory model and numerical solution of the Navier–Stokes equations using a finite volume method.

Figure 6

The minimum core radius R_min vs time t for a Newtonian fluid (solid line), an Oldroyd-B fluid with We=0.1 (dashed line) using the small We asymptotics of Sec. 3A, a Jeffreys fluid with We=0.1 (dash-dot line), and a Jeffreys fluid with We=1 (dash-dot-dot line). For the viscoelastic cases μ_s=0.5.

Figure 7

Evolution of a Newtonian liquid layer. The location of the air-liquid interface R(z,t) is plotted as a function of z, for 0≤z≤L∕2 at the times indicated in the figure legend. Here ε=0.13, A=0.01, L=2^3∕2π.

Figure 8

The order We correction to the location of the air-liquid interface as a function of z at the same times as in the previous figure and for the same ε using the small We Oldroyd-B model. Here μ_s=0.5.

Figure 9

Air liquid interface R(z,t) plotted as a function of z at different times (shown in the figure) during evolution toward closure for a Jeffreys fluid. Here ε=0.13, We=10, μ_s=0.5.

Figure 10

The wall shear stress τ_w as a function of z at different times (indicated in the figure). The parameters are the same as Fig. 9.

Figure 11

The film pressure p vs z for the same parameter values as in the previous figure.

Figure 12

Wall shear rate vs z for the same parameters as in Fig. 9.

Figure 13

The effect of the Weissenberg number We on the minimum core radius R_min=R(0,t), for two different solvent viscosities (a) μ_s=0.5 and (b) μ_s=0.01. The film thickness parameter is ε=0.13.

Figure 14

The effect of solvent viscosity μ_s on R_min using the Jeffreys model derived in Sec. 3B. Here ε=0.13, We=10.

Figure 15

The influence of We and ε on the closure time t_c for two different solvent viscosities (a) μ_s=0.5 and (b) μ_s=0.01.

Figure 16

Influence of the film thickness parameter ε and the Weissenberg number We_ref on the maximum wall shear stress τ_w, for two different solvent viscosities (a) μ_s=0.5 and (b) μ_s=0.01.