Mathematical Sciences: Surfactant Effects in Viscous Fingering

小行星9704746 该项目的目标是研究表面活性剂对高度非线性区域自由表面流动的影响。 这包括严重变形并包含许多激发波长的界面的时间依赖性演化。 注意力将集中在一个简单的模型系统，即位移的粘性流体由一个较小的粘性流体在一个Hele-Shaw细胞（细胞组成的两个紧密间隔的玻璃板）。接口 流体之间的流动受到Saffman-Taylor不稳定性的影响，并形成指状图案。 这个系统作为一个模型，通过多孔介质的流动。 此外，由于Saffman-Taylor不稳定性是原油开采中的不希望的现象，因此该问题在工业上是重要的。 主要重点将是研究表面活性剂对 稳定的和随时间变化的手指图案，以及它们在增强或抑制指尖分裂和侧支不稳定性中的作用。 这与大多数其他系统中的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仅轻微变形，或接近稳态构型。 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，在科学和工程的许多领域都很重要。许多现代技术过程涉及自由表面流动。实例包括涂覆工艺， 喷墨印刷工艺和油回收。 表面活性剂在自由表面上的存在可显著影响聚合物的性质。 面 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可以引起流体-流体界面处的表面张力值的显著变化。 这可能对自由表面的演化产生显著影响。 这种效应在材料加工过程中特别重要， 太空的微重力环境 了解表面张力变化在界面演化中的作用，将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.