February  2021, 14(2): 575-596. doi: 10.3934/dcdss.2020363

Instability of free interfaces in premixed flame propagation

1. 

School of Mathematical Sciences, Tongji University, 1239 Siping Rd., Shanghai 200092, China

2. 

Institut de Mathématiques de Bordeaux, Université de Bordeaux, 33405 Talence Cedex, France

3. 

Plesso di Matematica, Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy

* Corresponding author: claude-michel.brauner@u-bordeaux.fr

Dedicated to Michel Pierre on his 70th birthday, in friendship.

Received  December 2019 Revised  February 2020 Published  February 2021 Early access  May 2020

In this survey, we are interested in the instability of flame fronts regarded as free interfaces. We successively consider a classical Arrhenius kinetics (thin flame) and a stepwise ignition-temperature kinetics (thick flame) with two free interfaces. A general method initially developed for thin flame problems subject to interface jump conditions is proving to be an effective strategy for smoother thick flame systems. It relies on the elimination of the free interface(s) and reduction to a fully nonlinear parabolic problem. The theory of analytic semigroups is a key tool to study the linearized operators.

Citation: Claude-Michel Brauner, Luca Lorenzi. Instability of free interfaces in premixed flame propagation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (2) : 575-596. doi: 10.3934/dcdss.2020363
References:
[1]

D. AddonaC.-M. BraunerL. Lorenzi and W. Zhang, Instabilities in a combustion model with two free interfaces, J. Differential Equations, 268 (2020), 396-4016.  doi: 10.1016/j.jde.2019.10.015.

[2]

O. BaconneauC.-M. Brauner and A. Lunardi, Computation of bifurcated branches in a free boundary problem arising in combustion theory, ESAIM Math. Model. Numer. Anal., 34 (2000), 223-239.  doi: 10.1051/m2an:2000139.

[3]

I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem, Indiana Univ. Math. J., 50 (2001), 1077-1112.  doi: 10.1512/iumj.2001.50.1906.

[4]

I. BrailovskyP. V. GordonL. Kagan and G. I. Sivashinsky, Diffusive-thermal instabilities in premixed flames: Stepwise ignition-temperature kinetics, Combustion and Flame, 162 (2015), 2077-2086.  doi: 10.1016/j.combustflame.2015.01.006.

[5]

C. -M BraunerP. V. Gordon and W. Zhang, An ignition-temperature model with two free interfaces in premixed flames, Combust. Theory Model., 20 (2016), 976-994.  doi: 10.1080/13647830.2016.1220625.

[6]

C.-M. BraunerL. Hu and L. Lorenzi, Asymptotic analysis in a gas-solid combustion model with pattern formation, Chin. Ann. Math. Ser. B, 34 (2013), 65-88.  doi: 10.1007/s11401-012-0758-4.

[7]

C.-M. BraunerJ. HulshofL. Lorenzi and G. I. Sivashinsky, A fully nonlinear equation for the flame front in a quasi-steady combustion model, Discrete Contin. Dyn. Syst. Ser. A, 27 (2010), 1415-1446.  doi: 10.3934/dcds.2010.27.1415.

[8]

C.-M. BraunerJ. Hulshof and A. Lunardi, A general approach to stability in free boundary problems, J. Differential Equations, 164 (2000), 16-48.  doi: 10.1006/jdeq.1999.3734.

[9]

C.-M. Brauner and L. Lorenzi, Local existence in free interface problems with underlying second-order Stefan condition, Rev. Roumaine Math. Pures Appl., 63 (2018), 339-359. 

[10]

C.-M. BraunerL. LorenziG.I. Sivashinsky and C.-J. Xu, On a strongly damped wave equation for the flame front, Chin. Ann. Math. Ser. B, 31 (2010), 819-840.  doi: 10.1007/s11401-010-0616-1.

[11]

C.-M. Brauner, L. Lorenzi and M. Zhang, Stability analysis and Hopf bifurcation for large Lewis number in a combustion model with free interface, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2020), in press. doi: 10.1016/j.anihpc.2020.01.002.

[12]

C.-M. Brauner and A. Lunardi, Instabilities in a two-dimensional combustion model with free boundary, Arch. Ration. Mech. Anal., 154 (2000), 157-182.  doi: 10.1007/s002050000099.

[13] J. D. Buckmaster and G. S. S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge-New York, 1982. 
[14]

J. D. Buckmaster and G. S. S. Ludford, Lectures on Mathematical Combustion, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. doi: 10.1137/1.9781611970272.

