22:
171:
1559:
2241:
3052:
3319:
3700:
2402:
1833:
3939:
4607:
4435:
3157:
305:
3570:
2940:
2258:
2830:
161:
one can see how all these solutions are at the core of recent developments in theoretical physics such as the ads/cft duality, the SYK model, the physics of nematic liquids, strongly correlated systems and even to quark gluon plasmas.
2715:
3041:
1664:
4182:
3779:
1467:
1317:
797:
896:
1570:
The case of a vertical line emitting at a fixed rate a constant quantity of fluid Q per unit length is a line source. The problem has a cylindrical symmetry and can be treated in two dimensions on the orthogonal plane.
4093:
3146:
3509:
1548:
1031:
947:
3444:
90:
are basic flows that can be combined, using various techniques, to construct more complex flows. In this article the term "flow" is used interchangeably with the term "solution" due to historical reasons.
1952:
1200:
1146:
680:
2151:
1406:
626:
2023:
2219:
1070:
3768:
3376:
1653:
4446:
4269:
3314:{\displaystyle \oint \mathbf {v} \cdot d\mathbf {s} =\int _{0}^{2\pi }(v_{\theta }(r)\,\mathbf {e} _{\theta })\cdot (\mathbf {e} _{\theta }\,r\,d\theta )=\!2\pi \,r\,v_{\theta }(r)=\Gamma }
549:
2089:
1887:
3695:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \psi }{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial \theta ^{2}}}=0}
2224:
Given that the two results are the same a part from a minus sign we can treat transparently both line sources and line sinks with the same stream and potential functions permitting
192:
383:
3559:
2611:
507:
478:
1574:
Line sources and line sinks (below) are important elementary flows because they play the role of monopole for incompressible fluids (which can also be considered examples of
4254:
4218:
2836:
2397:{\displaystyle \psi (\mathbf {r} )=\psi _{Q}(\mathbf {r} -\mathbf {d} /2)-\psi _{Q}(\mathbf {r} +\mathbf {d} /2)\ \simeq \mathbf {d} \cdot \nabla \psi _{Q}(\mathbf {r} )}
443:
412:
1347:
827:
3990:
3966:
4866:
2726:
1090:
573:
338:
1828:{\displaystyle \int _{S}\mathbf {v} \cdot d\mathbf {S} =\int _{0}^{2\pi }(v_{r}(r)\,\mathbf {e} _{r})\cdot (\mathbf {e} _{r}\,r\,d\theta )=\!2\pi \,r\,v_{r}(r)=Q}
2622:
2951:
3934:{\displaystyle {\frac {r}{R(r)}}{\frac {d}{dr}}\left(r{\frac {dR(r)}{dr}}\right)=-{\frac {1}{\Theta (\theta )}}{\frac {d^{2}\Theta (\theta )}{d\theta ^{2}}}}
4099:
1412:
3063:
This is the case of a vortex filament rotating at constant speed, there is a cylindrical symmetry and the problem can be solved in the orthogonal plane.
1211:
691:
838:
3056:
2245:
1563:
175:
3998:
4796:
4775:
4754:
4731:
4689:
4633:
43:
3094:
3455:
1475:
958:
110:) of the different types of equations derived from the Navier-Stokes equations. Some of the flows reflect specific constraints such as
4646:
902:
4842:
4656:
3387:
65:
1898:
1152:
1098:
632:
2100:
1361:
581:
1963:
2162:
1039:
2252:
If we consider a line source and a line sink at a distance d we can reuse the results above and the stream function will be
3711:
3333:
1602:
4602:{\displaystyle \phi =\alpha _{0}-\beta _{0}\theta +\sum _{m\mathop {>} 0}{(\alpha _{m}r^{m}-\beta _{m}r^{-m})\cos {}}}
4430:{\displaystyle \psi =\alpha _{0}+\beta _{0}\ln r+\sum _{m>0}{\left(\alpha _{m}r^{m}+\beta _{m}r^{-m}\right)\sin {}}}
4821:
4813:
36:
30:
4679:
2037:
per unit length is a line sink. Everything is the same as the case of a line source a part from the negative sign.
4619:
520:
2043:
98:, by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or
47:
1844:
300:{\displaystyle \mathbf {v} =v_{0}\cos(\theta _{0})\,\mathbf {e} _{x}+v_{0}\sin(\theta _{0})\,\mathbf {e} _{y}}
4883:
1353:
95:
79:
343:
138:
130:
3528:
4817:
2416:
1590:
fields where the monopole is essentially the first non-trivial (e.g. constant) term of the expansion.
