# Trapping and displacement of liquid collars and plugs in rough-walled tubes

## TL;DR

This paper investigates how wall roughness in tubes affects the stability and displacement of liquid collars and plugs, providing threshold conditions and probabilistic analysis for their displacement under external forces.

## Contribution

It introduces a model for wall roughness affecting liquid structures in tubes and derives conditions for their stability and displacement, incorporating statistical roughness characteristics.

## Key findings

- Weak roughness can be characterized by azimuthally averaged radius.
- Conditions for equilibrium collar and plug locations are derived.
- Probability estimates for displacement under external forcing are provided.

## Abstract

A liquid film wetting the interior of a long circular cylinder redistributes under the action of surface tension to form annular collars or occlusive plugs. These equilibrium structures are invariant under axial translation within a perfectly smooth uniform tube and therefore can be displaced axially by very weak external forcing. We consider how this degeneracy is disrupted when the tube wall is rough, and determine threshold conditions under which collars or plugs resist displacement under forcing. Wall roughness is modelled as a non-axisymmetric Gaussian random field of prescribed correlation length and small variance, mimicking some of the geometric irregularities inherent in applications such as lung airways. The thin film coating this surface is modelled using lubrication theory. When the roughness is weak, we show how the locations of equilibrium collars and plugs can be identified in terms of the azimuthally averaged tube radius; we derive conditions specifying equilibrium collar locations under an externally imposed shear flow, and plug locations under an imposed pressure gradient. We use these results to determine the probability of external forcing being sufficient to displace a collar or plug from a rough-walled tube, when the tube roughness is defined only in statistical terms.