Mathematical justification of a nonlinear integro-differential equation for the propagation of spherical flames |
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Authors: | Email author" target="_blank">Claudia?LedermanEmail author Jean-Michel?Roquejoffre Noemi?Wolanski |
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Institution: | (1) Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina;(2) Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France |
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Abstract: | This paper is devoted to the justification of an asymptotic model for quasisteady three-dimensional spherical flames proposed by G. Joulin 17]. The paper 17] derives, by means of a three-scale matched asymptotics, starting from the classical thermo-diffusive model with high activation energies, an integro-differential equation for the flame radius. In the derivation, it is essential for the Lewis Number – i.e. the ratio between thermal and molecular diffusion – to be strictly less than unity. If is the inverse of the – reduced – activation energy, the idea underlying the construction of 17] is that (i) the time scale of the radius motion is -2, and that (ii) at each time step, the solution is -close to a steady solution.In this paper, we give a rigorous proof of the validity of this model under the restriction that the Lewis number is close to 1 – independently of the order of magnitude of the activation energy. The method used comprises three steps: (i) a linear stability analysis near a steady – or quasi-steady – solution, which justifies the fact that the relevant time scale is -2; (ii) the rigorous construction of an approximate solution; (iii) a nonlinear stability argument. Mathematics Subject Classification (2000) Primary 80A25, Secondary 35K57, 47G20 |
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Keywords: | half derivatives high activation energies linear and nonlinear stability combustion |
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