483:
454:
4846:
4838:
103:
4223:
2935:{\displaystyle v_{\theta }(r,\theta )={\frac {Qd}{2\pi }}{\frac {\sin(\theta -\theta _{0})}{r^{2}}}}
4784:
4190:
3078:
1579:
111:
421:
390:
158:
142:
4642:
2228:
to assume both positive and negative values and absorbing the minus sign into the definition of
1325:
805:
4860:
4792:
4771:
4750:
4727:
4685:
4652:
4629:
3074:
3067:
2825:{\displaystyle v_{r}(r,\theta )={\frac {Qd}{2\pi }}{\frac {\cos(\theta -\theta _{0})}{r^{2}}}}
154:
115:
3975:
3951:
4719:
4704:
1575:
150:
83:
2710:{\displaystyle \psi (r,\theta )=-{\frac {Qd}{2\pi }}{\frac {\sin(\theta -\theta _{0})}{r}}}
1075:
558:
316:
3036:{\displaystyle \phi (r,\theta )={\frac {Qd}{2\pi }}{\frac {\cos(\theta -\theta _{0})}{r}}}
552:
102:
and to be patched together. Elementary flows can be considered the basic building blocks (
3151:
Where the total circulation is constant for every closed line around the central vortex
1578:
i.e. divergence free fields). Generic flow patterns can be also de-composed in terms of
4740:
3082:
1587:
1583:
126:
119:
99:
4877:
4256:
In order to have a single-valued velocity (and also a single-valued stream function)
4177:{\displaystyle {\frac {d^{2}\Theta (\theta )}{d\theta ^{2}}}=-m^{2}\Theta (\theta )}
3066:
Dual to the case above of line sources, vortex lines play the role of monopoles for
1462:{\displaystyle v_{\theta }={\frac {1}{r}}{\frac {\partial \phi }{\partial \theta }}}
4623:
2033:
The case of a vertical line absorbing at a fixed rate a constant quantity of fluid
1312:{\displaystyle \phi =\phi _{0}-v_{0}\cos(\theta _{0})\,x-v_{0}\sin(\theta _{0})\,y}
792:{\displaystyle \psi =\psi _{0}-v_{0}\sin(\theta _{0})\,x+v_{0}\cos(\theta _{0})\,y}
134:
4744:
3522:
Given an incompressible two-dimensional flow which is also irrotational we have:
4763:
94:
The techniques involved to create more complex solutions can be for example by
891:{\displaystyle v_{r}=-{\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }}}
2240:
445:
is positive for angles measured in a counterclockwise sense from the positive
170:
4749:, CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages,
3051:
1558:
146:
4088:{\displaystyle r{\frac {d}{dr}}\left(r{\frac {d}{dr}}R(r)\right)=m^{2}R(r)}
141:
in general. To put it in perspective boundary layers can be interpreted as
107:
4681:
The Shaggy Steed of
Physics: Mathematical Beauty in the Physical World
122:, and some of the flows may be limited to the case of two dimensions.
1072:) so its velocity can be expressed in terms of a potential function,
4746:
Applied
Hydrodynamics: An Introduction to Ideal and Real Fluid Flows
3141:{\displaystyle \mathbf {v} =v_{\theta }(r)\,\mathbf {e} _{\theta }}
551:) and two-dimensional, its velocity can be expressed in terms of a
3050:
2239:
1557:
169:
3504:{\displaystyle \phi (r,\theta )=-{\frac {\Gamma }{2\pi }}\theta }
1543:{\displaystyle \phi =\phi _{0}-v_{0}\,r\cos(\theta -\theta _{0})}
1026:{\displaystyle \psi =\psi _{0}+v_{0}\,r\sin(\theta -\theta _{0})}
1593:
This flow pattern is also both irrotational and incompressible.
942:{\displaystyle v_{\theta }={\frac {\partial \psi }{\partial r}}}
4187:
The solution to the second equation is a linear combination of
3439:{\displaystyle \psi (r,\theta )={\frac {\Gamma }{2\pi }}\ln r}
149:, and considering fluid dynamics analogies and limit cases in
15:
3968:, the two parts must be equal to a constant independent from
1947:{\displaystyle \psi (r,\theta )=-{\frac {Q}{2\pi }}\theta }
1195:{\displaystyle v_{y}=-{\frac {\partial \phi }{\partial y}}}
1141:{\displaystyle v_{x}=-{\frac {\partial \phi }{\partial x}}}
675:{\displaystyle v_{y}=-{\frac {\partial \psi }{\partial x}}}
182:
For steady-state, spatially uniform flow of a fluid in the
2146:{\displaystyle \psi (r,\theta )={\frac {Q}{2\pi }}\theta }
1401:{\displaystyle v_{r}={\frac {\partial \phi }{\partial r}}}
621:{\displaystyle v_{x}={\frac {\partial \psi }{\partial y}}}
2018:{\displaystyle \phi (r,\theta )=-{\frac {Q}{2\pi }}\ln r}
414:
is the angle the velocity vector makes with the positive
2214:{\displaystyle \phi (r,\theta )={\frac {Q}{2\pi }}\ln r}
4837:(c) Aerospace, Mechanical & Mechatronic Engg. 2005
1065:{\displaystyle \nabla \times \mathbf {v} =\mathbf {0} }
4449:
4272:
4226:
4193:
4102:
4001:
3978:
3954:
3782:
3763:{\displaystyle \psi (r,\theta )=R(r)\Theta (\theta )}
3714:
3573:
3531:
3458:
3390:
3371:{\displaystyle v_{\theta }={\frac {\Gamma }{2\pi r}}}
3336:
3160:
3097:
2954:
2839:
2729:
2625:
2419:
2261:
2165:
2103:
2046:
1966:
1901:
1847:
1667:
1648:{\displaystyle \mathbf {v} =v_{r}(r)\mathbf {e} _{r}}
1605:
1478:
1415:
1364:
1328:
1214:
1155:
1101:
1078:
1042:
961:
905:
841:
808:
694:
635:
584:
561:
523:
486:
457:
424:
393:
346:
319:
195:
3514:
Which is dual to the previous case of a line source
3324:and is zero for any line not including the vortex.
