Linear stability of pathological detonations

In this paper we investigate the linear stability of detonations in which the underlying steady one-dimensional solutions are of the pathological type. Such detonations travel at a minimum speed, which is greater than the Chapman–Jouguet (CJ) speed, have an internal frozen sonic point at which the thermicity vanishes, and the unsupported wave is supersonic (i.e. weak) after the sonic point. Pathological detonations are possible when there are endothermic or dissipative effects present in the system. We consider a system with two consecutive irreversible reactions A→B→C, with an Arrhenius form of the reaction rates and the second reaction endothermic. We determine analytical asymptotic solutions valid near the sonic pathological point for both the one-dimensional steady equations and the equations for linearized perturbations. These are used as initial conditions for integrating the equations. We show that, apart from the existence of stable modes, the linear stability of the pathological detonation is qualitatively the same as for CJ detonations for both one- and two-dimensional disturbances. We also consider the stability of overdriven detonations for the system. We show that the frequency of oscillation for one-dimensional disturbances, and the cell size based on the wavenumber with the highest group velocity for two-dimensional disturbances, are both very sensitive to the detonation speed for overdriven detonations near the pathological speed. This dependence on the degree of overdrive is quite different from that obtained when the unsupported detonation is of the CJ type.