液體有很多特別的現象,譬如毛細現象、虹吸現象等。毛細現象是液體在
細管或多孔物體內部,由液體與物體的附著力和因液體分子間內聚力而產生的
表面張力組合而成,令液體在不需要施加外力的情況下,流向細管狀物體或細
縫的現象[6]。只存在於大氣狀態下的水位還要高,即表面張力所產生的能量與
水位高度差的能量相等。
我們發現將鋁環拉出水或甘油面後,環中間的液面會抵抗重力而提升到比
周圍的液面高,類似毛細現象。由於我們的實驗裝置並不具有細管狀物體,因
此希望釐清它背後的物理機制和一般毛細有什麼不同,並進而探討它獨有的特
色,例如如果鋁環上方從原本開放改成封閉,或是直接換成實心的鋁圓柱。我
們將系統地比較這三種不同的裝置下,液柱高度和外界拉力如何隨環或圓柱提
升高度改變,並關心液體柱從可逆到不可逆的轉換,以及最終斷裂的方式 – 當
它採取的是類似液滴截斷成兩顆的 pinch-off,我們使用高速攝影機仔細觀察其
細頸的最小半徑及附近的輪廓,如何隨倒數時間演化,討論它是否具備自我相
似及普遍性性質。
最終,我們認為這個仿毛細現象的研究,可以應用到微型機器人在潮濕地
面的行動,想像我們穿藍白拖鞋在雨天地面行走時,普塔普塔的拉力是如何隨
鞋底和地面高度而改變,這關係到微型機器人的平衡。這種能量耗損有異於我
們熟悉的靜與動摩擦力,它是否也具備類似的「只跟正向力有關,和接觸面積
無關」性質,也是我們這篇論文會討論的。

The liquid is a fascinating system with many special phenomena, such as capillary
and siphon. The former describes the elevation or movement of a liquid column inside
a thin tubular object or a porous object, which property has been understood to arise
from the combination of adhesion between liquid/solid wall and the surface tension
generated by the cohesive force between liquid molecules.
We observed a similar phenomenon by pulling an aluminum ring out of a liquid.
Inside the cylindrical liquid curtain, the liquid was found to rise above the level outside.
How does the length of this capillary-like column increase with the height of the ring?
What happens if we seal the top of the ring, i.e., prohibit the air inside the ring (or close
hollow cylinder) from flowing? Furthermore, what if we replace the ring or hollow
cylinder with a solid cylinder? Finally, how does the transition from reversible to
irreversible regions vary among these three different designs? As for the eventual
breakage of the liquid column and/or curtain, we were also interested in exploring and
comparing it with the famous pinch-off phenomenon, in particular, whether the profile
and minimum radius of the liquid cross-section exhibit self-similarity and universal
properties.
One interesting area comes to our mind for application. That is the movement of a
mini-robot on a wet surface. Aside from the common frictions, static and dynamic, the
robot needs to lift its foot from time to time. The water or whatever liquid that wets the
surface is going to drag its foot, while the foot tries to balance and enable the pulling
motion.

致謝 ·······································································································ii
摘要 ······································································································iii
Abstract ··································································································iv
目錄 ·······································································································v
圖目錄 ··································································································vii
1 簡介··································································································1
1.1 Fragmentation of stretched liquid ligaments [1] ········································2
1.2 How cats lap: Water uptake by Felis catus[2] ···········································6
1.3 Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge [3]9
1.4 Self-similar capillary pinchoff of an inviscid fluid[4] ································12
2 拉膜實驗··························································································15
2.1 實驗裝置···················································································15
2.2 實驗步驟···················································································16
2.3 邊界判定···················································································17
3 類掃描穿隧顯微鏡的實驗·····································································18
3.1 實驗裝置···················································································19
3.2 實驗步驟···················································································19
4 量測力的實驗····················································································20
4.1 實驗裝置···················································································20
5 實驗結果與討論·················································································21
5.1 環的單截斷與雙截斷····································································21
5.2 蓋子與圓柱的輪廓變化·································································23
5.3 環的h0與 H 的關係圖 ··································································23
5.4 環的三個不同的區段····································································25
5.5 環的普遍性質·············································································26
5.6 環的不可逆過程的自相似性質························································27

vi

5.7 環在可逆過程的特殊角度關係························································28
5.8 不同的系統的拉力·······································································29
6 理論································································································30
6.1 歐拉-拉格朗日方程式(Euler-Lagrange Equation) ···································30
6.2 針對我們實驗裝置的理論······························································31
6.3 環的中間高度理論·······································································33
7 結論································································································34
8 未來展望··························································································35
9 參考文獻··························································································36
10 附錄一:餘弦相似性(cosine-similarity) ·················································38
11 附錄二:實驗儀器及架設步驟 ···························································39
11.1 鋁環的表格················································································39
11.2 裝置側視圖················································································40
11.3 作動裝置說明·············································································41
11.3.1 Arduino 的程式設定·······························································44
11.3.2 步進馬達驅動器設定 ·····························································45
11.4 燈具使用說明·············································································46
11.5 高速攝影機使用簡介····································································48
11.6 邊界判斷程式·············································································51
11.6.1 程式使用步驟 ······································································53
11.7 Tracker 使用··············································································57
11.7.1 Tracker 程式判定速度的使用步驟 ·············································58

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