2407:The last approximation is to the first order in d.
4601:
4429:
4248:
4212:
4176:
4087:
3984:
3960:
3933:
3762:
3694:
3553:
3503:
3438:
3370:
3313:
3140:
3035:
2934:
2824:
2709:
2605:
2396:
2213:
2145:
2083:
2017:
1946:
1881:
1827:
1647:
1542:
1461:
1400:
1341:
1311:
1194:
1140:
1084:
1064:
1025:
941:
890:
821:
791:
674:
620:
567:
543:
501:
472:
437:
406:
377:
332:
299:
3705:We look for a solution with separated variables:
3274:
3088:This is characterized by a cylindrical symmetry:
1788:
1596:This is characterized by a cylindrical symmetry:
340:is the absolute magnitude of the velocity (i.e.,
4263:therefore the most generic solution is given by
3992:. The constant shall be positive. Therefore,
2236:Two-dimensional doublet or dipole line source
8:
4770:(6th ed.), Cambridge University Press,
133:, elementary flows are relevant not only to
4865:: CS1 maint: numeric names: authors list (
544:{\displaystyle \nabla \cdot \mathbf {v} =0}
517:Because this flow is incompressible (i.e.,
2084:{\displaystyle v_{r}=-{\frac {Q}{2\pi r}}}
1658:Where the total outgoing flux is constant
4684:. Springer Science & Business Media.
4585:
4565:
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3132:
3127:
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3110:
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3096:
3018:
2996:
2976:
2953:
2924:
2910:
2888:
2868:
2844:
2838:
2814:
2800:
2778:
2758:
2734:
2728:
2692:
2670:
2650:
2624:
2594:
2589:
2579:
2551:
2546:
2536:
2499:
2494:
2484:
2462:
2457:
2447:
2420:
2418:
2386:
2377:
2362:
2345:
2340:
2332:
2323:
2305:
2300:
2292:
2283:
2268:
2260:
2187:
2164:
2125:
2102:
2063:
2051:
2045:
1991:
1965:
1926:
1900:
1882:{\displaystyle v_{r}={\frac {Q}{2\pi r}}}
1861:
1852:
1846:
1804:
1799:
1795:
1775:
1771:
1765:
1760:
1744:
1739:
1737:
1722:
1706:
1701:
1689:
1678:
1672:
1666:
1639:
1634:
1618:
1606:
1604:
1531:
1508:
1502:
1489:
1477:
1439:
1429:
1420:
1414:
1378:
1369:
1363:
1333:
1327:
1305:
1296:
1277:
1266:
1257:
1238:
1225:
1213:
1172:
1160:
1154:
1118:
1106:
1100:
1077:
1057:
1049:
1041:
1014:
991:
985:
972:
960:
919:
910:
904:
868:
858:
846:
840:
813:
807:
785:
776:
757:
746:
737:
718:
705:
693:
652:
640:
634:
598:
589:
583:
560:
530:
522:
493:
488:
485:
464:
459:
456:
429:
423:
398:
392:
370:
365:
360:
351:
345:
324:
318:
291:
286:
284:
275:
256:
243:
238:
236:
227:
208:
196:
194:
66:Learn how and when to remove this message
3073:Also in this case the flow is also both
29:This article includes a list of general
4670:
3381:This is derived from a stream function
2094:This is derived from a stream function
1892:This is derived from a stream function
4858:
3518:Generic two-dimensional potential flow
3564:Which is in cylindrical coordinates
2248:for an ideal doublet, or dipole, line
7:
3948:and the right parts depends only on
3944:Given the left part depends only on
378:{\displaystyle v_{0}=|\mathbf {v} |}
4440:The potential is instead given by
4162:
4116:
3901:
3873:
3748:
3670:
3656:
3618:
3610:
3590:
3586:
3554:{\displaystyle \nabla ^{2}\psi =0}
3533:
3485:
3414:
3352:
3308:
2370:
1450:
1442:
1389:
1381:
1183:
1175:
1129:
1121:
1043:
930:
922:
879:
871:
663:
655:
609:
601:
524:
509:are the unit basis vectors of the
118:flows, or both, as in the case of
82:(and especially in the context of
35:it lacks sufficient corresponding
14:
4724:An introduction to fluid dynamics
1036:This flow is irrotational (i.e.,
3247:
3226:
3176:
3165:
3128:
3099:
2606:{\displaystyle \mathbf {d} =d=d}
2590:
2547:
