Abstract

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,  is smaller than a critical value there is no solution corresponding to steady rise. We explain this finding by studying the unsteady bubble motion for tube radius less than this critical value. We show the minimum spacing between the bubble and the tube goes to zero in infinite time. This leads to a rapid slow-down of the bubble’s mean speed proportional to the inverse-square of time giving the appearance of arrested motion. What may seem surprising is that this mean speed is negative: the bubble moves down rather than up. We explain this observation by the bubble’s expansion to the walls of the tube, pushing fluid in the direction opposite to gravity. Joint work with Jens Eggers