1480:
586:
1725:
1265:
1332:
2060:
455:
2367:
1491:
1140:
165:
2462:
2222:
759:
431:
2514:
856:
376:
326:
718:
1475:{\displaystyle U=S_{L}{\sqrt {1+\left({\frac {\partial f}{\partial x}}\right)^{2}}},\quad \Rightarrow \quad {\frac {\partial f}{\partial x}}={\frac {\sqrt {U^{2}-S_{L}^{2}}}{S_{L}}}}
276:
2570:
Kerstein, Alan R., William T. Ashurst, and Forman A. Williams. "Field equation for interface propagation in an unsteady homogeneous flow field." Physical Review A 37.7 (1988): 2728.
581:{\displaystyle {\frac {S_{T}}{S_{L}}}=1+{\mathcal {M}}_{c}\delta _{L}\nabla \cdot \mathbf {n} +{\mathcal {M}}_{s}\tau _{L}\mathbf {n} \cdot \nabla \mathbf {v} \cdot \mathbf {n} }
1918:
962:
686:
655:
59:
442:
2119:
1782:
1841:
1324:
193:
1132:
788:
2597:
Markstein, G. H., & Squire, W. (1955). On the stability of a plane flame front in oscillating flow. The
Journal of the Acoustical Society of America, 27(3), 416-424.
1890:
2253:
1720:{\displaystyle f(x)={\frac {\left(U^{2}-S_{L}^{2}\right)^{1/2}}{S_{L}}}|x|+C,\quad \Rightarrow \quad G(x,y)={\frac {\left(U^{2}-S_{L}^{2}\right)^{1/2}}{S_{L}}}|x|+y+C}
2245:
1092:
889:
2544:
1260:{\displaystyle U{\frac {\partial G}{\partial y}}=S_{L}{\sqrt {\left({\frac {\partial G}{\partial x}}\right)^{2}+\left({\frac {\partial G}{\partial y}}\right)^{2}}}}
994:
619:
222:
1072:
1046:
1020:
87:
1910:
922:
1134:, which is a good approximation except near the wedge corner where curvature effects will becomes important. In the steady case, the G equation reduces to
2378:
1048:
lies at the location of the mouth of the slot. When the mixture is ignited, a premixed flame develops from the mouth of the slot to a certain height
2127:
723:
2561:
Williams, F. A. (1985). Turbulent combustion. In The mathematics of combustion (pp. 97-131). Society for
Industrial and Applied Mathematics.
2579:
GH Markstein. (1951). Interaction of flow pulsations and flame propagation. Journal of the
Aeronautical Sciences, 18(6), 428-429.
2373:
904:
The G equation has an exact expression for a simple slot burner. Consider a two-dimensional planar slot burner of slot width
381:
2470:
795:
331:
281:
695:
231:
2630:
2055:{\displaystyle G(x,y)={\frac {\left(U^{2}-S_{L}^{2}\right)^{1/2}}{S_{L}}}\left(|x|-{\frac {b}{2}}\right)+y}
927:
660:
629:
27:
2068:
1733:
2635:
1787:
1273:
892:
70:
176:
2362:{\displaystyle \tan \alpha ={\frac {b/2}{L}}={\frac {S_{T}}{\left(U^{2}-S_{L}^{2}\right)^{1/2}}}}
1097:
924:. The premixed reactant mixture is fed through the slot from the bottom with a constant velocity
766:
62:
1846:
2588:
Markstein, G. H. (Ed.). (2014). Nonsteady flame propagation: AGARDograph (Vol. 75). Elsevier.
2230:
1077:
861:
689:
66:
65:
in 1985 in the study of premixed turbulent combustion. The equation is derived based on the
2522:
967:
597:
200:
160:{\displaystyle {\frac {\partial G}{\partial t}}+\mathbf {v} \cdot \nabla G=S_{T}|\nabla G|}
1051:
1025:
999:
1895:
907:
622:
2624:
446:
2457:{\displaystyle \tan ^{2}\alpha =\sin ^{2}\alpha /\left(1-\sin ^{2}\alpha \right)}
2519:
In fact, the above formula is often used to determine the planar burning speed
61:
field equation which describes the instantaneous flame position, introduced by
17:
2606:
Peters, Norbert. Turbulent combustion. Cambridge university press, 2000.
