198:
2107:
364:
1211:
766:
Suppose a radially symmetric hot source is deposited instantaneously in a reacting mixture. When the chemical time is comparable to the acoustic time, diffusion is neglected so that ignition is characterised by heat release by the chemical energy and cooling by the expansion waves. This problem is
49:
1950:
209:
1518:
2186:
Clarke, J. F. (1981), "Propagation of
Gasdynamic Disturbances in an Explosive Atmosphere", in Combustion in Reactive Systems, J.R. Bowen, R.I. Soloukhin, N. Manson, and A.K. Oppenheim (Eds), Progress in Astronautics and Aeronautics, pp.
1039:
1361:
193:{\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\left({\frac {\partial \theta }{\partial t}}-\gamma \delta e^{\theta }\right)=\nabla ^{2}\left({\frac {\partial \theta }{\partial t}}-\delta e^{\theta }\right)}
2102:{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right){\frac {\partial \theta }{\partial t}}=\left(\gamma {\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right)\delta \omega (\theta )}
359:{\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right){\frac {\partial \theta }{\partial t}}=\left(\gamma {\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2}\right)\delta e^{\theta }}
657:
538:
947:
2168:
Clarke, J. F. (1978). "A progress report on the theoretical analysis of the interaction between a shock wave and an explosive gas mixture", College of
Aeronautics report. 7801, Cranfield Inst. of Tech.
1848:
596:
488:
828:
2258:
Kapila, A. K., and J. W. Dold. "Evolution to detonation in a nonuniformly heated reactive medium." Asymptotic
Analysis and the Numerical Solution of Partial Differential Equations 130 (1991).
1907:
736:
1554:
1772:
1587:
2249:
Vázquez-Espí, C., & Liñán, A. (2001). Fast, non-diffusive ignition of a gaseous reacting mixture subject to a point energy source. Combustion Theory and
Modelling, 5(3), 485.
1739:
1706:
1673:
2218:
2139:
1407:
1399:
413:
2196:
Clarke, J. F. (1982). "Non-steady Gas
Dynamic Effects in the Induction Domain Behind a Strong Shock Wave", College of Aeronautics report. 8229, Cranfield Inst. of Tech.
1640:
1614:
1258:
756:
680:
437:
387:
1934:
855:
1031:
1011:
991:
967:
875:
1206:{\displaystyle t=0:\,-\theta =r^{2},\quad r=0:\,{\frac {\partial \theta }{\partial r}}=0,\quad r\gg 1:\,-\theta =r^{2}+(j+1){\frac {\gamma -1}{\gamma }}t^{2},}
1266:
2273:
39:
process, including both effects of constant-volume and constant-pressure processes, as well as the effects of adiabatic and isothermal
2177:
Clarke, J. F. (1978). Small amplitude gasdynamic disturbances in an exploding atmosphere. Journal of Fluid
Mechanics, 89(2), 343–355.
601:
493:
880:
1780:
543:
446:
2197:
28:
770:
1856:
685:
1033:
be the time, measured in units of acoustic time. In this case, the initial and boundary conditions are given by
659:
describes the wave propagation at isothermal sound speed. Molecular transports are neglected in the derivation.
