The motion of bubbles in Hele-Shaw cells

Thesis

In this thesis, we present and analyse models for the motion of bubbles in a Hele-Shaw cell. We consider the experimentally relevant regime in which the bubbles are large enough that they are flattened against the top and bottom of the channel and assume pancake-like shapes, but are small enough to remain approximately circular when in plan view. We begin in Chapter 2 by considering the motion of a single bubble in a Hele-Shaw cell with a uniform background flow. The theoretical prediction for the bubble velocity is found to agree well with experiments. Including the effect of channel walls can either increase or decrease the bubble velocity depending on whether the bubble is travelling slower or faster than the background flow speed, respectively. Next, in Chapter 3 we consider the motion of two bubbles. We find that for two bubbles of different radii with the larger initially behind, in some cases the bubbles rotate around one another, while in others the bubbles collide. The motion of an arbitrary number of bubbles is considered in Chapter 4. We begin by developing a numerical method to solve the full model. Then, we derive approximate models, which significantly reduce the computational complexity while still retaining the qualitative behaviour of the full model. In Chapter 5, we return to the motion of a single bubble, now focusing on the bubble shape. We find that the aspect ratio of the bubble varies non-monotonically with its size, with it initially flattening in the direction of motion for small bubbles and elongating for large bubbles. We then consider the deformations of a pair of bubbles and find that, for two bubbles aligned with the uniform background flow, the one in front flattens, while the one behind elongates. In Chapter 6, we consider the propagation of a bubble in a Hele-Shaw cell with a non-uniform background flow. We find that the bubble centre travels along a streamline of the background flow although, unlike a tracer particle, the bubble does not travel at the same speed as the background flow. Using the same methodology, we derive approximate solutions for the motion of a bubble in more complicated domains including walls or obstacles. Finally, in Chapter 7 we consider the effect of surfactant on the motion of a bubble. Surfactants are commonly used in practice to stabilize the bubbles against coalesence. We find that a surfactant-laden bubble travels slower than a surfactant-free bubble under the same flow conditions.

Authors: Booth, DJ

• DOI: 10.5287/ora-e2ezvxgjv
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: fluid mechanics; Hele-Shaw; low-Reynolds number
• Subjects: Applied mathematics; Fluid mechanics