2217:{\displaystyle L={\frac {b\left(U^{2}-S_{L}^{2}\right)^{1/2}}{2S_{L}}}}
754:{\displaystyle \mathbf {n} \cdot \nabla \mathbf {v} \cdot \mathbf {n} }
1074:
in the form of a two-dimensional wedge shape with a wedge angle
378:
is the region of unburnt gas. The normal vector to the flame is
1730:
Without loss of generality choose the flame location to be at
790:
are the laminar burning speed and thickness of a planar flame
667:
636:
533:
495:
1784:. Since the flame is attached to the mouth of the slot
2525:
2473:
2381:
2256:
2233:
2130:
2071:
1921:
1898:
1849:
1790:
1736:
1494:
1335:
1276:
1143:
1100:
1080:
1054:
1028:
1002:
970:
930:
910:
864:
798:
769:
726:
698:
663:
632:
600:
458:
384:
334:
284:
234:
203:
179:
90:
30:
449:, for small curvature and small strain, is given by
2538:
2508:
2456:
2361:
2239:
2216:
2113:
2054:
1904:
1884:
1835:
1776:
1719:
1474:
1318:
1259:
1126:
1086:
1066:
1040:
1014:
988:
956:
916:
883:
850:
782:
753:
712:
680:
649:
613:
580:
425:
370:
320:
270:
216:
187:
159:
73:, in a restrictive form for the burning velocity.
53:
761:corresponding to flow strain imposed on the flame
426:{\displaystyle \mathbf {n} =-\nabla G/|\nabla G|}
2615:Williams, Forman A. "Combustion theory." (1985).
2509:{\displaystyle \sin \alpha ={\frac {S_{L}}{U}}.}
2121:, which enable us to determine the flame height
1892:, which can be used to evaluate the constant
8:
851:{\displaystyle \tau _{L}=D_{T,u}/S_{L}^{2}}
278:which can be defined arbitrarily such that
371:{\displaystyle G(\mathbf {x} ,t)<G_{o}}
321:{\displaystyle G(\mathbf {x} ,t)>G_{o}}
2530:
2524:
2492:
2486:
2472:
2437:
2417:
2405:
2386:
2380:
2347:
2343:
2332:
2327:
2314:
2298:
2292:
2275:
2269:
2255:
2232:
2205:
2186:
2182:
2171:
2166:
2153:
2137:
2129:
2070:
2031:
2023:
2015:
2002:
1988:
1984:
1973:
1968:
1955:
1943:
1920:
1897:
1859:
1848:
1810:
1799:
1791:
1789:
1762:
1735:
1700:
1692:
1684:
1670:
1666:
1655:
1650:
1637:
1625:
1585:
1577:
1569:
1555:
1551:
1540:
1535:
1522:
1510:
1493:
1464:
1453:
1448:
1435:
1428:
1405:
1389:
1365:
1352:
1346:
1334:
1326:is introduced, then the equation becomes
1275:
1249:
1225:
1211:
1187:
1180:
1174:
1147:
1142:
1118:
1105:
1099:
1079:
1053:
1027:
1001:
969:
931:
929:
909:
869:
863:
842:
837:
828:
816:
803:
797:
774:
768:
746:
738:
727:
725:
713:{\displaystyle \nabla \cdot \mathbf {n} }
705:
697:
672:
666:
665:
662:
641:
635:
634:
631:
605:
599:
573:
565:
554:
548:
538:
532:
531:
522:
510:
500:
494:
493:
475:
465:
459:
457:
418:
407:
402:
385:
383:
362:
341:
333:
312:
291:
283:
262:
241:
233:
208:
202:
180:
178:
152:
141:
135:
114:
91:
89:
37:
29:
858:is the planar flame residence time with
2554:
271:{\displaystyle G(\mathbf {x} ,t)=G_{o}}
443:Matalon–Matkowsky–Clavin–Joulin theory
692:, associated with the curvature term
7:
69:. The equation was first studied by
1022:lies at the center of the slot and
1416:
1408:
1376:
1368:
1236:
1228:
1198:
1190:
1158:
1150:
957:{\displaystyle \mathbf {v} =(0,U)}
735:
699:
681:{\displaystyle {\mathcal {M}}_{s}}
650:{\displaystyle {\mathcal {M}}_{c}}
562:
516:
412:
396:
146:
122:
102:
94:
14:
54:{\displaystyle G(\mathbf {x} ,t)}
2546:, by measuring the wedge angle.
