31:
1068:
662:
932:
1193:
545:
346:
1083:
257:
1063:{\displaystyle \oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =\iint _{S}\nabla \times \mathbf {E} \cdot \mathrm {d} \mathbf {S} =-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .}
925:
847:
The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.
458:
this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken. It also implies that the vector field can be expressed as the
450:
657:{\displaystyle \Gamma =\oint _{\partial S}\mathbf {V} \cdot \mathrm {d} \mathbf {l} =\iint _{S}\nabla \times \mathbf {V} \cdot \mathrm {d} \mathbf {S} =\iint _{S}{\boldsymbol {\omega }}\cdot \mathrm {d} \mathbf {S} }
526:
735:
per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation, i.e. it can be expressed as the product of the circulation Î about the body, the fluid density
825:
929:
or that the circulation of the electric field around a loop is equal to the negative rate of change of the magnetic field flux through any surface spanned by the loop, by Stokes' theorem
1266:
874:
781:
754:
401:
1188:{\displaystyle \oint _{\partial S}\mathbf {B} \cdot \mathrm {d} \mathbf {l} =\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\text{enc}}.}
341:{\displaystyle \mathrm {d} \Gamma =\mathbf {V} \cdot \mathrm {d} \mathbf {l} =\left|\mathbf {V} \right|\left|\mathrm {d} \mathbf {l} \right|\cos \theta .}
492:
1815:
17:
871:
can be stated in two equivalent forms: that the curl of the electric field is equal to the negative rate of change of the magnetic field,
868:
1197:
For systems with electric fields that change over time, the law must be modified to include a term known as
Maxwell's correction.
786:
1956:
1872:
840:
where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the
698:
1928:
1808:
1834:
1839:
1222:
1781:
1482:
1319:
250:
length of a small element of a defined curve, the contribution of that differential length to circulation is
208:
2007:
1801:
1522:
1408:
852:
247:
1477:
1386:
1269:
726:
701:. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.
2012:
1393:
455:
1688:
1683:
1472:
1465:
1298:
2002:
1751:
1746:
1415:
759:
1829:
1303:
1726:
1344:
1073:
1948:
1864:
1564:
1381:
1361:
1349:
1293:
739:
200:
8:
1766:
1614:
1507:
1213:
673:
30:
1786:
1420:
1376:
1371:
920:{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}
531:
476:
138:
115:
to the closed loop and contribute to circulation, the perpendicular components do not.
1077:
1952:
1924:
1868:
1403:
1354:
683:
712:, all closed curves that enclose the vorticity have the same value for circulation.
1741:
1716:
1629:
1604:
1599:
1554:
464:
193:
151:
depicted as a helical arrow (not a literal representation). Note the projection of
1972:
1893:
1731:
1655:
1619:
1569:
1500:
1489:
1434:
1336:
841:
1858:
40:, around the boundary of an open curved surface with infinitesimal line element
1736:
1594:
1559:
1460:
1366:
705:
216:
189:
185:
1996:
1776:
1609:
837:
832:
This equation applies around airfoils, where the circulation is generated by
445:{\displaystyle \Gamma =\oint _{C}\mathbf {V} \cdot \mathrm {d} \mathbf {l} .}
395:
181:
1761:
1756:
1721:
1453:
732:
386:
204:
1894:"The Feynman Lectures on Physics Vol. II Ch. 3: Vector Integral Calculus"
1771:
1674:
62:
1973:"The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction"
1693:
1589:
1665:
1660:
1494:
709:
486:
219:
1644:
1549:
1529:
1515:
521:{\displaystyle {\boldsymbol {\omega }}=\nabla \times \mathbf {V} .}
460:
1398:
856:
112:
1539:
1859:
Robert W. Fox; Alan T. McDonald; Philip J. Pritchard (2003).
1443:
222:
102:
is split into components perpendicular (âĽ) parallel ( â ) to
862:
720:
535:
120:
1579:
1080:, proportional to the total current enclosed by the loop
27:
Line integral of the fluid velocity around a closed curve
1942:
756:, and the speed of the body relative to the free-stream
172:
may be in the negative sense, reducing the circulation.
855:
as an intermediate variable to calculate forces on an
1225:
1086:
935:
877:
789:
762:
742:
548:
495:
404:
260:
1260:
1187:
1062:
919:
819:
775:
748:
656:
542:is equal to the circulation around its perimeter,
520:
444:
340:
470:
1994:
196:, it can be the electric or the magnetic field.
