889:
590:
419:
884:{\displaystyle {\begin{aligned}{\frac {\partial p}{\partial x}}=\mu {\frac {\partial ^{2}v_{x}}{\partial z^{2}}},\quad {\frac {\partial p}{\partial y}}&=\mu {\frac {\partial ^{2}v_{y}}{\partial z^{2}}},\quad {\frac {\partial p}{\partial z}}=0,\\{\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}&=0,\\\end{aligned}}}
23:, who studied the problem in 1898. Various problems in fluid mechanics can be approximated to Hele-Shaw flows and thus the research of these flows is of importance. Approximation to Hele-Shaw flow is specifically important to micro-flows. This is due to manufacturing techniques, which creates shallow planar configurations, and the typically low
1167:
1896:
2220:
2586:
The term Hele-Shaw cell is commonly used for cases in which a fluid is injected into the shallow geometry from above or below the geometry, and when the fluid is bounded by another liquid or gas. For such flows the boundary conditions are defined by pressures and surface tensions.
959:
1307:
2540:
1725:
1411:
1479:
is a unit vector perpendicular to the side wall (note that on the side walls, non-slip boundary conditions cannot be imposed). The boundaries may also be regions exposed to constant pressure in which case a
Dirichlet boundary condition for
2037:
2340:
99:
2389:
2577:
1162:{\displaystyle {\begin{aligned}p&=p(x,y),\\v_{x}&=-{\frac {1}{2\mu }}{\frac {\partial p}{\partial x}}z(h-z),\\v_{y}&=-{\frac {1}{2\mu }}{\frac {\partial p}{\partial y}}z(h-z)\end{aligned}}}
1455:
964:
595:
2453:
1198:
1591:
2465:
1717:
1675:
1891:{\displaystyle \omega _{x}={\frac {1}{2\mu }}{\frac {\partial p}{\partial y}}(h-2z),\quad \omega _{y}=-{\frac {1}{2\mu }}{\frac {\partial p}{\partial x}}(h-2z),\quad \omega _{z}=0.}
398:
318:
1932:
1192:
is obtained from the continuity equation. Integrating the continuity equation from across the channel and imposing no-penetration boundary conditions at the walls, we have
271:
1477:
1322:
2215:{\displaystyle \Gamma =\oint _{C}v_{x}dx+v_{y}dy=-{\frac {1}{2\mu }}z(h-z)\oint _{C}\left({\frac {\partial p}{\partial x}}dx+{\frac {\partial p}{\partial y}}dy\right)=0}
225:
2271:
1986:
1534:
951:
536:
1500:
is appropriate. Similarly, periodic boundary conditions can also be used. It can also be noted that the vertical velocity component in the first approximation is
911:
182:
2029:
1955:
579:
2243:
2006:
1695:
1611:
1498:
1190:
556:
504:
484:
464:
444:
338:
162:
142:
122:
2279:
36:
921:
equations, except that there are no non-linear terms. In the first approximation, we then have, after imposing the non-slip boundary conditions at
1416:
This equation is supplemented by appropriate boundary conditions. For example, no-penetration boundary conditions on the side walls become:
2348:
19:
is defined as flow taking place between two parallel flat plates separated by a narrow gap satisfying certain conditions, named after
2548:
1419:
2632:
Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions
2596:
1302:{\displaystyle \int _{0}^{h}\left({\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}\right)dz=0,}
582:
2394:
1542:
2535:{\displaystyle -{\frac {12\mu }{h^{2}}}\mathbf {u} =\nabla p\quad {\text{with}}\quad \nabla \cdot \mathbf {u} =0.}
1700:
1616:
2797:
343:
1966:
20:
1313:
2666:
1406:{\displaystyle {\frac {\partial ^{2}p}{\partial x^{2}}}+{\frac {\partial ^{2}p}{\partial y^{2}}}=0.}
276:
2709:
2693:
1904:
230:
1460:
227:
need not always be small, but can be order unity or greater as long as it satisfies the condition
2774:
2606:
2601:
2253:
In a Hele-Shaw channel, one can define the depth-averaged version of any physical quantity, say
190:
1697:
direction, that is to say, streamline patterns at each level are similar. The vorticity vector
2635:
2256:
1971:
1506:
2766:
2674:
2611:
2751:
924:
509:
896:
185:
167:
24:
2670:
2011:
1937:
561:
2456:
2228:
1991:
1962:
1958:
1680:
1596:
1483:
1175:
918:
541:
489:
469:
449:
429:
407:
403:
323:
147:
127:
107:
2791:
2335:{\displaystyle \langle \varphi \rangle \equiv {\frac {1}{h}}\int _{0}^{h}\varphi dz.}
2778:
418:
2725:
2245:
is a single-valued function and the integration is done over a closed contour.
