Instabilities in smooth 2D and 3D isothermal flames

Chemical reactions exhibiting autocatalytic feedback support constant-form, constant-velocity reaction-diffusion wave fronts. The dimensionless velocity c of planar fronts depends on the ratio δ of the diffusion coefficients for the reactant A and autocatalytic species B and an approximate expression for c(δ) for δ1 (but 1 − δ−1 ⪡ 1) is presented. The implications of this, along with previous results for c appropriate to other ranges of δ, in terms of the dependence of the actual wave velocity dxdt as a function of the species diffusion coefficients DA and DB are discussed. An eikonal equation is then presented for the effect of curvature on the wave velocity for smooth circular or spherical waves. The coefficient that appears in the curvature term depends on δ: for δ less than some ‘critical’ value δ∗ ≈ 2.3, the coefficient is negative, indicating that the wave velocity increases towards the planar wave velocity c(δ) as the curvature decreases; for δδ∗, however, the coefficient is positive, indicating that the wave velocity decreases as the curvature decreases. This change in behaviour suggests that systems with δδ∗ will not exhibit a ‘critical radius’ below which wave propagation fails. The change in response to curvature also underlies the loss of stability of smooth waves to spatial perturbation transverse to the direction of propagation. For planar waves, the requirement for instability is δδ∗ with an additional condition on the wave number of the imposed perturbation (or, equivalently, on the width of the reaction zone). For spherical or circular waves, this latter condition is translated as a requirement on the radius of the smooth wave at the moment the perturbation is applied.