1439:. All three can be regarded as giving the ratio of advective to diffusive effects for a particular physical field and have the form of the product of a velocity and a length divided by a diffusivity. While the magnetic Reynolds number is related to the magnetic field in an magnetohydrodynamic flow, the Reynolds number is related to the fluid velocity itself and the PĂ©clet number is related to heat. The dimensionless groups arise in the non-dimensionalization of the respective governing equations: the induction equation, the
963:
is estimated to be of order 10 . Dissipation is more significant, but a magnetic field is supported by motion in the liquid iron outer core. There are other bodies in the solar system that have working dynamos, e.g. Jupiter, Saturn, and
Mercury, and others that do not, e.g. Mars, Venus and the Moon.
1047:
must be large enough such that induction outweighs diffusion. It is not the absolute magnitude of velocity that is important for induction, but rather the relative differences and shearing in the flow, which stretch and fold magnetic field lines . A more appropriate form for the magnetic
Reynolds
383:
194:
669:
1112:
738:
796:
1200:
300:
278:. When the magnetic Reynolds number is very large, however, diffusion and the dynamo are less of a concern, and in this case focus instead often rests on the influence of the magnetic field on the flow.
890:. Flux lines of the magnetic field are then advected with the fluid flow, until such time as gradients are concentrated into regions of short enough length scale that diffusion can balance advection.
1318:
79:
1401:
1002:
884:
844:
1146:
1045:
961:
927:
504:
2166:
846:, advection is relatively unimportant, and so the magnetic field will tend to relax towards a purely diffusive state, determined by the boundary conditions rather than the flow.
606:
1519:
558:
435:
411:
1649:
1605:
1268:
1566:
1480:
1364:
457:
264:
1544:
1229:
1421:
1341:
1004:. The generation of magnetic field by the motion of a conducting fluid has been achieved in only a handful of large experiments using mercury or liquid sodium.
578:
524:
242:
220:
2159:
1273:
Many studies of the generation of magnetic field by a flow consider the computationally-convenient periodic cube. In this case the minimum is found to be
614:
2511:
2152:
1054:
680:
1651:
the skin effect dominates and the braking torque decreases much slower with increasing speed than predicted by the induction motor model.
378:{\displaystyle {\frac {\partial \mathbf {B} }{\partial t}}=\nabla \times (\mathbf {u} \times \mathbf {B} )+\eta \nabla ^{2}\mathbf {B} }
929:, of order 10. Dissipative affects are generally small, and there is no difficulty in maintaining a magnetic field against diffusion.
749:
1154:
2175:
17:
2099:
1279:
1833:
Moncheaux, R.; et al. (2007). "Generation of a
Magnetic Field by Dynamo Action in a Turbulent Flow of Liquid Sodium".
189:{\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}~~\sim {\frac {\mathrm {induction} }{\mathrm {diffusion} }}}
1369:
970:
852:
812:
2134:
2115:
1117:
where S is a measure of strain. One of the most well known results is due to Backus which states that the minimum
1120:
1019:
935:
901:
478:
1696:
475:. The relative importance of these two terms can be found by taking their ratio, the magnetic Reynolds number
2521:
2321:
1440:
2064:
Ripper, M.D; Endean, V.G (Mar 1975). "Eddy-Current
Braking-Torque Measurements on a Thick Copper Disc".
583:
2138:
2119:
1488:
529:
468:
274:
The mechanism by which the motion of a conducting fluid generates a magnetic field is the subject of
1919:
2381:
418:
394:
2516:
2376:
1798:
Steiglitz, R.; U. Muller (2001). "Experimental demonstration of a homogeneous two-scale dynamo".
1621:
1577:
1270:
is the maximum strain rate. This bound has since been improved by approximately 25% by
Proctor.
2276:
1968:
Proctor, M. (1977). "On Backus' necessary condition for dynamo action in a conducting sphere".
1234:
46:
1906:
1737:
Gailitis, A.; et al. (2001). "Magnetic field saturation in the Riga dynamo experiment".
