### Abstract

We use the classical normal mode approach of hydrodynamic stability theory to define stability determinants (Evans functions) for multidimensional strong detonations in three commonly studied models of combustion: the full reactive Navier-Stokes (RNS) model, and the simpler Zeldovich-von Neumann-Döring (ZND) and Chapman-Jouguet (CJ) models. The determinants are functions of frequencies (λ, n), where A is a complex variable dual to the time variable, and η ∈ℝ^{d-1} is dual to the transverse spatial variables. The zeros of these determinants in λ > 0 correspond to perturbations that grow exponentially with time.The CJ determinant, Δ_{CJ}(λ, η), turns out to be explicitly computable. The RNS and ZND determinants are impossible to compute explicitly, but we are able to compute their first-order low-frequency expansions with an error term that is uniformly small with respect to all possible (λ, η) directions. Somewhat surprisingly, this computation yields an Equivalence Theorem: the leading coefficient in the expansions of both the RNS and ZND determinants is a constant multiple of Δ_{CJ}! In this sense the low-frequency stability conditions for strong detonations in all three models are equivalent. By computing Δ_{CJ} we are able to give low-frequency stability criteria valid for all three models in terms of the physical quantities: Mach number, Gruneisen coefficient, compression ratio, and heat release. The Equivalence Theorem and its surrounding analysis is a step toward the rigorous theoretical justification of the CJ and ZND models as approximations to the full RNS model.

Original language | English (US) |
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Pages (from-to) | 1-64 |

Number of pages | 64 |

Journal | Indiana University Mathematics Journal |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - Apr 28 2005 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Indiana University Mathematics Journal*,

*54*(1), 1-64. https://doi.org/10.1512/iumj.2005.54.2685