Surface-tension-driven evolution of a viscoplastic liquid coating the interior of a cylindrical tube

TL;DR

This study models the surface-tension-driven instability of a viscoplastic mucus layer in airways, revealing how yield stress influences the conditions for airway closure, with implications for diseases like cystic fibrosis.

Contribution

The paper develops a novel evolution equation for a viscoplastic liquid layer in a tube, incorporating yield stress effects into airway closure modeling.

Findings

Higher yield stress increases the critical thickness for airway closure.

Two static solution branches predict initial instability conditions.

Increasing Bingham number reduces final layer deformation.

Abstract

One mechanism for airway closure in the lung is the surface-tension-driven instability of the mucus layer which lines the airway wall. We study the instability of an axisymmetric layer of viscoplastic Bingham liquid coating the interior of a rigid tube, which is a simple model for an airway that takes into account the yield stress of mucus. An evolution equation for the thickness of the liquid layer is derived using long-wave theory, from which we also derive a simpler thin-film evolution equation. In the thin-film case, we show that two branches of marginally-yielded static solutions of the evolution equation can be used to both predict the size of initial perturbation required to trigger instability and quantify how increasing the capillary Bingham number (a parameter measuring yield stress relative to surface tension) reduces the final deformation of the layer. Using numerical solutions of the long-wave evolution equation, we quantify how the critical layer thickness required to form a liquid plug in the tube increases as the capillary Bingham number is increased. We discuss the significance of these findings for modelling airway closure in obstructive conditions such as cystic fibrosis, where the mucus layer is often thicker and has a higher yield stress.