American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 258-266. doi: 10.3934/proc.2015.0258

Stability of interacting traveling waves in reaction-convection-diffusion systems

Grigori Chapiro 1, , Lucas Furtado 2, , Dan Marchesin 3, and  Stephen Schecter 4,

1. 

Universidade Federal de Juiz de Fora, Juiz de Fora, MG 36036-900, Brazil
2. 

Columbia College, Columbia University, New York, NY 10027, United States
3. 

Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460-320, Brazil
4. 

Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205

Received  September 2014 Revised  January 2015 Published  November 2015

The stability of isolated combustion traveling waves has been exhaustively studied in the literature of reaction-diffusion systems. The analysis has been done mainly by neglecting other waves that are usually present in the solution and that can influence the stability of the combustion wave. In this paper, a numerical example on the influence of such interaction on wave stability are presented.
    The paper is illustrated through a simple model for the injection of air into a porous medium that contains a solid fuel. The model considered here reproduces a variety of observed phenomena and yet is simple enough to allow rigorous investigation. We refer on earlier work containing proofs of existence of traveling waves corresponding to combustion waves by phase plane analysis were presented; wave sequences that can occur as solutions of Riemann problems were identified.
Keywords: Riemann problem, ignition, Traveling wave, filtration combustion, stability, combustion in porous media, extinction..
Mathematics Subject Classification: Primary: 76S05, 35C07, 35B35; Secondary: 80A2.
Citation: Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258
References:
[1]

I. Akkutlu and Y. Yortsos, The dynamics of in-situ combustion fronts in porous media, J. of Combustion and Flame, 134 (2003), 229-247.

[2]

A. Aldushin and S. Kasparyan, Stability of stationary filtrational combustion waves, Combustion, Explosion, and Shock Waves, 17 (1981), 615-625.

[3]

A. Aldushin, I. Rumanov and B. Matkowsky, Maximal energy accumulation in a superadiabatic filtration combustion wave, J. of Combustion and Flame, 118 (1999), 76-90.

[4]

A. Bayliss, G. Leaf and B. Matkowsky, Pulsating and chaotic dynamics near the extinction limit, Combustion Science and Technology, 84 (1992), 253-278.

[5]

A. Bayliss and B. Matkowsky, Two routes to chaos in condensed phase combustion, SIAM J. Appl. Math., 50 (1990), 437-459.

[6]

A. Bayliss and B. Matkowsky, From traveling waves to chaos in combustion, SIAM Journal on Applied Mathematics, 54 (1994), 147-174.

[7]

I. Brailovsky and G. Sivashinsky, Chaotic dynamics in solid fuel combustion, Physica D: Nonlinear Phenomena, 65 (1993), 191-198.

[8]

J. Bruining, A. Mailybaev and D. Marchesin, Filtration combustion in wet porous medium, SIAM Journal on Applied Mathematics, 70 (2009), 1157-1177.

[9]

G. Chapiro, L. Furtado, D. Marchesin and S. Schecter, Numerical analysis of combustion waves and riemann solutions in light porous foam, 2014,, Preprint at http://preprint.impa.br/visualizar?id=5912., (). 

[10]

G. Chapiro, A. A. Mailybaev, A. Souza, D. Marchesin and J. Bruining, Asymptotic approximation of long-time solution for low-temperature filtration combustion, Comput. Geosciences, 16 (2012), 799-808.

[11]

G. Chapiro, D. Marchesin and S. Schecter, Combustion waves and Riemann solutions in light porous foam, Journal of Hyperbolic Differential Equations, 11 (2014), 295-328.

[12]

M. Decker and D. Schult, Dynamics of smoulder waves near extinction, Combustion Theory and Modelling, 8 (2004), 491-512.

[13]

A. Ghazaryan, Y. Latushkin, S. Schecter and A. de Souza, Stability of gasless combustion fronts in one-dimensional solids, Archive for Rational Mechanics and Analysis, 198 (2010), 981-1030.

[14]

L. W. Lake, Enhanced oil recovery, Old Tappan, NJ; Prentice Hall Inc., 1989.

[15]

A. Mailybaev, J. Bruining and D. Marchesin, Analysis of in situ combustion of oil with pyrolysis and vaporization, Combustion and Flame, 158 (2011), 1097-1108.

[16]

A. Mailybaev, D. Marchesin and J. Bruining, Resonance in low-temperature oxidation waves for porous media, SIAM Journal on Mathematical Analysis, 43 (2011), 2230-2252.

[17]

D. Marchesin and S. Schecter, Oxidation heat pulses in two-phase expansive flow in porous media, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 54 (2003), 48-83.

[18]

B. Matkowsky and G. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion, SIAM J. Appl. Math., 35 (1978), 465-478.

[19]

J. Mota and S. Schecter, Combustion fronts in a porous medium with two layers, Journal of Dynamics and Differential Equations, 18 (2006), 615-665.

[20]

J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part i-existence, SIAM J. on Appl. Math., 48 (1988), 155-169.

[21]

J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part ii-stability, SIAM J. on Appl. Math., 48 (1988), 374-392.

