We numerically study the dynamics of a hollow water droplet falling in the air under the action of gravity. The focus of our study is to investigate the effects of the difference in radii (thickness) of the hollow droplet, gravity and surface tension at the air–water interface on shape oscillations and the breakup dynamics of the hollow droplet. We found that the oscillations of the inner interface (inner air bubble) are mostly periodic, while the outer interface undergoes irregular oscillations due to the interaction with the surrounding air. Increasing the ‘thickness’ of the hollow droplet decreases the amplitude of oscillations which further decays with time for high surface tension. It is observed that for a fixed value of the ‘thickness’ and low surface tension, the hollow droplet undergoes transition from the oscillatory regime to the dripping regime as it falls. The velocity contours are used to explain the behaviour observed in the present study. The deformation and shape oscillations of the hollow droplet are also compared with those observed in the case of a normal droplet of equal liquid volume falling in air.

中文翻译：

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空心水滴落在空气中的数值研究

我们数值研究了在重力作用下落在空气中的空心水滴的动力学。我们研究的重点是研究空心液滴的半径（厚度）差异、空气-水界面处的重力和表面张力对空心液滴的形状振荡和破碎动力学的影响。我们发现内部界面（内部气泡）的振荡大多是周期性的，而外部界面由于与周围空气的相互作用而发生不规则振荡。增加空心液滴的“厚度”会降低振荡幅度，随着时间的推移，高表面张力会进一步衰减。据观察，对于固定值的“厚度”和低表面张力，空心液滴在下落时从振荡状态过渡到滴落状态。速度等值线用于解释本研究中观察到的行为。空心液滴的变形和形状振荡也与在空气中落入等量液体体积的正常液滴的情况下观察到的情况进行了比较。