| Summary: | 碩士 === 國立中正大學 === 物理所 === 95 === 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.
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