[15]

J. R. Cannon and J. Douglas Jr., The stability of the boundary in a Stefan problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1967), 83-91. 

[16]

F. Conrad and F. Issard-Roch, Loss of stability at turning points in nonlinear elliptic variational inequalities, Nonlinear Anal., 14 (1990), 329-356.  doi: 10.1016/0362-546X(90)90169-H.

[17]

F. ConradF. Issard-RochC.-M. Brauner and B. Nicolaenko, Nonlinear eigenvalue problems in elliptic variational inequalities: A local study, Comm. Partial Differential Equations, 10 (1985), 151-190.  doi: 10.1080/03605308508820375.

[18]

C. Elliot and J. R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman, Boston, Mass.-London, 1982.

[19]

M. Hadžić and S. Shkoller, Global stability and decay for the classical Stefan problem, Comm. Pure Appl. Math., 68 (2015), 689-757.  doi: 10.1002/cpa.21522.

[20]

M. Hadžić, G. Navarro and S. Shkoller, Local well-posedness and global stability of the two-phase Stefan problem, SIAM J. Math. Anal., 49 (2017), 4942–5006., doi: 10.1137/16M1083207.

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., Vol. 840, Springer-Verlag, Berlin-New York, 1981.

[22]

L. Lorenzi, Regularity and analyticity in a two-dimensional combustion model, Adv. Differential Equations, 7 (2002), 1343-1376. 

[23]

L. Lorenzi, A free boundary problem stemmed from combustion theory. Part I: Existence, uniqueness and regularity results, J. Math. Anal. Appl., 274 (2002), 505-535.  doi: 10.1016/S0022-247X(02)00271-8.

[24]

L. Lorenzi, A free boundary problem stemmed from combustion theory. Part II: Stability, instability and bifurcation results, J. Math. Anal. Appl., 275 (2002), 131-160.  doi: 10.1016/S0022-247X(02)00280-9.

[25]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995.

[26]

B. J. Matkowsky and G. I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J. Appl. Math., 37 (1979), 686-699.  doi: 10.1137/0137051.

[27]

J. R. Ockendon, Linear and nonlinear stability of a class of moving boundary problems, in Free Boundary Problems: Volume II, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980,443–478.

[28]

S. Serfaty and J. Serra, Quantitative stability of the free boundary in the obstacle problem, Anal. PDE, 11 (2018), 1803-1839.  doi: 10.2140/apde.2018.11.1803.

[29]

G. I. Sivashinsky, On flame propagation under condition of stoichiometry, SIAM J. Appl. Math., 39 (1980), 67-82.  doi: 10.1137/0139007.

[30]

G. I. Sivashinsky, Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech., 15 (1983), 179-199. 

show all references

References:
[1]

D. AddonaC.-M. BraunerL. Lorenzi and W. Zhang, Instabilities in a combustion model with two free interfaces, J. Differential Equations, 268 (2020), 396-4016.  doi: 10.1016/j.jde.2019.10.015.

[2]

O. BaconneauC.-M. Brauner and A. Lunardi, Computation of bifurcated branches in a free boundary problem arising in combustion theory, ESAIM Math. Model. Numer. Anal., 34 (2000), 223-239.  doi: 10.1051/m2an:2000139.

[3]

I. Blank, Sharp results for the regularity and stability of the free boundary in the obstacle problem, Indiana Univ. Math. J., 50 (2001), 1077-1112.  doi: 10.1512/iumj.2001.50.1906.

[4]

I. BrailovskyP. V. GordonL. Kagan and G. I. Sivashinsky, Diffusive-thermal instabilities in premixed flames: Stepwise ignition-temperature kinetics, Combustion and Flame, 162 (2015), 2077-2086.  doi: 10.1016/j.combustflame.2015.01.006.

[5]

C. -M BraunerP. V. Gordon and W. Zhang, An ignition-temperature model with two free interfaces in premixed flames, Combust. Theory Model., 20 (2016), 976-994.  doi: 10.1080/13647830.2016.1220625.

[6]

C.-M. BraunerL. Hu and L. Lorenzi, Asymptotic analysis in a gas-solid combustion model with pattern formation, Chin. Ann. Math. Ser. B, 34 (2013), 65-88.  doi: 10.1007/s11401-012-0758-4.

[7]

C.-M. BraunerJ. HulshofL. Lorenzi and G. I. Sivashinsky, A fully nonlinear equation for the flame front in a quasi-steady combustion model, Discrete Contin. Dyn. Syst. Ser. A, 27 (2010), 1415-1446.  doi: 10.3934/dcds.2010.27.1415.

