有界面Stokes方程之邊界積分方法

Abstract

本文主要的目的是研究二維不可壓縮Stokes方程的邊界積分方法。此數值方法是基於Poisson方程的基本解，和將界面速度表示為Green函數與奇異來源項之捲積。我們分析積分方程的奇異性以及將積分方程分解成為平滑部分和奇異部分。前者可以利用梯形法作積分計算，後者則利用quadrature形式來計算。兩個數值計算格式被提出用來模擬彈性界面的動態，一個是顯式的計算格式，另一個是隱式的計算格式。在數值實驗中，我們首先模擬橢圓彈性界面在靜止流體的動態，得到一個二階收斂的結果。第二個數值實驗是模擬單一vesicle在剪流中的動態，得到一系列與理論相對照的數值結果。

The essential purpose of this thesis is to study the boundary integral method for two dimensional incompressible Stokes flows. The method is inspired by the fundamental solution of Poisson equation, and presents the interfacial velocity in integral formulae, as convolution form of a Green’s function and a singular source term. Once the formulae are clear, we analyze the singularity in the integral equations and split it into a smooth part and a singular part. The former can be treated by the trapezoidal rule and the later is cured by quadrature form with specific weights. To simulate the dynamics of an elastic interface, two numerical schemes are proposed, one is the explicit scheme which a force in previous time step is equipped, the other is implicit so that a tension-like unknown is solved together with interfacial velocity. In numerical experiments, we first apply the method to an elliptic elastic material in a quiescent flow, and give a second-order convergence to the circular steady state. The second application is a vesicle suspended in a simple shear flow. A series of numerical studies about the tank-treading motion and the tumbling motion for a vesicle match previous works well.