當水滴從空中掉落水面時，一般人預期水滴會迅速與水面融合(coalesce)。事
實上不然，Lord Rayleigh遠在1899年就指出，當高度不大時，水滴會停留在水
面數十毫秒，這個現象稱作延遲融合(coalescence delay)，而那段時間就叫停留
時間(residence time)。在這份論文研究裡，我們發現在相同的實驗參數與初始條
件下，水滴的停留時間呈現雙峰分布，並藉由改變高度、水滴半徑、以及黏滯
度來尋找背後的機制，且建構理論模型除了成功解釋現象，且提供相關臨界參
數的解析式。
當顆粒填入摩擦係數非零的容器裡，簡森效應(Janssen effect)指出底部壓力
並不會像液體一樣線性增長，而會因摩擦力而達到飽和。在論文的第二部分，
我們報導一個有趣的實驗現象，亦即藉由插入一根細棍，竟然就能拉起裝滿顆
粒的容器。試想容器內壁遠大於細棍的面積，但是在簡森效應裡，前者提供
的摩擦力僅能削弱顆粒的部分重量，後者竟然能完全舉起全部的顆粒，因此
是個值得探討的現象。我們建構簡易的理論，成功預測最低可拉起的顆粒深
度，以及分別來自細棍與容器內壁的摩擦力大小與方向，討論反簡森(reverse
Janssen)效應起的作用，並且使用光彈性材料，即時觀測內部的應力分布如何隨
拉力改變。

(Part I for the dual residence time of droplet) When droplets approach a liquid surface, they have a tendency to merge in order to minimize surface energy. However, under certain conditions, they can exhibit a phenomenon called coalescence delay, where they remain separate for tens of milliseconds. This duration is known as the residence time or the non-coalescence time. Surprisingly, under identical parameters and initial conditions, the residence time for water droplets is not a constant value but exhibits dual peaks in its distribution. In this paper, we present the observation of the dual residence times through rigorous statistical analysis and investigate the quantitative variations in residence time by manipulating parameters such as droplet height, radius, and viscosity. Theoretical models and physical arguments are provided to explain their effects, particularly why a large viscosity or/and a small radius is detrimental to the appearance of the longer residence time peak.

(Part II for the lifting a the granular box with a half-buried rod) As the granules are packed into a frictional container, the Janssen effect dictates a nonlinear and saturating pressure with depth. We presented a surprising finding that a thin rod should be able to lift the granular box above the table. In Janssen's theory, the friction force provided by the wall can only support a partial weight of the granules. How come the thin rod with a much less contact area can support the full weight of granules and the container? By allowing the friction on the wall to change its direction, we revised and generalized the Janssen equation that described the force balance on each layer of granules, and successfully predicted the critical depth of granules to enable a successful lifting. In addition, a photoelasticity experiment is employed to allow direct visualization of the strain distribution as the pull force changes, particularly the directional change of friction force from the wall.

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