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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] R. C. Buck, Advanced Calculus, McGraw-Hill, New York. 1965
  • [2] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
  • [3] A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., 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 (Durham, 1982) 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. Liñán and F. A. Williams, Ignition of a reactive solid exposed to a step in surface temperature, SIAM J. Appl. Math. 36 (1979), no. 3, 587–603. MR 531615, https://doi.org/10.1137/0136042
  • [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] Peter Linz, Analytical and numerical methods for Volterra equations, SIAM Studies in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 796318
  • [10] Richard K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. 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. Olmstead, Ignition of a combustible half space, SIAM J. Appl. Math. 43 (1983), no. 1, 1–15. MR 687785, https://doi.org/10.1137/0143001
  • [13] F. A. Williams, Combustion Theory, 2nd Edition, Benjamin/Cummings Inc., Menlow Park, CA, 1985

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 80A25

Retrieve articles in all journals with MSC: 80A25


Additional Information

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


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website