1551:
2231:
2144:
2022:
1977:
1942:
1852:
1807:
1756:
1711:
1665:
1458:
1346:
460:
442:
287:
267:
249:
31:
2236:
1529:
8:
1208:
2301:
2026:
1981:
1946:
1856:
1811:
1760:
1715:
2485:
2271:
2211:
2046:
2012:
1876:
1842:
1780:
1746:
1612:
1406:
1326:
563:
509:
472:
291:
227:
205:
62:
54:
2440:
2405:
2186:
2130:
2111:
2095:
2038:
1954:
1895:
1868:
1772:
1013:
2050:
1880:
1697:"Constraints from material properties on the dynamics and evolution of Earth's core"
2490:
2361:
2346:
2311:
2221:
2073:
2034:
2030:
1985:
1950:
1860:
1815:
1764:
1719:
1660:
664:{\displaystyle \nabla \times (\mathbf {u} \times \mathbf {B} )\sim {\frac {UB}{L}}}
2366:
1864:
1784:
1675:
1432:
2455:
2435:
2393:
2388:
2216:
2103:
1670:
1436:
70:
66:
1768:
2465:
2445:
2430:
2425:
2371:
2356:
2331:
2326:
2316:
2296:
2286:
2251:
2196:
1933:
Backus, G. (1958). "A class of self-sustaining dissipative spherical dynamos".
58:
1989:
2505:
2475:
2470:
2460:
2450:
2415:
2410:
2351:
2291:
2281:
2261:
2256:
2226:
2191:
1444:
275:
1012:
In situations where permanent magnetisation is not possible, e.g. above the
2480:
2420:
2336:
2306:
2266:
2241:
2201:
2077:
2042:
1872:
1776:
1107:{\displaystyle \mathrm {\hat {R}} _{\mathrm {m} }={\frac {L^{2}S}{\eta }}}
2341:
2246:
2206:
1608:
1343:
is the root-mean-square strain over a scaled domain with sides of length
733:{\displaystyle \eta \nabla ^{2}\mathbf {B} \sim {\frac {\eta B}{L^{2}}}.}
1847:
1751:
1366:. If shearing over small length scales in the cube is ruled out, then
1819:
1723:
50:
1426:
1148:
for generation of a magnetic field by flow in a sphere is such that
1482:, is also used in cases where there is no physical fluid involved.
2017:
801:
1521:× (characteristic length) × (characteristic velocity)
791:{\displaystyle \mathrm {R} _{\mathrm {m} }={\frac {UL}{\eta }}.}
467:
The first term on the right hand side accounts for effects from
1195:{\displaystyle \mathrm {\hat {R}} _{\mathrm {m} }\geq \pi ^{2}}
2174:
1615:
torque follows the theoretical curve of an induction motor.
1431:
The magnetic
Reynolds number has a similar form to both the
471:
in the plasma and the second term accounts for effects from
2003:
Willis, A. (2012). "Optimization of the
Magnetic Dynamo".