836:; and around spinning objects experiencing the
538:of curl or vorticity vectors through a surface
829:This is known as the KuttaâJoukowski theorem.
49:along boundary, and through its interior with
1809:
228:
199:Circulation was first used independently by
184:of a vector field around a closed curve. In
1261:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
1816:
1802:
820:{\displaystyle L'=\rho v_{\infty }\Gamma }
1938:
1936:
863:Fundamental equations of electromagnetism
721:KuttaâJoukowski theorem in fluid dynamics
489:if the field is a fluid velocity field,
463:of a scalar function, which is called a
29:
637:
497:
14:
1995:
1933:
55:the infinitesimal surface element and
1915:
1913:
1888:
1886:
1884:
69:Circulation is the line integral of
1943:A.M. Kuethe; J.D. Schetzer (1959).
24:
1910:
1150:
1109:
1092:
1048:
1019:
1013:
995:
980:
958:
941:
908:
898:
878:
814:
809:
768:
666:Here, the closed integration path
645:
614:
599:
577:
560:
549:
504:
430:
405:
312:
281:
266:
262:
25:
2024:
1881:
356:is the angle between the vectors
1155:
1142:
1114:
1101:
1053:
1040:
1000:
987:
963:
950:
902:
885:
869:Maxwell-Faraday law of induction
676:or perimeter of an open surface
650:
619:
606:
582:
569:
511:
435:
422:
317:
298:
286:
273:
1861:Introduction to Fluid Mechanics
1835:BiotâSavart law in aerodynamics
1965:
1852:
682:, whose infinitesimal element
475:Circulation can be related to
471:Relation to vorticity and curl
111:, the parallel components are
34:Field lines of a vector field
13:
1:
1923:, section 3.16. McGraw-Hill.
1845:
851:Circulation is often used in
697:is oriented according to the
246:is a vector representing the
137:through the surface, and the
1921:Fundamentals of Aerodynamics
1840:Kelvin's circulation theorem
853:computational fluid dynamics
708:of a fluid with a region of
18:Circulation (fluid dynamics)
7:
1977:feynmanlectures.caltech.edu
1945:Foundations of Aerodynamics
1898:feynmanlectures.caltech.edu
1200:
776:{\displaystyle v_{\infty }}
485:and, more specifically, to
10:
2029:
1919:Anderson, John D. (1984),
724:
456:conservative vector field
229:Definition and properties
188:, the field is the fluid
1320:ClausiusâDuhem (entropy)
1270:Fick's laws of diffusion
867:In electrodynamics, the
211:. It is usually denoted
119:Circulation is also the
1478:NavierâStokes equations
1416:Material failure theory
731:In fluid dynamics, the
727:KuttaâJoukowski theorem
715:
65:normal to the surface.
1262:
1189:
1064:
921:
821:
777:
750:
658:
522:
446:
342:
239:is a vector field and
173:
1949:John Wiley & Sons
1473:Bernoulli's principle
1466:Archimedes' principle
1263:
1190:
1074:static magnetic field
1065:
922:
822:
778:
751:
749:{\displaystyle \rho }
659:
523:
447:
343:
75:around a closed loop
33:
1565:Cohesion (chemistry)
1387:Infinitesimal strain
1223:
1084:
933:
875:
787:
760:
740:
546:
493:
402:
258:
201:Frederick Lanchester
2008:Physical quantities
1830:Maxwell's equations
1483:Poiseuille equation
