Instabilities in chemically reacting mixtures

We shall study two different types of instability which arise in the theory of chemical and biochemical reactions [1], [2] and in the study of heat and mass transfer in porous catalysts [3], [4]. Peculiar physical effects involving local regions of oscillation and local instability (in a sense to be explained below) are observed experimentally. For each of the two different types of phenomena we believe that we have identified one possible mechanism for such occurrences. In § 2 we shall show that the sudden transition to localized temporal oscillation is reflected in a special situation first observed by N. Levinson [5]. The underlying chemistry and mathematics is introduced via a very simple initial value problem for a model system of reaction equations. A singular perturbation analysis clearly reveals the structure of the solution and also the mechanism which governs the occurrence of the oscillatory instabilities. In § 3 we consider the phenomenon of localized steady spatial oscillation for general reaction-diffusion equations. By combining singular perturbation and generalized WKB type methods we present a general technique for studying this type of phenomenon