1094:. For simplicity, let us assume
932:
747:
739:
728:
706:
574:
566:
555:
523:
386:
342:
292:
242:
181:
115:
38:
2114:{\displaystyle x=0,\ y=L,\ G=0}
1603:
1599:
1404:
1400:
328:is the region of burnt gas and
228:The flame location is given by
2024:
2016:
1937:
1925:
1873:
1853:
1800:
1792:
1777:{\displaystyle G(x,y)=G_{o}=0}
1752:
1740:
1701:
1693:
1619:
1607:
1600:
1586:
1578:
1504:
1498:
1401:
1313:
1307:
1292:
1280:
983:
971:
951:
939:
900:A simple example - Slot burner
445:, the burning velocity of the
419:
408:
352:
338:
302:
288:
252:
238:
153:
142:
48:
34:
1:
1836:{\displaystyle |x|=b/2,\ y=0}
1485:which upon integration gives
1319:{\displaystyle G(x,y)=y+f(x)}
224:is the local burning velocity
1843:, the boundary condition is
1270:If a separation of the form
188:{\displaystyle \mathbf {v} }
1912:. Thus the scalar field is
1127:{\displaystyle S_{T}=S_{L}}
895:in the unburnt gas mixture.
783:{\displaystyle \delta _{L}}
621:is the burning velocity of
2652:
2065:At the flame tip, we have
1885:{\displaystyle G(b/2,0)=0}
195:is the flow velocity field
81:The G equation reads as
77:Mathematical description
2240:{\displaystyle \alpha }
1087:{\displaystyle \alpha }
964:, where the coordinate
884:{\displaystyle D_{T,u}}
2540:
2510:
2458:
2374:trigonometric identity
2363:
2241:
2218:
2115:
2056:
1906:
1886:
1837:
1778:
1721:
1476:
1320:
1261:
1128:
1088:
1068:
1042:
1016:
990:
958:
918:
885:
852:
784:
755:
714:
682:
651:
615:
582:
437:Local burning velocity
427:
372:
322:
272:
218:
189:
161:
55:
2541:
2539:{\displaystyle S_{L}}
2511:
2459:
2364:
2242:
2219:
2116:
2057:
1907:
1887:
1838:
1779:
1722:
1477:
1321:
1262:
1129:
1089:
1069:
1043:
1017:
991:
989:{\displaystyle (x,y)}
959:
919:
886:
853:
785:
756:
715:
683:
652:
616:
614:{\displaystyle S_{L}}
583:
428:
373:
323:
273:
219:
217:{\displaystyle S_{T}}
190:
162:
56:
2523:
2471:
2379:
2254:
2231:
2227:and the flame angle
2128:
2069:
1919:
1896:
1847:
1788:
1734:
1492:
1333:
1274:
1141:
1098:
1078:
1052:
1026:
1000:
996:is chosen such that
968:
928:
908:
862:
796:
767:
724:
696:
661:
630:
598:
456:
382:
332:
282:
232:
201:
177:
88:
28:
2337:
2176:
1978:
1660:
1545:
1458:
1067:{\displaystyle y=L}
1041:{\displaystyle y=0}
1015:{\displaystyle x=0}
893:thermal diffusivity
847:
71:George H. Markstein
2536:
2506:
2454:
2359:
2323:
2237:
2214:
2162:
2111:
2052:
1964:
1902:
1882:
1833:
1774:
1717:
1646:
1531:
1472:
1444:
1316:
1257:
1124:
1084:
1064:
1038:
1012:
986:
954:
914:
881:
848:
833:
780:
751:
710:
678:
647:
611:
578:
423:
368:
318:
268:
214:
185:
157:
63:Forman A. Williams
51:
2501:
2357:
2287:
2212:
2101:
2086:
2039:
2008:
1905:{\displaystyle C}
1823:
1690:
1575:
1470:
1459:
1423:
1395:
1383:
1255:
1243:
1205:
1165:
917:{\displaystyle b}
891:representing the
690:Markstein numbers
623:unstretched flame
481:
109:
2643:
2616:
2613:
2607:
2604:
2598:
2595:
2589:
2586:
2580:
2577:
2571:
2568:
2562:
2559:
2545:
2543:
2542:
2537:
2535:
2534:
2515:
2513:
2512:
2507:
2502:
2497:
2496:
2487:
2463:
2461:
2460:
2455:
2453:
2449:
2442:
2441:
2421:
2410:
2409:
2391:
2390:
2368:
2366:
2365:
2360:
2358:
2356:
2355:
2351:
2342:
2338:
2336:
2331:
2319:
2318:
2303:
2302:
2293:
2288:
2283:
2279:
2270:
2246:
2244:
2243:
2238:
2223:
2221:
2220:
2215:
2213:
2211:
2210:
2209:
2196:
2195:
2194:
2190:
2181:
2177:
2175:
2170:
2158:
2157:
2138:
2120:
2118:
2117:
2112:
2099:
2084:
2061:
2059:
2058:
2053:
2045:
2041:
2040:
2032:
2027:
2019:
2009:
2007:
2006:
1997:
1996:
1992:
1983:
1979:
1977:
1972:
1960:
1959:
1944:
1911:
1909:
1908:
1903:
1891:
1889:
1888:
1883:
1863:
1842:
1840:
1839:
1834:
1821:
1814:
1803:
1795:
1783:
1781:
1780:
1775:
1767:
1766:
1726:
1724:
1723:
1718:
1704:
1696:
1691:
1689:
1688:
1679:
1678:
1674:
1665:
1661:
1659:
1654:
1642:
1641:
1626:
1589:
1581:
1576:
1574:
1573:
1564:
1563:
1559:
1550:
1546:
1544:
1539:
1527:
1526:
1511:
1481:
1479:
1478:
1473:
1471:
1469:
1468:
1457:
1452:
1440:
1439:
1430:
1429:
1424:
1422:
1414:
1406:
1396:
1394:
1393:
1388:
1384:
1382:
1374:
1366:
1353:
1351:
1350:
1325:
1323:
1322:
1317:
1266:
1264:
1263:
1258:
1256:
1254:
1253:
1248:
1244:
1242:
1234:
1226:
1216:
1215:
1210:
1206:
1204:
1196:
1188:
1181:
1179:
1178:
1166:
1164:
1156:
1148:
1133:
1131:
1130:
1125:
1123:
1122:
1110:
1109:
1093:
1091:
1090:
1085:
1073:
1071:
1070:
1065:
1047:
1045:
1044:
1039:
1021:
1019:
1018:
1013:
995:
993:
992:
987:
963:
961:
960:
955:
935:
923:
921:
920:
915:
890:
888:
887:
882:
880:
879:
857:
855:
854:
849:
846:
841:
832:
827:
826:
808:
807:
789:
787:
786:
781:
779:
778:
760:
758:
757:
752:
750:
742:
731:
719:
717:
716:
711:
709:
687:
685:
684:
679:
677:
676:
671:
670:
656:
654:
653:
648:
646:
645:
640:
639:
620:
618:
617:
612:
610:
609:
587:
585:
584:
579:
577:
569:
558:
553:
552:
543:
542:
537:
536:
526:
515:
