To neglect the influences by hydrodynamic instabilities such as
Darrieus–Landau instability,
Rayleigh–Taylor instability etc., the analysis usually neglects effects due to the thermal expansion of the gas mixture by assuming a constant density model. Such an approximation is referred to as
diffusive-thermal approximation or
thermo-diffusive approximation which was first introduced by
Grigory Barenblatt,
Yakov Zeldovich and A. G. Istratov in 1962. With a one-step chemistry model and assuming the perturbations to a steady planar flame in the form e^{i\mathbf{k}\cdot\mathbf{x}_\bot+\sigma t}, where \mathbf{x}_\bot is the transverse coordinate system perpendicular to flame, t is the time, \mathbf{k} is the perturbation wavevector and \sigma is the temporal growth rate of the disturbance, the dispersion relation \sigma(k) for one-reactant flames is given implicitly by :2\Gamma^2(\Gamma-1) + l (\Gamma-1 - 2 \sigma) = 0 where \Gamma=\sqrt{1+4\sigma+4k^2}, l\equiv (Le-1)/\beta, Le is the
Lewis number of the fuel and \beta is the
Zeldovich number. This relation provides in general three roots for \sigma in which the one with maximum \Re\{\sigma\} would determine the stability character. The stability margins are given by the following equations :8k^2 + l + 2 =0, \quad 256 k^4 +(-6l^2+32l+256)k^2 -l^2+8l + 32=0 describing two curves in the l vs. k plane. The first curve is associated with condition \Im\{\sigma\}=0, whereas on the second curve \Im\{\sigma\}\neq 0. The first curve separates the region of stable mode from the region corresponding to
cellular instability, whereas the second condition indicates the presence of
traveling and/or
pulsating instability. ==See also==