Abstract.
The evolution equation for the radius of an isolated premixed flame ball is derived in the framework of a new method that strongly simplifies previous ones and highlights that they are based on Gaussian modelling of diffusion. The main idea is to split the flame ball in two components: the inner kernel, which is driven by a Poisson-type equation with a general polynomial forcing term, and the outer part, which is driven by a generalized diffusion process valid for fractional diffusive media. The evolution equation for the radius of the flame ball is finally determined as the evolution equation for the interface that matches the solution of the inner spherical kernel and the solution of the outer diffusive part and it emerges to be a nonlinear fractional differential equation. The effects of fractional diffusion on stability of solution are also picked out.
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References
Ya.B. Zeldovich, Theory of Combustion and Detonation of Gases (USSR Academy of Sciences, Moscow, 1944)
P.D. Ronney, Combust. Flame 82, 1 (1990)
P.D. Ronney, K.N. Whaling, A. Abbud-Madrid, J.L. Gatto, V.L. Pisowicz, AIAA J. 32, 569 (1994)
P.D. Ronney, M.S. Wu, H.G. Pearlman, K.J. Weiland, AIAA J. 36, 1361 (1998)
G. Joulin, Combust. Sci. Tech. 185, 99 (1985)
J.D. Buckmaster, G. Joulin, P.D. Ronney, Combust. Flame 79, 381 (1990)
J.D. Buckmaster, G. Joulin, P.D. Ronney, Combust. Flame 84, 411 (1991)
V. Guyonne, P. Noble, J. SIAM Appl. Math. 67, 854 (2007)
G. Pagnini, in Proceedings of FDA10. The 4th IFAC Workshop Fractional Differentiation and its Applications, edited by I. Podlubny, B.M. Vinagre Jara, YQ. Chen, V. Feliu Batlle, I. Tejado Balsera (ISBN 9788055304878, 2010), Article no. FDA10-063
G. Pagnini, Fract. Calc. Appl. Anal. 14, 80 (2011)
J. Klafter, I.M. Sokolov, Physics World August, 29 (2005)
I.M. Sokolov, J. Klafter, A. Blumen, Physics Today November, 48 (2002)
A. Mura, M.S. Taqqu, F. Mainardi, Physica A 387, 5033 (2008) http://arxiv.org/abs/0712.0240
A. Mura, G. Pagnini, J. Phys. A 41, 285003 (2008) http://arxiv.org/abs/0801.4879
F. Mainardi, G. Pagnini, Fract. Calc. Appl. Anal. 4, 153 (2011)
R. Gorenflo, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, edited by edfnA. Carpinteri, F. Mainardi (Springer-Verlag, Wien and New York, 1997), p. 223, http://arxiv.org/abs/0805.3823
C.P. Li, Z.G. Zhao, Eur. Phys. J. Special Topics 193, 5 (2011)
J. Audounet, J.-M. Roquejoffre, in ESAIM: Proceedings Fractional Differential Systems: Models, Methods and Applications, edited by D. Matignon G. Montseny,Vol. 5 (SMAI, Paris, 1998), p. 15, http://www.emath.fr/proc/vol.5/
J. Audounet, J.-M. Roquejoffre, H. Rouzaud, Math. Modelling Numer. Anal. 36, 273 (2002)
F. Dubois, S. Mengué, Numerical Algorithms 34, 303 (2003)
K. Diethelm, M. Weilbeer, Fract. Calc. Appl. Anal. 7, 1 (2004)
A. Mura, Ph.D. thesis, University of Bologna, 2008 http://www.fracalmo.org/Mura/
F. Mainardi, Chaos, Solitons Fractals 7, 1461 (1996)
I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)
F. Mainardi, A. Mura, G. Pagnini, Int. J. Diff. Equations 2010, 104505 (2010) http://www.hindawi.com/journals/ijde/2010/104505.html
V. Kiryakova, Generalized Fractional Calculus and Applications (Longman, Harlow, 1994)
A. Mura, F. Mainardi, Integral Transforms Spec. Funct. 20, 185 (2009) http://arxiv.org/abs/0711.0665
E.K. Lenzi, M.K. Lenzi, L.R. Evangelista, L.C. Malacarne, R.S. Mendes, J. Stat. Mech., P02048 (2009)
J. Audounet, V. Giovangigli, J.-M. Roquejoffre, Physica D 121, 295 (1998)
H. Rouzaud, C.R. Acad. Sci. Paris Série I 332, 1083 (2001)
R. Gorenflo, S. Vessella, Abel Integral Equations. Analysis and Applications (Springer–Verlag, Berlin, 1991)
H. Rouzaud, Revista Matem. Compl. 16, 207 (2003)
J.-M. Roquejoffre, H. Rouzaud, J. Comput. Appl. Math. 190, 376 (2006)
C. Lederman, J.-M. Roquejoffre, N. Wolanski, C.R. Acad. Sci. Paris Série I 334, 569 (2002)
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Pagnini, G. The evolution equation for the radius of a premixed flame ball in fractional diffusive media. Eur. Phys. J. Spec. Top. 193, 105–117 (2011). https://doi.org/10.1140/epjst/e2011-01385-3
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DOI: https://doi.org/10.1140/epjst/e2011-01385-3