Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Critical behavior of an ignition model in chemical combustion

Author: Peter J. Tonellato
Journal: Quart. Appl. Math. 49 (1991), 795-812
MSC: Primary 80A25
DOI: https://doi.org/10.1090/qam/1134754
MathSciNet review: MR1134754
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Abstract: A model for the hot slab ignition problem is analyzed to determine critical conditions. The system is said to be super-critical if the solution of the reduced perturbation problem blows up in small finite time or sub-critical if the blow up time is large. Comparison principles for integral equations are used to construct upper and lower solutions of the equation. All solutions depend on two parameters $ {\varepsilon ^{ - 1}}$, the Zeĺdovitch number and $ \lambda $, the scaled hot slab size. Upper and lower bounds on a 'critical' curve $ {\lambda _c}\left( \epsilon \right)$ in the $ \left( {\epsilon, \lambda } \right)$ plane, separating the super-critical from the sub-critical region, are derived based upon the lower and upper solution behavior. Numerical results confirm the parameter space analysis.

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  • [1] R. C. Buck, Advanced Calculus, McGraw-Hill, New York. 1965
  • [2] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd Ed., Oxford University Press, Oxford, 1959 MR 0022294
  • [3] A. Erdelyi, Asymptotic Expansions, Dover, New York, 1956 MR 0078494
  • [4] D. Kershaw, Some results for Abel-Volterra integral equations of the second kind, Treatment of Integral Equations by Numerical Methods, edited by C. T. H. Baker and G. F. Miller, Academic Press, London, 1982, pp. 273-282 MR 755362
  • [5] C. K. Law and H. K. Law, Flat-plate ignition with reactant consumption, Combustion Science and Technology 25, 1-8 (1981)
  • [6] A. Liñan and F. A. Williams, Theory of ignition of a reactive solid by constant energy flux, Combustion Science and Technology 3, 91-98 (1971)
  • [7] A. Liñan and F. A. Williams, Ignition of a reactive solid exposed to a step in surface temperature, J. Appl. Math. SIAM 36, 587-603 (1979) MR 531615
  • [8] A. Liñan and M. Kindelan, Ignition of a reactive solid by an inert hot spot, Combustion in Reactive Systems, edited by J. Raymond Bowen, et al., Progress in Astronautics and Aeronautics, vol. 76 , American Institute of Aeronautics and Astronautics, New York, 1981, pp. 412-426
  • [9] N. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM Stud. Appl. Math., vol. 7, SIAM, Philadelphia, PA, 1985 MR 796318
  • [10] R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., Menlow Park, CA, 1971 MR 0511193
  • [11] A. G. Merzhanov and A. E. Averson, The present state of the thermal ignition theory: An invited review, Combustion and Flame 16, 89-124 (1971)
  • [12] W. E. Olmsted, Ignition of a combustable half space, SIAM J. Appl. Math. 43, 1-15 (1983) MR 687785
  • [13] F. A. Williams, Combustion Theory, 2nd Edition, Benjamin/Cummings Inc., Menlow Park, CA, 1985

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DOI: https://doi.org/10.1090/qam/1134754
Article copyright: © Copyright 1991 American Mathematical Society

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