Abstract In this study, we perform an experimental investigation on the dynamical behaviors of a liquid bridge formed during droplet coalescence. Notably, the two droplets are constrained respectively by two vertically aligned needles. Of particular interest is the pinch-off of the liquid bridge under this constrained condition. We observe three typical behaviors of the liquid bridge: (1) reaching a stable shape after damped oscillations (regime I); (2) pinching off at both sides in the first necking stage with the largest oscillation amplitude (regime II); and (3) pinching off at only the upper side after once or twice necking (regime III). To indicate the condition when the pinch-off of the liquid bridge may occur, we develop a phase diagram based on the radius ratio of the droplet to the needle R d / R n and the Bond number Bo, which respectively characterizes the relative importance of surface tension to the constraint force from needles and that of gravity to surface tension. In general, the dynamical behavior of the liquid bridge transitions from regime II to regime I with R d / R n decreasing, and the critical value for this transition is 1.6. This transition indicates that the constraint force keeps the liquid bridge stable while the surface tension promotes its instability. Surprisingly, once the Bond number is sufficiently high (e.g., Bo > 0.05), regime III (i.e., pinch-off at only the upper side) is observed when R d / R n becomes around the regime II to I transition (i.e., R d / R n = 1.6). It is because the effect of gravity becomes predominant over the surface tension and the constrained force from needles. Moreover, we analyze the pinch-off characteristics of the liquid bridge lying in regimes II and III, including its pinch-off time (i.e., the time interval from the formation and pinch-off of the liquid bridge) and its neck shape at the instant of pinching off.

中文翻译：

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约束条件下液滴聚结过程中液桥的夹断

摘要 在这项研究中，我们对液滴聚结过程中形成的液桥的动力学行为进行了实验研究。值得注意的是，这两个液滴分别受到两个垂直对齐的针的约束。特别令人感兴趣的是在这种约束条件下液桥的夹断。我们观察到液桥的三种典型行为：（1）在阻尼振荡后达到稳定的形状（状态 I）；(2) 在第一个颈缩阶段两侧夹断，振幅最大（第二阶段）；(3) 一次或两次颈缩后仅在上侧夹断（方式 III）。为了指示可能发生液桥夹断的条件，我们根据液滴与针的半径比 R d / R n 和键数 Bo 开发了相图，分别表征了表面张力对针约束力的相对重要性和重力对表面张力的相对重要性。一般来说，随着R d / R n 的减小，液桥的动力学行为从II 区过渡到I 区，该过渡的临界值为1.6。这种转变表明约束力保持液桥稳定，而表面张力促进其不稳定性。令人惊讶的是，一旦邦德数足够高（例如，Bo > 0.05），当 R d / R n 变得围绕区域 II 到 I 过渡（即，仅在上侧夹断）时，观察到区域 III（即， Rd/Rn = 1.6)。这是因为重力的影响超过了表面张力和针的约束力。而且，