514:
505:
504:
499:
498:
482:
480:
479:
470:
469:
460:
432:
430:
429:
424:
422:
411:
406:
389:
377:
375:
374:
369:
367:
366:
345:
327:
325:
324:
319:
317:
316:
295:
277:
275:
274:
269:
267:
266:
245:
223:
221:
220:
215:
213:
212:
194:
192:
191:
186:
184:
166:
164:
163:
158:
156:
145:
140:
139:
118:
110:
108:
100:
92:
67:Level-set method
60:
58:
57:
52:
41:
2651:
2650:
2646:
2645:
2644:
2642:
2641:
2640:
2621:
2620:
2619:
2614:
2610:
2605:
2601:
2596:
2592:
2587:
2583:
2578:
2574:
2569:
2565:
2560:
2556:
2552:
2526:
2521:
2520:
2488:
2469:
2468:
2433:
2426:
2422:
2401:
2382:
2377:
2376:
2310:
2309:
2305:
2304:
2294:
2271:
2252:
2251:
2229:
2228:
2201:
2197:
2149:
2148:
2144:
2143:
2139:
2126:
2125:
2067:
2066:
2014:
2010:
1998:
1951:
1950:
1946:
1945:
1917:
1916:
1894:
1893:
1845:
1844:
1786:
1785:
1758:
1732:
1731:
1680:
1633:
1632:
1628:
1627:
1565:
1518:
1517:
1513:
1512:
1490:
1489:
1460:
1431:
1415:
1407:
1375:
1367:
1361:
1360:
1342:
1331:
1330:
1272:
1271:
1235:
1227:
1221:
1220:
1197:
1189:
1183:
1182:
1170:
1157:
1149:
1139:
1138:
1114:
1101:
1096:
1095:
1076:
1075:
1050:
1049:
1024:
1023:
998:
997:
966:
965:
926:
925:
906:
905:
902:
865:
860:
859:
812:
799:
794:
793:
770:
765:
764:
722:
721:
694:
693:
664:
659:
658:
633:
628:
627:
601:
596:
595:
544:
530:
506:
492:
471:
461:
454:
453:
447:stretched flame
439:
380:
379:
358:
330:
329:
308:
280:
279:
258:
230:
229:
204:
199:
198:
175:
174:
131:
101:
93:
86:
85:
79:
26:
25:
12:
11:
5:
2649:
2647:
2639:
2638:
2633:
2631:Fluid dynamics
2623:
2622:
2618:
2617:
2608:
2599:
2590:
2581:
2572:
2563:
2553:
2551:
2548:
2533:
2529:
2517:
2516:
2505:
2500:
2495:
2491:
2485:
2482:
2479:
2476:
2452:
2448:
2445:
2440:
2436:
2432:
2429:
2425:
2420:
2416:
2413:
2408:
2404:
2400:
2397:
2394:
2389:
2385:
2370:
2369:
2354:
2350:
2346:
2341:
2335:
2330:
2326:
2322:
2317:
2313:
2308:
2301:
2297:
2291:
2286:
2282:
2278:
2274:
2268:
2265:
2262:
2259:
2236:
2225:
2224:
2208:
2204:
2200:
2193:
2189:
2185:
2180:
2174:
2169:
2165:
2161:
2156:
2152:
2147:
2142:
2136:
2133:
2110:
2107:
2104:
2098:
2095:
2092:
2089:
2083:
2080:
2077:
2074:
2063:
2062:
2051:
2048:
2044:
2038:
2035:
2030:
2026:
2022:
2018:
2013:
2005:
2001:
1995:
1991:
1987:
1982:
1976:
1971:
1967:
1963:
1958:
1954:
1949:
1942:
1939:
1936:
1933:
1930:
1927:
1924:
1901:
1881:
1878:
1875:
1872:
1869:
1866:
1862:
1858:
1855:
1852:
1832:
1829:
1826:
1820:
1817:
1813:
1809:
1806:
1802:
1798:
1794:
1773:
1770:
1765:
1761:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1728:
1727:
1716:
1713:
1710:
1707:
1703:
1699:
1695:
1687:
1683:
1677:
1673:
1669:
1664:
1658:
1653:
1649:
1645:
1640:
1636:
1631:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1602:
1598:
1595:
1592:
1588:
1584:
1580:
1572:
1568:
1562:
1558:
1554:
1549:
1543:
1538:
1534:
1530:
1525:
1521:
1516:
1509:
1506:
1503:
1500:
1497:
1483:
1482:
1467:
1463:
1456:
1451:
1447:
1443:
1438:
1434:
1427:
1421:
1418:
1413:
1410:
1403:
1399:
1392:
1387:
1381:
1378:
1373:
1370:
1364:
1359:
1356:
1349:
1345:
1341:
1338:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1268:
1267:
1252:
1247:
1241:
1238:
1233:
1230:
1224:
1219:
1214:
1209:
1203:
1200:
1195:
1192:
1186:
1177:
1173:
1169:
1163:
1160:
1155:
1152:
1146:
1121:
1117:
1113:
1108:
1104:
1083:
1063:
1060:
1057:
1037:
1034:
1031:
1011:
1008:
1005:
985:
982:
979:
976:
973:
953:
950:
947:
944:
941:
938:
934:
913:
901:
898:
897:
896:
878:
875:
872:
868:
845:
840:
836:
831:
825:
822:
819:
815:
811:
806:
802:
791:
777:
773:
762:
749:
745:
741:
737:
734:
730:
708:
704:
701:
675:
669:
644:
638:
625:
608:
604:
589:
588:
576:
572:
568:
564:
561:
557:
551:
547:
541:
535:
529:
525:
521:
518:
513:
509:
503:
497:
491:
488:
485:
478:
474:
468:
464:
438:
435:
421:
417:
414:
410:
405:
401:
398:
395:
392:
388:
365:
361:
357:
354:
351:
348:
344:
340:
337:
315:
311:
307:
304:
301:
298:
294:
290:
287:
265:
261:
257:
254:
251:
248:
244:
240:
237:
226:
225:
211:
207:
196:
183:
168:
167:
155:
151:
148:
144:
138:
134:
130:
127:
124:
121:
117:
113:
107:
104:
99:
96:
78:
75:
50:
47:
44:
40:
36:
33:
13:
10:
9:
6:
4:
3:
2:
2648:
2637:
2634:
2632:
2629:
2628:
2626:
2612:
2609:
2603:
2600:
2594:
2591:
2585:
2582:
2576:
2573:
2567:
2564:
2558:
2555:
2549:
2547:
2531:
2527:
2503:
2498:
2493:
2489:
2483:
2480:
2477:
2474:
2467:
2466:
2465:
2450:
2446:
2443:
2438:
2434:
2430:
2427:
2423:
2418:
2414:
2411:
2406:
2402:
2398:
2395:
2392:
2387:
2383:
2375:
2352:
2348:
2344:
2339:
2333:
2328:
2324:
2320:
2315:
2311:
2306:
2299:
2295:
2289:
2284:
2280:
2276:
2272:
2266:
2263:
2260:
2257:
2250:
2249:
2248:
2234:
2206:
2202:
2198:
2191:
2187:
2183:
2178:
2172:
2167:
2163:
2159:
2154:
2150:
2145:
2140:
2134:
2131:
2124:
2123:
2122:
2108:
2105:
2102:
2096:
2093:
2090:
2087:
2081:
2078:
2075:
2072:
2049:
2046:
2042:
2036:
2033:
2028:
2020:
2011:
2003:
1999:
1993:
1989:
1985:
1980:
1974:
1969:
1965:
1961:
1956:
1952:
1947:
1940:
1934:
1931:
1928:
1922:
1915:
1914:
1913:
1899:
1879:
1876:
1870:
1867:
1864:
1860:
1856:
1850:
1830:
1827:
1824:
1818:
1815:
1811:
1807:
1804:
1796:
1771:
1768:
1763:
1759:
1755:
1749:
1746:
1743:
1737:
1714:
1711:
1708:
1705:
1697:
1685:
1681:
1675:
1671:
1667:
1662:
1656:
1651:
1647:
1643:
1638:
1634:
1629:
1622:
1616:
1613:
1610:
1604:
1596:
1593:
1590:
1582:
1570:
1566:
1560:
1556:
1552:
1547:
1541:
1536:
1532:
1528:
1523:
1519:
1514:
1507:
1501:
1495:
1488:
1487:
1486:
1465:
1461:
1454:
1449:
1445:
1441:
1436:
1432:
1425:
1419:
1411:
1397:
1390:
1385:
1379:
1371:
1362:
1357:
1354:
1347:
1343:
1339:
1336:
1329:
1328:
1327:
1310:
1304:
1301:
1298:
1295:
1289:
1286:
1283:
1277:
1250:
1245:
1239:
1231:
1222:
1217:
1212:
1207:
1201:
1193:
1184:
1175:
1171:
1167:
1161:
1153:
1144:
1137:
1136:
1135:
1119:
1115:
1111:
1106:
1102:
1081:
1061:
1058:
1055:
1035:
1032:
1029:
1009:
1006:
1003:
980:
977:
974:
948:
945:
942:
936:
911:
899:
894:
876:
873:
870:
866:
843:
838:
834:
829:
823:
820:
817:
813:
809:
804:
800:
792:
775:
771:
763:
743:
732:
720:and the term
702:
691:
673:
642:
626:
624:
606:
602:
594:
593:
592:
570:
559:
549:
545:
539:
527:
519:
511:
507:
501:
489:
486:
483:
476:
472:
466:
462:
452:
451:
450:
448:
444:
441:According to
436:
434:
415:
403:
399:
393:
390:
363:
359:
355:
349:
346:
335:
313:
309:
305:
299:
296:
285:
263:
259:
255:
249:
246:
235:
209:
205:
197:
173:
172:
171:
149:
136:
132:
128:
125:
119:
111:
105:
97:
84:
83:
82:
76:
74:
72:
68:
64:
45:
42:
31:
23:
19:
2611:
2602:
2593:
2584:
2575:
2566:
2557:
2518:
2371:
2226:
2064:
1729:
1484:
1269:
903:
688:are the two
590:
440:
227:
169:
80:
24:is a scalar
21:
15:
2636:Combustion
2625:Categories
2550:References
2464:, we have
2372:Using the
22:G equation
18:Combustion
2481:α
2478:
2447:α
2444:
2431:−
2415:α
2412:
2396:α
2393:
2321:−
2264:α
2261:
2235:α
2160:−
2029:−
1962:−
1644:−
1601:⇒
1529:−
1442:−
1417:∂
1409:∂
1402:⇒
1377:∂
1369:∂
1237:∂
1229:∂
1199:∂
1191:∂
1159:∂
1151:∂
1082:α
801:τ
772:δ
744:⋅
736:∇
733:⋅
703:⋅
700:∇
571:⋅
563:∇
560:⋅
546:τ
520:⋅
517:∇
508:δ
413:∇
397:∇
394:−
147:∇
123:∇
120:⋅
103:∂
95:∂
2100:
2085:
1822:
591:where
170:where
657:and
356:<
306:>
2475:sin
2435:sin
2403:sin
2384:tan
2258:tan
433:.
16:In
2627::
2247:,
20:,
2532:L
2528:S
2504:.
