臺灣博碩士論文加值系統

在黏滯性流體中的拉普拉斯成長問題，我們選擇Hele-Shaw流體和量子霍爾液滴來探論。在Hele-Shaw流體中，由於流體的不可壓縮性，其流速滿足Darcy定律。考慮無表面張力的情況下，界面解可由弦方程式決定。雖然大多數情況的界面會形成"尖端(cusp)"，
但也存在少數穩態的指狀解。在加入表面張力修正後，將可避免尖端的形成，並可由保角映射定理得到界面隨時間演化的方程式。因一方面，量子霍爾液滴的形狀及成長和Hele-Shaw流體很像，其界面並無表面張力，所以隨著液滴的變大其形狀最後會形成尖端。利用Normal矩陣模型，量子霍爾液滴的形變及成長問頭，可視為磁性雜質中的A-B效應(Aharonove-Bohm effect)與量子化的Darcy定律。

Using Hele-Shaw fluid and quantum Hall droplet, we studied problems of the Laplacian growth in viscous fluids. Because of its incompressibility, the local velocity of the Hele-Shaw fluid must obey Darcy Law. With zero-surface tension, the interface equation can be determined by solving the string equation. Although the most of the solutions formed "cusp", there exist a few special solutions such as finger solutions that do not go to cusp. With nonzero-surface tension, the time evolution of the interface can be obtained by the conformal mapping theory without forming cusp. Besides, the shape and the growth of quantum Hall droplet are very much alike to the Hele-Shaw fluid
with zero-surface tension. In the end, the growth of the droplet will cause the cusp to form. The deformation and growth problems of the quantum Hall droplet can be regard as the A-B effect in magnetic impurities and quantized
Darcy Law.

致謝------------------------------------vii
摘要-------------------------------------ix
1 導論--------------------------------------1
2 二維黏滯性流體-Saffman-Taylor問題---------7
3 速度圖解法-------------------------------15
4 保角映射法-------------------------------27
5 矩陣模型---------------------------------37
6 整數量子霍爾流體-------------------------53
7 總結-------------------------------------67
附錄-------------------------------------71

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