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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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