Mathematical Sciences: Surfactant Effects in Viscous Fingering

[704746]西格尔提出的项目的目标是研究表面活性剂对高度非线性的自由表面流动的影响。这包括严重变形和包含许多激发波长的界面随时间的演变。注意将集中在一个简单的模型系统上，即Hele-Shaw单元（由两个紧密间隔的玻璃板组成的单元）中的粘性流体被粘性较低的流体置换。流体之间的界面受到Saffman-Taylor不稳定性的影响，形成了一个手指状的图案。该系统可作为多孔介质流动的模型。此外，由于Saffman-Taylor不稳定性在原油开采中是一种不良现象，因此该问题具有重要的工业意义。主要的重点将是检查表面活性剂对稳定和时间依赖的手指图案的影响，以及它们在增强或抑制指尖分裂和侧分支不稳定性方面的作用。这与大多数其他系统中表面活性剂效应的研究不同，因为通常假设自由表面在空间中是固定的，或者只是轻微变形，或者接近稳态构型。所采用的方法包括渐近分析和数值模拟。这些研究的实际目标是控制手指的大小和形状，以期提高石油采收率。自由表面运动，如水面上的波浪或液体中上升的气泡，在许多科学和工程领域都很重要。许多现代技术过程涉及自由表面流动。例子包括涂层工艺、喷墨印刷工艺和油回收。表面活性剂在自由表面上的存在会显著影响表面的性能。常见的表面活性剂包括洗涤剂、脂肪酸和醇。即使是非常少量的表面活性剂也会引起流体-流体界面表面张力值的显著变化。这可以对自由表面的演变产生巨大的影响。这种效应在空间微重力环境下的材料加工过程中尤为重要。了解表面张力变化在界面演化中的作用，有望在工程应用中更好地控制自由表面运动。

9704746 Siegel The goal of the proposed project is to study the effects of surfactant on free surface flows in their highly nonlinear regime. This includes the time-dependent evolution of an interface that is severely deformed and contains many excited wavelengths. Attention will be focused on a simple model system, namely the displacement of a viscous fluid by a less viscous fluid in a Hele-Shaw cell (a cell consisting of two closely spaced glass plates). The interface between the fluids is subject to the Saffman-Taylor instability and develops a finger-like pattern. This system serves as a model for flow through porous media. Moreover, since the Saffman-Taylor instability is an undesirable phenomena in crude oil recovery, the problem is industrially important. The main emphasis will be on examining the effects of surfactants on steady and time-dependent finger patterns, and their role in enhancing or suppressing finger-tip splitting and side-branching instabilities. This differs from studies of surfactant effects in most other systems, since typically the free surfaces are assumed to be fixed in space or only slightly deformed, or close to a steady state configuration. Methods employed will include asymptotic analysis and numerical simulation. The practical goal of such studies is to control the size and shape of fingers, in the hopes of enhancing oil recovery. Free surface motions, such as waves on water or bubbles rising in a liquid, are important in many areas of science and engineering. A number of modern technological processes involve free surface flows. Examples include coating processes, ink jet printing processes, and oil recovery. The presence of surfactant on a free surface can significantly affect the properties of the surface. Common surfactants include detergents, fatty acids, and alcohols. Even very small amounts of surfactant can cause significant variation in the value of surface tension at a fluid-fluid interface. Thi s can have a dramatic influence on the evolution of the free surface. Such effects are particularly important during materials processing in the microgravity environment of space. Understanding the role of surface tension variations in interfacial evolution will hopefully lead to better control of free surface motion in engineering applications.