流體界面問題之研究

Abstract

流體界面問題旨在探討兩種不相混合流體(或者空氣與水)的界面問 題。它在自然界的物理現象或者微流體工業技術中扮演了重要角色。譬如， 薄膜成長的技術，噴墨式印表機的液滴形成、溼化問題，甚至水面上行走 昆蟲的流力問題，皆是界面現象的貢獻。由於它的高度複雜性及應用性， 過去一直深受應用數學家廣泛的研究，這些努力也帶動了對流體界面問題 的更深一層的了解與認識。 在此計畫裡，我們將針對界面流體問題進行其數學模型與數值計算相 關的研究，延續前期計畫的研究議題，並將推進到考慮非牛頓流體及三維 問題，我們特別針對下列四個主題做深入的研究： (一) 二維及三維界面及不規則區域藕合之偏微分方程數值解及其應用至可 溶性界面活性劑問題 (二) 非牛頓流體之數值方法 (三) 黏性與內力對水泡動態之影響及三維問題之研究 (四) 三維乾型泡沫之模擬

The study of the incompressible flows with interfaces is of major interests among applied mathematics community. It plays an important role in numerous natural phenomena and industrial applications, especially, for the hydrodynamics of micro-fluidic systems. For instance, the thin film flows in coating devices, the dynamics of liquid drops in ink-jet printing, the wetting phenomena on substrate, or even the hydrodynamics of water-walking insect, just to name a few. Since the complexity of effects associated with capillarity gains a more wide applications during the last century, the research effort spent on this topic is overwhelming in applied mathematics community. These effort brought in many remarkable successes in the understanding of the fundamental physics and the quantitative modeling of capillary phenomena in different problems. In this proposal, we shall focus our research topics on the different issues of mathematical modeling and numerical methods for incompressible flows with interfaces. In our previous NSC projects, NSC-97-2628-M-009-007-MY3 and NSC-98-2115-M-009-014-MY3, we have developed a series of numerical methods and applications for 2D interfacial flows. Therefore, in this proposal, we are moving forward to viscoelastic (non-Newtonian) flows and some 3D problems. In addition, we will also continue investigating the interfacial flows with soluble surfactant in which a coupled surface-bulk convection-diffusion equations must be solved. We will need to develop an efficient and also conservative scheme for those equations. In this proposal, we shall focus our research topics on the issues of mathematical modeling and numerical methods for the interfacial flows with interfaces. In particular, our research topics will concentrate on four different directions; namely, (1) 2D and 3D numerical schemes for solving coupled surface-bulk convection-diffusion equations with applications; (2) Numerical methods for viscoelastic interfacial flows; (3) Viscosity and inertia effects on vesicle dynamics and efficient algorithms for 3D vesicle problems; (4) 3D dry foam simulations and von Neumann law.