94:{\displaystyle {\frac {h}{l}}\ll 1,\qquad {\frac {Uh}{\nu }}{\frac {h}{l}}\ll 1}
402:
The governing equation of Hele-Shaw flows is identical to that of the inviscid
2770:
2639:
914:
164:
is the characteristic length scale in directions parallel to the plate and
410:). It thus permits visualization of this kind of flow in two dimensions.
1539:
that follows from the continuity equation. While the velocity magnitude
2679:
2654:
2384:{\displaystyle \mathbf {u} \equiv \langle \mathbf {v} _{xy}\rangle }
581:-directions. Under the limits mentioned above, the incompressible
417:
2031:-plane), whether it encloses a solid object or not, is zero,
2572:{\displaystyle \langle {\boldsymbol {\omega }}\rangle =0.}
1450:{\displaystyle {\mathbf {\nabla } }p\cdot \mathbf {n} =0}
2345:
Then the two-dimensional depth-averaged velocity vector
2551:
2468:
2397:
2351:
2282:
2259:
2231:
2040:
2014:
1994:
1974:
1940:
1907:
1728:
1703:
1683:
1619:
1599:
1545:
1509:
1486:
1463:
1422:
1325:
1201:
1178:
962:
927:
899:
593:
564:
544:
512:
492:
472:
452:
432:
422:
A schematic description of a Hele-Shaw configuration.
346:
326:
279:
233:
193:
170:
150:
130:
110:
39:
558:be the relevant characteristic length scale in the
466:be the directions parallel to the flat plates, and
2571:
2534:
2447:
2383:
2334:
2265:
2237:
2214:
2023:
2000:
1980:
1949:
1926:
1890:
1711:
1689:
1669:
1605:
1585:
1528:
1492:
1471:
1449:
1405:
1301:
1184:
1161:
945:
905:
883:
573:
550:
530:
498:
478:
458:
438:
406:and to the flow of fluid through a porous medium (
392:
332:
312:
265:
219:
176:
156:
136:
116:
93:
2740:Acheson, D. J. (1991). Elementary fluid dynamics.
2448:{\displaystyle \mathbf {v} _{xy}=(v_{x},v_{y})}
2225:where the last integral is set to zero because
184:is the kinematic viscosity. Specifically, the
30:The conditions that needs to be satisfied are
1586:{\displaystyle {\sqrt {v_{x}^{2}+v_{y}^{2}}}}
8:
2560:
2552:
2378:
2360:
2289:
2283:
414:Mathematical formulation of Hele-Shaw flows
2736:
2734:
2678:
2555:
2550:
2521:
2509:
2494:
2486:
2472:
2467:
2436:
2423:
2404:
2399:
2396:
2369:
2364:
2352:
2350:
2314:
2309:
2295:
2281:
2258:
2230:
2175:
2146:
2135:
2098:
2080:
2061:
2051:
2039:
2013:
1993:
1973:
1939:
1912:
1906:
1876:
1830:
1815:
1803:
1757:
1742:
1733:
1727:
1704:
1702:
1682:
1658:
1649:
1643:
1624:
1618:
1613:direction, the velocity-vector direction
1598:
1575:
1570:
1557:
1552:
1546:
1544:
1514:
1508:
1485:
1464:
1462:
1436:
1424:
1423:
1421:
1388:
1370:
1363:
1351:
1333:
1326:
1324:
1262:
1252:
1232:
1222:
1211:
1206:
1200:
1177:
1117:
1102:
1086:
1037:
1022:
1006:
963:
961:
926:
898:
847:
837:
817:
807:
787:
777:
744:
731:
716:
706:
699:
669:
656:
641:
631:
624:
598:
594:
592:
563:
543:
511:
491:
471:
451:
431:
378:
366:
354:
345:
325:
302:
287:
278:
246:
232:
209:
192:
169:
149:
129:
109:
75:
60:
40:
38:
2622:
2556:
1712:{\displaystyle {\boldsymbol {\omega }}}
1705:
1670:{\displaystyle \tan ^{-1}(v_{y}/v_{x})}
2752:"Viscous fingering in Hele-Shaw cells"
2700:, 7th ed. New York: McGraw-Hill, 1979.
