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On Flames as Discontinuity Surfaces in Gasdynamic Flows

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A Celebration of Mathematical Modeling

Abstract

Viewed on a hydrodynamic scale, flames in experiments are often thin so that they may be described as gasdynamic discontinuities separating the dense, cold fresh mixture from the light, hot burned products. In addition to the fluid dynamical equations, the model of a flame as a discontinuity surface consists of a flame speed relation describing the evolution of the surface, and jump conditions across the surface which relate the fluid variables on the two sides of the surface. Models of flames as gasdynamic discontinuities exist, some merely postulated and others derived from the more general equations governing the reactive fluid dynamics. However, none is capable of both capturing all the relevant instabilities and exhibiting a high wave number cutoff for each. Furthermore, the derived models are appropriate only for restricted values of the Lewis number, the ratio of the thermal diffusivity of the mixture to the mass diffusivity of the deficient reactive component. In this paper, we derive a model consisting of a new flame speed relation and new jump conditions, which is valid for arbitrary Lewis numbers. It captures all the relevant instabilities, including the hydrodynamic, cellular and pulsating instabilities and exhibits a high wave number cutoff for each. The flame speed relation includes the effect of short wave lengths, not previously considered, which leads to stabilizing transverse surface diffusion terms. The surface to which the flame zone shrinks is here chosen to be located at a position different from that in previous theories, which leads to clear and simple physical interpretations of the jump conditions.

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Author information

Authors and Affiliations

  1. Northwestern University Evanston, IL, USA, 60208

    Bernard J. Matkowsky

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  1. Bernard J. Matkowsky
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Editor information

Editors and Affiliations

  1. Department of Aerospace Engineering, Technion — Israel Institute of Technology, Haifa, Israel

    Dan Givoli

  2. Department of Mathematics, University of Basel, Basel, Switzerland

    Marcus J. Grote

  3. Department of Mathematics, Stanford University, Stanford, California, USA

    George C. Papanicolaou

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Matkowsky, B.J. (2004). On Flames as Discontinuity Surfaces in Gasdynamic Flows. In: Givoli, D., Grote, M.J., Papanicolaou, G.C. (eds) A Celebration of Mathematical Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0427-4_8

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  • DOI: https://doi.org/10.1007/978-94-017-0427-4_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-6526-1

  • Online ISBN: 978-94-017-0427-4

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