[8]

C.-M. BraunerJ. Hulshof and A. Lunardi, A general approach to stability in free boundary problems, J. Differential Equations, 164 (2000), 16-48.  doi: 10.1006/jdeq.1999.3734.

[9]

C.-M. Brauner and L. Lorenzi, Local existence in free interface problems with underlying second-order Stefan condition, Rev. Roumaine Math. Pures Appl., 63 (2018), 339-359. 

[10]

C.-M. BraunerL. LorenziG.I. Sivashinsky and C.-J. Xu, On a strongly damped wave equation for the flame front, Chin. Ann. Math. Ser. B, 31 (2010), 819-840.  doi: 10.1007/s11401-010-0616-1.

[11]

C.-M. Brauner, L. Lorenzi and M. Zhang, Stability analysis and Hopf bifurcation for large Lewis number in a combustion model with free interface, Ann. Inst. H. Poincaré Anal. Non Linéaire, (2020), in press. doi: 10.1016/j.anihpc.2020.01.002.

[12]

C.-M. Brauner and A. Lunardi, Instabilities in a two-dimensional combustion model with free boundary, Arch. Ration. Mech. Anal., 154 (2000), 157-182.  doi: 10.1007/s002050000099.

[13] J. D. Buckmaster and G. S. S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge-New York, 1982. 
[14]

J. D. Buckmaster and G. S. S. Ludford, Lectures on Mathematical Combustion, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. doi: 10.1137/1.9781611970272.

[15]

J. R. Cannon and J. Douglas Jr., The stability of the boundary in a Stefan problem, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 21 (1967), 83-91. 

[16]

F. Conrad and F. Issard-Roch, Loss of stability at turning points in nonlinear elliptic variational inequalities, Nonlinear Anal., 14 (1990), 329-356.  doi: 10.1016/0362-546X(90)90169-H.

[17]

F. ConradF. Issard-RochC.-M. Brauner and B. Nicolaenko, Nonlinear eigenvalue problems in elliptic variational inequalities: A local study, Comm. Partial Differential Equations, 10 (1985), 151-190.  doi: 10.1080/03605308508820375.

[18]

C. Elliot and J. R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman, Boston, Mass.-London, 1982.

[19]

M. Hadžić and S. Shkoller, Global stability and decay for the classical Stefan problem, Comm. Pure Appl. Math., 68 (2015), 689-757.  doi: 10.1002/cpa.21522.

[20]

M. Hadžić, G. Navarro and S. Shkoller, Local well-posedness and global stability of the two-phase Stefan problem, SIAM J. Math. Anal., 49 (2017), 4942–5006., doi: 10.1137/16M1083207.

[21]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., Vol. 840, Springer-Verlag, Berlin-New York, 1981.

[22]

L. Lorenzi, Regularity and analyticity in a two-dimensional combustion model, Adv. Differential Equations, 7 (2002), 1343-1376. 

[23]

L. Lorenzi, A free boundary problem stemmed from combustion theory. Part I: Existence, uniqueness and regularity results, J. Math. Anal. Appl., 274 (2002), 505-535.  doi: 10.1016/S0022-247X(02)00271-8.

[24]

L. Lorenzi, A free boundary problem stemmed from combustion theory. Part II: Stability, instability and bifurcation results, J. Math. Anal. Appl., 275 (2002), 131-160.  doi: 10.1016/S0022-247X(02)00280-9.

[25]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995.

[26]

B. J. Matkowsky and G. I. Sivashinsky, An asymptotic derivation of two models in flame theory associated with the constant density approximation, SIAM J. Appl. Math., 37 (1979), 686-699.  doi: 10.1137/0137051.

[27]

J. R. Ockendon, Linear and nonlinear stability of a class of moving boundary problems, in Free Boundary Problems: Volume II, Ist. Naz. Alta Mat. Francesco Severi, Rome, 1980,443–478.

[28]

S. Serfaty and J. Serra, Quantitative stability of the free boundary in the obstacle problem, Anal. PDE, 11 (2018), 1803-1839.  doi: 10.2140/apde.2018.11.1803.

[29]

G. I. Sivashinsky, On flame propagation under condition of stoichiometry, SIAM J. Appl. Math., 39 (1980), 67-82.  doi: 10.1137/0139007.

[30]

G. I. Sivashinsky, Instabilities, pattern formation and turbulence in flames, Ann. Rev. Fluid Mech., 15 (1983), 179-199. 

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