1412:
1699:
630:
807:
1543:
1275:
1811:
The circulation around those closed streamlines is not zero (unless the velocity at each point of the streamline is zero with a possible discontinuous vorticity jump across the streamline) . The only way the above equation can be satisfied is only if
1558:
480:
175:
985:
530:
1806:
249:
can have infinite number of possibilities, all of which satisfies the equation and the boundary condition. This is not true if no streamline is closed, in which case, every streamline can be traced back to the boundary
1864:
i.e., vorticity is not changing across these closed streamlines, thus proving the theorem. Of course, the theorem is not valid inside the boundary layer regime. This theorem cannot be derived from the Euler equations.
1236:
712:
1063:
670:
1267:
911:
322:
are prescribed. The difficulty arises only when there are some closed streamlines inside the domain that does not connect to the boundary and one may suppose that at high
Reynolds numbers,
1859:
1407:{\displaystyle {\frac {1}{\mathrm {Re} }}\oint _{C}\nabla \omega \cdot \mathbf {n} \ dl={\frac {1}{\mathrm {Re} }}\oint _{C}{\frac {d\omega }{d\psi }}\nabla \psi \cdot \mathbf {n} \ dl=0.}
704:
1457:
1118:
1745:
1907:
Feynman, R. P., & Lagerstrom, P. A. (1956). Remarks on high
Reynolds number flows in finite domains. In Proc. IX International Congress on Applied Mechanics (Vol. 3, pp. 342-343).
1026:
407:
1694:{\displaystyle \Gamma =-\oint _{C}\mathbf {u} \cdot d\mathbf {l} =-\int _{S}\omega d\mathbf {S} =\int _{S}\nabla ^{2}\psi d\mathbf {S} =\oint _{C}\nabla \psi \cdot \mathbf {n} dl}
418:
1449:
378:
349:
320:
247:
1143:
832:
271:
1083:
198:
291:
93:
101:
855:
919:
875:
523:
503:
218:
1880:
Prandtl, L. (1904). Über
Flussigkeitsbewegung bei sehr kleiner Reibung. Verhandl. III, Internat. Math.-Kong., Heidelberg, Teubner, Leipzig, 1904, 484–491.
625:{\displaystyle \int _{S}\mathbf {u} \cdot \nabla \mathbf {\omega } \,d\mathbf {S} ={\frac {1}{\mathrm {Re} }}\int _{S}\nabla ^{2}\omega \,d\mathbf {S} .}
1753:
351:
is not uniquely defined in regions where closed streamlines occur. The
Prandtl–Batchelor theorem, however, asserts that this is not the case and
1925:
Lagerstrom, P. A. (1975). Solutions of the Navier–Stokes equation at large
Reynolds number. SIAM Journal on Applied mathematics, 28(1), 202-214.
1889:
Batchelor, G. K. (1956). On steady laminar flow with closed streamlines at large
Reynolds number. Journal of Fluid Mechanics, 1(2), 177–190.
877:
is taken to be one of the closed streamlines since then the velocity vector projected normal to the contour will be zero, that is to say
802:{\displaystyle \oint _{C}\omega \mathbf {u} \cdot \mathbf {n} dl={\frac {1}{\mathrm {Re} }}\oint _{C}\nabla \omega \cdot \mathbf {n} dl.}
1148:
69:
1031:
638:
1241:
880:
1818:
1538:{\displaystyle {\frac {1}{\mathrm {Re} }}{\frac {d\omega }{d\psi }}\oint _{C}\nabla \psi \cdot \mathbf {n} \ dl=0.}
675:
1088:
990:
This expression is true for finite but large
Reynolds number since we did not neglect the viscous term before.
1711:
1238:, and this small corrections become smaller and smaller as we increase the Reynolds number. Thus, in the limit
1940:
996:
220:-direction of the vorticity vector. As it stands, the problem is ill-posed since the vorticity distribution
383:
1916:
Wood, W. W. (1957). Boundary layers whose streamlines are closed. Journal of Fluid
Mechanics, 2(1), 77-87.
475:{\displaystyle \mathbf {u} \cdot \nabla \mathbf {\omega } ={\frac {1}{\mathrm {Re} }}\nabla ^{2}\omega .}
1549:
1420:
354:
325:
296:
223:
1898:
Davidson, P. A. (2016). Introduction to magnetohydrodynamics (Vol. 55). Cambridge university press.
