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

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

where ${\displaystyle \nu }$ 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]

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

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

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

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

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

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

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

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

Dold–Joulin equation

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. The Dold–Joulin equation is given by

${\displaystyle {\frac {\partial ^{2}\varphi }{\partial t^{2}}}+{\mathcal {I}}\left({\frac {\partial \varphi }{\partial t}}-{\frac {1}{2}}(\nabla \varphi )^{2}-\nu \nabla ^{2}\varphi -\nu {\mathcal {I}}(\varphi )\right)=0,}$

where ${\displaystyle {\mathcal {I}}(e^{i\mathbf {k} \cdot \mathbf {x} })=|\mathbf {k} |e^{i\mathbf {k} \cdot \mathbf {x} }}$ corresponds to the non-local integral operator.

Joulin–Cambray equation

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

${\displaystyle {\frac {\partial \phi }{\partial t}}+{\frac {S_{L}}{2}}\left(1+{\frac {\epsilon }{2}}\right)|\nabla \phi |^{2}+\epsilon {\frac {S_{L}}{4}}\langle |\nabla \phi |^{2}\rangle ={\frac {S_{L}\epsilon }{2}}\left(1+{\frac {\epsilon }{2}}\right)\left(\nu \nabla ^{2}\phi +I(\phi ,\mathbf {x} )\right).}$

See also

• Kuramoto–Sivashinsky equation

References

1. 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.
2. 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.
3. Matalon, Moshe. "Intrinsic flame instabilities in premixed and nonpremixed combustion." Annu. Rev. Fluid Mech. 39 (2007): 163-191.
4. Rakib, Z., & Sivashinsky, G. I. (1987). Instabilities in upward propagating flames. Combustion science and technology, 54(1-6), 69-84.
5. 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.
6. Frisch, Uriel, and Rudolf Morf. "Intermittency in nonlinear dynamics and singularities at complex times." Physical review A 23, no. 5 (1981): 2673.
7. Joulin, Guy. "Nonlinear hydrodynamic instability of expanding flames: Intrinsic dynamics." Physical Review E 50, no. 3 (1994): 2030.
8. 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.
9. Clavin, Paul, and Geoff Searby. Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge University Press, 2016.
10. 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.
11. Dold, J. W., & Joulin, G. (1995). An evolution equation modeling inversion of tulip flames. Combustion and flame, 100(3), 450-456.
12. Joulin, G., & Cambray, P. (1992). On a tentative, approximate evolution equation for markedly wrinkled premixed flames. Combustion science and technology, 81(4-6), 243-256.
13. Sivashinsky, G. I., & Clavin, P. (1987). On the nonlinear theory of hydrodynamic instability in flames. Journal de Physique, 48(2), 193-198.