标题： 关于尖端有气泡的Saffman-Taylor指形流体的动力学振荡研究 显示英文标题

标题： On the oscillatory dynamics of a Saffman--Taylor finger with a bubble at its tip

摘要： 高速驱动进入Hele-Shaw通道的气液界面表现出复杂的动态行为，这种行为可能源于振荡动力学；然而，导致界面振荡的物理机制和动力学机制仍然不清楚。 我们将Couder的实验\textit{等。}（\textit{物理评论 A}，第34卷，1986年，第5175页）扩展，以系统地研究当一个小空气泡被放置在Hele-Shaw通道中一个稳定传播的空气指状物尖端时所产生的动力学。 该系统可以表现出稳定和振荡行为，我们证明了这些不同的行为分别发生在由流量和气泡大小定义的相空间中的明确区域。 对于足够大的流量，周期性的指状物振荡会转变为无序的动力学，表现为指状物尖端的不规则摆动。 我们演示了，在固定流量下，当气泡尺寸增加到一定程度，使气泡尖端的平面内曲率与指状物尖端的平面内曲率相匹配时，振荡开始出现。 这两个平面内曲率之间的相等导致驱动指状物的气泡横截面压力梯度消失，从而使指状物容易受到横向扰动的影响。 在扰动下，指状物和气泡恢复的时间尺度差异允许指状物-气泡系统中持续振荡的发展。 当气泡足够大以至于可以充当复合指状物的尖端时，振荡停止。 在高流量下的无序动力学与不稳定周期状态的瞬态探索一致，这表明类似的动力学可能在粘性指状物观察到的无序动力学中起作用。 显示英文摘要

摘要： The complex behaviour of air-liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet, both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder \textit{et. al.} (\textit{Phys. Rev. A}, vol. 34, 1986, p. 5175) to present a systematic investigation of the dynamics that result when a small air bubble is placed at the tip of a steadily propagating air finger in a Hele-Shaw channel. The system can exhibit steady and oscillatory behaviour, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger's tip. We demonstrate that at a fixed flow rate, the oscillations commence when the bubble size is increased sufficiently so that the decreased in-plane curvature of the bubble tip matches the in-plane curvature of the finger tip. The equality between the two in-plane curvatures causes the axial pressure gradient across the bubble, which drives the finger, to vanish, thus rendering the finger susceptible to lateral perturbations. Differing timescales for finger and bubble restoral under perturbation allow sustained oscillations to develop in the finger-bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a compound finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie the observed disordered dynamics in viscous fingering.