Existence of Detonations for Large Values of the Rate Parameter

Recent investigations of idealized Navier-Stokes detonation structure have left unresolved the existence and nature of flow solutions in the case of very fast reactions. By formulating the problem explicitly in terms of a specified initial state and a specified hot boundary mass velocity wP (relative to the unreacted explosive), it is shown that suitable ``steady'' flows exist in all cases, providing that a rarefaction wave or a slower moving shock wave is allowed to follow the detonation front. An initial-state parameter Γ0, involving the reaction-rate ``pre-exponential factor'' and the thermal conductivity, is defined, as well as an associated critical value Γu(CJ), which depends on the heat capacity ratio, and the enthalpy and activation energy of the reaction. If Γ0 < Γu(CJ), then for small wP the predicted flow is the Chapman-Jouguet (CJ) detonation followed by a rarefaction. As wP is increased through the CJ mass velocity wC J, the flow changes continuously to increasingly overdriven detonations. If Γ0 > Γu(CJ), then for small wP the predicted flow is a weak detonation whose velocity D depends on Γ0, followed by a rarefaction. As wP is increased through the corresponding weak detonation mass velocity w(Γ0), the flow becomes the same weak detonation followed by a slower moving shock wave. With continued increase of wP this shock velocity increases until, at a certain value of wP, it becomes equal to the weak detonation velocity. At still larger wP, the flow changes continuously into an overdriven strong detonation. Both unimolecular and bimolecular kinetics are considered. So-called ``reasonable'' values of the unimolecular parameters lead to the possibility of the flows involving weak detonations, but as noted by Hirschfelder, Curtiss, et al., the conditions are then such as to require a quasi-unimolecular mechanism to become bimolecular, due for example to depletion of the activated state. When bimolecular kinetics are considered, reasonable parameter values appear to exclude the weak detonation flows.