2495:
2458:
2421:
2387:
2363:
2341:
2333:
2301:
2293:
2269:
1761:
1740:
1690:
1679:
1635:
1607:
1058:
1050:
531:
502:{\displaystyle \mathbf {e} _{y}}
489:
473:{\displaystyle \mathbf {e} _{x}}
460:
366:
287:
239:
197:
125:Due to the relationship between
20:
4726:, Cambridge University Press,
4651:, Cambridge university press,
4594:
4591:
4572:
4566:
4556:
4507:
4422:
4419:
4400:
4394:
4249:{\displaystyle e^{-im\theta }}
4171:
4165:
4125:
4119:
4082:
4076:
4052:
4046:
3910:
3904:
3882:
3876:
3842:
3836:
3798:
3792:
3757:
3751:
3745:
3739:
3730:
3718:
3474:
3462:
3406:
3394:
3302:
3296:
3268:
3242:
3236:
3220:
3214:
3201:
3122:
3116:
3024:
3005:
2970:
2958:
2916:
2897:
2862:
2850:
2806:
2787:
2752:
2740:
2698:
2679:
2641:
2629:
2600:
2585:
2566:
2542:
2523:
2514:
2505:
2490:
2477:
2453:
2440:
2431:
2391:
2383:
2353:
2329:
2313:
2289:
2273:
2265:
2181:
2169:
2119:
2107:
1982:
1970:
1917:
1905:
1816:
1810:
1782:
1756:
1750:
1734:
1728:
1715:
1630:
1624:
1537:
1518:
1302:
1289:
1263:
1250:
1020:
1001:
782:
769:
743:
730:
371:
361:
281:
268:
233:
220:
186:plane, the velocity vector is
1:
4648:Fluid Dynamics for Physicists
4260:shall be a positive integer.
4213:{\displaystyle e^{im\theta }}
3449:or from a potential function
2156:or from a potential function
1957:or from a potential function
78:In the larger context of the
4843:"Elements of Potential Flow"
4678:Oliver, David (2013-03-14).
1582:, in the same manner as for
832:In cylindrical coordinates:
166:Two-dimensional uniform flow
4822:University of Texas, Austin
4814:University of Texas, Austin
3047:Two-dimensional vortex line
1554:Two-dimensional line source
438:{\displaystyle \theta _{0}}
407:{\displaystyle \theta _{0}}
4900:
4625:Theoretical fluid dynamics
2945:And the potential instead
4789:Theoretical hydrodynamics
2029:Two-dimensional line sink
1342:{\displaystyle \phi _{0}}
822:{\displaystyle \psi _{0}}
178:for an ideal uniform flow
3081:and therefore a case of
3059:for an ideal vortex line
1566:for an ideal line source
4791:(5th ed.), Dover,
3985:{\displaystyle \theta }
3961:{\displaystyle \theta }
1354:cylindrical coordinates
80:Navier-Stokes equations
50:more precise citations.
4603:
4431:
4250:
4214:
4178:
4089:
3986:
3962:
3935:
3764:
3696:
3555:
3505:
3440:
3372:
3315:
3142:
3060:
3037:
2936:
2826:
2711:
2607:
2398:
2249:
2215:
2147:
2085:
2019:
1948:
1883:
1829:
1649:
1567:
1544:
1463:
1402:
1343:
1313:
1196:
1142:
1086:
1066:
1027:
943:
892:
823:
793:
676:
622:
569:
545:
503:
474:
439:
408:
379:
334:
301:
179:
106:, local solutions and
4604:
4432:
4251:
4215:
4179:
4090:
3987:
3963:
3936:
3765:
3697:
3556:
3506:
3441:
3373:
3316:
3143:
3054:
3038:
2937:
2827:
2720:The velocity is then
2712:
2608:
2399:
2243:
2216:
2148:
2086:
2020:
1949:
1884:
1830:
1650:
1561:
1545:
1464:
1403:
1344:
1314:
1197:
1143:
1087:
1085:{\displaystyle \phi }
1067:
1028:
944:
893:
824:
794:
677:
623:
570:
568:{\displaystyle \psi }
546:
504:
475:
440:
409:
380:
335:
333:{\displaystyle v_{0}}
302:
173:
104:fundamental solutions
4847:University of Sydney
4839:University of Sydney
4812:Richard Fitzpatrick
4620:Fitzpatrick, Richard
4447:
4270:
4224:
4191:
4100:
3999:
3976:
3952:
3780:
3712:
3571:
3529:
3456:
3388:
3334:
3158:
3095:
2952:
2837:
2727:
2623:
2417:
2259:
2163:
2101:
2044:
1964:
1899:
1845:
1665:
1603:
1580:multipole expansions
1476:
1413:
1362:
1326:
1212:
1153:
1099:
1076:
1040:
959:
903:
839:
806:
692:
633:
582:
559:
521:
484:
455:
422:
391:
344:
317:
193:
4785:Milne-Thomson, L.M.
