The Erpenbeck High Frequency Instability Theorem for Zeldovitch–von Neumann–Döring Detonations

Abstract

The rigorous study of spectral stability for strong detonations was begun by Erpenbeck (Phys. Fluids 5:604–614 1962). Working with the Zeldovitch–von Neumann–Döring (ZND) model (more precisely, Erpenbeck worked with an extension of ZND to general chemistry and thermodynamics), which assumes a finite reaction rate but ignores effects such as viscosity corresponding to second order derivatives, he used a normal mode analysis to define a stability function ${V(\tau,\epsilon)}$ whose zeros in ${\mathfrak{R}\tau > 0}$ correspond to multidimensional perturbations of a steady detonation profile that grow exponentially in time. Later in a remarkable paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966; Stability of detonations for disturbances of small transverse wavelength, 1965) he provided strong evidence, by a combination of formal and rigorous arguments, that for certain classes of steady ZND profiles, unstable zeros of V exist for perturbations of sufficiently large transverse wavenumber ${\epsilon}$ , even when the von Neumann shock, regarded as a gas dynamical shock, is uniformly stable in the sense defined (nearly 20 years later) by Majda. In spite of a great deal of later numerical work devoted to computing the zeros of ${V(\tau,\epsilon)}$ , the paper (Erpenbeck in Phys. Fluids 9:1293–1306, 1966) remains one of the few works we know of [another is Erpenbeck (Phys. Fluids 7:684–696, 1964), which considers perturbations for which the ratio of longitudinal over transverse components approaches ∞] that presents a detailed and convincing theoretical argument for detecting them. The analysis in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) points the way toward, but does not constitute, a mathematical proof that such unstable zeros exist. In this paper we identify the mathematical issues left unresolved in Erpenbeck (Phys. Fluids 9:1293–1306, 1966) and provide proofs, together with certain simplifications and extensions, of the main conclusions about stability and instability of detonations contained in that paper. The main mathematical problem, and our principal focus here, is to determine the precise asymptotic behavior as ${\epsilon\to\infty}$ of solutions to a linear system of ODEs in x, depending on ${\epsilon}$ and a complex frequency τ as parameters, with turning points x _* on the half-line [0,∞).