144:is the characteristic velocity scale,
585:, in the first approximation becomes
506:being the gap between the plates (at
393:{\displaystyle Re_{l}(h/l)^{2}\ll 1.}
124:is the gap width between the plates,
7:
2515:
2502:
2186:
2178:
2157:
2149:
2041:
1975:
1841:
1833:
1768:
1760:
1425:
1381:
1367:
1344:
1330:
1270:
1255:
1240:
1225:
1128:
1120:
1048:
1040:
855:
840:
825:
810:
795:
780:
755:
747:
724:
703:
680:
672:
649:
628:
609:
601:
486:the perpendicular direction, with
14:
1934:, the streamline patterns in the
917:. These equations are similar to
2750:Saffman, P. G. (21 April 2006).
2522:
2495:
2400:
2365:
2353:
1465:
1437:
273:In terms of the Reynolds number
2653:Hele-Shaw, H. S. (1 May 1898).
2514:
2508:
1871:
1798:
743:
668:
59:
2442:
2416:
2128:
2116:
1865:
1850:
1792:
1777:
1664:
1636:
1152:
1140:
1072:
1060:
992:
980:
375:
360:
313:{\displaystyle Re_{l}=Ul/\nu }
254:
240:
1:
2597:Diffusion-limited aggregation
1927:{\displaystyle \omega _{z}=0}
266:{\displaystyle Re(h/l)\ll 1.}
1961:(irrotational flow). Unlike
1472:{\displaystyle \mathbf {n} }
2814:
2759:Journal of Fluid Mechanics
2716:. Dover Publications, Inc.
2630:Shaw, Henry S. H. (1898).
1988:around any closed contour
1957:-plane thus correspond to
220:{\displaystyle Re=Uh/\nu }
2771:10.1017/s0022112086001088
2714:Theoretical Hydrodynamics
2266:{\displaystyle \varphi }
340:, the condition becomes
2728:, Hydrodynamics (1934).
1981:{\displaystyle \Gamma }
1529:{\displaystyle v_{z}=0}
583:Navier–Stokes equations
2573:
2536:
2449:
2385:
2336:
2267:
2239:
2216:
2025:
2002:
1982:
1951:
1928:
1892:
1713:
1691:
1671:
1607:
1587:
1530:
1494:
1473:
1451:
1407:
1303:
1186:
1163:
947:
907:
885:
575:
552:
532:
500:
480:
460:
440:
423:
394:
334:
314:
267:
221:
178:
158:
138:
118:
95:
2698:Boundary Layer Theory
2574:
2537:
2450:
2386:
2337:
2268:
2240:
2217:
2026:
2003:
1983:
1952:
1929:
1893:
1714:
1692:
1672:
1608:
1588:
1531:
1495:
1474:
1452:
1408:
1304:
1187:
1164:
948:
946:{\displaystyle z=0,h}
908:
886:
576:
553:
533:
531:{\displaystyle z=0,h}
501:
481:
461:
441:
421:
395:
335:
315:
268:
222:
179:
159:
139:
119:
96:
21:Henry Selby Hele-Shaw
2549:
2466:
2395:
2349:
2280:
2257:
2229:
2038:
2012:
1992:
1972:
1938:
1905:
1726:
1701:
1681:
1617:
1597:
1543:
1507:
1484:
1461:
1420:
1323:
1199:
1176:
960:
925:
906:{\displaystyle \mu }
897:
591:
562:
542:
510:
490:
470:
450:
430:
344:
324:
277:
231:
191:
177:{\displaystyle \nu }
168:
148:
128:
108:
37:
2710:L. M. Milne-Thomson
2694:Hermann Schlichting
2671:1898Natur..58...34H
2655:"The Flow of Water"
2319:
2249:Depth-averaged form
1719:has the components
1580:
1562:
1312:which leads to the
1216:
2607:Thin-film equation
2602:Lubrication theory
2569:
2532:
2445:
2381:
2332:
2305:
2263:
2235:
2212:
2024:{\displaystyle xy}
2021:
1998:
1978:
1950:{\displaystyle xy}
1947:
1924:
1888:
1709:
1687:
1677:is independent of
1667:
1603:
1583:
1566:
1548:
1526:
1490:
1469:
1447:
1403:
1299:
1202:
1182:
1159:
1157:
943:
903:
881:
879:
574:{\displaystyle xy}
571:
548:
528:
496:
476:
456:
436:
424:
390:
330:
310:
263:
217:
174:
154:
134:
114:
91:
2512:
2492:
2303:
2238:{\displaystyle p}
2193:
2164:
2111:
2008:(parallel to the
2001:{\displaystyle C}
1848:
1828:
1775:
1755:
1690:{\displaystyle z}
1606:{\displaystyle z}
1581:
1493:{\displaystyle p}
1395:
1358:
1277:
1247:
1185:{\displaystyle p}
1172:The equation for
1135:
1115:
1055:
1035:
862:
832:
802:
762:
738:
687:
663:
616:
551:{\displaystyle l}
499:{\displaystyle h}
479:{\displaystyle z}
459:{\displaystyle y}
439:{\displaystyle x}
333:{\displaystyle l}
157:{\displaystyle l}
137:{\displaystyle U}
117:{\displaystyle h}
83:
73:
48:
2805:
2783:
2782:
2756:
2747:
2741:
2738:
2729:
2723:
2717:
2707:
2701:
2691:
2685:
2684:
2682:
2680:10.1038/058034a0
2650:
2644:
2643:
2627:
2612:Hele-Shaw clutch
2578:
2576:
2575:
2570:
2559:
2541:
2539:
2538:
2533:
2525:
2513:
2510:
2498:
2493:
2491:
2490:
2481:
2473:
2455:, satisfies the
2454:
2452:
2451:
2446:
2441:
2440:
2428:
2427:
2412:
2411:
2403:
2390:
2388:
2387:
2382:
2377:
2376:
2368:
2356:
2341:
2339:
2338:
2333:
2318:
2313:
2304:
2296:
2272:
2270:
2269:
2264:
2244:
2242:
2241:
2236:
2221:
2219:
2218:
2213:
2205:
2201:
2194:
2192:
2184:
2176:
2165:
2163:
2155:
2147:
2140:
2139:
2112:
2110:
2099:
2085:
2084:
2066:
2065:
2056:
2055:
2030:
2028:
2027:
2022:
2007:
2005:
2004:
1999:
1987:
1985:
1984:
1979:
1956:
1954:
1953:
1948:
1933:
1931:
1930:
1925:
1917:
1916:
1897:
1895:
1894:
1889:
1881:
1880:
1849:
1847:
1839:
1831:
1829:
1827:
1816:
1808:
1807:
1776:
1774:
1766:
1758:
1756:
1754:
1743:
1738:
1737:
1718:
1716:
1715:
1710:
1708:
1696:
1694:
1693:
1688:
1676:
1674:
1673:
1668:
1663:
1662:
1653:
1648:
1647:
1632:
1631:
1612:
1610:
1609:
1604:
1592:
1590:
1589:
1584:
1582:
1579:
1574:
1561:
1556:
1547:
1535:
1533:
1532:
1527:
1519:
1518:
1499:
1497:
1496:
1491:
1478:
1476:
1475:
1470:
1468:
1456:
1454:
1453:
1448:
1440:
1429:
1428:
1412:
1410:
1409:
1404:
1396:
1394:
1393:
1392:
1379:
1375:
1374:
1364:
1359:
1357:
1356:
1355:
1342:
1338:
1337:
1327:
1314:Laplace Equation
1308:
1306:
1305:
1300:
1283:
1279:
1278:
1276:
1268:
1267:
1266:
1253:
1248:
1246:
1238:
1237:
1236:
1223:
1215:
1210:
1191:
1189:
1188:
1183:
1168:
1166:
1165:
1160:
1158:
1136:
1134:
1126:
1118:
1116:
1114:
1103:
1091:
1090:
1056:
1054:
1046:
1038:
1036:
1034:
1023:
1011:
1010:
952:
950:
949:
944:
912:
910:
909:
904:
890:
888:
887:
882:
880:
863:
861:
853:
852:
851:
838:
833:
831:
823:
822:
821:
808:
803:
801:
793:
792:
791:
778:
763:
761:
753:
745:
739:
737:
736:
735:
722:
721:
720:
711:
710:
700:
688:
686:
678:
670:
664:
662:
661:
660:
647:
646:
645:
636:
635:
625:
617:
615:
607:
599:
580:
578:
577:
572:
557:
555:
554:
549:
537:
535:
534:
529:
505:
503:
502:
497:
485:
483:
482:
477:
465:
463:
462:
457:
445:
443:
442:
437:
399:
397:
396:
391:
383:
382:
370:
359:
358:
339:
337:
336:
331:
319:
317:
316:
311:
306:
292:
291:
272:
270:
269:
264:
250:
226:
224:
223:
218:
213:
183:
181:
180:
175:
163:
161:
160:
155:
143:
141:
140:
135:
123:
121:
120:
115:
100:
98:
97:
92:
84:
76:
74:
69:
61:
49:
41:
27:of micro-flows.