1123:
815:
48:
unaware of this work proved the theorem in 1956. The problem was also studied in the same year by
253:
26:
if in a two-dimensional laminar flow at high
Reynolds number closed streamlines occur, then the
1068:
183:
170:{\displaystyle \nabla ^{2}\psi =-\omega (\psi ),\quad \psi =\psi _{o}{\text{ on }}\partial D}
65:
45:
37:
505:
lying entirely in the region where we have closed streamlines, bounded by a closed contour
276:
78:
980:{\displaystyle {\frac {1}{\mathrm {Re} }}\oint _{C}\nabla \omega \cdot \mathbf {n} \ dl=0}
73:
53:
49:
837:
1705:
860:
508:
488:
203:
41:
33:
17:
1934:
32:. A similar statement holds true for axisymmetric flows. The theorem is named after
380:
is uniquely defined in such cases, through an examination of the limiting process
1451:
is constant for a given streamline, we can take that term outside the integral,
1801:{\displaystyle {\frac {\Gamma }{\mathrm {Re} }}{\frac {d\omega }{d\psi }}=0.}
27:
1269:, in the first approximation (neglecting the small corrections), we have
412:
The steady, non-dimensional vorticity equation in our case reduces to
1231:{\displaystyle \omega =\omega (\psi )+{\rm {small\ corrections}}}
857:. The left-hand side integrand can be made zero if the contour
44:
in his celebrated 1904 paper stated this theorem in arguments,
834:
is the outward unit vector normal to the contour line element
635:
The integrand in the left-hand side term can be written as
68:, the two-dimensional problem governed by two-dimensional
1821:
1756:
1714:
1561:
1460:
1423:
1278:
1244:
1151:
1126:
1091:
1071:
1034:
999:
922:
883:
863:
840:
818:
715:
678:
641:
533:
511:
491:
421:
386:
357:
328:
299:
279:
256:
226:
206:
186:
104:
81:
1548:
One may notice that the integral is negative of the
1853:
1800:
1739:
1693:
1537:
1443:
1406:
1261:
1230:
1137:
1112:
1077:
1058:{\displaystyle \mathbf {u} \cdot \nabla \omega =0}
1057:
1020:
979:
905:
869:
849:
826:
801:
698:
665:{\displaystyle \nabla \cdot (\omega \mathbf {u} )}
664:
624:
517:
497:
474:
401:
372:
343:
314:
285:
265:
241:
212:
192:
169:
87:
30:in the closed streamline region must be a constant
993:Unlike the two-dimensional inviscid flows, where
1262:{\displaystyle \mathrm {Re} \rightarrow \infty }
906:{\displaystyle \mathbf {u} \cdot \mathbf {n} =0}
200:is the only non-zero vorticity component in the
1065:with no restrictions on the functional form of
1854:{\displaystyle {\frac {d\omega }{d\psi }}=0,}
8:
699:{\displaystyle \nabla \cdot \mathbf {u} =0}
1113:{\displaystyle \omega \neq \omega (\psi )}
293:and therefore its corresponding vorticity
1822:
1820:
1772:
1762:
1757:
1755:
1740:{\displaystyle \omega =-\nabla ^{2}\psi }
1728:
1713:
1680:
1665:
1653:
1641:
1631:
1619:
1607:
1592:
1581:
1575:
1560:
1515:
1500:
1476:
1466:
1461:
1459:
1430:
1422:
1384:
1355:
1349:
1335:
1330:
1313:
1298:
1284:
1279:
1277:
1245:
1243:
1174:
1173:
1150:
1127:
1125:
1090:
1070:
1035:
1033:
998:
957:
942:
928:
923:
921:
892:
884:
882:
862:
839:
819:
817:
785:
770:
756:
751:
737:
729:
720:
714:
685:
677:
654:
640:
614:
610:
601:
591:
577:
572:
564:
560:
555:
544:
538:
532:
510:
490:
460:
446:
441:
433:
422:
420:
385:
356:
327:
298:
278:
255:
225:
205:
185:
156:
150:
109:
103:
80:
1873:
1021:{\displaystyle \omega =\omega (\psi )}
485:Integrate the equation over a surface
706:. By divergence theorem, one obtains
402:{\displaystyle Re\rightarrow \infty }
7:
72:reduce to solving a problem for the
1766:
1763:
1759:
1725:
1671:
1638:
1562:
1506:
1470:
1467:
1375:
1339:
1336:
1304:
1288:
1285:
1256:
1249:
1246:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1187:
1184:
1181:
1178:
1175:
1131:
1128:
1043:
948:
932:
929:
776:
760:
757:
679:
642:
598:
581:
578:
552:
457:
450:
447:
430:
396:
257:
161:
106:
14:
1681:
1654:
1620:
1593:
1582:
1516:
1385:
1314:
1036:
958:
893:
885:
820:
786:
738:
730:
686:
655:
615:
565:
545:
423:
1444:{\displaystyle d\omega /d\psi }
139:
1253:
1167:
1161:
1107:
1101:
1015:
1009:
659:
648:
393:
373:{\displaystyle \omega (\psi )}
367:
361:
344:{\displaystyle \omega (\psi )}
338:
332:
315:{\displaystyle \omega (\psi )}
309:
303:
242:{\displaystyle \omega (\psi )}
236:
230:
133:
127:
1:
1138:{\displaystyle \mathrm {Re} }
827:{\displaystyle \mathbf {n} }
1120:. But for large but finite
1957:
266:{\displaystyle \partial D}
56:and by W.W. Wood in 1957.
