Active control of propagating bubbles in Hele-Shaw channels

We explore the capabilities of active control to stabilise and manipulate propagating bubbles in the confined geometry of a rectangular Hele-Shaw channel. Several steadily-propagating solution branches exist in this system, with only one linearly stable and subsequent branches featuring increasingly deformed bubble shapes and increasing numbers of unstable eigenmodes. Our aim is to use feedback control and control-based continuation to detect and stabilise at least the first of these unstable branches. This system is an appealing prototype for control: the low Reynolds number and strongly confined geometry means the system state is essentially encapsulated in the interface shape, recent experimental realisations of this system are in good agreement with depth-averaged models, and the system responds well to actuation via fluid injection, but nonetheless practical implementation of feedback control presents significant challenges. In this talk, we use a depth-averaged model to illustrate how control would work in this system, including the design of a suitable gain matrix, the impact of control on the bifurcation structure, the complexities of controlling a propagating bubble moving past a fixed array of injection points, and how this simulation relates to experimental reality.