Viscoplasticity can stabilise liquid collar motion on vertical cylinders

TL;DR

This paper explores how viscoplastic properties influence the formation, stability, and translation of liquid collars on vertical cylinders, revealing novel asymptotic behaviors and stabilizing effects absent in Newtonian fluids.

Contribution

It introduces a new asymptotic model for viscoplastic collars, showing how yield stress modifies instability and steady translation, contrasting with Newtonian fluid behavior.

Findings

Viscoplasticity stabilizes collar translation against small perturbations.

A critical film thickness determines whether collars reach steady translation.

The asymptotic model captures novel behaviors at collar ends.

Abstract

Liquid films coating vertical cylinders can form annular liquid collars which translate downwards under gravity. We investigate the dynamics of a thin viscoplastic liquid film coating the interior or exterior of a vertical cylindrical tube, quantifying how the yield stress modifies both the Rayleigh-Plateau instability leading to collar formation and the translation of collars down the tube. We use thin-film theory to derive an evolution equation for the layer thickness, which we solve numerically to examine the nonlinear dynamics. Instability and collar formation occur when gravity is sufficiently strong to make the fluid yield initially. We use matched asymptotics to derive a model describing the quasi-steady translation of a slender liquid collar when the Bond number is small. The structure of the asymptotic solution for a viscoplastic collar shares some features with the Newtonian version, but there are several novel asymptotic regions that emerge at the two ends of the collar. The global force balance, which determines the collar's speed, is modified by a leading-order contribution from viscous drag in the collar when the liquid is viscoplastic. We use the asymptotic model to describe slow changes in collar volume when the film thicknesses ahead of, and behind, the collar are unequal. When the film thickness ahead of the collar is less than a critical value that we determine, viscoplastic collars adjust their volume and reach a steadily-translating state. This contrasts with the Newtonian problem, where the only state in which steady translation occurs is unstable to small changes in the film thickness.