886:, diffusion is relatively unimportant on the length scale
506:. If it is assumed that both terms share the scale length
2118:) (Ed. G.K. Batchelor, H.K. Moffatt & M.G. Worster)
1450:
1313:{\displaystyle \mathrm {\hat {R}} _{\mathrm {m} }=2.48}
967:
The human length scale is very small so that typically
290:, the magnetic Reynolds number can be derived from the
1624:
1580:
1554:
1532:
1491:
1461:
1409:
1372:
1349:
1329:
1282:
1237:
1211:
1157:
1123:
1057:
1022:
973:
938:
904:
855:
815:
752:
683:
617:
586:
566:
532:
512:
481:
445:
421:
397:
303:
252:
230:
208:
82:
1797:
1643:
1599:
1560:
1538:
1513:
1474:
1415:
1395:
1358:
1335:
1312:
1262:
1223:
1194:
1140:
1106:
1039:
996:
955:
921:
878:
838:
790:
732:
663:
600:
572:
552:
518:
498:
451:
429:
405:
377:
258:
236:
214:
188:
1427:Relationship to Reynolds number and PĂ©clet number
2503:
1396:{\displaystyle \mathrm {R} _{\mathrm {m} }=1.73}
997:{\displaystyle \mathrm {R} _{\mathrm {m} }\ll 1}
879:{\displaystyle \mathrm {R} _{\mathrm {m} }\gg 1}
839:{\displaystyle \mathrm {R} _{\mathrm {m} }\ll 1}
1970:Geophysical & Astrophysical Fluid Dynamics
27:Dimensionless quantity in magnetohydrodynamics
2160:
802:General characteristics for large and small R
2063:
1455:The dimensionless magnetic Reynolds number,
61:by the motion of a conducting medium to the
1141:{\displaystyle \mathrm {R} _{\mathrm {m} }}
1040:{\displaystyle \mathrm {R} _{\mathrm {m} }}
956:{\displaystyle \mathrm {R} _{\mathrm {m} }}
922:{\displaystyle \mathrm {R} _{\mathrm {m} }}
499:{\displaystyle \mathrm {R} _{\mathrm {m} }}
2167:
2153:
2016:
1846:
1832:
1750:
2512:Dimensionless numbers of fluid mechanics
2176:Dimensionless numbers in fluid mechanics
1736:
743:The ratio of the two terms is therefore
222:is a typical velocity scale of the flow,
2127:An Introduction to Magnetohydrodynamics
1967:
1893:
608:, the induction term can be written as
49:that estimates the relative effects of
14:
2504:
2002:
1932:
1694:
244:is a typical length scale of the flow,
2148:
1896:"Reflections on Magnetohydrodynamics"
65:. It is the magnetic analogue of the
1451:Relationship to eddy current braking
2096:Reflections on Magnetohydrodynamics
24:
2088:
1381:
1375:
1298:
1287:
1173:
1162:
1132:
1126:
1073:
1062:
1031:
1025:
982:
976:
947:
941:
913:
907:
893:
864:
858:
824:
818:
761:
755:
688:
618:
601:{\displaystyle \mathbf {u} \sim U}
533:
490:
484:
361:
329:
317:
307:
180:
177:
174:
171:
168:
165:
162:
159:
156:
151:
148:
145:
142:
139:
136:
133:
130:
127:
91:
85:
25:
2533:
1514:{\displaystyle R_{m}=\mu \sigma }
1048:number in this case is therefore
1695:Davies, C.; et al. (2015).
1231:is the radius of the sphere and
698:
636:
628:
588:
423:
399:
371:
347:
339:
311:
1568:is the electrical conductivity.
1423:is the root-mean-square value.
1016:, to maintain a magnetic field
553:{\displaystyle \nabla \sim 1/L}
2108:Perspectives in Fluid Dynamics
2057:
2035:10.1103/PhysRevLett.109.251101
1996:
1961:
1926:
1887:
1826:
1791:
1730:
1688:
1290:
1165:
1065:
640:
624:
351:
335:
13:
1:
1865:10.1103/PhysRevLett.98.044502