3200:
1714:
143:topological defects
4599:
4505:
4427:
4329:
4246:
4210:
4174:
4085:
3982:
3958:
3931:
3760:
3692:
3551:
3501:
3436:
3368:
3311:
3183:
3138:
3068:irrotational flows
3061:
3033:
2932:
2822:
2707:
2603:
2394:
2250:
2211:
2143:
2081:
2015:
1944:
1879:
1825:
1697:
1645:
1568:
1540:
1459:
1398:
1339:
1309:
1192:
1138:
1082:
1062:
1023:
939:
888:
819:
789:
672:
618:
565:
541:
513:coordinate system.
499:
470:
435:
404:
375:
330:
297:
180:
159:general relativity
4818:"Fluid Mechanics"
4798:978-0-486-68970-8
4777:978-0-521-45868-9
4756:978-0-415-49271-3
4733:978-0-521-09817-5
4691:978-1-4757-4347-0
4635:978-0-7503-1554-8
4485:
4314:
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4018:
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2886:
2820:
2776:
2705:
2668:
2358:
2200:
2138:
2079:
2004:
1939:
1877:
1576:solenoidal fields
1457:
1437:
1396:
1190:
1136:
937:
886:
866:
670:
616:
155:quantum mechanics
76:
75:
68:
4891:
4870:
4864:
4856:
4854:
4853:
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4759:
4736:
4707:
4705:Laplace operator
4702:
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4638:
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4605:
4600:
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3312:
3295:
3294:
3256:
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3235:
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3213:
3212:
3199:
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3131:
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3102:
3042:
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2126:
2090:
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2056:
2055:
2024:
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2016:
2005:
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1992:
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1199:
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1024:
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989:
977:
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928:
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828:
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798:
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795:
790:
781:
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762:
761:
742:
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723:
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710:
709:
681:
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673:
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645:
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492:
479:
477:
476:
471:
469:
468:
463:
448:
444:
442:
441:
436:
434:
433:
417:
413:
411:
410:
405:
403:
402:
384:
382:
381:
376:
374:
369:
364:
356:
355:
339:
337:
336:
331:
329:
328:
306:
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298:
296:
295:
290:
280:
279:
261:
260:
248:
247:
242:
232:
231:
213:
212:
200:
185:
151:electromagnetism
88:elementary flows
84:potential theory
71:
64:
60:
57:
51:
46:this article by
37:inline citations
24:
23:
16:
4899:
4898:
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4826:
4824:
4811:
4808:
4799:
4783:
4778:
4762:
4757:
4739:
4734:
4720:Batchelor, G.K.
4718:
4715:
4713:Further reading
4710:
4703:
4699:
4692:
4677:
4676:
4672:
4659:
4641:
4636:
4628:, IOP science,
4618:
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3949:
3918:
3914:
3891:
3890:
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3788:
3778:
3777:
3710:
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3599:
3589:
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3488:
3454:
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3417:
3386:
3385:
3355:
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3332:
3331:
3286:
3245:
3224:
3204:
3156:
3155:
3126:
3106:
3093:
3092:
3055:Potential flow
3049:
3014:
2998:
2986:
2978:
2950:
2949:
2920:
2906:
2890:
2878:
2870:
2840:
2835:
2834:
2810:
2796:
2780:
2768:
2760:
2730:
2725:
2724:
2688:
2672:
2660:
2652:
2621:
2620:
2588:
2575:
2545:
2532:
2493:
2480:
2456:
2443:
2415:
2414:
2373:
2319:
2279:
2257:
2256:
2244:Potential flow
2238:
2192:
2161:
2160:
2130:
2099:
2098:
2068:
2047:
2042:
2041:
2031:
1996:
1962:
1961:
1931:
1897:
1896:
1866:
1848:
1843:
1842:
1800:
1759:
1738:
1718:
1668:
1663:
1662:
1633:
1614:
1601:
1600:
1562:Potential flow
1556:
1527:
1498:
1485:
1474:
1473:
1449:
1441:
1416:
1411:
1410:
1388:
1380:
1365:
1360:
1359:
1349:is a constant.
1329:
1324:
1323:
1292:
1273:
1253:
1234:
1221:
1210:
1209:
1182:
1174:
1156:
1151:
1150:
1128:
1120:
1102:
1097:
1096:
1074:
1073:
1038:
1037:
1010:
981:
968:
957:
956:
929:
921:
906:
901:
900:
878:
870:
842:
837:
836:
829:is a constant.
