Two Kinds Of Typical Initial Value Problems For The Simplest Combustion Model

This dissertation is concerned with two kinds of initial value problems for the simplest combustion model. The first one is to consider the generalized Riemann problem for the simplest Chapman-Jouguet combustion model. The second one is to discuss the inteaction problem of shock and chemical reaction zone for the simplest Zeldovich-von Neumann-D(o|¨)ring combustion model.First of all, with the method of characteristic analysis, we consider the generalized Riemann problem for the simplest Chapman-Jouguet combustion model—the perturbation on initial binding energy. The problem is able to exhibit the instability for combustion wave due to the perturbation on initial binding energy. Under the entropy conditions given by [65] and [77], the solutions are obtained constructively in a neighborhood of the origin (t is small enough) on the physical plane(x-t plane). We accomplish the discussion according to the convexity and nonconvexity of flux function. Comparing the structure of the solutions to the generalized Riemann problem and the corresponding Riemann problem, we obtain the following results. Combustion waves in the corresponding Riemann solutions are able to retain their forms after perturbation on initial binding energy. However, the perturbation may bring essential changes to the combustion waves for some typical cases, such as the transition between different types of detonation and the transition between different types of deflagration. Furthermore, we can observe two important phenomena. The one is the transition from detonation to deflagration, which also appears in the numerical solutions in [44]. The other is the transition from deflagration to detonation (DDT), which has been one of the core problems in gas dynamic combustion.Second, with the refined mathematical analysis and characteristic analysis, we consider the interaction problem of shock and chemical reaction zone for the simplest Zeldovich-von Neumann-D(o|¨)ring combustion model. The problem helps us understand the influence of the shock on reaction processes on the basis of internal mechanism. Our work consists of two parts. In partⅠ, by analyzing the characteristics in chemical reaction zone, we obtain the global solutions of the interactions of shock and detonation zone of finite width constructively and find that the shock speeds up the reaction front in some cases. Moreover, we prove that the limits of the global solutions as reactive rate k tends to infinity are just the solutions of interactions of shock and detonation for the simplest Chapman-Jouguet combustion model. In partⅡ, the interactions of shock and deflagration zone of finite width are considered. The global solutions of the problem arc constructed by analyzing characteristics in the chemical reaction zone. We observe some burning gas in the reaction zone will be extinguished at a finite time in some cases. By studying the limits of the solutions as the reactive rate goes to infinity, we obtain that the limits are the solutions of the corresponding initial value problem for the simplest Chapman-Jouguet combustion model.