1214:Continuum mechanics
1208:Part of a series on
1689:Magnetorheological
1684:Electrorheological
1421:Fracture mechanics
1258:
1185:
1060:
917:
817:
773:
746:
654:
518:
479:of a vector field
442:
379:of a vector field
338:
174:
1958:978-0-471-50952-3
1874:978-0-471-20231-8
1826:
1825:
1701:
1700:
1635:
1634:
1404:Contact mechanics
1327:
1326:
1256:
1179:
1072:Circulation of a
1027:
915:
209:Nikolay Zhukovsky
96:, then sum. Here
16:(Redirected from
2020:
2013:Electromagnetism
1987:
1986:
1984:
1983:
1969:
1963:
1962:
1940:
1931:
1917:
1908:
1907:
1905:
1904:
1890:
1879:
1878:
1856:
1818:
1811:
1804:
1650:
1649:
1615:Gay-Lussac's law
1605:Combined gas law
1555:Capillary action
1440:
1439:
1283:
1282:
1267:
1265:
1264:
1259:
1257:
1255:
1247:
1239:
1205:
1204:
1194:
1192:
1191:
1186:
1181:
1180:
1177:
1171:
1170:
1158:
1153:
1145:
1140:
1139:
1130:
1129:
1117:
1112:
1104:
1099:
1098:
1069:
1067:
1066:
1061:
1056:
1051:
1043:
1038:
1037:
1028:
1026:
1022:
1016:
1011:
1003:
998:
990:
979:
978:
966:
961:
953:
948:
947:
926:
924:
923:
918:
916:
914:
906:
905:
896:
888:
826:
824:
823:
818:
813:
812:
797:
782:
780:
779:
774:
772:
771:
755:
753:
752:
747:
696:
681:
671:
663:
661:
660:
655:
653:
648:
640:
635:
634:
622:
617:
609:
598:
597:
585:
580:
572:
567:
566:
527:
525:
524:
519:
514:
500:
484:
451:
449:
448:
443:
438:
433:
425:
420:
419:
393:
384:
378:
368:
361:
355:
347:
345:
344:
339:
325:
321:
320:
315:
305:
301:
289:
284:
276:
265:
253:
245:
238:
214:
171:
165:
156:
146:
136:
110:
101:
95:
86:
80:
74:
60:
54:
48:
39:
21:
2028:
2027:
2023:
2022:
2021:
2019:
2018:
2017:
1993:
1992:
1991:
1990:
1981:
1979:
1971:
1970:
1966:
1959:
1941:
1934:
1918:
1911:
1902:
1900:
1892:
1891:
1882:
1875:
1857:
1853:
1848:
1822:
1793:
1792:
1791:
1711:
1703:
1702:
1656:Viscoelasticity
1647:
1637:
1636:
1624:
1574:
1570:Surface tension
1534:
1437:
1435:Fluid mechanics
1427:
1426:
1425:
1339:
1337:Solid mechanics
1329:
1328:
1280:
1272:
1248:
1240:
1238:
1224:
1221:
1220:
1203:
1176:
1172:
1166:
1162:
1154:
1149:
1141:
1135:
1131:
1125:
1121:
1113:
1108:
1100:
1091:
1087:
1085:
1082:
1081:
1052:
1047:
1039:
1033:
1029:
1018:
1017:
1012:
1010:
999:
994:
986:
974:
970:
962:
957:
949:
940:
936:
934:
931:
930:
907:
901:
897:
895:
884:
876:
873:
872:
865:
859:or other body.
842:Kutta condition
808:
804:
790:
788:
785:
784:
767:
763:
761:
758:
757:
741:
738:
737:
729:
723:
718:
699:right-hand rule
686:
677:
667:
649:
644:
636:
630:
626:
618:
613:
605:
593:
589:
581:
576:
568:
559:
555:
547:
544:
543:
532:Stokes' theorem
510:
496:
494:
491:
490:
480:
473:
434:
429:
421:
415:
411:
403:
400:
399:
389:
380:
376:
363:
357:
351:
316:
311:
310:
306:
297:
293:
285:
280:
272:
261:
259:
256:
255:
251:
240:
234:
231:
212:
194:electrodynamics
167:
158:
152:
142:
124:
103:
97:
88:
82:
76:
70:
56:
50:
41:
35:
28:
23:
22:
15:
12:
11:
5:
2026:
2016:
2015:
2010:
2005:
2003:Fluid dynamics
1989:
1988:
1964:
1957:
1947:(2 ed.).
1932:
1909:
1880:
1873:
1863:(6 ed.).
