text: If the disturbances to the steady planar flame sheet are of the form , where  is the transverse coordinate system that lies on the undisturbed stationary flame sheet,  is the time,  is the wavevector of the disturbance and  is the temporal growth rate of the disturbance, then the dispersion relation is given by
  where  is the laminar burning velocity (or, the flow velocity far upstream of the flame in a frame that is fixed to the flame),  and  is the ratio of unburnt to burnt gas density. In combustion  always and therefore the growth rate  for all wavenumbers. This implies that a plane sheet of flame with a burning velocity  is unstable for all wavenumbers. In fact, Amable Liñán and Forman A. Williams quote in their book that in view of laboratory observations of stable, planar, laminar flames, publication of their theoretical predictions required courage on the part of Darrieus and Landau.
  If the buoyancy forces are taken into account (in others words, accounts of Rayleigh–Taylor instability are considered) for planar flames that are perpendicular to the gravity vector, then some level of stability can be anticipated for flames propagating vertically downwards (or flames that held stationary by a vertically upward flow) since in these cases, the denser unburnt gas lies beneath the lighter burnt gas mixture. Of course, flames that are propagating vertically upwards or those that are held stationary by a vertically downward flow, both the Darrieus–Landau mechanism and the Rayleigh–Taylor mechanism contributes to the destabilizing effect. The dispersion relation when buoyance forces are included becomes
  where  corresponds to gravitational acceleration for flames propagating downwards and  corresponds to gravitational acceleration for flames propagating upwards. The above dispersion implies that gravity introduces stability for downward propagating flames when , where  is a characteristic buoyancy length scale.
  Darrieus and Landau's analysis treats the flame as a plane sheet to investigate its stability with the neglect of diffusion effects, whereas in reality, the flame has a definite thickness, say the laminar flame thickness , where  is the thermal diffusivity, wherein diffusion effects cannot be neglected. Accounting for the flame structure, are found to stabilize the flames for small wavelengths , except when fuel diffusion coefficient and thermal diffusivity differ from each other significantly leading to the so-called (Turing) diffusive-thermal instability. 
  Darrieus–Landau instability manifests in the range  for downward propagating flames and  for upward propagating flames.