Bounds and critical parameters for a combustion problem. (English) Zbl 1088.35022
Summary: A model from combustion theory consisting of a nonlinear elliptic equation and boundary conditions of Dirichlet type, is considered. Upper and lower solutions for the problem are obtained by solving linear elliptic equations. These solutions are used to obtain analytical bounds for the extinction and ignition limits. Numerical results are presented for the slab, cylindrical and spherical geometries. Results compare very well with existing ones in the literature.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 80A25 | Combustion |
References:
| [1] | M. Al-Refai, Existence, uniqueness and bounds for a problem in combustion theory, J. Comput. Appl. Math. 167 (2004) 255-269.; M. Al-Refai, Existence, uniqueness and bounds for a problem in combustion theory, J. Comput. Appl. Math. 167 (2004) 255-269. · Zbl 1042.80005 |
| [2] | Aris, R., The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, vol. I (1975), Clarendon Press: Clarendon Press Oxford · Zbl 0315.76051 |
| [3] | Aris, R., The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, vol. II (1975), Clarendon Press: Clarendon Press Oxford · Zbl 0315.76051 |
| [4] | Drott, D. W.; Aris, R., Communications on the theory of diffusion and reaction-Ia complete parametric study of the first-order, irreversible exothermic reaction in a flat slap of catalyst, Chem. Eng. Sci., 24, 541-551 (1969) |
| [5] | Jackson, R., A simple geometric condition for instability in catalysts at unit Lewis number, Chem. Eng. Sci., 28, 1355-1358 (1973) |
| [6] | Kapila, A. K.; Matkowsky, B. J., Reactive-diffuse systems with Arrhenius kineticsmultiple solutions, ignition and extinction, SIAM J. Appl. Math., 36, 2, 373-389 (1979) · Zbl 0406.76070 |
| [7] | Kapila, A. K.; Matkowsky, B. J.; Vega, J., Reactive-diffuse systems with Arrhenius kinetics: peculiarities of the spherical geometry, SIAM J. Appl. Math., 38, 3, 382-402 (1980) · Zbl 0451.60074 |
| [8] | Michelsen, M. L.; Villadsen, J., Diffusion and reaction on spherical catalyst pelletssteady-state- and local stability analysis, Chem. Eng. Sci., 27, 751-762 (1972) |
| [9] | Protter, M. H.; Weinberger, H. F., Maximum Principles in Differential Equations (1984), Springer: Springer New York · Zbl 0153.13602 |
| [10] | Tam, K. K., Criticality dependence on data and parameters for a problem in combustion theory, J. Austral. Math. Soc. Ser. B, 27, 416-441 (1986) · Zbl 0604.35038 |
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