Stabilisation of propagating bubbles using control-based continuation

We explore the potential of feedback control to stabilise and detect unstable steady propagation modes for confined, deformable bubbles moving through a fluid-filled Hele-Shaw channel. We use a depth-averaged model as a numerical experiment to develop and test the capabilities of control-based continuation (CBC) in this system, with our work being the first use of CBC in free-surface fluid dynamics. We construct a suitable stabilising feedback gain, with control actuation delivered via fluid injection at the sides of the channel and determined from real-time observations of the bubble shape as viewed from above. We show how this feedback gain can be used to stabilise, detect and observe unstable solutions, without prior knowledge of their behaviour. As the steady solutions in fact involve steady bubble propagation along the channel length, we investigate both the idealised case of co-moving actuators, as well as the more plausible scenario when actuation is delivered through an array of equally-spaced injection points.