25:Reynolds numbers
2813:
2812:
2808:
2807:
2806:
2804:
2803:
2802:
2788:
2787:
2786:
2754:
2749:
2748:
2744:
2739:
2732:
2724:
2720:
2708:
2704:
2692:
2688:
2665:(1489): 34–36.
2652:
2651:
2647:
2629:
2628:
2624:
2620:
2593:
2584:
2547:
2546:
2482:
2474:
2464:
2463:
2432:
2419:
2398:
2393:
2392:
2363:
2347:
2346:
2278:
2277:
2255:
2254:
2251:
2227:
2226:
2185:
2177:
2156:
2148:
2145:
2141:
2131:
2103:
2076:
2057:
2047:
2036:
2035:
2010:
2009:
1990:
1989:
1970:
1969:
1936:
1935:
1908:
1903:
1902:
1872:
1840:
1832:
1820:
1799:
1767:
1759:
1747:
1729:
1724:
1723:
1699:
1698:
1679:
1678:
1654:
1639:
1620:
1615:
1614:
1595:
1594:
1541:
1540:
1510:
1505:
1504:
1482:
1481:
1459:
1458:
1418:
1417:
1384:
1380:
1366:
1365:
1347:
1343:
1329:
1328:
1321:
1320:
1269:
1258:
1254:
1239:
1228:
1224:
1221:
1217:
1197:
1196:
1174:
1173:
1156:
1155:
1127:
1119:
1107:
1092:
1082:
1079:
1078:
1047:
1039:
1027:
1012:
1002:
999:
998:
970:
958:
957:
923:
922:
895:
894:
878:
877:
864:
854:
843:
839:
824:
813:
809:
794:
783:
779:
774:
773:
754:
746:
727:
723:
712:
702:
701:
689:
679:
671:
652:
648:
637:
627:
626:
608:
600:
589:
588:
560:
559:
540:
539:
508:
507:
488:
487:
468:
467:
448:
447:
428:
427:
416:
374:
350:
342:
341:
322:
321:
283:
275:
274:
229:
228:
189:
188:
186:Reynolds number
166:
165:
146:
145:
126:
125:
106:
105:
62:
35:
34:
12:
11:
5:
2811:
2809:
2801:
2800:
2798:Fluid dynamics
2790:
2789:
2785:
2784:
2742:
2730:
2718:
2702:
2686:
2645:
2621:
2619:
2616:
2615:
2614:
2609:
2604:
2599:
2592:
2589:
2583:
2582:Hele-Shaw cell
2580:
2568:
2565:
2562:
2558:
2554:
2543:
2542:
2531:
2528:
2524:
2520:
2517:
2507:
2504:
2501:
2497:
2489:
2485:
2480:
2477:
2471:
2444:
2439:
2435:
2431:
2426:
2422:
2418:
2415:
2410:
2407:
2402:
2380:
2375:
2372:
2367:
2362:
2359:
2355:
2343:
2342:
2331:
2328:
2325:
2322:
2317:
2312:
2308:
2302:
2299:
2294:
2291:
2288:
2285:
2262:
2250:
2247:
2234:
2223:
2222:
2211:
2208:
2204:
2200:
2197:
2191:
2188:
2183:
2180:
2174:
2171:
2168:
2162:
2159:
2154:
2151:
2144:
2138:
2134:
2130:
2127:
2124:
2121:
2118:
2115:
2109:
2106:
2102:
2097:
2094:
2091:
2088:
2083:
2079:
2075:
2072:
2069:
2064:
2060:
2054:
2050:
2046:
2043:
2020:
2017:
1997:
1977:
1963:potential flow
1959:potential flow
1946:
1943:
1923:
1920:
1915:
1911:
1899:
1898:
1887:
1884:
1879:
1875:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1846:
1843:
1838:
1835:
1826:
1823:
1819:
1814:
1811:
1806:
1802:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1773:
1770:
1765:
1762:
1753:
1750:
1746:
1741:
1736:
1732:
1707:
1686:
1666:
1661:
1657:
1652:
1646:
1642:
1638:
1635:
1630:
1627:
1623:
1602:
1593:varies in the
1578:
1573:
1569:
1565:
1560:
1555:
1551:
1537:
1536:
1525:
1522:
1517:
1513:
1489:
1467:
1446:
1443:
1439:
1435:
1432:
1427:
1414:
1413:
1402:
1399:
1391:
1387:
1383:
1378:
1373:
1369:
1362:
1354:
1350:
1346:
1341:
1336:
1332:
1310:
1309:
1298:
1295:
1292:
1289:
1286:
1282:
1275:
1272:
1265:
1261:
1257:
1251:
1245:
1242:
1235:
1231:
1227:
1220:
1214:
1209:
1205:
1181:
1170:
1169:
1154:
1151:
1148:
1145:
1142:
1139:
1133:
1130:
1125:
1122:
1113:
1110:
1106:
1101:
1098:
1095:
1093:
1089:
1085:
1081:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1053:
1050:
1045:
1042:
1033:
1030:
1026:
1021:
1018:
1015:
1013:
1009:
1005:
1001:
1000:
997:
994:
991:
988:
985:
982:
979:
976:
973:
971:
969:
966:
965:
942:
939:
936:
933:
930:
919:boundary layer
902:
876:
873:
870:
867:
865:
860:
857:
850:
846:
842:
836:
830:
827:
820:
816:
812:
806:
800:
797:
790:
786:
782:
776:
775:
772:
769:
766:
760:
757:
752:
749:
742:
734:
730:
726:
719:
715:
709:
705:
698:
695:
692:
690:
685:
682:
677:
674:
667:
659:
655:
651:
644:
640:
634:
630:
623:
620:
614:
611:
606:
603:
597:
596:
570:
567:
547:
527:
524:
521:
518:
515:
495:
475:
455:
435:
415:
412:
404:potential flow
389:
386:
381:
377:
373:
369:
365:
362:
357:
353:
349:
329:
309:
305:
301:
298:
295:
290:
286:
282:
262:
259:
256:
253:
249:
245:
242:
239:
236:
216:
212:
208:
205:
202:
199:
196:
173:
153:
133:
113:
102:
101:
90:
87:
82:
79:
72:
68:
65:
58:
55:
52:
47:
44:
17:Hele-Shaw flow
13:
10:
9:
6:
4:
3:
2:
2810:
2799:
2796:
2795:
2793:
2780:
2776:
2772:
2768:
2764:
2760:
2753:
2746:
2743:
2737:
2735:
2731:
2727:
2722:
2719:
2715:
2711:
2706:
2703:
2699:
2695:
2690:
2687:
2681:
2676:
2672:
2668:
2664:
2660:
2656:
2649:
2646:
2641:
2637:
2634:. Inst. N.A.