1681:
281:
73:and is typically defined by:
1955:10.1016/0003-4916(58)90054-X
1546:is the magnetic permeability
430:{\displaystyle \mathbf {u} }
406:{\displaystyle \mathbf {B} }
7:
1769:10.1103/PhysRevLett.86.3024
1654:
1644:{\displaystyle R_{m}>30}
10:
2538:
2139:Cambridge University Press
2120:Cambridge University Press
2094:Moffatt, H. Keith, 2000, "
1600:{\displaystyle R_{m}<1}
674:and the diffusion term as
2182:
1990:10.1080/03091927708242317
1263:{\displaystyle S=e_{max}}
1007:
36:magnetic Reynolds number
18:Magnetic Reynolds Number
2005:Physical Review Letters
1835:Physical Review Letters
1739:Physical Review Letters
1561:{\displaystyle \sigma }
1441:Navier–Stokes equations
560:and the scale velocity
2125:P. A. Davidson, 2001,
2078:10.1049/piee.1975.0080
1914:Cite journal requires
1645:
1611:is negligible and the
1601:
1562:
1540:
1515:
1476:
1417:
1403:is the minimum, where
1397:
1360:
1337:
1314:
1264:
1225:
1196:
1142:
1108:
1041:
998:
957:
923:
880:
840:
792:
734:
665:
602:
574:
554:
520:
500:
453:
437:is the fluid velocity,
431:
413:is the magnetic field,
407:
379:
260:
238:
216:
190:
47:dimensionless quantity
1646:
1602:
1563:
1541:
1516:
1477:
1475:{\displaystyle R_{m}}
1418:
1398:
1361:
1359:{\displaystyle 2\pi }
1338:
1315:
1265:
1226:
1197:
1143:
1109:
1042:
999:
958:
924:
881:
841:
793:
735:
666:
603:
575:
555:
521:
501:
454:
452:{\displaystyle \eta }
432:
408:
380:
261:
259:{\displaystyle \eta }
239:
217:
191:
2522:Magnetohydrodynamics
1894:Moffatt, K. (2000).
1666:Magnetohydrodynamics
1622:
1613:eddy current braking
1578:
1552:
1539:{\displaystyle \mu }
1530:
1489:
1459:
1407:
1370:
1347:
1327:
1280:
1235:
1209:
1155:
1121:
1055:
1020:
971:
936:
902:
898:The Sun has a large
853:
813:
750:
681:
615:
584:
564:
530:
510:
479:
461:magnetic diffusivity
443:
419:
395:
301:
288:magnetohydrodynamics
268:magnetic diffusivity
250:
228:
206:
80:
32:magnetohydrodynamics
2027:2012PhRvL.109y1101W
1982:1977GApFD...9...89P
1947:1958AnPhy...4..372B
1857:2007PhRvL..98d4502M
1812:2001PhFl...13..561S
1761:2001PhRvL..86.3024G
1716:2015NatGe...8..678D
1224:{\displaystyle L=a}
2322:Keulegan–Carpenter
2102:2007-09-29 at the
1641:
1597:
1558:
1536:
1511:
1472:
1413:
1393:
1356:
1333:
1310:
1260:
1221:
1192:
1138:
1104:
1037:
994:
953:
919:
876:
836:
788:
730:
661:
598:
570:
550:
516:
496:
473:magnetic diffusion
469:magnetic induction
449:
427:
403:
375:
292:induction equation
256:
234:
212:
186:
63:magnetic diffusion
2499:
2498:
1820:10.1063/1.1331315
1800:Physics of Fluids
1745:(14): 3024–3027.
1704:Nature Geoscience
1416:{\displaystyle U}
1336:{\displaystyle S}
1293:
1168:
1102:
1068:
1014:Curie temperature
783:
725:
659:
573:{\displaystyle U}
519:{\displaystyle L}
324:
286:In the theory of
237:{\displaystyle L}
215:{\displaystyle U}
184:
120:
117:
113:
16:(Redirected from
2529:
2169:
2162:
2155:
2146:
2145:
2082:
2081:
2061:
2055:
2054:
2020:
2000:
1994:
1993:
1965:
1959:
1958:
1930:
1924:
1923:
1917:
1912:
1910:
1902:
1900:
1891:
1885:
1884:
1850:
1830:
1824:
1823:
1795:
1789:
1788:
1754:
1734:
1728:
1727:
1724:10.1038/ngeo2492
1701:
1692:
1661:Lundquist number
1650:
1648:
1647:
1642:
1634:
1633:
1606:
1604:
1603:
1598:
1590:
1589:
1567:
1565:
1564:
1559:
1545:
1543:
1542:
1537:
1520:
1518:
1517:
1512:
1501:
1500:
1481:
1479:
1478:
1473:
1471:
1470:
1422:
1420:
1419:
1414:
1402:
1400:
1399:
1394:
1386:
1385:
1384:
1378:
1365:
1363:
1362:
1357:
1342:
1340:
1339:
1334:
1319:
1317:
1316:
1311:
1303:
1302:
1301:
1295:
1294:
1286:
1269:
1267:
1266:
1261:
1259:
1258:
1230:
1228:
1227:
1222:
1201:
1199:
1198:
1193:
1191:
1190:
1178:
1177:
1176:
1170:
1169:
1161:
1147:
1145:
1144:
1139:
1137:
1136:
1135:
1129:
1113:
1111:
1110:
1105:
1103:
1098:
1094:
1093:
1083:
1078:
1077:
1076:
1070:
1069:
1061:
1046:
1044:
1043:
1038:
1036:
1035:
1034:
1028:
1003:
1001:
1000:
995:
987:
986:
985:
979:
962:
960:
959:
954:
952:
951:
950:
944:
928:
926:
925:
920:
918:
917:
916:
910:
885:
883:
882:
877:
869:
868:
867:
861:
845:
843:
842:
837:
829:
828:
827:
821:
797:
795:
794:
789:
784:
779:
771:
766:
765:
764:
758:
739:
737:
736:
731:
726:
724:
723:
714:
706:
701:
696:
695:
670:
668:
667:
662:
660:
655:
647:
639:
631:
607:
605:
604:
599:
591:
579:
577:
576:
571:
559:
557:
556:
551:
546:
525:
523:
522:
517:
505:
503:
502:
497:
495:
494:
493:
487:
458:
456:
455:
450:
436:
434:
433:
428:
426:
412:
410:
409:
404:
402:
384:
382:
381:
376:
374:
369:
368:
350:
342:
325:
323:
315:
314:
305:
265:
263:
262:
257:
243:
241:
240:
235:
221:
219:
218:
213:
195:
193:
192:
187:
185:
183:
154:
125:
118:
115:
114:
109:
101:
96:
95:
94:
88:
21:
2537:
2536:
2532:
2531:
2530:
2528:
2527:
2526:
2502:
2501:
2500:
2495:
2178:
2173:
2104:Wayback Machine
2091:
2089:Further reading
2086:
2085:
2062:
2058:
2001:
1997:
1966:
1962:
1931:
1927:
1915:
1913:
1904:
1903:
1898:
1892:
1888:
1848:physics/0701075
1831:
1827:
1796:
1792:
1752:physics/0010047
1735:
1731:
1699:
1693:
1689:
1684:
1671:Reynolds number
1657:
1629:
1625:
1623:
1620:
1619:
1585:
1581:
1579:
1576:
1575:
1553:
1550:
1549:
1531:
1528:
1527:
1496:
1492:
1490:
1487:
1486:
1466:
1462:
1460:
1457:
1456:
1453:
1437:Reynolds number
1429:
1408:
1405:
1404:
1380:
1379:
1374:
1373:
1371:
1368:
1367:
1348:
1345:
1344:
1328:
1325:
1324:
1297:
1296:
1285:
1284:
1283:
1281:
1278:
1277:
1248:
1244:
1236:
1233:
1232:
1210:
1207:
1206:
1186:
1182:
1172:
1171:
1160:
1159:
1158:
1156:
1153:
1152:
1131:
1130:
1125:
1124:
1122:
1119:
1118:
1089:
1085:
1084:
1082:
1072:
1071:
1060:
1059:
1058:
1056:
1053:
1052:
1030:
1029:
1024:
1023:
1021:
1018:
1017:
1010:
981:
980:
975:
974:
972:
969:
968:
946:
945:
940:
939:
937:
934:
933:
932:For the Earth,
912:
911:
906:
905:
903:
900:
899:
896:
894:Range of values
863:
862:
857:
856:
854:
851:
850:
823:
822:
817:
816:
814:
811:
810:
807:
805:
772:
770:
760:
759:
754:
753:
751:
748:
747:
719:
715:
707:
705:
697:
691:
687:
682:
679:
678:
648:
646:
635:
627:
616:
613:
612:
587:
585:
582:
581:
565:
562:
561:
542:
531:
528:
527:
511:
508:
507:
489:
488:
483:
482:
480:
477:
476:
444:
441:
440:
422:
420:
417:
416:
398:
396:
393:
392:
370:
364:
360:
346:
338:
316:
310:
306:
304:
302:
299:
298:
284:
251:
248:
247:
229:
226:
225:
207:
204:
203:
155:
126:
124:
102:
100:
90:
89:
84:
83:
81:
78:
77:
71:fluid mechanics
67:Reynolds number
43:
28:
23:
22:
15:
12:
11:
5:
2535:
2525:
2524:
2519:
2517:Fluid dynamics
2514:
2497:
2496:
2494:
2493:
2488:
2483:
2478:
2473:
2468:
2463:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2402:
2401:
2391:
2386:
2385:
2384:
2379:
2369:
2364:
2359:
2354:
2349:
2344:
2339:
2334:
2329:
2324:
2319:
2314:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2244:
2239:
2234:
2229:
2224:
2219:
2214:
2209:
2204:
2199:
2194:
2189:
2183:
2180:
2179:
2172:
2171:
2164:
2157:
2149:
2143:
2142:
2123:
2090:
2087:
2084:
2083:
2072:(3): 301–302.
2056:
2011:(25): 251101.
1995:
1960:
1941:(4): 372–447.
1925:
1916:|journal=
1886:
1825:
1806:(3): 561–564.
1790:
1729:
1710:(9): 678–685.
1686:
1685:
1683:
1680:
1679:
1678:
1673:
1668:
1663:
1656:
1653:
1640:
1637:
1632:
1628:
1596:
1593:
1588:
1584:
1572:
1571:
1570:
1569:
1557:
1547:
1535:
1525:
1510:
1507:
1504:
1499:
1495:
1469:
1465:
1452:
1449:
1428:
1425:
1412:
1392:
1389:
1383:
1377:
1355:
1352:
1332:
1321:
1320:
1309:
1306:
1300:
1292:
1289:
1257:
1254:
1251:
1247:
1243:
1240:
1220:
1217:
1214:
1203:
1202:
1189:
1185:
1181:
1175:
1167:
1164:
1134:
1128:
1115:
1114:
1101:
1097:
1092:
1088:
1081:
1075:
1067:
1064:
1033:
1027:
1009:
1006:
993:
990:
984:
978:
949:
943:
915:
909:
895:
892:
875:
872:
866:
860:
835:
832:
826:
820:
806:
803:
800:
799:
798:
787:
782:
778:
775:
769:
763:
757:
741:
740:
729:
722:
718:
713:
710:
704:
700:
694:
690:
686:
672:
671:
658:
654:
651:
645:
642:
638:
634:
630:
626:
623:
620:
597:
594:
590:
569:
549:
545:
541:
538:
535:
515:
492:
486:
465:
464:
448:
438:
425:
414:
401:
386:
385:
373:
367:
363:
359:
356:
353:
349:
345:
341:
337:
334:
331:
328:
322:
319:
313:
309:
283:
280:
272:
271:
255:
245:
233:
223:
211:
197:
196:
182:
179:
176:
173:
170:
167:
164:
161:
158:
153:
150:
147:
144:
141:
138:
135:
132:
129:
123:
112:
108:
105:
99:
93:
87:
59:magnetic field
41:
26:
9:
6:
4:
3:
2:
2534:
2523:
2520:
2518:
2515:
2513:
2510:
2509:
2507:
2492:
2489:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2400:
2397:
2396:
2395:
2392:
2390:
2387:
2383:
2380:
2378:
2375:
2374:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2323:
2320:
2318:
2315:
2313:
2310:
2308:
2305:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2255:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2235:
2233:
2232:Chandrasekhar
2230:
2228:
2225:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2193:
2190:
2188:
2185:
2184:
2181:
2177:
2170:
2165:
2163:
2158:
2156:
2151:
2150:
2147:
2140:
2136:
2135:0-521-79487-0
2132:
2128:
2124:
2121:
2117:
2116:0-521-53169-1
2113:
2109:
2105:
2101:
2097:
2093:
2092:
2079:
2075:
2071:
2067:
2060:
2052:
2048:
2044:
2040:
2036:
2032:
2028:
2024:
2019:
2014:
2010:
2006:
1999:
1991:
1987:
1983:
1979:
1975:
1971:
1964:
1956:
1952:
1948:
1944:
1940:
1936:
1929:
1921:
1908:
1897:
1890:
1882:
1878:
1874:
1870:
1866:
1862:
1858:
1854:
1849:
1844:
1841:(4): 044502.
