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A nonlinear evolution equation for pulsating Chapman–Jouguet detonations with chain-branching kinetics

A nonlinear evolution equation for pulsating Chapman–Jouguet detonations with chain-branching kinetics is derived. We consider a model reaction system having two components: a thermally neutral chain-branching induction zone governed by an Arrhenius reaction, terminating at a location where conversion of fuel into chain radical occurs; and a longer exothermic main reaction layer or chain-recombination zone having a temperature-independent reaction rate. The evolution equation is derived under the assumptions of a large activation energy in the induction zone and a slow evolution time based on the particle transit time through the induction zone, and is autonomous and second-order in time in the shock velocity perturbation. It describes both stable and unstable solutions, the latter leading to stable periodic limit cycles, as the ratio of the length of the chain-recombination zone to chain-induction zone, the exothermicity of reaction, and the specific heats ratio are varied. These dynamics correspond remarkably well with numerical solutions conducted earlier for a model three-step chain-branching reaction.