22:Prandtl–Batchelor theorem
1085:, in the viscous flows,
1078:{\displaystyle \omega }
193:{\displaystyle \omega }
1855:
1802:
1741:
1695:
1539:
1445:
1408:
1263:
1232:
1139:
1114:
1079:
1059:
1022:
981:
907:
871:
851:
828:
803:
700:
666:
626:
519:
499:
476:
403:
374:
345:
316:
287:
267:
243:
214:
194:
171:
89:
1856:
1803:
1742:
1696:
1540:
1446:
1409:
1264:
1233:
1140:
1115:
1080:
1060:
1023:
982:
908:
872:
852:
829:
804:
701:
667:
627:
520:
500:
477:
404:
375:
346:
317:
288:
286:{\displaystyle \psi }
268:
244:
215:
195:
172:
90:
88:{\displaystyle \psi }
1819:
1754:
1712:
1708:for circulation and
1559:
1458:
1421:
1276:
1242:
1149:
1124:
1089:
1069:
1032:
997:
920:
881:
861:
838:
816:
713:
676:
639:
531:
509:
489:
419:
384:
355:
326:
297:
277:
254:
224:
204:
184:
102:
79:
913:. Thus one obtains
1851:
1798:
1737:
1704:where we used the
1691:
1535:
1441:
1404:
1259:
1228:
1135:
1110:
1075:
1055:
1018:
977:
903:
867:
850:{\displaystyle dl}
847:
824:
799:
696:
662:
622:
515:
495:
472:
399:
370:
341:
312:
283:
263:
239:
210:
190:
167:
95:, which satisfies
85:
60:Mathematical proof
1840:
1790:
1770:
1522:
1494:
1474:
1391:
1373:
1343:
1320:
1292:
1192:
964:
936:
870:{\displaystyle C}
764:
585:
518:{\displaystyle C}
498:{\displaystyle S}
454:
213:{\displaystyle z}
159:
1948:
1926:
1923:
1917:
1914:
1908:
1905:
1899:
1896:
1890:
1887:
1881:
1878:
1860:
1858:
1857:
1852:
1841:
1839:
1831:
1823:
1807:
1805:
1804:
1799:
1791:
1789:
1781:
1773:
1771:
1769:
1758:
1747:. Thus, we have
1746:
1744:
1743:
1738:
1733:
1732:
1700:
1698:
1697:
1692:
1684:
1670:
1669:
1657:
1646:
1645:
1636:
1635:
1623:
1612:
1611:
1596:
1585:
1580:
1579:
1544:
1542:
1541:
1536:
1520:
1519:
1505:
1504:
1495:
1493:
1485:
1477:
1475:
1473:
1462:
1450:
1448:
1447:
1442:
1434:
1413:
1411:
1410:
1405:
1389:
1388:
1374:
1372:
1364:
1356:
1354:
1353:
1344:
1342:
1331:
1318:
1317:
1303:
1302:
1293:
1291:
1280:
1268:
1266:
1265:
1260:
1252:
1237:
1235:
1234:
1229:
1227:
1226:
1190:
1144:
1142:
1141:
1136:
1134:
1119:
1117:
1116:
1111:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1039:
1027:
1025:
1024:
1019:
986:
984:
983:
978:
962:
961:
947:
946:
937:
935:
924:
912:
910:
909:
904:
896:
888:
876:
874:
873:
868:
856:
854:
853:
848:
833:
831:
830:
825:
823:
808:
806:
805:
800:
789:
775:
774:
765:
763:
752:
741:
733:
725:
724:
705:
703:
702:
697:
689:
671:
669:
668:
663:
658:
631:
629:
628:
623:
618:
606:
605:
596:
595:
586:
584:
573:
568:
559:
548:
543:
542:
524:
522:
521:
516:
504:
502:
501:
496:
481:
479:
478:
473:
465:
464:
455:
453:
442:
437:
426:
408:
406:
405:
400:
379:
377:
376:
371:
350:
348:
347:
342:
321:
319:
318:
313:
292:
290:
289:
284:
272:
270:
269:
264:
248:
246:
245:
240:
219:
217:
216:
211:
199:
197:
196:
191:
176:
174:
173:
168:
160:
157:
155:
154:
114:
113:
94:
92:
91:
86:
66:Reynolds numbers
46:George Batchelor
38:George Batchelor
1956:
1955:
1951:
1950:
1949:
1947:
1946:
1945:
1931:
1930:
1929:
1924:
1920:
1915:
1911:
1906:
1902:
1897:
1893:
1888:
1884:
1879:
1875:
