Sal'nikov's reaction in a compressible gas

This thesis analyses a combustion reaction modelled by Sal'nikov's scheme in three geometries and contexts. Chapter 1 considers the reaction as it occurs through a pipe of infnite length, so that the gas properties vary only in one dimension along the pipe's axis. Various asymptotic techniques are applied to reduce the governing equations to an approximate, weakly non-linear system to which travelling wave solutions are sought. These solutions are then compared to the results of a numerical scheme, which models the full, highly non-linear governing equations. Finally, the numerical scheme is used to explore behaviours beyond the scope of the asymptotic analysis. Chapter 2 models the reaction in a spherically-symmetric cloud of gas, in which the gas properties are dependent only on the radial distance from the centre of the reaction. The governing partial differential equations are replaced by an approximate linear system, and solutions are found either as travelling waves or by using integral transforms, depending on the value of a bifurcation parameter which is also shown to determine the reaction stability. Again, these approximate solutions are compared to a numerical scheme to verify their accuracy. In chapter 3, a similar radially-symmetric geometry is considered, but the reaction is now assumed to take place behind a shock. In this regime, the governing partial differential equations only hold between the centre of the reaction and the shock face, while the strength and propagation speed of the shock must satisfy a system of algebraic jump conditions. A novel numerical scheme is used to model the propagation of the shock, and its accuracy is compared to that of a classic shock-capturing method.