American Institute of Mathematical Sciences

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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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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., ().   Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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., ().   Google Scholar

[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. Google Scholar

[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. Google Scholar

[12]

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

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

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

[24]

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

[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. Google Scholar

[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. Google Scholar

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