Viscous froth model applied to multiple topological transformations of bubbles flowing in a channel: three-bubble case

Abstract

The flow of just three bubbles along a channel [1] captures many of the features of foam flow in porous media. Here a situation is considered in which bubbles are arranged in a staircase fashion zig-zagging across a channel (two bubbles attaching to one channel wall, but just a single bubble attaching to the other). The resulting topological asymmetry also implies asymmetry in the drag forces associated with foam film motion. When the system is driven fast enough, the imbalance in drag can cause the staircase structure to break. Bubbles then exchange neighbours during a so called T1 topological transformation. Previous work [2,3] has shown that the three-bubble system is sufficiently complex that it admits different “flavours” of T1 transformation, variously called T1c, T1u, T1l, and so on. Which flavour of T1 is selected depends on bubble sizes relative to channel size and also upon imposed driving pressure. All that previous work however focussed solely on the first T1 that the three-bubble system encountered [2,3]. The present contribution therefore examines the entire sequence of T1 transformations that a three-bubble system can undergo [1]. It is revealed that that the daughter states produced after the first T1 tend themselves to be unstable, meaning they are short-lived intermediates which break again via additional T1 transformations. Eventually the three bubble system reaches a stable final configuration that can then simply flow along. However, like the T1 transformations that produced them, these final configurations themselves come in different flavours. Which flavour is selected depends on bubble size relative to channel size and upon imposed driving pressure. A feature common to all the final configurations however is topological symmetry: in the final flowing structure, equal numbers foam films attach to either channel wall. One configuration which seems to be particularly favoured is a state in which just two bubbles stacked across the width of the channel continue flowing along, with a third bubble left behind altogether. This two-bubble configuration is found to have high mobility: not just higher mobility than the original parent three-bubble configuration, but also higher than any other competitor final configuration.