Control-based methods to unravel complexity for propagating Hele-Shaw bubbles.

Feedback control (FBC) can be used to force a system into a particular configuration, and also to help explore its nonlinear behaviour, including direct observation of unstable states. Importantly, Control-Based Continuation (CBC) can be performed directly in experiments, with no need for a mathematical model. We discuss how FBC and CBC can be implemented for the propagation of bubbles within a Hele-Shaw channel. Here the depth-averaged Darcy model predicts an infinite sequence of steadily-propagating solutions, one which is linearly stable. It is not clear to what extent the predicted unstable states persist into the real 3D system. We develop a system of FBC for the propagating bubble, with feedback delivered via fluid injection at the sides of the channel at amplitudes determined from real-time observation of the bubble shape. We use the model to design a suitable control gain and overcome the complexities of controlling a propagating bubble moving past a fixed array of actuators. For CBC, the target state is unknown a priori, but detected so that the control amplitudes are zero at steady state (non-invasive control). We explore the effect of noise and delay in this system, with particular reference to our experiments in progress for a deformable but stationary bubble.