2633:
2626:
2623:
2617:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2594:
2590:
2588:
2581:
2579:
2566:
2563:
2529:
2526:
2518:
2505:
2499:
2487:
2483:
2478:
2475:
2469:
2462:
2461:
2460:
2458:
2437:
2433:
2429:
2424:
2420:
2413:
2408:
2405:
2373:
2370:
2357:
2329:
2326:
2323:
2320:
2315:
2310:
2306:
2300:
2297:
2292:
2286:
2276:
2275:
2274:
2260:
2248:
2246:
2232:
2209:
2206:
2202:
2198:
2195:
2189:
2181:
2172:
2169:
2166:
2160:
2152:
2142:
2136:
2132:
2125:
2122:
2119:
2113:
2107:
2104:
2100:
2095:
2092:
2089:
2086:
2081:
2077:
2073:
2070:
2067:
2062:
2058:
2052:
2048:
2044:
2034:
2033:
2032:
2018:
2015:
1995:
1968:
1964:
1960:
1944:
1941:
1921:
1918:
1913:
1909:
1885:
1882:
1877:
1873:
1868:
1862:
1859:
1856:
1853:
1844:
1836:
1824:
1821:
1817:
1812:
1809:
1804:
1800:
1795:
1789:
1786:
1783:
1780:
1771:
1763:
1751:
1748:
1744:
1739:
1734:
1730:
1722:
1721:
1720:
1684:
1659:
1655:
1650:
1644:
1640:
1633:
1628:
1625:
1621:
1600:
1576:
1571:
1567:
1563:
1558:
1553:
1549:
1523:
1520:
1515:
1511:
1503:
1502:
1501:
1487:
1444:
1441:
1433:
1430:
1400:
1397:
1389:
1385:
1376:
1371:
1360:
1352:
1348:
1339:
1334:
1319:
1318:
1317:
1315:
1296:
1293:
1290:
1287:
1284:
1280:
1273:
1263:
1259:
1249:
1243:
1233:
1229:
1218:
1212:
1207:
1203:
1195:
1194:
1193:
1179:
1149:
1146:
1143:
1137:
1131:
1123:
1111:
1108:
1104:
1099:
1096:
1094:
1087:
1083:
1075:
1069:
1066:
1063:
1057:
1051:
1043:
1031:
1028:
1024:
1019:
1016:
1014:
1007:
1003:
995:
989:
986:
983:
977:
974:
972:
967:
956:
955:
954:
940:
937:
934:
931:
928:
920:
916:
900:
891:
874:
871:
868:
866:
858:
848:
844:
834:
828:
818:
814:
804:
798:
788:
784:
770:
767:
764:
758:
750:
740:
732:
728:
717:
713:
707:
696:
693:
691:
683:
675:
665:
657:
653:
642:
638:
632:
621:
618:
612:
604:
586:
584:
568:
565:
545:
525:
522:
519:
516:
513:
493:
473:
453:
433:
420:
413:
411:
409:
405:
400:
387:
384:
379:
371:
367:
363:
355:
351:
347:
327:
307:
303:
299:
296:
293:
288:
284:
280:
260:
257:
251:
247:
243:
237:
234:
214:
210:
206:
203:
200:
197:
194:
187:
171:
151:
131:
111:
88:
85:
80:
77:
70:
66:
63:
56:
53:
50:
45:
42:
33:
32:
31:
28:
26:
22:
18:
2762:
2758:
2745:
2721:
2713:
2705:
2697:
2689:
2662:
2658:
2648:
2631:
2625:
2585:
2544:
2344:
2252:
2224:
1900:
1538:
1415:
1311:
1171:
892:
587:
425:
401:
103:
29:
16:
15:
2726:Horace Lamb
2457:Darcy's law
1967:circulation
1965:, here the
408:Darcy's law
2618:References
2765:: 73–94.
2561:⟩
2557:ω
2553:⟨
2545:Further,
2519:⋅
2516:∇
2503:∇
2479:μ
2470:−
2379:⟩
2361:⟨
2358:≡
2321:φ
2307:∫
2293:≡
2290:⟩
2287:φ
2284:⟨
2261:φ
2187:∂
2179:∂
2158:∂
2150:∂
2133:∮
2123:−
2108:μ
2096:−
2049:∮
2042:Γ
1976:Γ
1910:ω
1874:ω
1857:−
1842:∂
1834:∂
1825:μ
1813:−
1801:ω
1784:−
1769:∂
1761:∂
1752:μ
1731:ω
1706:ω
1634:
1626:−
1434:⋅
1426:∇
1382:∂
1368:∂
1345:∂
1331:∂
1271:∂
1256:∂
1241:∂
1226:∂
1204:∫
1147:−
1129:∂
1121:∂
1112:μ
1100:−
1067:−
1049:∂
1041:∂
1032:μ
1020:−
915:viscosity
901:μ
856:∂
841:∂
826:∂
811:∂
796:∂
781:∂
756:∂
748:∂
725:∂
704:∂
697:μ
681:∂
673:∂
650:∂
629:∂
622:μ
610:∂
602:∂
385:≪
320:based on
308:ν
258:≪
215:ν
172:ν
86:≪
71:ν
51:≪
2792:Category
2779:17003612
2712:(1996).
