In combustion, Michelson–Sivashinsky equation describes the evolution of a premixed flame front, subjected to the Darrieus–Landau instability, in the small heat release approximation.
The equation was derived by Gregory Sivashinsky in 1977,[1] who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year.
[2] Let the planar flame front, in a uitable frame of reference be on the
-plane, then the evolution of this planar front is described by the amplitude function
) describing the deviation from the planar shape.
Incorporating also the Rayleigh–Taylor instability of the flame, one obtains the Rakib–Sivashinsky equation (named after Z. Rakib and Gregory Sivashinsky),[4] where
denotes the spatial average of
The equations, in the absence of gravity, admits an explicit solution, which is called as the N-pole solution since the equation admits a pole decomposition,as shown by Olivier Thual, Uriel Frisch and Michel Hénon in 1988.
(which appear in complex conjugate pairs) are poles in the complex plane.
, the it is sufficient to consider poles whose real parts lie between the interval
In this case, we have These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.
[10] In 1995,[11] John W. Dold and Guy Joulin generalised the Michelson–Sivashinsky equation by introducing the second-order time derivative, which is consistent with the quadratic nature of the dispersion relation for the Darrieus–Landau instability.
corresponds to the non-local integral operator.
In 1992,[12] Guy Joulin and Pierre Cambray extended the Michelson–Sivashinsky equation to include higher-order correction terms, following by an earlier incorrect attempt to derive such an equation by Gregory Sivashinsky and Paul Clavin.
[13] The Joulin–Cambray equation, in dimensional form, reads as