Second Order Theory of Unsteady Burner-Anchored Flames With Arbitrary Lewis Number

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Abstract
The stability of burner flames for arbitrary Lewis number is considered on the basis of large activation energy modelling. Previous leading-order-only approaches are now extended to second order in Math (the inverse activation energy). The assumption that the unsteady perturbations are small (order ε) means that one must discuss the distinguished limit implicit in the product Math. lt is demonstrated here that different governing equations (and in particular the inner zone equation) are obtained in the two cases, θ1ε → 0 and θ1ε → ∞. It is shown that there are two complex frequency relations governing the behaviour of flames near burners. It is found that for free flames a dispersion relationship specifically dependent on activation energy is obtained which is similar though not identical to the classical relationship. For near-free flames, the complex frequency obeys a "composite" dispersion relation, the solution to which clearly indicates the destabilizing effect of heat losses to the burner. However, it is shown that there are regions well away from Lewis number unity where the response is on a slowly varying time scale and the predictions of slowly varying flame theory are confirmed.

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