[22]

S. Schecter and D. Marchesin, Geometric singular perturbation analysis of oxidation heat pulses for two-phase flow in porous media. dedicated to constantine dafermos on his 60th birthday, Bulletin of the Brazilian Mathematical Society, 32 (2001), 237-270.

[23]

D. Schult, B. Matkowsky, V. Volpert and A. Fernandez-Pello, Forced forward smolder combustion, Combustion and Flame, 104 (1996), 1-26.

[24]

R. Seydel, Practical bifurcation and stability analysis, Springer, 2010.

[25]

R. Weber, G. Mercer, H. Sidhu and B. Gray, Combustion waves for gases ($Le= 1$) and solids ($Le\to\infty$), Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453 (1997), 1105-1118.

[26]

Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The mathematical theory of combustion and explosion, Consultants Bureau, New York, 1985.

show all references

References:
[1]

I. Akkutlu and Y. Yortsos, The dynamics of in-situ combustion fronts in porous media, J. of Combustion and Flame, 134 (2003), 229-247.

[2]

A. Aldushin and S. Kasparyan, Stability of stationary filtrational combustion waves, Combustion, Explosion, and Shock Waves, 17 (1981), 615-625.

[3]

A. Aldushin, I. Rumanov and B. Matkowsky, Maximal energy accumulation in a superadiabatic filtration combustion wave, J. of Combustion and Flame, 118 (1999), 76-90.

[4]

A. Bayliss, G. Leaf and B. Matkowsky, Pulsating and chaotic dynamics near the extinction limit, Combustion Science and Technology, 84 (1992), 253-278.

[5]

A. Bayliss and B. Matkowsky, Two routes to chaos in condensed phase combustion, SIAM J. Appl. Math., 50 (1990), 437-459.

[6]

A. Bayliss and B. Matkowsky, From traveling waves to chaos in combustion, SIAM Journal on Applied Mathematics, 54 (1994), 147-174.

[7]

I. Brailovsky and G. Sivashinsky, Chaotic dynamics in solid fuel combustion, Physica D: Nonlinear Phenomena, 65 (1993), 191-198.

[8]

J. Bruining, A. Mailybaev and D. Marchesin, Filtration combustion in wet porous medium, SIAM Journal on Applied Mathematics, 70 (2009), 1157-1177.

[9]

G. Chapiro, L. Furtado, D. Marchesin and S. Schecter, Numerical analysis of combustion waves and riemann solutions in light porous foam, 2014,, Preprint at http://preprint.impa.br/visualizar?id=5912., (). 

[10]

G. Chapiro, A. A. Mailybaev, A. Souza, D. Marchesin and J. Bruining, Asymptotic approximation of long-time solution for low-temperature filtration combustion, Comput. Geosciences, 16 (2012), 799-808.

[11]

G. Chapiro, D. Marchesin and S. Schecter, Combustion waves and Riemann solutions in light porous foam, Journal of Hyperbolic Differential Equations, 11 (2014), 295-328.

[12]

M. Decker and D. Schult, Dynamics of smoulder waves near extinction, Combustion Theory and Modelling, 8 (2004), 491-512.

[13]

A. Ghazaryan, Y. Latushkin, S. Schecter and A. de Souza, Stability of gasless combustion fronts in one-dimensional solids, Archive for Rational Mechanics and Analysis, 198 (2010), 981-1030.

[14]

L. W. Lake, Enhanced oil recovery, Old Tappan, NJ; Prentice Hall Inc., 1989.

[15]

A. Mailybaev, J. Bruining and D. Marchesin, Analysis of in situ combustion of oil with pyrolysis and vaporization, Combustion and Flame, 158 (2011), 1097-1108.

[16]

A. Mailybaev, D. Marchesin and J. Bruining, Resonance in low-temperature oxidation waves for porous media, SIAM Journal on Mathematical Analysis, 43 (2011), 2230-2252.

[17]

D. Marchesin and S. Schecter, Oxidation heat pulses in two-phase expansive flow in porous media, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 54 (2003), 48-83.

[18]

B. Matkowsky and G. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion, SIAM J. Appl. Math., 35 (1978), 465-478.

[19]

J. Mota and S. Schecter, Combustion fronts in a porous medium with two layers, Journal of Dynamics and Differential Equations, 18 (2006), 615-665.

[20]

J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part i-existence, SIAM J. on Appl. Math., 48 (1988), 155-169.

[21]

J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part ii-stability, SIAM J. on Appl. Math., 48 (1988), 374-392.

[22]

S. Schecter and D. Marchesin, Geometric singular perturbation analysis of oxidation heat pulses for two-phase flow in porous media. dedicated to constantine dafermos on his 60th birthday, Bulletin of the Brazilian Mathematical Society, 32 (2001), 237-270.

[23]

D. Schult, B. Matkowsky, V. Volpert and A. Fernandez-Pello, Forced forward smolder combustion, Combustion and Flame, 104 (1996), 1-26.

[24]

R. Seydel, Practical bifurcation and stability analysis, Springer, 2010.

[25]

R. Weber, G. Mercer, H. Sidhu and B. Gray, Combustion waves for gases ($Le= 1$) and solids ($Le\to\infty$), Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453 (1997), 1105-1118.

[26]

Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The mathematical theory of combustion and explosion, Consultants Bureau, New York, 1985.

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