2150:
1526:
2278:
1744:
1559:
1260:, respectively, corresponds to the planar, cylindrical and spherical problems. Let us define a new variable
1711:
1678:
1645:
1513:{\displaystyle \varphi =\gamma \delta te^{-r^{2}}+{\frac {1}{2}}(\gamma \delta t)^{2}e^{-2r^{2}}+\cdots }
32:
2115:
440:
2283:
416:
1369:
392:
1619:
1592:
1219:
741:
665:
422:
372:
2227:
1912:
833:
36:
16:
Equation that calculates gas diffusion across membranes in biological and medical contexts
2198:
https://repository.tudelft.nl/view/aereports/uuid%3A9c064b5f-97b4-4527-a97e-a805d5e1abd7
1016:
996:
976:
952:
860:
762:
Example: Fast, non-diffusive ignition by deposition of a radially symmetric hot source
2267:
1401:
from its distant values. Then, at small times, the asymptotic solution is given by
1013:
denote the distance from the center, measured in units of initial hot core size and
2209:
970:
1356:{\displaystyle \varphi (r,t)=\theta +r^{2}+(j+1){\frac {\gamma -1}{\gamma }}t^{2}}
40:
20:
2232:
2213:
540:
describes the thermal explosion at constant volume. Similarly, the term
598:
describes the wave propagation at adiabatic sound speed and the term
490:
describes the thermal explosion at constant pressure and the term
652:{\displaystyle \gamma \partial ^{2}/\partial t^{2}-\nabla ^{2}}
533:{\displaystyle \partial \theta /\partial t-\gamma e^{\theta }}
942:{\displaystyle \varepsilon T_{m}\equiv RT_{m}^{2}/E\ll T_{m}}
1944:
For generalised form for the reaction term, one may write
1843:{\displaystyle \varphi =-\ln(1-\gamma \delta te^{-r^{2}})}
758:
may also appear in the initial and boundary conditions.
591:{\displaystyle \partial ^{2}/\partial t^{2}-\nabla ^{2}}
1774:, the solution in the first approximation is given by
682:
can be removed from the equation by the transformation
483:{\displaystyle \partial \theta /\partial t-e^{\theta }}
2141:
is arbitrary function representing the reaction term.
1523:
As time progresses, a steady state is approached when
2118:
1953:
1915:
1859:
1783:
1747:
1714:
1681:
1648:
1622:
1595:
1562:
1529:
1410:
1372:
1269:
1222:
1042:
1019:
999:
979:
955:
883:
863:
836:
773:
744:
688:
668:
604:
546:
496:
449:
425:
395:
375:
212:
52:
2219:
Biographical
Memoirs of Fellows of the Royal Society
823:{\displaystyle \theta =(T_{m}-T)/\varepsilon T_{m}}
2133:
2101:
1928:
1901:
1842:
1766:
1733:
1700:
1667:
1634:
1608:
1581:
1548:
1512:
1393:
1355:
1252:
1205:
1025:
1005:
985:
961:
941:
869:
849:
822:
750:
730:
674:
651:
590:
532:
482:
431:
407:
381:
358:
192:
2214:"John Frederick Clarke 1 May 1927 – 11 June 2013"
389:is the non-dimensional temperature perturbation,
1902:{\displaystyle t=t_{i}\equiv 1/(\gamma \delta )}
1556:and a thermal explosion is found to occur when
1853:which shows that thermal explosion occurs at
993:is the activation energy). Furthermore, let
8:
731:{\displaystyle (x,t)\to (\delta x,\delta t)}
2231:
2117:
2073:
2057:
2043:
2037:
2006:
1995:
1979:
1965:
1959:
1952:
1920:
1914:
1882:
1870:
1858:
1829:
1821:
1782:
1758:
1746:
1719:
1713:
1686:
1680:
1653:
1647:
1621:
1600:
1594:
1573:
1561:
1540:
1528:
1496:
1485:
1475:
1449:
1438:
1430:
1409:
1371:
1347:
1325:
1301:
1268:
1221:
1194:
1172:
1148:
1134:
1092:
1091:
1069:
1055:
1041:
1018:
998:
978:
954:
933:
918:
912:
907:
891:
882:
862:
841:
835:
814:
802:
787:
772:
743:
687:
667:
643:
630:
618:
612:
603:
582:
569:
557:
551:
545:
524:
503:
495:
474:
456:
448:
424:
394:
374:
350:
332:
316:
302:
296:
265:
254:
238:
224:
218:
211:
179:
149:
138:
120:
87:
73:
59:
53:
51:
2245:
2243:
2161:
1616:is the Frank-Kamenetskii parameter; if
1549:{\displaystyle \delta \leq \delta _{c}}
767:governed by the Clarke's equation with
1767:{\displaystyle \delta \gg \delta _{c}}
1582:{\displaystyle \delta >\delta _{c}}
949:is the Frank-Kamenetskii temperature (
738:, it is, however, retained here since
7:
857:is the maximum initial temperature,
35:in 1978. The equation describes the
1734:{\displaystyle \delta _{c}=0.91448}
1701:{\displaystyle \delta _{c}=0.73583}
1668:{\displaystyle \delta _{c}=0.50340}
2070:
2050:
2040:
2017:
2009:
1992:
1972:
1962:
1103:
1095:
640:
623:
609:
579:
562:
548:
508:
497:
461:
450:
329:
309:
299:
276:
268:
251:
231:
221:
160:
152:
135:
98:
90:
66:
56:
14:
662:It may appear that the parameter
2134:{\displaystyle \omega (\theta )}
1121:
1078:
2274:Partial differential equations
2128:
2122:
2096:
2090:
1896:
1887:
1837:
1799:
1472:
1459:
1388:
1376:
1322:
1310:
1285:
1273:
1247:
1229:
1169:
1157:
799:
780:
725:
707:
704:
701:
689:
1:
29:partial differential equation
1708:in the cylindrical case and
1394:{\displaystyle \theta (r,t)}
408:{\displaystyle \gamma >1}
1741:in the spherical case. For
1635:{\displaystyle \gamma =1.4}
1609:{\displaystyle \delta _{c}}
27:is a third-order nonlinear
2300:
1366:which is the increment of
1253:{\displaystyle j=(0,1,2)}
2151:Frank-Kamenetskii theory
43:. The equation reads as
1936:is the ignition time.
877:is the temperature and
751:{\displaystyle \delta }
675:{\displaystyle \delta }
432:{\displaystyle \delta }
382:{\displaystyle \theta }
2233:10.1098/rsbm.2014.0012
2135:
2103:
1930:
1903:
1844:
1768:
1735:
1702:
1669:
1636:
1610:
1583:
1550:
1514:
1395:
1357:
1254:
1207:
1027:
1007:
987:
963:
943:
871:
851:
824:
752:
732:
676:
653:
592:
534:
484:
433:
409:
383:
360:
194:
2136:
2104:
1931:
1929:{\displaystyle t_{i}}
1904:
1845:
1769:
1736:
1703:
1670:
1637:
1611:
1584:
1551:
1515:
1396:
1358:
1255:
1208:
1028:
1008:
988:
964:
944:
872:
852:
850:{\displaystyle T_{m}}
825:
753:
733:
677:
654:
593:
535:
485:
434:
410:
384:
361:
195:
33:John Frederick Clarke
2212:; Riley, N. (2014).
2116:
1951:
1913:
1857:
1781:
1745:
1712:
1679:
1675:in the planar case,
1646:
1620:
1593:
1560:
1527:
1408:
1370:
1267:
1220:
1040:
1017:
997:
977:
953:
881:
861:
834:
771:
742:
686:
666:
602:
544:
494:
447:
423:
393:
373:
210:
50:
917:
417:specific heat ratio
31:, first derived by
2131:
2099:
1926:
1899:
1840:
1764:
1731:
1698:
1665:
1632:
1606:
1579:
1546:
1510:
1391:
1353:
1250:
1203:
1023:
1003:
983:
959:
939:
903:
867:
847:
820:
748:
728:
672:
649:
588:
530:
480:
429:
405:
379:
356:
203:or, alternatively
190:
2064:
2024:
1986:
1457:
1341:
1188:
1110:
1026:{\displaystyle t}
1006:{\displaystyle r}
986:{\displaystyle E}
962:{\displaystyle R}
870:{\displaystyle T}
323:
283:
245:
167:
105:
80:
37:thermal explosion
25:Clarke's equation
2291:
2259:
2256:
2250:
2247:
2238:
2237:
2235:
2206:
2200:
2194:
2188:
2184:
2178:
2175:
2169:
2166:
2140:
2138:
2137:
2132:
2108:
2106:
2105:
2100:
2083:
2079:
2078:
2077:
2065:
2063:
2062:
2061:
2048:
2047:
2038:
2025:
2023:
2015:
2007:
2005:
2001:
2000:
1999:
1987:
1985:
1984:
1983:
1970:
1969:
1960:
1940:Generalised form
1935:
1933:
1932:
1927:
1925:
1924:
1908:
1906:
1905:
1900:
1886:
1875:
1874:
1849:
1847:
1846:
1841:
1836:
1835:
1834:
1833:
1773:
1771:
1770:
1765:
1763:
1762:
1740:
1738:
1737:
1732:
1724:
1723:
1707:
1705:
1704:
1699:
1691:
1690:
1674:
1672:
1671:
1666:
1658:
1657:
1641:
1639:
1638:
1633:
1615:
1613:
1612:
1607:
1605:
1604:
1588:
1586:
1585:
1580:
1578:
1577:
1555:
1553:
1552:
1547:
1545:
1544:
1519:
1517:
1516:
1511:
1503:
1502:
1501:
1500:
1480:
1479:
1458:
1450:
1445:
1444:
1443:
1442:
1400:
1398:
1397:
1392:
1362:
1360:
1359:
1354:
1352:
1351:
1342:
1337:
1326:
1306:
1305:
1259:
1257:
1256:
1251:
1212:
1210:
1209:
1204:
1199:
1198:
1189:
1184:
1173:
1153:
1152:
1111:
1109:
1101:
1093:
1074:
1073:
1032:
1030:
1029:
1024:
1012:
1010:
1009:
1004:
992:
990:
989:
984:
968:
966:
965:
960:
948:
946:
945:
940:
938:
937:
922:
916:
911:
896:
895:
876:
874:
873:
868:
856:
854:
853:
848:
846:
845:
829:
827:
826:
821:
819:
818:
806:
792:
791:
757:
755:
754:
749:
737:
735:
734:
729:
681:
679:
678:
673:
658:
656:
655:
650:
648:
647:
635:
634:
622:
617:
616:
597:
595:
594:
589:
587:
586:
574:
573:
561:
556:
555:
539:
537:
536:
531:
529:
528:
507:
489:
487:
486:
481:
479:
478:
460:
441:Damköhler number
439:is the relevant
438:
436:
435:
430:
414:
412:
411:
406:
388:
386:
385:
380:
365:
363:
362:
357:
355:
354:
342:
338:
337:
336:
324:
322:
321:
320:
307:
306:
297:
284:
282:
274:
266:
264:
260:
259:
258:
246:
244:
243:
242:
229:
228:
219:
199:
197:
196:
191:
189:
185:
184:
183:
168:
166:
158:
150:
143:
142:
130:
126:
125:
124:
106:
104:
96:
88:
81:
79:
78:
77:
64:
63:
54:
2299:
2298:
2294:
2293:
2292:
2290:
2289:
2288:
2264:
2263:
2262:
2257:
2253:
2248:
2241:
2208:
2207:
2203:
2195:
2191:
2185:
2181:
2176:
2172:
2167:
2163:
2159:
2147:
2114:
2113:
2069:
2053:
2049:
2039:
2033:
2029:
2016:
2008:
1991:
1975:
1971:
1961:
1958:
1954:
1949:
1948:
1942:
1916:
1911:
1910:
1866:
1855:
1854:
1825:
1817:
1779:
1778:
1754:
1743:
1742:
1715:
1710:
1709:
1682:
1677:
1676:
1649:
1644:
1643:
1618:
1617:
1596:
1591:
1590:
1569:
1558:
1557:
1536:
1525:
1524:
1492:
1481:
1471:
1434:
1426:
1406:
1405:
1368:
1367:
1343:
1327:
1297:
1265:
1264:
1218:
1217:
1190:
1174:
1144:
1102:
1094:
1065:
1038:
1037:
1015:
1014:
995:
994:
975:
974:
951:
950:
929:
887:
879:
878:
859:
858:
837:
832:
831:
810:
783:
769:
768:
764:
740:
739:
684:
683:
664:
663:
639:
626:
608:
600:
599:
578:
565:
547:
542:
541:
520:
492:
491:
470:
445:
444:
421:
420:
391:
390:
371:
370:
346:
328:
312:
308:
298:
292:
288:
275:
267:
250:
234:
230:
220:
217:
213:
208:
207:
175:
159:
151:
148:
144:
134:
116:
97:
89:
86:
82:
69:
65:
55:
48:
47:
17:
12:
11:
5:
2297:
2295:
2287:
2286:
2281:
2279:Fluid dynamics
2276:
2266:
2265:
2261:
2260:
2251:
2239:
2210:Bray, K. N. C.