1840:
1836:
1829:
1821:
1817:
1813:
1809:
1805:
1801:
1794:
1786:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1753:
1748:
1744:
1740:
1733:
1725:
1721:
1717:
1713:
1709:
1705:
1698:
1691:
1687:
1677:
1676:PĂ©clet number
1674:
1672:
1669:
1667:
1664:
1662:
1659:
1658:
1652:
1638:
1635:
1630:
1626:
1616:
1614:
1610:
1594:
1591:
1586:
1582:
1555:
1548:
1533:
1526:
1523:
1522:
1508:
1505:
1502:
1497:
1493:
1485:
1484:
1483:
1467:
1463:
1448:
1446:
1445:heat equation
1442:
1438:
1434:
1433:PĂ©clet number
1424:
1410:
1390:
1387:
1353:
1350:
1330:
1307:
1304:
1276:
1275:
1274:
1271:
1255:
1252:
1249:
1245:
1241:
1238:
1218:
1215:
1212:
1187:
1183:
1179:
1151:
1150:
1149:
1099:
1095:
1090:
1086:
1079:
1051:
1050:
1049:
1015:
1005:
991:
988:
965:
930:
891:
889:
873:
870:
847:
833:
830:
785:
780:
776:
773:
767:
746:
745:
744:
727:
720:
716:
711:
708:
702:
692:
684:
677:
676:
675:
656:
652:
649:
643:
632:
621:
611:
610:
609:
595:
592:
567:
547:
543:
539:
536:
513:
474:
470:
462:
446:
439:
415:
391:
390:
389:
365:
357:
354:
343:
332:
326:
320:
297:
296:
295:
293:
289:
279:
277:
276:dynamo theory
269:
253:
246:
231:
224:
209:
202:
201:
200:
121:
110:
106:
103:
97:
76:
75:
74:
72:
68:
64:
60:
56:
52:
48:
44:
37:
33:
19:
2398:
2126:
2122:, p 347–391.
2107:
2069:
2065:
2059:
2008:
2004:
1998:
1976:(1): 89–93.
1973:
1969:
1963:
1938:
1934:
1928:
1907:cite journal
1889:
1838:
1834:
1828:
1803:
1799:
1793:
1742:
1738:
1732:
1707:
1703:
1690:
1617:
1573:
1454:
1430:
1322:
1272:
1204:
1116:
1011:
966:
931:
897:
887:
848:
808:
742:
673:
466:
387:
285:
273:
198:
39:
35:
29:
2486:Weissenberg
1609:skin effect
2506:Categories
2406:Richardson
2187:Archimedes
1901:: 347–391.
