Effective Rheology of Bubbles Moving in a Capillary Tube

TL;DR

This paper models the flow of bubbles in a capillary tube, deriving a relation between flux and pressure drop, and analyzing bubble configurations using physics-based equations.

Contribution

It introduces a novel derivation of bubble motion equations in capillaries and links flow behavior to a pendulum analogy, providing new insights into rheology.

Findings

Flux versus pressure drop follows a square-root relation.

Derived equations of motion from capillary and viscous forces.

Calculated probability distribution of bubble positions.

Abstract

We calculate the average volumetric flux versus pressure drop of bubbles moving in a single capillary tube with varying diameter, finding a square-root relation from mapping the flow equations onto that of a driven overdamped pendulum. The calculation is based on a derivation of the equation of motion of a bubble train from considering the capillary forces and the entropy production associated with the viscous flow. We also calculate the configurational probability of the positions of the bubbles.