2201:
2189:
2179:
2170:
2160:
2158:
2155:
2154:
2153:
2146:
2143:
2130:
2127:
2124:
2121:
2110:
2109:
2098:
2095:
2092:
2089:
2086:
2082:
2076:
2072:
2068:
2060:
2056:
2052:
2046:
2042:
2036:
2032:
2028:
2022:
2019:
2014:
2011:
2004:
1998:
1994:
1990:
1982:
1978:
1974:
1968:
1964:
1957:
1941:
1938:
1923:
1919:
1898:
1895:
1892:
1889:
1885:
1881:
1878:
1873:
1869:
1865:
1862:
1851:
1850:
1839:
1832:
1828:
1824:
1820:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1761:
1757:
1753:
1750:
1730:
1727:
1722:
1718:
1697:
1694:
1689:
1685:
1664:
1661:
1656:
1652:
1631:
1628:
1625:
1603:
1599:
1576:
1572:
1568:
1565:
1543:
1539:
1535:
1532:
1521:
1520:
1509:
1506:
1499:
1495:
1491:
1488:
1484:
1478:
1474:
1470:
1467:
1464:
1461:
1456:
1453:
1448:
1441:
1437:
1433:
1429:
1425:
1422:
1419:
1416:
1413:
1390:
1387:
1384:
1381:
1378:
1375:
1364:
1363:
1350:
1346:
1340:
1336:
1333:
1330:
1324:
1321:
1318:
1315:
1312:
1309:
1304:
1300:
1296:
1293:
1290:
1287:
1284:
1281:
1278:
1275:
1272:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1214:
1213:
1202:
1197:
1193:
1187:
1183:
1180:
1177:
1171:
1168:
1165:
1162:
1159:
1156:
1151:
1147:
1143:
1140:
1137:
1133:
1130:
1127:
1124:
1120:
1117:
1114:
1108:
1105:
1100:
1097:
1090:
1087:
1084:
1081:
1077:
1072:
1068:
1064:
1061:
1058:
1054:
1051:
1048:
1045:
1022:
1002:
982:
958:
936:
932:
928:
925:
921:
915:
910:
906:
902:
899:
894:
890:
886:
866:
844:
840:
817:
813:
809:
805:
801:
798:
795:
790:
786:
782:
779:
776:
763:
760:
747:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
697:
694:
691:
671:
646:
642:
638:
633:
629:
625:
621:
615:
611:
607:
585:
581:
577:
572:
568:
564:
560:
554:
550:
527:
523:
519:
516:
513:
510:
506:
502:
499:
477:
473:
469:
466:
463:
459:
455:
452:
428:
404:
401:
398:
378:
367:
366:
353:
349:
345:
341:
335:
331:
327:
319:
315:
311:
305:
301:
295:
291:
287:
281:
278:
273:
270:
263:
257:
253:
249:
241:
237:
233:
227:
223:
216:
201:
200:
188:
182:
178:
174:
171:
165:
162:
157:
154:
147:
141:
137:
133:
129:
123:
119:
115:
112:
109:
103:
100:
95:
92:
85:
76:
72:
68:
62:
58:
15:
13:
10:
9:
6:
4:
3:
2:
2296:
2285:
2282:
2280:
2277:
2275:
2272:
2271:
