臺灣博碩士論文加值系統

利用高精密度的數值方法模擬可互溶液滴，於均質的 Hele-Shaw流場中之運動現象。對影響參數，包括黏滯度差異、無因次流量（Peclet number, Pe）和液滴大小以及Korteweg應力，作一系統式分析，以瞭解各參數之重要性及影響效果。
Hele-Shaw流場為兩塊透明薄板，夾以極小的間隙接合而成的二維流場。由於其第三維的高度極小於其他兩維的長度，因此可將第三維的速度加以平均起來，視Hele-Shaw流場為一非旋轉的勢流場（potential flow）。計算模擬乃將統御方程式轉換為渦度-流函數之關係式，深入研究因濃度梯度所造成的速度散度，對流場之影響。當液滴黏滯度較環境流體低時，在下游部分的液滴前沿，指狀化現象受流場的控制參數影響極大，一旦黏滯度差異愈大，或是Pe值愈高時，指狀化情形將更為劇烈。至於在液滴黏滯度較環境流體高時，可溶液滴移動較慢且較為穩定，並在液滴後緣處產生一個尾巴外型。在考慮到Korteweg應力的影響時，Hu and Joseph在1992年曾利用線性穩定分析結果預測負的 值，可避免不穩定的發生；而從我們的模擬結果亦顯示這時的液滴，前沿將因Korteweg應力的影響，呈現穩定的狀態，產生相當類似Kopf-Sill and Homsy實驗觀察到的尾巴形液滴（tailed-bubbles）。這時的液滴外型可類比不可溶液滴對表面張力分佈的說法來預測之，且這些尾巴形液滴的移動速度均介於實驗所量測的範圍。

Time-dependent numerical simulations of the motion of miscible droplets in homogeneous porous medium are presented. Both the influences of conventional flow parameters, such as the mobility ratios, dimensionless flow rates and droplet size (Peclet number) and dynamic Korteweg stresses are analyzed systematically. Vorticity-streamfunction formulation is applied to make the effects of velocity divergence caused by the concentration gradient implicitly in the governing equations. If the droplet is less viscous than the surrounding fluids, various fingering patterns, which depend strongly on the flow control parameters, occur only on the leading front where the mobility ratio is unfavorable. The droplet always moves faster than the surrounding fluid. Vigorous fingerings are observed at higher mobility ratio, Peclet number. A slower and stable droplet with a tail on the leading front is formed for a more viscous droplet in a less viscous environment. If the dynamic Korteweg stresses are considered, a negative Korteweg stresses constant stabilizes the droplet significantly, as predicted by the stability analysis (Hu and Joseph, 1992). Tailed droplets, which are similar to the experimental shapes by Kopf-Sill and Homsy (1987), are found if the leading fronts are totally stabilized by Korteweg stresses. The droplet shape can be predicted by the distribution of surface tension similar to immiscible bubble. The moving velocities of these rear-tailed droplets are within the ranges measured in experiments.

