The Life and fate of a bubble in a Hele-Shaw channel

Motivated by a desire to understand complex transient behaviour in fluid flows, we study the propagation of an air bubble through a fluid-filled, geometrically-perturbed Hele-Shaw channel; a system which supports several stable modes of bubble propagation. Using experiments and numerical simulations of a depth-averaged lubrication model, we investigate the evolution of a centred bubble of fixed volume as a function of flow rate and initial shape. During its transient evolution, a bubble may undergo several topological changes in the form of breakup and coalescence, depending on parameter values. Long-term, either a single bubble is recovered or else multiple bubbles remain, whose relative separation increases with time. Despite its apparent complexity the bubbles’ transient behaviour is organised by a number of weakly unstable invariant solutions of the system, so-called edge states. An unusual feature of the system is that changes in topology due to bubble break-up or coalescence lead to changes in the invariant-solution structure during temporal evolution. Families of two-bubble and single-bubble invariant solutions can be related in the sense that their propagation speeds are identical, but there are also two-bubble solutions that do not have single-bubble equivalents. We explore how the bubble becomes increasingly sensitive to initial conditions and roughness of the channel as the flow rate increases with a view to establish the origin of the long and disordered transients observed.