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High-frequency stability of detonations and turning points at infinity. (English) Zbl 1321.80004
The authors consider the spectral stability of strong detonations, extending previous works by J. J. Erpenbeck [Phys. Fluids 9, 1293–1306 (1966; Zbl 0144.47204)]. They prove that for type D detonations there exists a uniform cutoff magnitude for stability independent of the frequency direction. They start with the Zeldovitch-von Neumann-Döring (ZND) evolution system \begin{aligned} & \partial _{t}v+\mathbf{u}\cdot \nabla v-v\nabla \cdot \mathbf{u}=0, \\ & \partial _{t}\mathbf{u}+\mathbf{u}\cdot \nabla u+v\nabla p=0, \\ &\partial _{t}S+\mathbf{u}\cdot \nabla S=-r\Delta F/T, \\ &\partial _{t}\lambda +\mathbf{u}\cdot \nabla \lambda =r, \end{aligned} which couples the compressible Euler equations for a reacting gas to a reaction equation. Here $$v$$ is the specific volume, $$\mathbf{u}$$ is the particle velocity, $$\lambda$$ is the mass fraction of reactant, $$S$$ is the entropy, $$p=p(v,S,\lambda )$$ is the pressure, $$T$$ is the temperature, $$\Delta F$$ is the free energy increment and $$r(v,S,\lambda )$$ is the reaction rate function. The authors look for a steady planar strong detonation profile that is a weak solution depending only on $$x$$ with a jump at $$x=0$$, where the solution must satisfy a Rankine-Hugoniot condition. The detonation solution is given a profile $$P(x)=(v,u,0,0,S,\lambda )$$, where $$u>0$$ is the $$x$$-component of the particle velocity. The spectral stability of a ZND profile is governed by the nonautonomous $$5\times 5$$ system of linear ordinary differential equations $$\frac{d\theta }{dx}=-\mathcal{G}^{t}(x,\tau ,\varepsilon )\theta$$ on the half-line $$x\geq 0$$. Here $$\mathcal{G}^{t}$$ is the transpose of the matrix $$\mathcal{G}(x,\tau ,\varepsilon )$$ obtained by linearizing the ZND system about the profile $$P(x)$$ and taking the Laplace transform in time and the Fourier transform in the transverse spatial variables $$(y,z)$$. The authors also define the notion of a stability function. Analyzing the matrix $$\mathcal{ G}$$, they rewrite the nonautonomous system and they define a detonation profile of type D. The main result of the paper proves the existence of $$h_{0}>0$$ such that for all $$\zeta \geq 0$$, the stability function $$V(\zeta ,h)$$ vanishes in this D case for every $$0<h\leq h_{0}$$, under assumptions on the data. For the proof, the authors mainly study the above nonautonomous system and they establish properties of the decaying solution $$\theta$$.

##### MSC:
 80A25 Combustion 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 76N15 Gas dynamics (general theory)
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##### References:
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