809:
804:
803:
772:
753:
733:
714:
701:
690:
689:
662:
654:
636:
631:
630:
608:
600:
585:
580:
579:
557:
556:
553:stream function
519:
518:
510:
487:
482:
481:
458:
453:
452:
446:
425:
420:
419:
415:
394:
389:
388:
347:
342:
341:
320:
315:
314:
285:
271:
252:
237:
223:
204:
191:
190:
183:
174:Potential flow
168:
72:
61:
55:
52:
42:Please help to
41:
25:
21:
12:
11:
5:
4897:
4895:
4887:
4886:
4884:Fluid dynamics
4876:
4875:
4872:
4871:
4833:
4832:
4807:
4806:External links
4804:
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3163:
3149:
3148:
3135:
3130:
3124:
3121:
3118:
3113:
3109:
3105:
3101:
3083:potential flow
3079:incompressible
3048:
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3026:
3021:
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3013:
3010:
3007:
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467:
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222:
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199:
167:
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127:fluid dynamics
120:potential flow
112:incompressible
100:boundary layer
74:
73:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
4896:
4885:
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4779:
4773:
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4768:Hydrodynamics
4765:
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2110:
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2040:
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2038:
2036:
2028:
2012:
2009:
2006:
2000:
1997:
1993:
1988:
1985:
1979:
1976:
1973:
1967:
1960:
1959:
1958:
1941:
1935:
1932:
1928:
1923:
1920:
1914:
1911:
1908:
1902:
1895:
1894:
1893:
1873:
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1849:
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1471:
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1426:
1421:
1417:
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1384:
1375:
1370:
1366:
1358:
1357:
1356:
1355:
1350:
1334:
1330:
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1297:
1293:
1286:
1283:
1278:
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1169:
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1015:
1011:
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998:
995:
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986:
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973:
969:
965:
962:
955:
954:
953:
933:
925:
916:
911:
907:
899:
882:
874:
863:
860:
855:
852:
847:
843:
835:
834:
833:
830:
814:
810:
786:
777:
773:
766:
763:
758:
754:
750:
747:
738:
734:
727:
724:
719:
715:
711:
706:
702:
698:
695:
688:
687:
686:
666:
658:
649:
646:
641:
637:
629:
612:
604:
595:
590:
586:
578:
577:
576:
562:
554:
538:
535:
527:
494:
465:
451:
430:
426:
399:
395:
387:
357:
352:
348:
325:
321:
313:
312:
311:
292:
276:
272:
265:
262:
257:
253:
249:
244:
228:
224:
217:
214:
209:
205:
201:
189:
188:
187:
177:
172:
165:
163:
160:
156:
152:
148:
144:
140:
136:
132:
128:
123:
121:
117:
113:
109:
105:
101:
97:
96:superposition
92:
89:
85:
81:
70:
67:
59:
56:February 2018
49:
45:
39:
38:
32:
27:
18:
17:
4850:. Retrieved
4825:. Retrieved
4788:
4767:
4745:
4723:
4700:
4680:
4673:
4647:
4624:
4439:
4262:
4257:
4186:
3969:
3945:
3943:
3773:which gives
3772:
3704:
3563:
3521:
3513:
3448:
3380:
3326:
3323:
3150:
3087:
3075:irrotational
3072:
3065:
3062:
2944:
2719:
2615:
2409:
2406:
2251:
2229:
2225:
2223:
2155:
2093:
2034:
2032:
1956:
1891:
1837:
1657:
1595:
1592:
1573:
1569:
1351:
1321:
1204:
1035:
951:
831:
801:
684:
516:
309:
181:
139:field theory
135:aerodynamics
131:field theory
124:
116:irrotational
93:
87:
77:
62:
53:
34:
4741:Chanson, H.
4643:Faber, T.E.
