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 (where ) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as [3]
where is a constant. 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 , which is a time-dependent function and is another constant.
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. [5] [6] [7] [8] Consider the 1d equation
where is the Fourier transform of . This has a solution of the form [5] [9]
where (which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity , the it is sufficient to consider poles whose real parts lie between the interval and . 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 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 (where ) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as [3]
where is a constant. 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 , which is a time-dependent function and is another constant.
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. [5] [6] [7] [8] Consider the 1d equation
where is the Fourier transform of . This has a solution of the form [5] [9]
where (which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity , the it is sufficient to consider poles whose real parts lie between the interval and . 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]