Self-similar solutions to the detonation equations in nonhomogeneous media

Using the method of invariance under Lie groups of transformations simplified forms of the equation governing detonations are obtained. In the specific case of an irreversible chemical reaction with rate function proportional to temperature, reduction to a two-dimensional phase plane is accomplished. Critical and singular points are noted in phase space and trajectories are found for different values of a modelling parameter k proportional to the ratio of chemical to kinetic energy. The presence of a line of singularities for large values of k is observed, and a perturbed solution of the governing equations is obtained for k large. The problem is generalized in a natural way by considering detonations in a non-homogeneous medium. All initial density distributions that permit self-similar solutions are found to be power laws with exponent $\eta$. For certain values of $\eta$ and k closed form solutions for the density, pressure, and particle velocity are obtained. In these cases the existence of singularities can be explained in terms of the physical conditions in the reaction zone. The case of non-homogeneous initial density also allows consideration of more realistic reaction rate functions including those with depletion factors and Arrhenius exponential-type factors. A numerical examination of these is included. The desire to study detonations in the case of constant initial density that involve more realistic reaction rates leads to the consideration of approximate forms, particularly the Langmuir-Hinshelwood reaction rate function. Once again singularities are found and discussed and perturbation methods are applied to obtain solutions