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.
