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A minimal model for curved detonations is studied, illustrating the role ofthe reaction rate on the detonation speed and its propagation limits. The modelis based on a simple extension of the minimal Fickett toy model for detonationsbased on the kinematic wave equation. The use of a simple depletion rateconditioned on the shock speed serves to illustrate its role in thequasi-steady structure of curved waves and their initiation from a strong blastwave. Calculations of strong initiation from a self-similar explosionillustrate the various asymptotic regimes of the transition to self-sustenanceand their link to the steady wave structure. We recover the asymptotic regimesof detonation formation suggested by He and Clavin and modelled in the contextof Detonation Shock Dynamics by Stewart and collaborators. Following ananalysis using the shock change equation, we identify a unique criterion thatpermits to infer the critical energy for initiation from the competitionbetween energy release and geometric decay.

A minimal model for the role of the reaction rate on the initiation and self-sustenance of curved detonations