1850:
1849:
1847:
1844:
1843:
1842:
1837:
1832:
1824:
1823:
1821:
1820:
1813:
1806:
1798:
1795:
1794:
1790:
1789:
1784:
1779:
1774:
1769:
1764:
1759:
1754:
1749:
1744:
1739:
1734:
1729:
1724:
1719:
1713:
1712:
1709:
1708:
1705:
1704:
1699:
1698:
1697:
1696:
1691:
1686:
1678:
1677:
1671:
1670:
1669:
1668:
1663:
1658:
1648:
1643:
1642:
1639:
1638:
1633:
1632:
1626:
1625:
1623:
1622:
1617:
1612:
1607:
1602:
1597:
1592:
1586:
1583:
1582:
1576:
1575:
1573:
1572:
1567:
1562:
1560:Chromatography
1557:
1552:
1546:
1543:
1542:
1536:
1535:
1533:
1532:
1513:
1512:
1511:
1492:
1480:
1475:
1463:
1450:
1447:
1446:
1438:
1433:
1432:
1429:
1428:
1424:
1423:
1418:
1413:
1412:
1411:
1401:
1396:
1391:
1390:
1389:
1384:
1374:
1369:
1364:
1359:
1358:
1357:
1347:
1341:
1340:
1335:
1334:
1331:
1330:
1325:
1324:
1323:
1322:
1314:
1313:
1309:
1308:
1307:
1306:
1301:
1296:
1288:
1287:
1281:
1278:
1277:
1274:
1273:
1268:
1254:
1251:
1246:
1243:
1237:
1234:
1231:
1228:
1217:
1216:
1210:
1209:
1202:
1199:
1184:
1175:
1169:
1165:
1161:
1157:
1152:
1148:
1144:
1138:
1134:
1128:
1124:
1120:
1116:
1111:
1107:
1103:
1097:
1094:
1090:
1059:
1055:
1050:
1046:
1042:
1036:
1032:
1025:
1021:
1015:
1009:
1006:
1002:
997:
993:
989:
985:
982:
977:
973:
969:
965:
960:
956:
952:
946:
943:
939:
913:
910:
904:
900:
894:
891:
887:
883:
880:
864:
861:
834:airfoil action
816:
811:
807:
803:
800:
796:
793:
770:
766:
745:
725:Main article:
722:
719:
717:
714:
706:potential flow
652:
647:
643:
639:
633:
629:
625:
621:
616:
612:
608:
604:
601:
596:
592:
588:
584:
579:
575:
571:
565:
562:
558:
554:
551:
517:
513:
509:
506:
503:
499:
472:
469:
441:
437:
432:
428:
424:
418:
414:
410:
407:
337:
334:
331:
328:
324:
319:
314:
309:
304:
300:
296:
292:
288:
283:
279:
275:
271:
268:
264:
230:
227:
190:velocity field
186:fluid dynamics
26:
9:
6:
4:
3:
2:
2025:
2014:
2011:
2009:
2006:
2004:
2001:
2000:
1998:
1978:
1974:
1968:
1960:
1954:
1950:
1946:
1939:
1937:
1930:
1929:0-07-001656-9
1926:
1922:
1916:
1914:
1899:
1895:
1889:
1887:
1885:
1876:
1870:
1866:
1862:
1855:
1851:
1841:
1838:
1836:
1833:
1831:
1828:
1827:
1819:
1814:
1812:
1807:
1805:
1800:
1799:
1797:
1796:
1788:
1785:
1783:
1780:
1778:
1775:
1773:
1770:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1720:
1718:
1715:
1714:
1707:
1706:
1695:
1692:
1690:
1687:
1685:
1682:
1681:
1680:
1679:
1676:
1673:
1672:
1667:
1664:
1662:
1659:
1657:
1654:
1653:
1652:
1651:
1646:
1641:
1640:
1631:
1628:
1627:
1621:
1618:
1616:
1613:
1611:
1608:
1606:
1603:
1601:
1600:Charles's law
1598:
1596:
1593:
1591:
1588:
1587:
1585:
1584:
1581:
1578:
1577:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1547:
1545:
1544:
1541:
1538:
1537:
1531:
1528:
1524:
1521:
1517:
1514:
1509:
1508:non-Newtonian
1506:
1502:
1498:
1497:
1496:
1493:
1491:
1488:
1484:
1481:
1479:
1476:
1474:
1471:
1467:
1464:
1462:
1459:
1455:
1452:
1451:
1449:
1448:
1445:
1442:
1441:
1436:
1431:
1430:
1422:
1419:
1417:
1414:
1410:
1407:
1406:
1405:
1402:
1400:
1397:
1395:
1394:Compatibility
1392:
1388:
1385:
1383:
1382:Finite strain
1380:
1379:
1378:
1375:
1373:
1370:
1368:
1365:
1363:
1360:
1356:
1353:
1352:
1351:
1348:
1346:
1343:
1342:
1338:
1333:
1332:
1321:
1318:
1317:
1316:
1315:
1311:
1310:
1305:
1302:
1300:
1297:
1295:
1292:
1291:
1290:
1289:
1286:Conservations
1285:
1284:
1276:
1275:
1271:
1252:
1249:
1244:
1241:
1235:
1232:
1229:
1226:
1219:
1218:
1215:
1212:
1211:
1207:
1206:
1198:
1195:
1182:
1173:
1167:
1163:
1159:
1146:
1136:
1132:
1126:
1122:
1118:
1105:
1095:
1088:
1079:
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