Experimental analysis of bubble dynamics in Hele-Shaw channels using Control-based Continuation

Abstract

Understanding the transient behaviour of nonlinear systems can be approached from a dynamical systems perspective. The fundamental principle is that the system's evolution is guided by stable and unstable invariant states, which locally attract and repel the system's trajectories in phase space. Experimentally, only stable equilibria can be observed in experiments long enough for measurements of the system to be taken. Unstable equilibria may only be accessed briefly as the system continuously evolves. Control-based continuation (CBC) is an experimental procedure that uses active feedback control to locate and stabilize unstable equilibria, so that they can be observed in experiments. Continuation techniques are used to track solution branches of nonlinear systems as a system parameter is varied, enabling bifurcation diagrams to be constructed. The procedure is systematic and does not involve a mathematical model of the system. Stabilized controlled equilibria are achieved using noninvasive control, such that the control signal decays to zero as the system approaches the target state, ensuring the controlled equilibria is equivalent to the original (uncontrolled) system. Previous applications of CBC have been very successful in uncovering unstable equilibria of various nonlinear systems. All of these applications were conducted in experiments in which spatial dynamics played no role. In this thesis we present the first application of CBC to spatially extended systems. We investigate two nonlinear systems, both consisting of an inviscid (air) bubble confined in a Hele-Shaw channel, filled with viscous fluid. We first applied CBC to a bubble placed in a straining flow, where no stable equilibria exist. By controlling the bubble's position and shape we were able to track two unstable solution branches of distinct equilibria shapes, where the two branches are connected by a saddle-node bifurcation. We then studied the case of a bubble propagating through the channel with the inclusion of a small-depth perturbation (rail). With the experimental parameters we used, no on-rail stable equilibria exist, the bubble will always propagate steadily off-rail, however unstable on-rail equilibria do exist. To achieve actuation which moved with the bubble, we used an array of actuators, which activated/deactivated as the bubble entered their local vicinity. This allowed us to investigate time-dependent control signals, which had never been encountered using CBC but are vital to the analysis of noninvasive control. As these control signals are not equal to zero for all time (to be noninvasive), we found that a simple linear regression of time-averaged data points close to non-invasive control accurately estimated the true measurement of the equilibrium. By using positional and shape control, we were able to find two distinct unstable solution branches of on-rail equilibria, connected by a saddle-node bifurcation. We constructed two bifurcation diagrams, mapping the bubble's shape and velocity to the injection flow rate which drove the bulk flow.

Date of Award	13 Aug 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Anne Juel (Supervisor) & Alice Thompson (Supervisor)