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January  2012, 11(1): 275-305. doi: 10.3934/cpaa.2012.11.275

Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion

Kota Ikeda 1, and  Masayasu Mimura 2,

1. 

Graduate School of Advanced Mathematical Science, Meiji University, Kawasaki, 214-8571
2. 

Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, Kawasaki, 214-8571

Received  August 2010 Revised  January 2011 Published  September 2011

We are concerned with a reaction diffusion model describing slow smoldering combustion. The process consists of a sheet of paper ignited on one side and in the presence of a flow of air confined in a narrow gap above the paper. It is observed that thermal-diffusion instability generates diverse spatial patterns in combustion front propagation, depending on flow velocity of gas supply. Particularly, if the velocity is rather fast, planar front propagating with almost constant velocity appears. Motivated by this observation, we discuss the existence and stability of $1$ dimensional traveling wave solutions of the model.
Keywords: traveling wave solution, stability., fingering pattern, Combustion model.
Mathematics Subject Classification: Primary: 35B25, 35B35; Secondary: 35K57, 35P1.
Citation: Kota Ikeda, Masayasu Mimura. Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion. Communications on Pure and Applied Analysis, 2012, 11 (1) : 275-305. doi: 10.3934/cpaa.2012.11.275
References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics, Lecture Notes in Math., Springer, 446 (1975), 5-49. doi: 10.1007/BFb0070595.

[2]

A. Fasano, M. Mimura and M. Primicerio, Modelling a slow smoldering combustion process, Mathematical Methods in the Applied Science, 33 (2010), 1211-1220. doi: 10.1002/mma.1301.

[3]

A. Fasano, H. Izuhara, M. Mimura and M. Primicerio, Traveling wave solutions in a simplified smoldering combustion model, manuscript.

[4]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.

[5]

K. Ikeda and M. Mimura, Mathematical treatment of a model for smoldering combustion, Hiroshima Math. J., 38 (2008), 349-361.

[6]

C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44-118. doi: 10.1007/BFb0095239.

[7]

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser Verlag, 1995.

[8]

S. L. Olson, H. R. Baum and T. Kashiwagi, Finger-like smoldering over thin cellulosic sheets in microgravity, The Combustion Institute, (1998), 2525-2533. doi: 10.1016/S0082-0784(98)80104-5.

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 44, 1983.

[10]

L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions, European J. Appl. Math., 16 (2005), 741-765. doi: 10.1017/S0956792505006431.

[11]

E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104. doi: 10.1007/BF00276548.

[12]

O. Zik, Z. Olami and E.Moses, Fingering instability in combustion, Phys. Rev. Lett., 81 (1998), 3868-3871. doi: 10.1103/PhysRevLett.81.3868.

show all references

References:
[1]

D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics, Lecture Notes in Math., Springer, 446 (1975), 5-49. doi: 10.1007/BFb0070595.

[2]

A. Fasano, M. Mimura and M. Primicerio, Modelling a slow smoldering combustion process, Mathematical Methods in the Applied Science, 33 (2010), 1211-1220. doi: 10.1002/mma.1301.

[3]

A. Fasano, H. Izuhara, M. Mimura and M. Primicerio, Traveling wave solutions in a simplified smoldering combustion model, manuscript.

[4]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.

[5]

K. Ikeda and M. Mimura, Mathematical treatment of a model for smoldering combustion, Hiroshima Math. J., 38 (2008), 349-361.

[6]

C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994), Lecture Notes in Math., Springer, Berlin, 1609 (1995), 44-118. doi: 10.1007/BFb0095239.

[7]

A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Birkhäuser Verlag, 1995.

[8]

S. L. Olson, H. R. Baum and T. Kashiwagi, Finger-like smoldering over thin cellulosic sheets in microgravity, The Combustion Institute, (1998), 2525-2533. doi: 10.1016/S0082-0784(98)80104-5.

[9]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Springer-Verlag, New York, 44, 1983.

[10]

L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions, European J. Appl. Math., 16 (2005), 741-765. doi: 10.1017/S0956792505006431.

[11]

E. Yanagida, Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104. doi: 10.1007/BF00276548.

[12]

O. Zik, Z. Olami and E.Moses, Fingering instability in combustion, Phys. Rev. Lett., 81 (1998), 3868-3871. doi: 10.1103/PhysRevLett.81.3868.

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