不可溶界面活性劑之流體模擬數值方法

Abstract

本論文之主要目的在於發展一個簡易且精確的數值方法，來處理含有不可溶界面活性劑的界面流問題。長久以來，界面流問題的數值模擬已經成為了解各種相關流體之現象的熱門管道。在這個論文中，我們先介紹一個有關界面流問題(包含moving contact line problems)的數學模型，並且提出一種immersed boundary method來處理二維流體中帶有不可溶界面活性劑之界面的數值模擬。這個數學模型可以寫成一般常見的immersed boundary method的公式，包含Eulerian座標系下的流體方程式以及建立在Lagrangian座標系中有關界面的變數，而這兩個座標系之間各個變數的轉換，則是藉由Dirac delta function來連結。界面上的作用力主要依靠表面張力的影響，而界面上的表面張力則隨著界面活性劑的分佈而有所不同。對於moving contact line problems，我們必需在contact line附近額外提供一個unbalanced Young force來趨動界面。利用Lagrangian markers來追蹤界面，我們可以導出一個簡單的界面活性劑方程。整個數值方法主要可分為幾個部分，首先計算界面所提供給流體的力量，再利用投影法算出流體的速度並內插求得界面移動的速度；算出新的界面位置之後，在界面的切線方向引入人工的速度場以達到界面上網格的均勻分佈；此間，界面活性劑方程也會受到這個人工切線速度的影響，所以活性劑方程需要做一些調整，而活性劑在界面上的濃度則經由這個調整過後的方程式來決定。在研究界面活性劑影響界面流問的過程中，最重要的一個關鍵在於保持界面活性劑的不可溶特性，而本論文主要的貢獻在於提出一個新的對稱的數值離散方法，來處理界面活性劑方程式，基於這個方法，活性劑在數值模擬過程中可以完全的被保持住。在數值結果方面，包括剪切流中水泡的形變及附著在固態物質上液滴等。本論文提出的數值方法可以有效的處理有表面活性劑的moving contact line problems。

Numerical simulations of the interfacial flows have been a popular way to study a variety of fluid-world phenomena for a long time. In this dissertation, a mathematical model for interfacial flow problems (including the moving contact line problems) is demonstrated and an immersed boundary method is proposed for the simulation of two-dimensional fluid interfaces with insoluble surfactant. The governing equations are written in a usual immersed boundary formulation where a mixture of Eulerian flow and Lagrangian interfacial variables are used and the linkage between these two set of variables is provided by the Dirac delta function. The immersed boundary force comes from the surface tension which is affected by the distribution of surfactant along the interface. In particular, the unbalanced Young force should be applied in the moving contact line problems to derive the interface movement near moving contact lines. By tracking the interface in the Lagrangian manner, a simplified surfactant transport equation is derived. The numerical method involves solving the Navier-Stokes equations on a staggered grid by a semi-implicit pressure increment projection method where the immersed interfacial forces are calculated at the beginning of each time step. Once the velocity field and interfacial configurations are obtained, an equi-distributed technique of the Lagrangian markers is applied to force the markers to reach a uniform distribution in physical space. Meantime, the surfactant transport equation should be modified due to the effect of the tangential velocity arising from the equi-distributed process. Then the surfactant concentration is updated using the modified transport equation. The essential purpose of this dissertation is to study the effects of insoluble surfactants in the interfacial flow problems. Since it is important to maintain the insolubility of the surfactant concentration, the main contribution of this work is to propose a new symmetric discretization for the surfactant concentration equation such that the total mass of surfactant is conserved numerically. In numerical experiments, a bubble rises in a gravitational field, a vesicle deforms in a shear flow, and a hydrophilic or hydrophobic drop adheres to a solid substrate, are typical examples to observe the effects of the surfactant. To our best knowledge, the numerical method we propose here provides a wonderful chance to simulate moving contact line problems with insoluble surfactant.