模擬可溶性界面活性劑問題之沉浸邊界法

Abstract

本論文之第一部分在探討沉浸邊界法中求解壓力項或是指示函數之精度，我們提供一維的理論證明與二維的數值結果來說明L1範數為一階收斂，L2範數為半階收斂，而L∞範數則存在O(1)的誤差。我們也將討論帶有其他種類界面奇異項的帕松方程式(Poisson Equation)，求解時的精度預測。

第二部份我們探討兩項流中分子兩端極性不同的界面活性劑，這些分子通常喜好駐於兩種液體的界面上，而且能透過吸收與釋放等過程跟可溶於液體中之界面活性劑交流。這類問題牽涉到在可變形界面上或是複雜區域內求解偏微分方程，因此在界面變動時如何精確計算界面與外在區域耦合之對流擴散方程實為本問題重點所在。我們首先改寫可溶的複雜區域內界面活性劑濃度方程，透過前述指示函數讓該方程鑲嵌於規則空間以方便計算，此外界面與外在區域之間的界面活性劑交流，例如吸收與釋放等過程，則可視為界面的奇異項導入外在區域之濃度方程中。在沉浸邊界法的模型之下，我們發展守衡數值格式求解界面與外在區域耦合之濃度方程，在數值計算下依舊保持界面活性劑之總質量守衡。我們做了一系列的數值測試來驗證我們提出的數值方法的正確性。我們也將過去針對不可溶性界面活性劑的研究工作拓展到可溶性界面活性劑，並且將探討界面活性劑的可溶性對於液體界面的形變造成的影響。

In the first part of this thesis, we provide a simplified one-dimensional analysis and two-dimensional numerical experiments to predict that the overall accuracy for the pressure problem or indicator function in immersed boundary calculations is first-order accurate in L1 norm, half-order accurate in L2 norm, but has O(1) error in L∞ norm. We also discuss the accuracy for another type of source terms for solving Poisson problems with singular conditions on the interface. In the second part, we consider the surfactant, which is an amphiphilic molecular, under multi-phase fluids. These particles usually favor the presence in the fluid interface, and they may couple with the surfactant soluble in one of bulk domains through adsorption and desorption processes. This type of problem needs to solve partial differential equations in deformable interfaces or complex domains. Thus, it is important to accurately solve coupled surface-bulk convection-diffusion equations especially when the interface is moving. We first rewrite the original bulk concentration equation in an irregular domain (soluble region) into a regular computational domain via the usage of the indicator function, which is described in previous part, so that the concentration flux across the interface due to adsorption and desorption processes can be termed as a singular source in the modified equation. Based on the immersed boundary formulation, we then develop a new conservative scheme for solving this coupled surface-bulk concentration equations which the total surfactant mass is conserved in discrete sense. A series of numerical tests has been conducted to validate the present scheme. As an application, we extend our previous work to the soluble case and investigate the effect of solubility on drop deformations.