2269:
2255:
2252:
2246:
2244:
2240:
2234:
2229:
2225:
2221:
2220:
2215:
2211:
2205:
2202:
2199:
2193:
2190:
2183:
2180:
2174:
2171:
2165:
2162:
2156:
2152:
2149:
2148:
2144:
2142:
2125:
2119:
2093:
2087:
2084:
2080:
2074:
2066:
2058:
2054:
2044:
2034:
2030:
2026:
2020:
2012:
2002:
1996:
1988:
1980:
1976:
1966:
1955:
1947:
1946:
1945:
1939:
1937:
1921:
1917:
1893:
1890:
1883:
1879:
1876:
1871:
1867:
1863:
1860:
1830:
1826:
1822:
1818:
1814:
1811:
1808:
1805:
1802:
1796:
1793:
1790:
1787:
1784:
1777:
1776:
1775:
1759:
1755:
1751:
1748:
1728:
1725:
1720:
1716:
1695:
1692:
1687:
1683:
1662:
1659:
1654:
1650:
1629:
1626:
1623:
1601:
1597:
1574:
1570:
1566:
1563:
1541:
1537:
1533:
1530:
1507:
1504:
1497:
1493:
1489:
1486:
1482:
1476:
1468:
1465:
1462:
1454:
1451:
1446:
1439:
1435:
1431:
1427:
1423:
1420:
1417:
1414:
1411:
1404:
1403:
1402:
1385:
1382:
1379:
1373:
1348:
1344:
1338:
1334:
1331:
1328:
1319:
1316:
1313:
1307:
1302:
1298:
1294:
1291:
1288:
1282:
1279:
1276:
1270:
1263:
1262:
1261:
1244:
1241:
1238:
1235:
1232:
1226:
1223:
1200:
1195:
1191:
1185:
1181:
1178:
1175:
1166:
1163:
1160:
1154:
1149:
1145:
1141:
1138:
1135:
1131:
1128:
1125:
1122:
1118:
1115:
1112:
1106:
1098:
1088:
1085:
1082:
1079:
1075:
1070:
1066:
1062:
1059:
1056:
1052:
1049:
1046:
1043:
1036:
1035:
1034:
1020:
1000:
980:
972:
956:
934:
930:
926:
923:
919:
913:
908:
904:
900:
897:
892:
888:
884:
864:
842:
838:
815:
811:
807:
803:
796:
793:
788:
784:
777:
774:
761:
759:
745:
722:
719:
716:
713:
710:
698:
695:
692:
669:
660:
644:
636:
631:
627:
619:
613:
605:
583:
575:
570:
566:
558:
552:
525:
521:
517:
514:
511:
504:
500:
475:
471:
467:
464:
457:
453:
442:
426:
418:
402:
399:
396:
376:
351:
347:
343:
339:
333:
325:
317:
313:
303:
293:
289:
285:
279:
271:
261:
255:
247:
239:
235:
225:
214:
206:
205:
204:
186:
180:
176:
172:
169:
163:
155:
145:
139:
131:
127:
121:
117:
113:
110:
107:
101:
93:
83:
74:
70:
60:
46:
45:
44:
42:
38:
34:
30:
26:
22:
2254:
2223:
2217:
2204:
2192:
2182:
2173:
2164:
2111:
1943:
1852:
1522:
1365:
1215:
971:gas constant
765:
661:
368:
202:
41:sound speeds
24:
18:
443:. The term
2284:Combustion
2268:Categories
2226:: 87–106.
