Existence of Detonations for Small Values of the Rate Parameter

The simplified, one-dimensional, steady-state Navier-Stokes detonation problem formulated by Hirschfelder, Curtiss, et al. is examined with respect to the existence of solutions when the reaction rate is small compared to the molecular collision rate. The latter condition is necessary for the assumed unimolecular mechanism to be valid, but is also the condition where these authors were unable to find solutions by numerical integration of the differential equations. It is shown that solutions indeed exist for arbitrarily small values of the reaction rate, and that these solutions approach the von Neumann model of a shock preceding a deflagration. An asymptotic approximation to the solution is given which should be adequate for such small reaction rates. In order to make the steady-state problem mathematically well defined, the usual difficulty of the very small reaction rate in the initial state is removed by imposing an ignition temperature. It is shown that if the reaction rate and Mach number are held fixed while the ignition temperature is raised slightly above the von Neumann spike temperature, the solution changes to a flame having the same hot boundary state, but with the detonation von Neumann spike state as its initial state.