2499:U
2494:L
2490:S
2484:=
2451:)
2439:2
2428:1
2424:(
2419:/
2407:2
2399:=
2388:2
2353:2
2349:/
2345:1
2340:)
2334:2
2329:L
2325:S
2316:2
2312:U
2307:(
2300:T
2296:S
2290:=
2285:L
2281:2
2277:/
2273:b
2267:=
2207:L
2203:S
2199:2
2192:2
2188:/
2184:1
2179:)
2173:2
2168:L
2164:S
2155:2
2151:U
2146:(
2141:b
2135:=
2132:L
2109:0
2106:=
2103:G
2097:,
2094:L
2091:=
2088:y
2082:,
2079:0
2076:=
2073:x
2050:y
2047:+
2043:)
2037:2
2034:b
2025:|
2021:x
2017:|
2012:(
2004:L
2000:S
1994:2
1990:/
1986:1
1981:)
1975:2
1970:L
1966:S
1957:2
1953:U
1948:(
1941:=
1938:)
1935:y
1932:,
1929:x
1926:(
1923:G
1900:C
1880:0
1877:=
1874:)
1871:0
1868:,
1865:2
1861:/
1857:b
1854:(
1851:G
1831:0
1828:=
1825:y
1819:,
1816:2
1812:/
1808:b
1805:=
1801:|
1797:x
1793:|
1772:0
1769:=
1764:o
1760:G
1756:=
1753:)
1750:y
1747:,
1744:x
1741:(
1738:G
1715:C
1712:+
1709:y
1706:+
1702:|
1698:x
1694:|
1686:L
1682:S
1676:2
1672:/
1668:1
1663:)
1657:2
1652:L
1648:S
1639:2
1635:U
1630:(
1623:=
1620:)
1617:y
1614:,
1611:x
1608:(
1605:G
1597:,
1594:C
1591:+
1587:|
1583:x
1579:|
1571:L
1567:S
1561:2
1557:/
1553:1
1548:)
1542:2
1537:L
1533:S
1524:2
1520:U
1515:(
1508:=
1505:)
1502:x
1499:(
1496:f
1466:L
1462:S
1455:2
1450:L
1446:S
1437:2
1433:U
1426:=
1420:x
1412:f
1398:,
1391:2
1386:)
1380:x
1372:f
1363:(
1358:+
1355:1
1348:L
1344:S
1340:=
1337:U
1314:)
1311:x
1308:(
1305:f
1302:+
1299:y
1296:=
1293:)
1290:y
1287:,
1284:x
1281:(
1278:G
1251:2
1246:)
1240:y
1232:G
1223:(
1218:+
1213:2
1208:)
1202:x
1194:G
1185:(
1176:L
1172:S
1168:=
1162:y
1154:G
1145:U
1120:L
1116:S
1112:=
1107:T
1103:S
1062:L
1059:=
1056:y
1036:0
1033:=
1030:y
1010:0
1007:=
1004:x
984:)
981:y
978:,
975:x
972:(
952:)
949:U
946:,
943:0
940:(
937:=
933:v
912:b
877:u
874:,
871:T
867:D
844:2
839:L
835:S
830:/
824:u
821:,
818:T
814:D
810:=
805:L
776:L
748:n
740:v
729:n
707:n
674:s
668:M
643:c
637:M
607:L
603:S
575:n
567:v
556:n
550:L
540:s
534:M
528:+
524:n
512:L
502:c
496:M
490:+
487:1
484:=
477:L
473:S
467:T
463:S
420:|
416:G
409:|
404:/
400:G
391:=
387:n
364:o
360:G
353:)
350:t
347:,
343:x
339:(
336:G
314:o
310:G
303:)
300:t
297:,
293:x
289:(
286:G
264:o
260:G
256:=
253:)
250:t
247:,
243:x
239:(
236:G
210:T
206:S
182:v
154:|
150:G
143:|
137:T
133:S
129:=
126:G
116:v
112:+
106:t
98:G
49:)
46:t
43:,
39:x
35:(
32:G
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