2157:References
21:combustion
2126:θ
2120:ω
2094:θ
2088:ω
2085:δ
2071:∇
2067:−
2051:∂
2041:∂
2035:γ
2018:∂
2013:θ
2010:∂
1993:∇
1989:−
1973:∂
1963:∂
1894:δ
1891:γ
1877:≡
1823:−
1812:δ
1809:γ
1806:−
1797:
1791:−
1785:φ
1756:δ
1752:≫
1749:δ
1717:δ
1684:δ
1651:δ
1624:γ
1598:δ
1571:δ
1564:δ
1538:δ
1534:≤
1531:δ
1508:⋯
1487:−
1466:δ
1463:γ
1432:−
1421:δ
1418:γ
1412:φ
1374:θ
1339:γ
1332:−
1329:γ
1292:θ
1271:φ
1186:γ
1179:−
1176:γ
1139:θ
1136:−
1126:≫
1104:∂
1099:θ
1096:∂
1060:θ
1057:−
927:≪
898:≡
885:ε
808:ε
794:−
775:θ
746:δ
720:δ
711:δ
705:→
670:δ
641:∇
637:−
624:∂
610:∂
606:γ
580:∇
576:−
563:∂
549:∂
526:θ
518:γ
515:−
509:∂
501:θ
498:∂
476:θ
468:−
462:∂
454:θ
451:∂
427:δ
397:γ
377:θ
352:θ
344:δ
330:∇
326:−
310:∂
300:∂
294:γ
277:∂
272:θ
269:∂
252:∇
248:−
232:∂
222:∂
181:θ
173:δ
170:−
161:∂
156:θ
153:∂
136:∇
122:θ
114:δ
111:γ
108:−
99:∂
94:θ
91:∂
67:∂
57:∂
2187:383-402.
2145:See also
1909:, where
1589:, where
830:, where
1729:0.91448
1696:0.73583
1663:0.50340
1642:, then
969:is the
415:is the
2112:where
1216:where
369:where
1567:>
973:and
419:and
400:>
2228:doi
1630:1.4
19:In
2270::
2242:^
2224:60
2222:.
2216:.
1794:ln
23:,
2236:.
2230::
2129:)
2123:(
2097:)
2091:(
2081:)
2075:2
2059:2
2055:t
2045:2
2031:(
2027:=
2021:t
2003:)
1997:2
1981:2
1977:t
1967:2
1956:(
1922:i
1918:t
1897:)
1888:(
1884:/
1880:1
1872:i
1868:t
1864:=
1861:t
1838:)
1831:2
1827:r
1819:e
1815:t
1803:1
1800:(
1788:=
1760:c
1726:=
1721:c
1693:=
1688:c
1660:=
1655:c
1627:=
1602:c
1575:c
1542:c
1505:+
1498:2
1494:r
1490:2
1483:e
1477:2
1473:)
1469:t
1460:(
1455:2
1452:1
1447:+
1440:2
1436:r
1428:e
1424:t
1415:=
1389:)
1386:t
1383:,
1380:r
1377:(
1349:2
1345:t
1335:1
1323:)
1320:1
1317:+
1314:j
1311:(
1308:+
1303:2
1299:r
1295:+
1289:=
1286:)
1283:t
1280:,
1277:r
1274:(
1248:)
1245:2
1242:,
1239:1
1236:,
1233:0
1230:(
1227:=
1224:j
1201:,
1196:2
1192:t
1182:1
1170:)
1167:1
1164:+
1161:j
1158:(
1155:+
1150:2
1146:r
1142:=
1132::
1129:1
1123:r
1119:,
1116:0
1113:=
1107:r
1089::
1086:0
1083:=
1080:r
1076:,
1071:2
1067:r
1063:=
1053::
1050:0
1047:=
1044:t
1021:t
1001:r
981:E
957:R
935:m
931:T
924:E
920:/
914:2
909:m
905:T
901:R
893:m
889:T
865:T
843:m
839:T
816:m
812:T
804:/
800:)
797:T
789:m
785:T
781:(
778:=
726:)
723:t
717:,
714:x
708:(
702:)
699:t
696:,
693:x
690:(
645:2
632:2
628:t
620:/
614:2
584:2
571:2
567:t
559:/
553:2
522:e
512:t
505:/
472:e
465:t
458:/
403:1
348:e
340:)
334:2
318:2
314:t
304:2
290:(
286:=
280:t
262:)
256:2
240:2
236:t
226:2
215:(
187:)
177:e
164:t
146:(
140:2
132:=
128:)
118:e
102:t
84:(
75:2
71:t
61:2
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