Air-driven dynamics of viscoplastic liquid layers

TL;DR

The paper develops a depth-averaged long-wave model for a viscoplastic liquid film driven by turbulent air, introducing key nondimensional groups and a plastocapillarity number $\mathcal{J}=B\mathcal{S}^3$ to capture yield-stress effects. Coupled theory and experiments show that small depth perturbations can trigger localized yielding, producing isolated nonlinear waves that can explosively grow by consuming upstream fluid and depositing a thinner layer behind, potentially causing blow-out. In Newtonian cases, multiple surface waves form and may blow out at high flow or thick layers, while viscoplastic layers exhibit yield-threshold–induced transitions, hysteresis, and, in some regimes, residual rigid waves. The results highlight the crucial role of mucus yield stress in air-driven clearance dynamics and point to extensions to elastoviscoplastic rheologies for more physiologically relevant cough models.

Abstract

Airway clearance by coughing is a key mechanism for mucus transport, particularly in obstructive lung diseases associated with altered mucus rheology. We investigate the dynamics of a viscoplastic liquid film driven by flow in a turbulent air layer, which is a model for air-driven mucus transport that incorporates yield-stress effects. Our theoretical analysis is based on a long-wave model for the liquid film flow, and we complement this with experiments, in which layers of Newtonian and yield-stress liquids are exposed to air flow in a rectangular duct. We demonstrate how perturbations to the layer depth can lead to localised yielding and wave generation. Rapid wave growth occurs when the fluid ahead of the oncoming wave is unyielded, so that as the wave propagates, it consumes this static fluid while depositing a much thinner film behind. This mechanism causes dramatic "blow-out" events in experiments, where liquid hits the roof of the tank. By contrast, in Newtonian thin films, multiple surface waves typically form, and blow-out only occurs in experiments when a Newtonian film is sufficiently thick.

Figures (21)

• Figure 1: Sketch of the long-wave model geometry.
• Figure 2: (a) Growth rate, $\lambda_r$, for $\mathcal{S}=10$, $\bar{h}=0.25$ and $B=\{0,2,2.2,2.4,2.6,2.8,3\}$. (b) Boundary between regions of convective and absolute instability for $B=\{0,0.5,1,1.5,2,2.5,3\}$. Instability is absolute if $\mathop{\mathrm{\mathcal{S}}}\nolimits>\mathop{\mathrm{\mathcal{S}}}\nolimits_\mathrm{crit}$ for given $B$ and $\bar{h}$. Where the lines terminate in (b) corresponds to the minimum $\bar{h}$ for which a uniform layer is in motion for the given $B$.
• Figure 3: Numerical solution of the long-wave model for $\bar{h}=0.25$, $\mathcal{S}=10$, $\mathop{\mathrm{\mathcal{J}}}\nolimits=2500$, $L=2\pi/k_m$, $\mathop{\mathrm{\mathcal{G}}}\nolimits=0$ and initial conditions \ref{['ICsin']}. (a) Time series of the deviation of the maximum layer height from the mean thickness. (b) Time series of $x_{\max}$, the location of the peak in $h$, superposed on a density plot of $h(x,t)$. The dashed red line and triangle indicate the linear growth rate \ref{['LSA_omr']} and phase speed \ref{['LSA_omi']}. (c) Snapshots, at the times indicated, of $h$ (blue), $Y_-$ (dashed red), and $u$ (as a density plot on the $(x,y)-$plane). In each snapshot, $Y_+=h$.
• Figure 4: Numerical solution of the long-wave model for $\bar{h}=0.25$, $\mathcal{S}=15$, $\mathop{\mathrm{\mathcal{J}}}\nolimits=8438$, $L=2\pi/k_m$, $\mathop{\mathrm{\mathcal{G}}}\nolimits=0$ and initial conditions \ref{['ICsin']}. (a) Time series of the deviation of the maximum layer height from the mean thickness. (b) Time series of $x_{\max}$, the location of the peak in $h$, superposed on a density plot of $h(x,t)$. The dashed red line and triangle indicate the linear growth rate \ref{['LSA_omr']} and phase speed \ref{['LSA_omi']}. The star in (a) indicates the time at which the computation ended due to the near blow-up of the solution. (c) Snapshot (at $t=6.16$) of $h$ (blue), $Y_-$ (dashed red), and $u$ (as a density plot on the $(x,y)-$plane); $Y_+=h$.
• Figure 5: Data from numerical computations with (a) $\mathcal{J}=0$ and (b) $\mathcal{J}=10^4$, all with $\mathop{\mathrm{\mathcal{G}}}\nolimits=0$ and $L=2\pi/k_m$. Each point represents one computation. Colour indicates the mean horizontal liquid flux, $\bar{q}$, scaled by $\bar{h}^2$, at the end time of the simulation. Grey crosses indicate solutions that were stopped prior to finite-time blow-up. Otherwise, computations were run to a maximum time of $150/\lambda_r$, where $\lambda_r$ is the Newtonian growth rate \ref{['LSA_omr']}. The dot-dashed line in (b) shows the yielding threshold for the base state, $\mathop{\mathrm{\mathcal{S}}}\nolimits^3(1+\bar{h})=\mathop{\mathrm{\mathcal{J}}}\nolimits(1-\bar{h})^3$. The dashed line shows the large-$\mathcal{S}$ prediction for the critical $\bar{h}$ for blow-up (see \ref{['app:largeS']}).