649:
356:
1588:
1218:
399:
138:
1331:
944:
644:{\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 }
884:
351:{\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,}
1583:{\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}}}
1213:{\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}}}
1712:
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.
1730:
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.
1685:
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.
1336:
949:
1655:
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.
737:
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,
32:
in 1977, who along the Daniel M. Michelson, presented the numerical solutions of the equation in the same year. Let the planar flame front, in a uitable frame of reference be on the
687:
130:
90:
1721:
Clavin, Paul, and Geoff Searby. Combustion waves and fronts in flows: flames, shocks, detonations, ablation fronts and explosion of stars. Cambridge
University Press, 2016.
916:
1257:
727:
1323:
1280:
379:
751:
53:
1300:
936:
707:
1694:
Frisch, Uriel, and Rudolf Morf. "Intermittency in nonlinear dynamics and singularities at complex times." Physical review A 23, no. 5 (1981): 2673.
1673:
Rakib, Z., & Sivashinsky, G. I. (1987). Instabilities in upward propagating flames. Combustion science and technology, 54(1-6), 69-84.
1602:
1703:
Joulin, Guy. "Nonlinear hydrodynamic instability of expanding flames: Intrinsic dynamics." Physical Review E 50, no. 3 (1994): 2030.
1664:
Matalon, Moshe. "Intrinsic flame instabilities in premixed and nonpremixed combustion." Annu. Rev. Fluid Mech. 39 (2007): 163-191.
382:
25:
1751:
1621:
Sivashinsky, G.I. (1977). "Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations".
1593:
These poles are interesting because in physical space, they correspond to locations of the cusps forming in the flame front.
1259:(which appear in complex conjugate pairs) are poles in the complex plane. In the case periodic solution with periodicity
1766:
1756:
657:
95:
58:
1761:
390:
29:
892:
132:) describing the deviation from the planar shape. The Michelson–Sivashinsky equation, reads as
1638:
1226:
712:
1630:
879:{\displaystyle u_{t}+uu_{x}-\nu u_{xx}=\int _{-\infty }^{+\infty }e^{ikx}{\hat {u}}(k,t)dk,}
1305:
1262:
364:
742:
55:-plane, then the evolution of this planar front is described by the amplitude function
35:
1285:
921:
692:
1745:
1634:
1282:, the it is sufficient to consider poles whose real parts lie between the interval
738:
17:
1642:
28:, in the small heat release approximation. The equation was derived by
24:
describes the evolution of a premixed flame front, subjected to the
1334:
1308:
1288:
1265:
1229:
947:
924:
895:
754:
715:
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402:
367:
141:
98:
61:
38:
1582:
1317:
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1274:
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878:
721:
701:
681:
643:
373:
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124:
84:
47:
8:
667:
661:
623:
617:
1567:
1555:
1541:
1523:
1510:
1503:
1485:
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1413:
1400:
1385:
1374:
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1307:
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1228:
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1182:
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1140:
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1098:
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948:
946:
923:
897:
896:
894:
841:
840:
828:
815:
807:
791:
775:
759:
753:
714:
709:, which is a time-dependent function and
694:
659:
591:
582:
565:
545:
536:
525:
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507:
502:
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477:
465:
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229:
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188:
165:
142:
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99:
97:
68:
60:
37:
1613:
381:is a constant. Incorporating also the
1681:
1679:
7:
682:{\displaystyle \langle u\rangle (t)}
1559:
1556:
1548:
1545:
1542:
1186:
1183:
1175:
1172:
1169:
938:. This has a solution of the form
819:
811:
745:in 1988. Consider the 1d equation
462:
439:
414:
406:
201:
178:
153:
145:
125:{\displaystyle \mathbf {x} =(x,y)}
14:
85:{\displaystyle u(\mathbf {x} ,t)}
592:
583:
566:
546:
537:
526:
508:
331:
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305:
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276:
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247:
100:
69:
689:denotes the spatial average of
640:
1573:
1552:
1425:
1419:
1354:
1342:
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1200:
1179:
1038:
1032:
967:
955:
902:
864:
852:
846:
676:
670:
576:
562:
554:
533:
513:
503:
446:
436:
385:of the flame, one obtains the
315:
301:
293:
272:
252:
242:
185:
175:
119:
107:
79:
65:
22:Michelson–Sivashinsky equation
1:
1603:Kuramoto–Sivashinsky equation
1635:10.1016/0094-5765(77)90096-0
918:is the Fourier transform of
383:Rayleigh–Taylor instability
26:Darrieus–Landau instability
1783:
911:{\displaystyle {\hat {u}}}
389:(named after Z. Rakib and
387:Rakib–Sivashinsky equation
1325:. In this case, we have
1252:{\displaystyle z_{n}(t)}
722:{\displaystyle \gamma }
1752:Differential equations
1584:
1393:
1319:
1296:
1276:
1253:
1214:
1129:
1009:
932:
912:
880:
723:
703:
683:
645:
375:
352:
126:
86:
49:
1585:
1370:
1320:
1318:{\displaystyle 2\pi }
1297:
1277:
1275:{\displaystyle 2\pi }
1254:
1215:
1094:
986:
933:
913:
881:
729:is another constant.
724:
704:
684:
646:
376:
353:
127:
87:
50:
1629:(11–12): 1177–1206.
1332:
1306:
1286:
1263:
1227:
945:
922:
893:
752:
713:
693:
658:
400:
374:{\displaystyle \nu }
365:
139:
96:
59:
36:
823:
391:Gregory Sivashinsky
30:Gregory Sivashinsky
1580:
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348:
122:
82:
48:{\displaystyle xy}
45:
1623:Acta Astronautica
1533:
1481:
1466:
1432:
1295:{\displaystyle 0}
1160:
1076:
1042:
931:{\displaystyle u}
905:
849:
702:{\displaystyle u}
497:
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1767:1977 in science
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824:
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771:
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749:
735:
733:N-pole solution
711:
710:
691:
690:
656:
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1757:Fluid dynamics
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743:Michel Hénon
739:Uriel Frisch
736:
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386:
360:
21:
15:
1762:Combustion
1746:Categories
1609:References
18:combustion
1643:0094-5765
1536:−
1517:−
1501:
1490:≠
1483:∑
1479:ν
1476:−
1407:−
1398:
1390:π
1372:∑
1368:ν
1365:−
1313:π
1270:π
1163:−
1147:−
1115:≠
1096:∑
1092:ν
1086:−
1020:−
988:∑
984:ν
978:−
903:^
847:^
820:∞
812:∞
809:−
805:∫
785:ν
782:−
717:γ
668:⟩
662:⟨
624:⟩
618:⟨
615:−
604:γ
542:−
531:⋅
500:∫
488:π
475:−
463:∇
459:ν
456:−
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415:∂
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369:ν
281:−
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239:∫
227:π
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154:∂
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