[论文解读] Propagation of a finite bubble in a Hele-Shaw channel of variable depth

Andrés Franco-Gómez, Alice Thompson+2 more|arXiv (Cornell University)|Nov 7, 2017

Pickering emulsions and particle stabilization被引用 1

一句话总结

本研究探讨了在具有中心深度凸起（轨道）的Hele-Shaw通道中有限气泡的稳定性，表明在低毛细数（Ca）条件下，粘性力可稳定对称、居中气泡的传播；而较高Ca值则会破坏对称模式，导致气泡偏移中心。深度平均模型能准确预测稳态分岔及瞬态动力学，包括高流速下的气泡破裂和拓扑结构变化。

ABSTRACT

We study the propagation of finite bubbles in a Hele-Shaw channel, where a centred occlusion (termed a rail) is introduced to provide a small axially-uniform depth constriction. For bubbles wide enough to span the channel, the system's behaviour is similar to that of semi-infinite fingers and a symmetric static solution is stable. Here, we focus on smaller bubbles, in which case the symmetric static solution is unstable and the static bubble is displaced towards one of the deeper regions of the channel on either side of the rail. Using a combination of experiments and numerical simulations of a depth-averaged model, we show that a bubble propagating axially due to a small imposed flow rate can be stabilised in a steady symmetric mode centred on the rail through a subtle interaction between stabilising viscous forces and destabilising surface tension forces. However, for sufficiently large capillary numbers Ca, the ratio of viscous to surface tension forces, viscous forces in turn become destabilising thus returning the bubble to an off-centred propagation regime. With decreasing bubble size, the range of Ca for which steady centred propagation is stable decreases, and eventually vanishes through the coalescence of two supercritical pitchfork bifurcations. The depth-averaged model is found to accurately predict all the steady modes of propagation observed experimentally, and provides a comprehensive picture of the underlying steady bifurcation structure. However, for sufficiently large imposed flow rates, we find that initially centred bubbles do not converge onto a steady mode of propagation. Instead they transiently explore weakly unstable steady modes, an evolution which results in their break-up and eventual settling into a steady propagating state of changed topology.

研究动机与目标

• 理解具有中心深度凸起（轨道）的Hele-Shaw通道中有限气泡的稳定性。
• 确定粘性力与表面张力如何共同作用，使对称、居中气泡传播稳定或失稳。
• 通过与实验观测的稳态和瞬态气泡动力学对比，验证深度平均模型的准确性。
• 研究导致对称性破缺及气泡形状拓扑变化的分岔结构。

提出的方法

• 通过在Hele-Shaw池中设置中心轨道，实现轴向深度均匀性与局部狭窄，开展实验。
• 建立深度平均润滑模型，模拟不同流速和毛细数下气泡界面的演化。
• 数值求解深度平均方程，预测不同毛细数范围内的稳态模式及其稳定性。
• 应用分岔分析，识别不稳定性起始点及随着气泡尺寸减小，超临界叉子分岔的合并现象。
• 分析瞬态动力学，理解初始居中气泡无法收敛至稳态时的演化过程。
• 将模型预测与实验观测对比，验证深度平均方法的准确性。

实验结果

研究问题

• RQ1 在何种条件下，有限气泡能在变深度Hele-Shaw通道的轨道上保持对称居中？
• RQ2 粘性力与表面张力如何相互作用，以稳定或破坏对称气泡传播？
• RQ3 毛细数（Ca）在决定对称稳态气泡模式稳定性方面起什么作用？
• RQ4 随着气泡尺寸减小，对称传播稳定的Ca范围如何变化？
• RQ5 当初始居中气泡无法收敛至稳态时会发生什么？其最终拓扑结构由什么决定？

主要发现

• 在低毛细数（Ca）条件下，粘性力主导，可稳定对称、居中气泡的传播，此时表面张力效应较弱。
• 当Ca足够大时，粘性力转为破坏性作用，导致气泡偏离中心并发生非对称传播。
• 随着气泡尺寸减小，支持对称传播的Ca范围逐渐缩小，最终因两个超临界叉子分岔的合并而完全消失。
• 深度平均模型能准确预测所有观测到的稳态模式，包括对称性破缺转变。
• 在高施加流速下，初始居中的气泡无法达到稳态，而是在破裂前短暂探索弱不稳定模式，最终进入新的拓扑状态。
• 该模型成功捕捉了瞬态动力学及拓扑变化，这些现象仅靠稳态分析无法预测。