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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[30]

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

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