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.
R. C. Buck, Advanced Calculus, McGraw-Hill, New York. 1965
- H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford, at the Clarendon Press, 1947. MR 0022294
- A. Erdélyi, Asymptotic expansions, Dover Publications, Inc., New York, 1956. MR 0078494
- 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
C. K. Law and H. K. Law, Flat-plate ignition with reactant consumption, Combustion Science and Technology 25, 1–8 (1981)
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)
- 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, DOI https://doi.org/10.1137/0136042
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
- 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
- Richard K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series. MR 0511193
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)
- W. E. Olmstead, Ignition of a combustible half space, SIAM J. Appl. Math. 43 (1983), no. 1, 1–15. MR 687785, DOI https://doi.org/10.1137/0143001
F. A. Williams, Combustion Theory, 2nd Edition, Benjamin/Cummings Inc., Menlow Park, CA, 1985
R. C. Buck, Advanced Calculus, McGraw-Hill, New York. 1965
H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd Ed., Oxford University Press, Oxford, 1959
A. Erdelyi, Asymptotic Expansions, Dover, New York, 1956
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
C. K. Law and H. K. Law, Flat-plate ignition with reactant consumption, Combustion Science and Technology 25, 1–8 (1981)
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)
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)
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
N. P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM Stud. Appl. Math., vol. 7, SIAM, Philadelphia, PA, 1985
R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin Inc., Menlow Park, CA, 1971
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)
W. E. Olmsted, Ignition of a combustable half space, SIAM J. Appl. Math. 43, 1–15 (1983)
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
Article copyright:
© Copyright 1991
American Mathematical Society