由於液滴在許多領域的應用廣泛，需要研究其原理以了解其中物理機制。本研究使用數值模擬，探討液滴在微流道內因溫度梯度所形成液滴與空氣介面的表面張力梯度，而使液滴移動之物理機制，運用等為函數法(Level set Method)，ALE運動描述法(Arbitrary Lagrangian Eulerian method, ALE method)以及連續表面力學法，利用有限元素法求解Navier-Stoke方程式與能量方程式。
本研究模擬液滴在微流道中移動時，表面張力(surface tension)受到溫度梯度作用而產生變化，液滴內部會產生兩個不對稱的熱毛細(thermocapillary)渦旋，液滴在較熱端的熱毛細渦漩大於較冷端的熱毛細渦漩，因此淨動量會將液滴由熱邊往冷邊推動。當液滴上板邊界條件為室溫時，液滴內部兩個渦漩尺寸會液滴移動而達到平衡，當上板邊界條件為絕熱時，由於液滴內部溫度分布類似於底板溫度分布，因此只會造成一個渦旋，這也使其熱毛細動量降低造成遷移速度較慢，而真實情況應介在兩者之間。上板邊界離液滴的距離越近，則對液滴的影響越大，上板邊界條件為室溫時，液滴的近穩定速度越快，而上板邊界條件為絕熱時，起始速度會變快，但近穩定速度沒有太大差距。溫度梯度越大所造成的液滴近穩態速度增加，也會使其近穩態的前進接觸角與後退接觸角的差值變大。而在不同起始靜接觸角的情況下，由於須保持質量守恆，所以會造成起始靜接觸角較小(大)的液滴，其足跡半徑較大(小)，所受到的熱毛細力較大(小)，而靜態接觸角小(大)於90˚的液滴，其毛細力方向相反(相同)於液滴移動方向，但馬蘭哥尼數(Marangoni number, Ma)較大的液滴移動速度較快，可以知道熱毛細力主導液滴移動機制。

The migration of a liquid droplet in a microchannel has widespread attention from many researchers. In this dissertation, a proper computational model is developed for investigating the transient migration of a liquid droplet in a microchannel. Numerical calculations are carried out by solving the Navier-Stokes equations coupled with the energy equation through the finite element method (FEM). The conservative level set method, the arbitrary Lagrangian-Eulerian (ALE) method, and the continuum surface force (CSF) method are employed to treat the movement and deformation of the droplet/air interface and the surface tension force during the motion process. The density and surface tension coefficient are dependent on temperature.
The study indicates that when a liquid droplet is of small size, two asymmetric thermocapillary vortices are generated inside the droplet. The thermocapillary vortex on the hot side is always larger in size than that appearing on the cold one. The net momentum of the thermocapillary convection inside droplet pushes the droplet moves from the larger vortex (hot side) to the smaller one (cold side). When the upper boundary condition is ambient temperature, the variation of the size of the thermocapillary vortex during the movement causes the speed of the droplet to initially increase and then decrease slowly until approaching constant value. When the upper boundary is insulation, the vortex inside the droplet becomes to one, and the speed of the droplet decrease rapidly after reaching the max value. The real phenomenon is between of ambient temperature and insulation. Microchannel of smaller height leads the upper boundary effects the migration of the droplet more. When the upper boundary condition is ambient temperature and smaller height microchannel, the stagnation point on the droplet/air interface is closer to the apex of the droplet. The temperature of the stagnation point in small height channel is lower than large channel. If the temperature difference at droplet/air interface is larger, the thermocapillary force is stronger, and the droplet migration speed is faster. When the upper boundary condition is insulation, smaller height channel leads the droplet moves faster at the beginning but also reaches a tiny quasi-steady migration. A higher imposed temperature gradient leads the droplet velocity to reach the maximal value earlier and have a higher final speed. If the static contact angle of the droplet is less (or higher) than 90 degree, the droplet speed is higher (or lower) since the net thermocapillary momentum is larger (or smaller).

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