Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-30T16:14:53.039Z Has data issue: false hasContentIssue false
  • English
  • Français

On roll, square and hexagonal cellular flames

Published online by Cambridge University Press:  26 September 2008

R. Kuske  and
B.J. Matkowsky
R. Kuske
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
B.J. Matkowsky
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the behaviour of a premixed flame anchored on a flat burner. For Lewis numbers L < L* < 1, stationary spatially periodic solutions corresponding to stationary cellular flames bifurcate from the basic solution which corresponds to a steady planar flame. We study the existence and stability of two-dimensional patterns which correspond to certain imposed symmetries by considering the evolution of N pairs of wave vectors, each of which is separated from the next by angle π/N. In the neighbourhood of the critical Lewis number L*, we derive evolution equations for the amplitudes corresponding to N = 2, which corresponds to square patterns, and N = 3, which corresponds to triangular or hexagonal patterns. We determine existence and stability results in terms of m∈(0, 1), the flow rate of the fuel, and K > 2/e, the scaled heat loss to the burner. Square patterns exist for L < L* and are stable for values of m and K above a stability boundary in the m−K plane, which has a maximum at K = K* ∼ 4.77, so that for K > K* square patterns are stable for all m. The stability of the square patterns does not vary with L. Hexagonal patterns exist for L < LH, where 1 > LH > L*. The size of the stability region increases with decreasing L < L*. For a range of values of L there is bistability, that is, for given parameter values rolls and hexagons are simultaneously stable, each with its own domain of attraction.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 1 , March 1994 , pp. 65 - 93
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Matkowsky, B. J. & Olagunju, D. 1981 Pulsations in a burner stabilized premixed plane flame. SIAM, J. Appl. Math. 40: 551–562.CrossRefGoogle Scholar
[2]Olagunju, D. & Matkowsky, B. J. 1990 Burner stabilized cellular flames. Quart. Appl. Math. 48: 645–664.Google Scholar
[3]Carrier, G. F., Fendell, F. E. & Bush, W. B. 1978 Stoichiometry and flameholder effects on a one-dimensional flame. Comb. Sci. Tech. 18: 33–46.CrossRefGoogle Scholar
[4]Newell, A. C. 1989 In Lectures in the Sciences of Complexity, Lectures I, (Stein, D. L., ed.). Addison-Wesley.Google Scholar
[5]Manneville, P. 1990 Dissipative Structures and Weak Turbulence. Academic Press.Google Scholar
[6]Swift, J. W. 1984 Convection in a rotating fluid layer. Cont. Math. 28: 435–448.Google Scholar
[7]Shtilman, L. & Sivashinsky, G. 1990 On the hexagonal structure of cellular flames. Can. J. Phys. 68: 768–771.Google Scholar
[8]Wollkind, D. J., Sriranganathan, R. & Oulton, D. B. 1984 Interfacial patterns during plane front alloy solidification. Physica 12D: 215–240.Google Scholar