2640:17929897
2591:See also
2391:, where
1457:, where
2667:Bibcode
913:is the
2777:
2659:Nature
2638:
1901:Since
893:where
538:) and
104:where
2775:S2CID
2755:(PDF)
2636:OCLC
2511:with
426:Let
2767:doi
2763:173
2675:doi
2273:by
1622:tan
953:,
2794::
2773:.
2761:.
2757:.
2733:^
2673:.
2663:58
2661:.
2657:.
2567:0.
2530:0.
2476:12
2459:,
1886:0.
1401:0.
1316::
446:,
388:1.
261:1.
2781:.
2769::
2696:,
2683:.
2677::
2669::
2642:.
2564:=
2527:=
2523:u
2506:p
2500:=
2496:u
2488:2
2484:h
2443:)
2438:y
2434:v
2430:,
2425:x
2421:v
2417:(
2414:=
2409:y
2406:x
2401:v
2374:y
2371:x
2366:v
2354:u
2330:.
2327:z
2324:d
2316:h
2311:0
2301:h
2298:1
2233:p
2210:0
2207:=
2203:)
2199:y
2196:d
2190:y
2182:p
2173:+
2170:x
2167:d
2161:x
2153:p
2143:(
2137:C
2129:)
2126:z
2120:h
2117:(
2114:z
2105:2
2101:1
2093:=
2090:y
2087:d
2082:y
2078:v
2074:+
2071:x
2068:d
2063:x
2059:v
2053:C
2045:=
2019:y
2016:x
1996:C
1945:y
1942:x
1922:0
1919:=
1914:z
1883:=
1878:z
1869:,
1866:)
1863:z
1860:2
1854:h
1851:(
1845:x
1837:p
1822:2
1818:1
1810:=
1805:y
1796:,
1793:)
1790:z
1787:2
1781:h
1778:(
1772:y
1764:p
1749:2
1745:1
1740:=
1735:x
1685:z
1665:)
1660:x
1656:v
1651:/
1645:y
1641:v
1637:(
1629:1
1601:z
1577:2
1572:y
1568:v
1564:+
1559:2
1554:x
1550:v
1524:0
1521:=
1516:z
1512:v
1488:p
1466:n
1445:0
1442:=
1438:n
1431:p
1398:=
1390:2
1386:y
1377:p
1372:2
1361:+
1353:2
1349:x
1340:p
1335:2
1297:,
1294:0
1291:=
1288:z
1285:d
1281:)
1274:y
1264:y
1260:v
1250:+
1244:x
1234:x
1230:v
1219:(
1213:h
1208:0
1180:p
1153:)
1150:z
1144:h
1141:(
1138:z
1132:y
1124:p
1109:2
1105:1
1097:=
1088:y
1084:v
1076:,
1073:)
1070:z
1064:h
1061:(
1058:z
1052:x
1044:p
1029:2
1025:1
1017:=
1008:x
1004:v
996:,
993:)
990:y
987:,
984:x
981:(
978:p
975:=
968:p
941:h
938:,
935:0
932:=
929:z
875:,
872:0
869:=
859:z
849:z
845:v
835:+
829:y
819:y
815:v
805:+
799:x
789:x
785:v
771:,
768:0
765:=
759:z
751:p
741:,
733:2
729:z
718:y
714:v
708:2
694:=
684:y
676:p
666:,
658:2
654:z
643:x
639:v
633:2
619:=
613:x
605:p
569:y
566:x
546:l
526:h
523:,
520:0
517:=
514:z
494:h
474:z
454:y
434:x
380:2
376:)
372:l
368:/
364:h
361:(
356:l
352:e
348:R
328:l
304:/
300:l
297:U
294:=
289:l
285:e
281:R
255:)
252:l
248:/
244:h
241:(
238:e
235:R
211:/
207:h
204:U
201:=
198:e
195:R
152:l
132:U
112:h
89:1
81:l
78:h
67:h
64:U
57:,
54:1
46:l
43:h
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