3327:Therefore,
3057:streamlines
2616:It remains
2246:streamlines
1838:Therefore,
1564:streamlines
176:streamlines
145:on generic
137:but to all
48:introducing
4852:2019-04-19
4827:2018-02-07
4613:References
449:axis); and
31:references
4787:(1996) ,
4766:(1994) ,
4583:θ
4579:−
4576:θ
4563:
4549:−
4535:β
4531:−
4512:α
4499:
4487:∑
4480:θ
4471:β
4467:−
4458:α
4451:ϕ
4411:θ
4407:−
4404:θ
4391:
4375:−
4361:β
4338:α
4316:∑
4306:
4294:β
4281:α
4274:ψ
4242:θ
4233:−
4206:θ
4169:θ
4163:Θ
4150:−
4135:θ
4123:θ
4117:Θ
3980:θ
3956:θ
3920:θ
3908:θ
3902:Θ
3880:θ
3874:Θ
3865:−
3755:θ
3749:Θ
3728:θ
3716:ψ
3675:θ
3671:∂
3666:ψ
3657:∂
3619:∂
3614:ψ
3611:∂
3591:∂
3587:∂
3543:ψ
3534:∇
3499:θ
3493:π
3486:Γ
3481:−
3472:θ
3460:ϕ
3431:
3422:π
3415:Γ
3404:θ
3392:ψ
3360:π
3353:Γ
3343:θ
3309:Γ
3292:θ
3279:π
3266:θ
3253:θ
3240:⋅
3232:θ
3210:θ
3197:π
3185:∫
3170:⋅
3162:∮
3134:θ
3112:θ
3016:θ
3012:−
3009:θ
3003:
2991:π
2968:θ
2956:ϕ
2908:θ
2904:−
2901:θ
2895:
2883:π
2860:θ
2846:θ
2798:θ
2794:−
2791:θ
2785:
2773:π
2750:θ
2690:θ
2686:−
2683:θ
2677:
2665:π
2648:−
2639:θ
2627:ψ
2596:θ
2577:θ
2573:−
2570:θ
2564:
2534:θ
2530:−
2527:θ
2521:
2482:θ
2475:
2445:θ
2438:
2375:ψ
2371:∇
2368:⋅
2360:≃
2321:ψ
2317:−
2298:−
2281:ψ
2263:ψ
2206:
2197:π
2179:θ
2167:ϕ
2141:θ
2135:π
2117:θ
2105:ψ
2073:π
2061:−
2010:
2001:π
1989:−
1980:θ
1968:ϕ
1942:θ
1936:π
1924:−
1915:θ
1903:ψ
1871:π
1793:π
1780:θ
1754:⋅
1711:π
1699:∫
1684:⋅
1670:∫
1529:θ
1525:−
1522:θ
1516:
1496:−
1487:ϕ
1480:ϕ
1454:θ
1451:∂
1446:ϕ
1443:∂
1422:θ
1390:∂
1385:ϕ
1382:∂
1331:ϕ
1294:θ
1287:
1271:−
1255:θ
1248:
1232:−
1223:ϕ
1216:ϕ
1184:∂
1179:ϕ
1176:∂
1170:−
1130:∂
1125:ϕ
1122:∂
1116:−
1080:ϕ
1047:×
1044:∇
1012:θ
1008:−
1005:θ
999:
970:ψ
963:ψ
931:∂
926:ψ
923:∂
912:θ
883:θ
880:∂
875:ψ
872:∂
856:−
811:ψ
774:θ
767:
735:θ
728:
712:−
703:ψ
696:ψ
664:∂
659:ψ
656:∂
650:−
610:∂
605:ψ
602:∂
563:ψ
528:⋅
525:∇
427:θ
396:θ
273:θ
266:
225:θ
218:
147:manifolds
4878:Category
4861:cite web
4841:(2005).
4816:(2017).
4764:Lamb, H.
4743:(2009),
4722:(1973),
4666:Specific
4645:(1995),
4622:(2017),
1588:magnetic
1584:electric
108:solitons
44:improve
4795:
4774:
4753:
4730:
4688:
4655:
4632:
2410:Given
2357:
1205:where
685:where
418:axis (
310:where
33:, but
4867:link
4793:ISBN
4772:ISBN
4751:ISBN
4728:ISBN
4686:ISBN
4653:ISBN
4630:ISBN
4495:>
4323:>
4220:and
3972:and
3077:and
1586:and
1322:and
952:and
802:and
480:and
157:and
129:and
4560:cos
4388:sin
3000:cos
2892:sin
2782:cos
2674:sin
2561:sin
2518:cos
2472:sin
2435:cos
1513:cos
1352:In
1284:sin
1245:cos
996:sin
764:cos
725:sin
263:sin
215:cos
114:or
86:),
4880::
4863:}}
4859:{{
4845:.
4820:.
4303:ln
3428:ln
3085:.
3070:.
2232:.
2203:ln
2007:ln
1092::
575::
555:,
511:xy
385:);
184:xy
153:,
4869:)
4855:.
4830:.
4694:.
