The influence of van der Waals interactions on a bubble moving in a tube

TL;DR

This paper investigates how van der Waals forces influence the stability of a thin film around a bubble in a tube, predicting rupture conditions based on intermolecular forces and flow parameters.

Contribution

It introduces a combined numerical and asymptotic analysis of thin-film rupture caused by van der Waals interactions in bubble flows within capillary tubes.

Findings

Identifies a critical capillary number for film rupture.

Shows the effect of intermolecular forces on film stability.

Provides time-scale estimates for rupture events.

Abstract

We consider the unsteady thin-film dynamics of a long bubble of negligible viscosity that advances at a uniform speed in a cylindrical capillary tube. The bubble displaces a viscous, nonwetting fluid, creating a thin film between its interface and the tube walls. The film is considered thin enough that intermolecular forces in the form of van der Waals attractions are significant and thin-film rupture is possible. In the case of negligible intermolecular forces, a steady-state solution exits where a film of uniform thickness is deposited in the annular region between the bubble interface and the tube walls. However, once intermolecular interactions are important, the interface is perturbed out of its steady state and either (i) the perturbation grows sufficiently before reaching the rear meniscus of the bubble such that rupture occurs; or (ii) the perturbation remains small due to weak intermolecular forces until it leaves the bubble interface through the rear meniscus. We obtain, both numerically and asymptotically, the time-scale over which rupture occurs, and thus, we find a critical capillary number, depending on the bubble length and the strength of the intermolecular forces, below which the film is predicted to rupture.