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Enhancement of the traveling front speeds in reaction-diffusion equations with advection. (English) Zbl 1002.35069
The authors investigate the following question: which characteristic of the ambient fluid flow is responsible for burning rate enhancement? Note that the bulk burning rate \[ V(t)=\int_\Omega T_t(x,y,t) {dxdy \over H} \] and its time average \[ \langle V\rangle_t={1\over t}\int^t_0 V(s)ds, \] where \(T\) denotes the temperature, satisfy the corresponding PDE. Here they consider a general class of reaction rates \(f(T)\) that are either of the ignition or general KPP type, and establish lower bounds for \(V(t)\) for percolating flows that are periodic in space. Another interesting result of the paper concerns cellular flows with closed streamlines.

35K57 Reaction-diffusion equations
80A32 Chemically reacting flows
35K40 Second-order parabolic systems
35B45 A priori estimates in context of PDEs
35B10 Periodic solutions to PDEs
Full Text: DOI Numdam EuDML arXiv
[1] Audoly B., Berestycki H., Pomeau Y., Reaction-diffusion in fast steady flow, C.R. Acad. Sci., Ser. IIB, 328, 255-262 · Zbl 0992.76097
[2] Bensoussan, A; Lions, J.L; Papanicolaou, G, Asymptotic analysis for periodic structures, (1978), North-Holland Amsterdam · Zbl 0411.60078
[3] Berestycki H., Hamel F., Propagation of flames through vortical cells, preprint
[4] Berestycki, H; Larrouturou, B; Lions, P.L, Multi-dimensional traveling wave solutions of a flame propagation model, Arch. rational mech. anal, 111, 33-49, (1990) · Zbl 0711.35066
[5] Berestycki, H; Larrouturou, B; Roquejoffre, J.-M, Stability of traveling fronts in a model for flame propagation I: linear stability, Arch. rational mech. anal., 117, 97-117, (1992) · Zbl 0763.76033
[6] Berestycki, H; Nirenberg, L, Some qualitative properties of solutions of semilinear equations in cylindrical domains, (), 115-164 · Zbl 0705.35004
[7] Berestycki, H; Nirenberg, L, Traveling fronts in cylinders, Annales de l’IHP, analyse non linéare, 9, 497-572, (1992) · Zbl 0799.35073
[8] Clavin, P; Williams, F.A, Theory of pre-mixed flame propagation in large-scale turbulence, J. fluid. mech., 90, 589-604, (1979) · Zbl 0434.76052
[9] Constantin, P; Kiselev, A; Oberman, A; Ryzhik, L, Bulk burning rate in passive-reactive diffusion, Arch. rat. mech. anal., 154, 53-91, (2000) · Zbl 0979.76093
[10] Embid, P; Majda, A; Souganidis, P, Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. fluids., 7, 2052-2060, (1995) · Zbl 1039.80504
[11] Embid, P; Majda, A; Souganidis, P, Effective geometric from dynamics for premixed turbulent combustion with separated velocity scales, Combust. sci. tech., 103, 85, (1994)
[12] Fannjiang, A; Papanicolaou, G, Convection enhanced diffusion for periodic flows, SIAM J. appl. math., 54, 333-408, (1994) · Zbl 0796.76084
[13] Fisher, R, The wave of advance of advantageous genes, Ann. eugenics, 7, 355-369, (1937) · JFM 63.1111.04
[14] Freidlin, M; Gärtner, J, On the propagation of concentration waves in periodic and random media, Soviet math. dokl., 20, 1282-1286, (1979) · Zbl 0447.60060
[15] Freidlin, M, Geometric optics approach to reaction-diffusion equations, SIAM J. appl. math., 46, 222-232, (1986) · Zbl 0626.35047
[16] Freidlin, M, Limit theorems for large deviations and reaction-diffusion equations, Ann. probab., 13, 639-675, (1985) · Zbl 0576.60070
[17] Hamel, F, Formules MIN-MAX pour LES vitesses d’ondes progressives multidimensionnelles, Ann. fac. sci., Toulouse math. (6), 8, 259-280, (1999) · Zbl 0956.35041
[18] Heinze S., Papanicolaou G., Stevens A., Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math., to appear · Zbl 0995.35031
[19] Kerstein, A; Ashurst, W, Phys. rev. lett., 68, 934, (1992)
[20] Kerstein, A, Simple derivation of Yakhot’s turbulent premixed flame speed formula, Combust. sci. tech., 60, 163-165, (1988)
[21] Kagan, L; Sivashinsky, G; Makhviladze, G, On flame extinction by a spatially periodic shear flow, Combust. theor. model., 2, 399-404, (1998) · Zbl 0944.76093
[22] Kagan, L; Sivashinsky, G, On flame propagation and extinction in large scale vortical flows, Combustion and flame, 120, 222-232, (2000)
[23] Kolmogorov, A.N; Petrovskii, I.G; Piskunov, N.S, Étude de I’équation de la chaleurde matiére et son application à un problème biologique, Bull. moskov. GoS. univ. mat. mekh., 1, 1-25, (1937), (see [27] pp. 105-130 for an English transl.)
[24] Majda, A; Souganidis, P, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7, 1-30, (1994) · Zbl 0839.76093
[25] Mallordy, J.-F; Roquejoffre, J.-M, A parabolic equation of the KPP type in higher dimensions, SIAM J. math. anal., 26, 1-20, (1995) · Zbl 0813.35041
[26] McLaughlin, R; Zhu, J, The effect of finite front thickness on the enhanced speed of propagation, Combust. sci. tech., 129, 89-112, (1997)
[27] ()
[28] Protter, M; Weinberger, H, Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602
[29] Ronney, P, Some open issues in premixed turbulent combustion, ()
[30] Roquejoffre, J.-M, Stability of traveling fronts in a model for flame propagation II: nonlinear stability, Arch. rat. mech. anal., 117, 119-153, (1992) · Zbl 0763.76034
[31] Xin, J, Existence of planar flame fronts in convective-diffusive periodic media, Arch. rat. mech. anal., 121, 205-233, (1992) · Zbl 0764.76074
[32] Xin, J, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. stat. phys., 73, 893-926, (1993) · Zbl 1102.35340
[33] Xin, J, Analysis and modeling of front propagation in heterogeneous media, SIAM review, 42, 161-230, (2000)
[34] Yakhot V., Propagation velocity of premixed turbulent flames, Combust. Sci. Tech. 60, 191-214
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