Mathematical Sciences: The Mathematics of Reacting and Diffusing Systems

计划项目涉及电化学、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处理扩散和/或对流传输，以及诸如化学反应之类的相互作用过程。有许多电泳技术通常用于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与等速电泳和等电聚焦（也可能是连续流动电泳）中模型的稳态和瞬态行为有关。在两相连续体的静力学和动力学研究中，各种相场模型作为物理上的提示和数值上的方便的方法出现了。部分建议是扩展这些模型的先前分析，由PI和Caginalp，在(1)更高（比二阶）阶模型的方向上，这将在某种意义上更接近潜在的离散晶格理论，(2)考虑到材料与边界的相互作用，(3)稳定性考虑。为了充分理解和模拟火焰中的燃烧，必须知道重要的化学反应，以及它们在燃烧过程中的参与程度。除此之外，还需要了解反应物的浓度，从而了解反应速率和（至少近似的）火焰速度。PI最近开发了一种策略来获取这些知识，使用已知的反应网络和速率常数数据，并成功地将其应用于贫氢-氧火焰。该方法需要进一步的开发和简单的计算机程序实现。提出了一些非标准问题的研究方向。本课题属于反应扩散现象数学理论研究的一般范畴。该理论试图用数学术语解释溶液中化学物质的燃烧、电泳、火焰传播和扩散等现象。这是当今一个非常活跃的研究领域。

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.