Analysis of Steady-State Supported One-Dimensional Detonations and Shocks

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Abstract
Consideration is given to the possible steady one-dimensional flows which can occur in a medium in which an arbitrary number of chemical reactions proceed behind an initiating shock, and the stability of solutions to the chemical rate equations is investigated. The theoretical apparatus is that of irreversible thermodynamics and nonlinear mechanics, with neglect of transport processes. Most of the discussion is concerned with detonations, but the analysis applies to all such reacting systems. For detonations, it is shown that under suitable conditions on the rate functions, there are stable solutions resulting in an equilibrium final state for detonation velocities equal to or greater than the ``equilibrium Chapman-Jouguet (C-J)'' value corresponding to tangency of the Rayleigh line and the equilibrium Hugoniot. The final state in such a flow is the high-pressure intersection of the Rayleigh line and the equilibrium Hugoniot. We suggest that these solutions correspond to piston-supported detonations after decay of initiation transients, and we further suggest that the equilibrium C-J detonation is stable with respect to removal of the piston support at sufficiently late times. The ``normal frozen C-J condition,'' corresponding to attainment of chemical equilibrium at a point where the flow velocity is sonic with respect to the ``frozen'' or high frequency sound speed, is shown to result in an unstable solution. Solutions corresponding to ``pathological detonations,'' in which the region of steady flow terminates at a point of incomplete reaction, are identified, but the conditions necessary or sufficient for their realization have not been obtained, nor has the nature of the subsequent time-dependent flow been elucidated. Thus their physical significance remains somewhat doubtful.