由於液滴在水平之固體表面的移動之不同應用，具備在工業、化學與生物學上的前景，最近正吸引許多學者與工程師的注意。本研究為使用數值模擬，探討固體表面溫度梯度所形成液滴與空氣界面表面張力梯度，而促成液滴移動之物理機制。運用有限元素法、等位函數法 ( Level Set Method )、 ALE 運 動 描 述 法（arbitrary Lagrangian Eulerian method, ALE method ） 與 連 續 表 面 力 學 法（continuum surface force method, CFS method），求解 Navier-Stokes 方 程 式 與能量方程式。
本研究指出當液滴為小尺寸時，液滴內會產生兩個不對稱的熱毛細力( ther-mocapillary force） 渦 漩。 在 液 滴 熱 邊 的 熱 毛 細 力 渦 漩 尺 寸 總 是 大 於 在 冷邊者 。液滴內 熱毛細力 渦漩之淨 動量驅動 液滴由熱 邊 （大 渦 漩） 向冷邊（小渦漩）移動。移動過程中熱毛細力渦漩尺寸的變化，造成液滴的速度於初期增加而後緩慢降低，直至接近一個近穩定速度之定值。增加溫度梯度，將導致液滴較早達到近穩定速度，及較高近穩定速度。在水平方向的熱毛細力與毛細力相互作用下，當液滴之靜接觸角小於（或大於）90度時，熱毛細力與毛細力具有抵制（或相乘）效果，使液滴的近穩定速度降低（或增加）。較短的滑動長度導致較低的液滴移動速度。而淨熱毛細力使液滴之近穩定移動速度與其尺寸呈線性關係。重力的效果不明顯而熱毛細力對流才是主導移動的主因。本研究模擬結果可驗證其他學者已發表之實驗結果。
當液滴尺寸變大，重力的影響愈重要。液滴內熱毛細力與浮力生成之對流，產生複雜的流體動力行為。對於中尺寸的液滴，它的近穩定移動速度達到最大，但隨液滴尺寸增大，浮力抑制熱毛細力淨動量，使近穩定移動速逐漸降低。對於大尺寸液滴，液滴內存在兩對對流渦漩，這是浮力對流與熱毛細力對流所產生的結果。由於較強浮力對流使淨熱毛細力動量降低，液滴的近穩定移動速度很快減低。上述模擬之液滴速度趨勢與其他學者已發表實驗結果有很好的一致性。

Recently, the migration of a liquid droplet on a horizontal solid surface has attracted widespread attention from many researchers and engineers because of its promising prospective in a variety of applications in biology, chemistry, and industry. In this dissertation, a proper computational model is developed for investigating the transient migration of a liquid droplet on a horizontal solid surface subjected to uniform temperature gradients. 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. Some physical properties of fluids dependent on temperature are also considered.
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). 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 a constant value. 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 degrees, the droplet speed is lower (or higher) since the net thermocapillary momentum in the horizontal direction is diminished (or enhanced) by the presence of capillary force. The lower slip length leads to the smaller droplet speed. In addition, the quasisteady migration speed of a small droplet is linearly proportional to its size due to the stronger net thermocapillary momentum. The effect of gravity is insignificant and the thermocapillary convection is dominated. The computational model is verified by comparing to the previous experimental results.
When the droplet turns into larger, the influence of gravity becomes important. The combined thermocapillary and buoyancy force driven convection produce complex dynamic behavior of fluid motion inside the droplet. In the middle size regime, the quasisteady migration speed of the droplet reaches a maximum, but this is gradually reduced as the droplet size increases due to the suppression of the net thermocapillary momentum by the buoyancy force. In the large droplet size regime, two pairs of convection vortices exist inside the droplet as a result of the appearance of the buoyancy-driven convection accompanying the thermocapillary convection. The quasisteady migration speed quickly diminishes, mainly due to the reduction of the net thermocapillary momentum from the stronger buoyancy convection. The droplet speed tendency is found to be a good agreement with the experimental results.

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