Initial Value Problems For Gas Dynamic Combustion Model

This paper is concerned with the initial value problem for the simplest Chapman-Jouguet (CJ) combustion model, the generalized Riemann problem for the gas dy-namic combustion model and the Riemann problem for the selfsimilar Zeldovich-von Neumann-Doring (ZND) combustion model for the pressure-gradient system.By the study of the ignition problem, the simplest CJ model is able to exihibit instability for combustible gas due to changing of combustion waves. In chapter 3, under the assumption that the flux function f(u) is nonconvex, we consider the instability of the unburnt gas for the scalar CJ combustion model. Under the entropy conditions in [65], letting the perturbation vanish in the result of the interactions of the combustion waves and noncombustion waves, we constructively obtain the solution when the ignited perturbation vanishes. We observe that the unburnt gas will be ignited which shows that the unburnt gas is unstable when the binding energy is larger than a critical value. While when the binding energy is less than the critical value, the solution as the ignited perturbation vanishes contains no combustion wave which indicates that the unburnt gas is stable. Furthermore, the transitions between deflagration and detonation appear.In chapter 4, the generalized Riemann problem for the pressure-gradient sys-tem is considered. Under the modified entropy conditions, we obtain the perturbed Riemann solution constructively in a neighborhood of the origin (t> 0) on the (x,t) plane. Compared with the corresponding Riemann solution, it can be found that for most of the cases, the combustion waves in the corresponding Riemann solutions are able to retain their forms after perturbation. We find that, for some cases, there are intrinsical differences between the structures of the perturbed Riemann solutions and the corresponding Riemann solutions. Especially, a strong detonation in the corresponding Riemann solution may transform into a weak deflagration coalescing with pre-compression shock wave after perturbation. And although there is no com-bustion wave of the corresponding Riemann solution, the combustion wave occurs after perturbation which shows the instability for unburnt gases.In chapter 5, we construct the selfsimilar solutions of the Riemann problem for the selfsimilar ZND (SZND) combustion model for the pressure-gradient sys-tem and study the limit behaviors of the Riemann solutions. By investigating the temperatures along the SZND burning solutions, we can determine when the Rie-mann solutions of the corresponding C J model are exactly the limits of the Riemann solutions of SZND model as the reaction rate tends to infinity.