Michelson–Sivashinsky equation

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 $xy$ -plane, then the evolution of this planar front is described by the amplitude function $u(\mathbf {x} ,t)$ (where $\mathbf {x} =(x,y)$ ) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as^[3]

{\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x} ;,t)d\mathbf {k} d\mathbf {x} '=0,

where $\nu$ 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],

{\frac {\partial u}{\partial t}}+{\frac {1}{2}}(\nabla u)^{2}-\nu \nabla ^{2}u-{\frac {1}{8\pi ^{2}}}\int |\mathbf {k} |e^{i\mathbf {k} \cdot (\mathbf {x} -\mathbf {x} ')}u(\mathbf {x} ;,t)d\mathbf {k} d\mathbf {x} '+\gamma \left(u-\langle u\rangle \right)=0,\quad

where $\langle u\rangle (t)$ denotes the spatial average of $u$ , which is a time-dependent function and $\gamma$ is another constant.

N-pole solution

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

u_{t}+uu_{x}-\nu u_{xx}=\int _{-\infty }^{+\infty }e^{ikx}{\hat {u}}(k,t)dk,

where ${\hat {u}}$ is the Fourier transform of $u$ . This has a solution of the form^[5]^[9]

{\begin{aligned}u(x,t)&=-2\nu \sum _{n=1}^{2N}{\frac {1}{x-z_{n}(t)}},\\{\frac {dz_{n}}{dt}}&=-2\nu \sum _{l=1,l\neq n}^{2N}{\frac {1}{z_{n}-z_{l}}}-i\mathrm {sgn} (\mathrm {Im} z_{n}),\end{aligned}}

where $z_{n}(t)$ (which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity $2\pi$ , the it is sufficient to consider poles whose real parts lie between the interval $0$ and $2\pi$ . In this case, we have

{\begin{aligned}u(x,t)&=-\nu \sum _{n=1}^{2\pi }\cot {\frac {x-z_{n}(t)}{2}},\\{\frac {dz_{n}}{dt}}&=-\nu \sum _{l\neq n}\cot {\frac {z_{n}-z_{l}}{2}}-i\mathrm {sgn} (\mathrm {Im} z_{n})\end{aligned}}

These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.^[10]

References

^ Sivashinsky, G.I. (1977). "Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations". Acta Astronautica. 4 (11–12): 1177–1206. doi:10.1016/0094-5765(77)90096-0. ISSN 0094-5765.
^ Michelson, Daniel M., and Gregory I. Sivashinsky. "Nonlinear analysis of hydrodynamic instability in laminar flames—II. Numerical experiments." Acta astronautica 4, no. 11-12 (1977): 1207-1221.
^ Matalon, Moshe. "Intrinsic flame instabilities in premixed and nonpremixed combustion." Annu. Rev. Fluid Mech. 39 (2007): 163-191.
^ Rakib, Z., & Sivashinsky, G. I. (1987). Instabilities in upward propagating flames. Combustion science and technology, 54(1-6), 69-84.
^ ^a ^b Thual, O., U. Frisch, and M. Henon. "Application of pole decomposition to an equation governing the dynamics of wrinkled flame fronts." In Dynamics of curved fronts , pp. 489-498. Academic Press, 1988.
^ Frisch, Uriel, and Rudolf Morf. "Intermittency in nonlinear dynamics and singularities at complex times." Physical review A 23, no. 5 (1981): 2673.
^ Joulin, Guy. "Nonlinear hydrodynamic instability of expanding flames: Intrinsic dynamics." Physical Review E 50, no. 3 (1994): 2030.
^ Matsue, K., & Matalon, M. (2023). Dynamics of hydrodynamically unstable premixed flames in a gravitational field–local and global bifurcation structures. Combustion Theory and Modelling, 27(3), 346-374.
^ Clavin, Paul, and Geoff Searby. Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press, 2016.
^ Vaynblat, Dimitri, and Moshe Matalon. "Stability of pole solutions for planar propagating flames: I. Exact eigenvalues and eigenfunctions." SIAM Journal on Applied Mathematics 60, no. 2 (2000): 679-702.

This page was last edited on 16 June 2024, at 09:53

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N-pole solution

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References