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

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 the 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.

Surfactant amplifies yield-stress effects in the capillary instability of a film coating a tube

“…(The scalings (2.22) correct those printed in (2.11) of Shemilt et al. (2022) but the equations derived there are still correct and consistent with what we derive here.) We then truncate the governing equations (2.10)–(2.21) at leading order in , with the exception of (2.16) where we retain the exact curvature, which is a commonly used device to improve accuracy in near-static regions of the flow (Gauglitz & Radke 1988; Halpern & Grotberg 1992, 1993; Halpern et al.…”

“…2010; Ogrosky 2021; Shemilt et al. 2022). The leading-order governing equations and boundary conditions are given in Appendix A, where we also detail the derivation of the long-wave evolution equations, which are presented in the remainder of this section.…”

“…The evolution equation for the interface position, , is where the axial volume flux is given by with If , or equivalently if there is no surfactant, , then and (2.27)–(2.29) reduce to the evolution equation for the surfactant-free problem (correcting a typographical sign error in the expression for given in (2.16) of Shemilt et al. 2022). The surfactant transport equation is where the surface velocity is with Equations (2.27)–(2.32) are solved subject to the boundary conditions which completes the long-wave system of equations.…”

“…Shemilt et al. (2022) derived reduced-order models to study the effect of viscoplastic rheology on the dynamics of thin films and of thick films in the lead-up to plug formation. It was found that increasing the capillary Bingham number, (a parameter proportional to the liquid yield stress), can suppress instability by rigidification of the layer, reduce deformation when there is instability and significantly increase the critical layer thickness required for plug formation to occur.…”

“…Our aim is to quantify the effects of surfactant on the capillary instability and on the previously established effects of the liquid's yield stress (Shemilt et al. 2022). We will derive evolution equations for the layer height and surfactant concentration using long-wave theory, and subsequently deduce a simpler version of these equations that is valid in the thin-film limit.…”