目 錄
封面內頁
簽名頁
授權書1………………………………….…………………….………iii
授權書2……………………………….……………………….………iv
中文摘要………………………………….…………………….………v
英文摘要……………………...………….…………………….………vi
誌謝…………………………………… ………………………..….....vii
目錄………………………………….… …………… ………………viii
圖目錄………………………………………………………….……....x符號說明………………………………………………………...…....xiii第一章、 緒論…………………………………..………………...…….1
1.1 研究背景…..…..…….……………………………….……...1
1.2 研究目的…….……………………………………….……...6
第二章、 物理問題與統御方程式…………………………..…...…….7
第三章、 研究結果……………………...…..………..……………….11
3.1 參考範例……….…………………………….…..………….11
3.2 不加入Korteweg應力的影響………….………....………..15
3.2.1 Pe的影響……………………………………....………..15
3.2.2 黏滯比控制參數R的影響…………………..…..…….17
3.3 加入Korteweg應力的影響……………….…..……..……..18
3.3.1 無因次Korteweg控制係數δ的影響…....……..……..18
3.3.2 Pe的影響………………………………..………..……..20
3.3.3黏滯比控制參數R的影響…………………..…..……...22
第四章、 結論……………………………………….……….…….…24
參考文獻………………………………………………………..……..27
附錄一、論文架構流程圖..……………………………………..……..68
附錄二、統御方程式的推導…………………..……………….……..69
圖 目 錄
圖1： 可溶液滴於Hele-Shaw 流場之示意圖……….…….………...31
圖2：Hele-Shaw 裝置簡圖……………………………….…….…….32
圖3：實際實驗範例…………….……………..………...…..….……...33
圖4：各種Hele-Shaw流場…..………………………..…..…….…...34
圖5：6種液滴外型及U/V與Ca的關係圖…..……….……...………35
圖6：參考範例：Pe=2,000, R=-2.5 與 δ=0，在時間 t=0.2, 1, 2.5 與
3.12的濃度圖…...………………………………......…..………36
圖7：參考範例：在時間 t=1與2.0的渦度圖及流線圖.…..……....37
圖8：R=-2.5與δ=0最大渦度隨時間的成長情形……….…….……38
圖9：R=-2.5與δ=0，加權速度Uw隨時間的演變….………….…….39
圖10：R=-2.5與δ=0，液滴界面長度L隨時間的演變….…….…….40
圖11：R=-2.5與δ=0，液滴擴散長度ld隨時間的演變…….…...…41
圖12：參考範例：圖2中平均濃度的剖面圖…………….….……..42
圖13：加入初始擾動的參考範例，在t=0.2與1.0時的濃度圖.…...43
圖14：參考範例取寬徑比H/d=4在t=1的濃度圖……………...…….44
圖15：Pe=1,000, R=-2.5與δ=0，在t= 1時的濃度及渦度圖…..……45
圖16：Pe=200和500，R=-2.5與δ=0，在t= 1時的濃度圖…….…….46
圖17：Pe=4,000，R=-2.5與δ=0，在t= 1.2與2.8時的濃度及渦度
圖……………………………………………………….……..47
圖18：Pe=6,000, R=-2.5與δ=0，在t= 0.6與1時的濃度圖….……...48
圖19：Pe=2,000, R=-1與δ=0。在t=2與3.6時的濃度圖…….……...49
圖20：Pe=2,000與δ=0。在t= 0.2與1時，R=-3和-3.5的濃度
圖……………...……………….………………………..……..50
圖21：Pe=2,000與δ=0，最大渦度隨時間的成長情形…….……..51
圖22：Pe=2,000與δ=0，加權速度Uw隨時間的演變………….....52
圖23：Pe=2,000與δ=0，液滴介面長度L，隨時間的演變……….53
圖24：Pe=2,000與δ=0 ，液滴擴散長度ld隨時間的演變……….54
圖25：Pe=2,000, R=5與δ=0，在t=3時的濃度圖及流線圖..……...55
圖26：Pe=2,000，R=-2.5與δ= -10-6，在 t=0.2與1時的濃度
圖………………………………………………………..……..56
圖27：Pe=2,000，R=-2.5與δ = -10-5，在t=3時的濃度圖及流線
圖……………………………………………………….……..57
圖28：Pe=2,000，R=-2.5與δ = -10-5，液滴在 t=0.4, 1.2, 2與3.2
時的濃度0.99等高線圖………………………………..…….58
圖29：Pe=2,000，R=-2.5與δ=-10-5，不同時間所對應液滴中心線
的濃度梯度……………………………………………..……..59
圖30：R=-2.5與δ=-10-5，液滴對Pe=200, 500 , 1,000與4,000時間t=3.2的外型…..…………..…………..…………………….....60
圖31：R=-2.5與δ=-10-5，液滴整體速度U'與不同Pe的關係
圖…………………………………………..…………………..61
圖32：R=-2.5與δ=-10-5，加權速度與不同Pe 的關係圖……..….62
圖33：R=-2.5與δ=-10-5，液滴介面長度與不同Pe 的關係圖……63
圖34：R=-2.5與δ=-10-5，液滴擴散長度與不同Pe 的關係圖……64
圖35：Pe=2,000與δ=-10-5，液滴在R=-1, -2, -3.5，時間3.2時的
外形……………………………………………………………65
圖36：Pe=2,000與δ =-10-5，液滴整體速度U'與不同R的關係
圖………………………………………………………………66
圖37：液滴整體速度U'與不同Ca'的關係圖…………...……………67

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