IUMJ

Title: Existence and stability of curved multidimensional detonation fronts

Authors: Mark Williams, Gregory D. Lyng, Kris Jenssen and Nicola Costanzino

Issue: Volume 56 (2007), Issue 3, 1405-1462

Abstract:

The rigorous study of spectral stability for ZND detonations was begun by J.J. Erpenbeck in \emph{Stability of steady-state equilibrium detonations}, Physics of Fluids \textbf{5} (1962), 604--614. He used a normal mode analysis to define a stability function $V(\lambda, \eta)$, whose zeros in $\mbox{\upshape Re} \lambda > 0$ correspond to multidimensional perturbations of a steady planar profile that grow exponentially with time. In [J.J. Erpenbeck, \emph{Detonation stability for disturbances of small transverse wavelength}, Physics of Fluids \textbf{9} (1966), 1293-1306], he was able to prove that for large classes of steady ZND profiles, unstable zeros of $V$ always exist in the high frequency regime, \emph{even when} the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense (later) defined by Majda; subsequent numerical work has shown that unstable zeros usually exist in the medium frequency regime as well.\par In this paper we begin a rigorous study of the implications for \emph{nonlinear stability} of the spectral instabilities just described. We show that in spite of the existence of unstable zeros of $V(\lambda, \eta)$, one can prove the finite (but arbitrarily long) time existence of slightly curved, nonsteady, multidimensional detonation fronts for ideal polytropic gases in both the ZND and Chapman-Jouguet models. In the ZND case we show that this nonlinear stability problem is actually governed by a different stability function, $\Delta_{\mathrm{ZND}}(\hat{\lambda}, \hat{\eta})$, which turns out to coincide with the high frequency limit of $V(\lambda, \eta)/ |\lambda, \eta|$ in $\mbox{\upshape Re} \hat{\lambda} > 0$. Moreover, the above nonlinear stability result for ideal polytropic gases holds more generally in any situation where $\Delta_{\mathrm{ZND}}(\hat{\lambda}, \hat{\eta})$ is bounded away from zero in $\mbox{\upshape Re} \hat{\lambda} > 0$. We also revisit the argument of Erpenbeck (1966) in order to simplify and complete some of the analysis in the proof of the main result there.