1682:References
1443:, and the
580:such that
526:such that
282:Derivation
2491:Womersley
2382:turbulent
2362:Ohnesorge
2347:Marangoni
2312:Iribarren
2237:Damköhler
2222:Capillary
2018:1209.1559
1935:Ann. Phys
1556:σ
1534:μ
1509:σ
1506:μ
1354:π
1291:^
1184:π
1180:≥
1166:^
1100:η
1066:^
989:≪
871:≫
831:≪
781:η
709:η
703:∼
689:∇
685:η
644:∼
633:×
622:×
619:∇
593:∼
537:∼
534:∇
447:η
362:∇
358:η
344:×
333:×
330:∇
318:∂
308:∂
254:η
122:∼
111:η
55:induction
51:advection
2466:Suratman
2456:Strouhal
2436:Sherwood
2399:magnetic
2394:Reynolds
2389:Rayleigh
2377:magnetic
2217:Brinkman
2100:Archived
2066:Proc IEE
2051:23466555
2043:23368443
1881:21114816
1873:17358779
1777:11290098
1655:See also
1435:and the
2446:Stanton
2441:Shields
2431:Scruton
2426:Schmidt
2372:Prandtl
2357:Nusselt
2332:Laplace
2327:Knudsen
2317:Kapitza
2302:Görtler
2297:Grashof
2287:Galilei
2252:Deborah
2197:Bagnold
2023:Bibcode
1978:Bibcode
1943:Bibcode
1853:Bibcode
1808:Bibcode
1757:Bibcode
1712:Bibcode
459:is the
266:is the
45:) is a
2476:Ursell
2471:Taylor
2461:Stuart
2451:Stokes
2416:Rossby
2411:Roshko
2367:PĂ©clet
2352:Morton
2292:Graetz
2282:Froude
2272:Eötvös
2262:Eckert
2257:Dukhin
2227:Cauchy
2192:Atwood
2133:
2114:
2106:. In:
2049:
2041:
1879:
1871:
1785:638748
1783:
1775:
1323:where
1205:where
1008:Bounds
388:where
199:where
119:
116:
34:, the
2481:Weber
2421:Rouse
2337:Lewis
2307:Hagen
2277:Euler
2267:Ekman
2242:Darcy
2202:Bejan
2047:S2CID
2013:arXiv
1899:(PDF)
1877:S2CID
1843:arXiv
1781:S2CID
1747:arXiv
1700:(PDF)
1524:where
57:of a
2342:Mach
2247:Dean
2212:Bond
2207:Biot
2131:ISBN
2112:ISBN
2039:PMID
1920:help
1869:PMID
1773:PMID
1636:>
1618:For
1607:the
1592:<
1574:For
1391:1.73
1308:2.48
849:For
809:For
2137:),
2074:doi
2070:122
2031:doi
2009:109
1986:doi
1951:doi
1861:doi
1816:doi
1765:doi
1720:doi
69:in
53:or
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2037:.
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1743:86
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1706:.
1702:.
1639:30
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294::
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2161:t
2154:v
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2129:(
2110:(
2080:.
2076::
2053:.
2033::
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1992:.
1988::
1980::
1974:9
1957:.
1953::
1945::
1939:4
1922:)
1918:(
1883:.
1863::
1855::
1845::
1822:.
1818::
1810::
1787:.
1767::
1759::
1749::
1726:.
1722::
1714::
1708:8
1631:m
1627:R
1595:1
1587:m
1583:R
1503:=
1498:m
1494:R
1468:m
1464:R
1411:U
1388:=
1382:m
1376:R
1351:2
1331:S
1305:=
1299:m
1288:R
1256:x
1253:a
1250:m
1246:e
1242:=
1239:S
1219:a
1216:=
1213:L
1188:2
1174:m
1163:R
1133:m
1127:R
1096:S
1091:2
1087:L
1080:=
1074:m
1063:R
1032:m
1026:R
992:1
983:m
977:R
948:m
942:R
914:m
908:R
888:L
874:1
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859:R
834:1
825:m
819:R
804:m
786:.
777:L
774:U
768:=
762:m
756:R
728:.
721:2
717:L
712:B
699:B
693:2
657:L
653:B
650:U
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637:B
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625:(
596:U
589:u
568:U
548:L
544:/
540:1
514:L
491:m
485:R
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424:u
400:B
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366:2
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340:u
336:(
327:=
321:t
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232:L
210:U
181:n
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172:s
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