TY - JOUR

T1 - On roll, square and hexagonal cellular flames

AU - Kuske, R.

AU - Matkowsky, Bernard J

PY - 1994/1/1

Y1 - 1994/1/1

N2 - 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.

AB - 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.

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U2 - 10.1017/S0956792500001303

DO - 10.1017/S0956792500001303

M3 - Article

AN - SCOPUS:0343538564

VL - 5

SP - 65

EP - 93

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 1

ER -