Mathematical Sciences: The Mathematics of Reacting and Diffusing Systems

数学科学：反应和扩散系统的数学

• 批准号: 8703247
• 负责人: Paul Fife
• 金额: $ 7.14万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1988-09-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8703247&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Mathematics; Reacting; Diffusing

The proposed projects involve mathematical analysis and modeling in the fields of electro-chemistry, two-phase media and flame theory. The common tie among these problems is that they all deal with diffusive and/or convective transport, and interactive processes such as chemical reactions. There are many electrophoretic techniques commonly used to separate charged particles in solution. They all give rise to mathematical models in need of qualitative analysis. The specific projects proposed here have to do with the steady-state and transient behavior of models in isotachophoresis and isoelectric focussing (possibly also continuous flow electrophoresis). In the study of the statics and dynamics of two-phase continua, phase field models of various kinds are emerging as physically suggestive and numerically convenient approaches. Part of the proposal is to extend the previous analyses of these models, by the PI and Caginalp, in the directions of (1) higher (than second) order models, which will be closer in some sense to the underlying discrete lattice theories, (2) taking into account the interaction of the material with boundaries, and (3) stability considerations. To fully understand and model combustion in flames, the important chemical reactions should be known, as well as the extent of their participation in the combustion process. Along with this , a knowledge of the concentrations of the reactants, hence of the reaction rates and the ( at least approximate) flame speed, is desirable. The PI has very recently developed a strategy for obtaining this knowledge, using known reaction networks and rate constant data, ans successfully applied it to lean hydrogen-oxygen flames. The method needs further development and implementation with simple computer routines. It is proposed to investigate aspects of some nonstandard problems. This project falls into the general area of research on mathematical theory of reaction - diffusion phenomena. This theory attempts to explain in mathematical terms the phenomena of combustion, electrophoresis, flame propagation and diffusion of chemicals in a solution. It is a very active area of research nowadays.

拟议的项目涉及电化学、两相介质和火焰理论领域的数学分析和建模。这些问题之间的共同联系是，它们都涉及扩散和/或对流输送，以及化学反应等相互作用的过程。有许多常用的电泳技术来分离溶液中的带电粒子。它们都产生了需要定性分析的数学模型。这里提出的具体项目涉及等速电泳法和等电聚焦(也可能是连续流动电泳法)模型的稳态和瞬变行为。在两相连续介质的静力学和动力学研究中，各种相场模型作为一种物理暗示和数值方便的方法不断涌现。建议的一部分是将PI和Caginalp之前对这些模型的分析扩展到(1)更高(比二阶)模型在某种意义上更接近基本离散晶格理论的方向，(2)考虑材料与边界的相互作用，以及(3)稳定性考虑。为了充分理解火焰中的燃烧并建立模型，应该知道重要的化学反应，以及它们参与燃烧过程的程度。此外，还需要了解反应物的浓度，从而了解反应速度和(至少是近似的)火焰速度。PI最近开发了一种利用已知的反应网络和速率常数数据来获得这种知识的策略，并成功地将其应用于稀氢-氧气火焰。该方法需要用简单的计算机程序进行进一步的开发和实现。建议研究一些非标准问题的方方面面。本项目属于反应扩散现象数学理论研究的一般领域。这一理论试图用数学术语来解释化学物质在溶液中的燃烧、电离、火焰传播和扩散现象。这是当今一个非常活跃的研究领域。