# Arrested Bubble Rise in a Narrow Tube

## Abstract

If a long air bubble is placed inside a vertical tube closed at the top it can rise by displacing the fluid above it. However, Bretherton found that if the tube radius, *R*, is smaller than a critical value \(R_{c}=0.918 \; \ell _c\), where \(\ell _c=\sqrt{\gamma /\rho g}\) is the capillary length, there is no solution corresponding to steady rise. Experimentally, the bubble rise appears to have stopped altogether. Here we explain this observation by studying the unsteady bubble motion for \(R<R_{c}\). We find that the minimum spacing between the bubble and the tube goes to zero in limit of large *t* like \(t^{-4/5}\), leading to a rapid slow-down of the bubble’s mean speed \(U \propto t^{-2}\). As a result, the total bubble rise in infinite time remains very small, giving the appearance of arrested motion.

## Keywords

Singularities Thin film flow Lubrication theory Surface tension## Notes

### Acknowledgments

We are grateful to Howard Stone for pointing out to us the paradox of the stuck bubble, and for enlightening discussions. Discussions with John Kolinski and Hyoungsoo Kim on the possibility of experiments are also gratefully acknowledged.

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