Arrested bubble 'rise' in a narrow tube

A long air bubble placed inside a vertical tube closed at the top rises by displacing the fluid above it. Bretherton, however, 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. We explain this finding by studying the unsteady bubble motion for $R