1871:
1832:
1824:
1817:
1816:
1782:
1774:
1752:
1751:
1724:
1710:
1709:
1661:
1637:
1627:
1603:
1571:
1557:
1556:
1496:
1486:
1478:
1456:
1455:
1419:
1418:
1365:
1357:
1345:
1294:
1274:
1273:
1240:
1239:
1147:
1146:
1145:, we can write
1122:
1121:
1087:
1086:
1067:
1066:
1030:
1029:
995:
994:
938:
918:
917:
879:
878:
859:
858:
836:
835:
814:
813:
766:
716:
711:
710:
674:
673:
637:
636:
597:
587:
534:
529:
528:
507:
506:
487:
486:
456:
417:
416:
382:
381:
353:
352:
324:
323:
295:
294:
275:
274:
252:
251:
222:
221:
202:
201:
182:
181:
146:
105:
100:
99:
77:
76:
74:stream function
70:Euler equations
62:
54:Paco Lagerstrom
50:Richard Feynman
12:
11:
5:
1954:
1952:
1944:
1943:
1941:Fluid dynamics
1933:
1932:
1928:
1927:
1918:
1909:
1900:
1891:
1882:
1872:
1870:
1867:
1862:
1861:
1850:
1847:
1844:
1838:
1835:
1830:
1827:
1809:
1808:
1797:
1794:
1788:
1785:
1780:
1777:
1768:
1765:
1761:
1736:
1731:
1727:
1723:
1720:
1717:
1706:Stokes theorem
1702:
1701:
1690:
1687:
1683:
1679:
1676:
1673:
1668:
1664:
1660:
1656:
1652:
1649:
1644:
1640:
1634:
1630:
1626:
1622:
1618:
1615:
1610:
1606:
1602:
1599:
1595:
1591:
1588:
1584:
1578:
1574:
1570:
1567:
1564:
1546:
1545:
1534:
1531:
1528:
1525:
1518:
1514:
1511:
1508:
1503:
1499:
1492:
1489:
1484:
1481:
1472:
1469:
1465:
1440:
1437:
1433:
1429:
1426:
1415:
1414:
1403:
1400:
1397:
1394:
1387:
1383:
1380:
1377:
1371:
1368:
1363:
1360:
1352:
1348:
1341:
1338:
1334:
1329:
1326:
1323:
1316:
1312:
1309:
1306:
1301:
1297:
1290:
1287:
1283:
1258:
1255:
1251:
1248:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1189:
1186:
1183:
1180:
1177:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1133:
1130:
1109:
1106:
1103:
1100:
1097:
1094:
1074:
1054:
1051:
1048:
1045:
1042:
1038:
1017:
1014:
1011:
1008:
1005:
1002:
988:
987:
976:
973:
970:
967:
960:
956:
953:
950:
945:
941:
934:
931:
927:
902:
899:
895:
891:
887:
866:
846:
843:
822:
810:
809:
798:
795:
792:
788:
784:
781:
778:
773:
769:
762:
759:
755:
750:
747:
744:
740:
736:
732:
728:
723:
719:
695:
692:
688:
684:
681:
661:
657:
653:
650:
647:
644:
633:
632:
621:
617:
613:
609:
604:
600:
594:
590:
583:
580:
576:
571:
567:
563:
558:
554:
551:
547:
541:
537:
514:
494:
483:
482:
471:
468:
463:
459:
452:
449:
445:
440:
436:
432:
429:
425:
398:
395:
392:
389:
369:
366:
363:
360:
340:
337:
334:
331:
311:
308:
305:
302:
282:
262:
259:
238:
235:
232:
229:
209:
189:
178:
177:
166:
163:
158: on
153:
149:
145:
142:
138:
135:
132:
129:
126:
123:
120:
117:
112:
108:
84:
61:
58:
34:Ludwig Prandtl
18:fluid dynamics
13:
10:
9:
6:
4:
3:
2:
1953:
1942:
1939:
1938:
1936:
1922:
1919:
1913:
1910:
1904:
1901:
1895:
1892:
1886:
1883:
1877:
1874:
1868:
1866:
1848:
1845:
1842:
1836:
1833:
1828:
1825:
1815:
1814:
1813:
1795:
1792:
1786:
1783:
1778:
1775:
1750:
1749:
1748:
1734:
1729:
1721:
1718:
1715:
1707:
1688:
1685:
1677:
1674:
1666:
1662:
1658:
1650:
1647:
1642:
1632:
1628:
1624:
1616:
1613:
1608:
1604:
1600:
1597:
1589:
1586:
1576:
1572:
1568:
1565:
1555:
1554:
1553:
1551:
1532:
1529:
1526:
1523:
1512:
1509:
1501:
1497:
1490:
1487:
1482:
1479:
1463:
1454:
1453:
1452:
1438:
1435:
1431:
1427:
1424:
1401:
1398:
1395:
1392:
1381:
1378:
1369:
1366:
1361:
1358:
1350:
1346:
1332:
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1324:
1321:
1310:
1307:
1299:
1295:
1281:
1272:
1271:
1270:
1170:
1164:
1158:
1155:
1152:
1104:
1098:
1095:
1092:
1072:
1052:
1049:
1046:
1040:
1012:
1006:
1003:
1000:
991:
974:
971:
968:
965:
954:
951:
943:
939:
925:
916:
915:
914:
900:
897:
889:
864:
844:
841:
796:
793:
790:
782:
779:
771:
767:
753:
748:
745:
742:
734:
726:
721:
717:
709:
708:
707:
693:
690:
682:
651:
645:
619:
611:
607:
602:
592:
588:
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569:
561:
556:
549:
539:
535:
527:
526:
525:
512:
492:
469:
466:
461:
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438:
434:
427:
415:
414:
413:
410:
390:
387:
364:
358:
335:
329:
306:
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280:
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233:
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207:
187:
164:
151:
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124:
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118:
115:
110:
98:
97:
96:
82:
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71:
67:
59:
57:
55:
51:
47:
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39:
35:
31:
29:
23:
19:
1921:
1912:
1903:
1894:
1885:
1876:
1863:
1810:
1703:
1547:
1416:
992:
989:
811:
634:
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411:
179:
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24:states that
21:
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1550:circulation
1869:References
409:properly.
1837:ψ
1829:ω
1787:ψ
1779:ω
1760:Γ
1735:ψ
1726:∇
1722:−
1716:ω
1678:⋅
1675:ψ
1672:∇
1663:∮
1648:ψ
1639:∇
1629:∫
1614:ω
1605:∫
1601:−
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1573:∮
1569:−
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1513:⋅
1510:ψ
1507:∇
1498:∮
1491:ψ
1483:ω
1439:ψ
1428:ω
1382:⋅
1379:ψ
1376:∇
1370:ψ
1362:ω
1347:∮
1311:⋅
1308:ω
1305:∇
1296:∮
1257:∞
1254:→
1165:ψ
1159:ω
1153:ω
1105:ψ
1099:ω
1096:≠
1093:ω
1073:ω
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1013:ψ
1007:ω
1001:ω
955:⋅
952:ω
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940:∮
890:⋅
783:⋅
780:ω
777:∇
768:∮
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428:⋅
397:∞
394:→
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307:ψ
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234:ψ
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148:ψ
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122:−
116:ψ
107:∇
83:ψ
28:vorticity
1935:Category
64:At high
42:Prandtl
1552:since
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1417:Since
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1191:
1028:since
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672:since
273:where
180:where
52:and
36:and
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