4595:]
4592:)
4587:m
4573:(
4570:m
4567:[
4557:)
4552:m
4545:r
4539:m
4526:m
4522:r
4516:m
4508:(
4502:0
4491:m
4483:+
4475:0
4462:0
4454:=
4423:]
4420:)
4415:m
4401:(
4398:m
4395:[
4384:)
4378:m
4371:r
4365:m
4357:+
4352:m
4348:r
4342:m
4333:(
4326:0
4320:m
4312:+
4309:r
4298:0
4290:+
4285:0
4277:=
4258:m
4239:m
4236:i
4229:e
4203:m
4200:i
4196:e
4172:)
4166:(
4158:2
4154:m
4147:=
4139:2
4131:d
4126:)
4120:(
4112:2
4108:d
4083:)
4080:r
4077:(
4074:R
4069:2
4065:m
4061:=
4057:)
4053:)
4050:r
4047:(
4044:R
4038:r
4035:d
4031:d
4026:r
4022:(
4015:r
4012:d
4008:d
4003:r
3970:r
3946:r
3924:2
3916:d
3911:)
3905:(
3897:2
3893:d
3883:)
3877:(
3870:1
3862:=
3858:)
3851:r
3848:d
3843:)
3840:r
3837:(
3834:R
3831:d
3825:r
3821:(
3814:r
3811:d
3807:d
3799:)
3796:r
3793:(
3790:R
3786:r
3758:)
3752:(
3746:)
3743:r
3740:(
3737:R
3734:=
3731:)
3725:,
3722:r
3719:(
3690:0
3687:=
3679:2
3661:2
3646:2
3642:r
3638:1
3633:+
3629:)
3622:r
3605:r
3601:(
3594:r
3580:r
3577:1
3549:0
3546:=
3538:2
3490:2
3478:=
3475:)
3469:,
3466:r
3463:(
3434:r
3419:2
3410:=
3407:)
3401:,
3398:r
3395:(
3363:r
3357:2
3348:=
3339:v
3306:=
3303:)
3300:r
3297:(
3288:v
3283:r
3276:2
3272:=
3269:)
3263:d
3259:r
3248:e
3243:(
3237:)
3227:e
3221:)
3218:r
3215:(
3206:v
3202:(
3194:2
3189:0
3181:=
3177:s
3173:d
3166:v
3129:e
3123:)
3120:r
3117:(
3108:v
3104:=
3100:v
3029:r
3025:)
3020:0
3006:(
2988:2
2983:d
2980:Q
2974:=
2971:)
2965:,
2962:r
2959:(
2926:2
2922:r
2917:)
2912:0
2898:(
2880:2
2875:d
2872:Q
2866:=
2863:)
2857:,
2854:r
2851:(
2842:v
2816:2
2812:r
2807:)
2802:0
2788:(
2770:2
2765:d
2762:Q
2756:=
2753:)
2747:,
2744:r
2741:(
2736:r
2732:v
2703:r
2699:)
2694:0
2680:(
2662:2
2657:d
2654:Q
2645:=
2642:)
2636:,
2633:r
2630:(
2601:]
2591:e
2586:)
2581:0
2567:(
2558:+
2553:r
2548:e
2543:)
2538:0
2524:(
2515:[
2512:d
2509:=
2506:]
2501:y
2496:e
2491:)
2486:0
2478:(
2469:+
2464:x
2459:e
2454:)
2449:0
2441:(
2432:[
2429:d
2426:=
2422:d
2392:)
2388:r
2384:(
2379:Q
2364:d
2354:)
2351:2
2347:/
2342:d
2338:+
2334:r
2330:(
2325:Q
2314:)
2311:2
2307:/
2302:d
2294:r
2290:(
2285:Q
2277:=
2274:)
2270:r
2266:(
2230:Q
2226:Q
2209:r
2194:2
2190:Q
2185:=
2182:)
2176:,
2173:r
2170:(
2132:2
2128:Q
2123:=
2120:)
2114:,
2111:r
2108:(
2076:r
2070:2
2066:Q
2058:=
2053:r
2049:v
2035:Q
2013:r
1998:2
1994:Q
1986:=
1983:)
1977:,
1974:r
1971:(
1933:2
1929:Q
1921:=
1918:)
1912:,
1909:r
1906:(
1874:r
1868:2
1864:Q
1859:=
1854:r
1850:v
1823:Q
1820:=
1817:)
1814:r
1811:(
1806:r
1802:v
1797:r
1790:2
1786:=
1783:)
1777:d
1773:r
1767:r
1762:e
1757:(
1751:)
1746:r
1741:e
1735:)
1732:r
1729:(
1724:r
1720:v
1716:(
1708:2
1703:0
1695:=
1691:S
1687:d
1680:v
1674:S
1641:r
1636:e
1631:)
1628:r
1625:(
1620:r
1616:v
1612:=
1608:v
1538:)
1533:0
1519:(
1510:r
1504:0
1500:v
1491:0
1483:=
1435:r
1432:1
1427:=
1418:v
1393:r
1376:=
1371:r
1367:v
1335:0
1307:y
1303:)
1298:0
1290:(
1279:0
1275:v
1268:x
1264:)
1259:0
1251:(
1240:0
1236:v
1227:0
1219:=
1187:y
1167:=
1162:y
1158:v
1133:x
1113:=
1108:x
1104:v
1059:0
1055:=
1051:v
1021:)
1016:0
1002:(
993:r
987:0
983:v
979:+
974:0
966:=
934:r
917:=
908:v
864:r
861:1
853:=
848:r
844:v
815:0
787:y
783:)
778:0
770:(
759:0
755:v
751:+
748:x
744:)
739:0
731:(
720:0
716:v
707:0
699:=
667:x
647:=
642:y
638:v
613:y
596:=
591:x
587:v
539:0
536:=
532:v
495:y
490:e
466:x
461:e
447:x
431:0
416:x
400:0
372:|
367:v
362:|
358:=
353:0
349:v
326:0
322:v
293:y
288:e
282:)
277:0
269:(
258:0
254:v
250:+
245:x
240:e
234:)
229:0
221:(
210:0
206:v
202:=
198:v
69:)
63:(
58:)
54:(
40:.
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