3336:
2982:
2554:
3331:{\displaystyle {\begin{aligned}{\frac {\partial {\hat {u}}}{\partial \chi }}+{\frac {\partial {\hat {v}}}{\partial \xi }}&=0,\\{\hat {u}}{\frac {\partial {\hat {u}}}{\partial \chi }}+{\hat {v}}{\frac {\partial {\hat {u}}}{\partial \xi }}&=-\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {A''({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}+{\frac {\partial ^{2}{\hat {u}}}{\partial \xi ^{2}}}.\end{aligned}}}
3586:
1865:
2238:
2275:
1599:
3347:
372:
1670:
2549:{\displaystyle P(\chi )=\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {A'({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}\quad {\text{and}}\quad A'(\chi )=-\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {P({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}.}
1999:
376:
The lower deck is characterized by viscous, rotational disturbances, whereas the middle deck (same thickness as the boundary-layer thickness) is characterized by inviscid, rotational disturbances. The upper deck, which extends into the potential flow region, is characterized by inviscid, irrotational
443:
The most important aspect of the triple-deck formulation is that pressure is not prescribed, and so it has to be solved as part of the boundary-layer problem. This coupling between velocity and pressure reintroduces ellipticity to the problem, which is in contrast to the parabolic nature of the
1181:
1450:
3693:
926:
2716:
3581:{\displaystyle \xi =0:\,\,\,{\begin{cases}{\hat {u}}={\hat {v}}=0,\,\,\chi \leq 0,\\{\frac {\partial {\hat {u}}}{\partial \xi }}={\hat {v}}=0,\,\,\chi >0,\end{cases}}\quad {\text{and}}\quad \xi \to \infty :{\hat {u}}\to {\frac {\xi +A(\chi )}{\sqrt {2\alpha ^{3}}}},}
218:
1860:{\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+\varepsilon A(\chi )f_{B}'\left({\frac {\eta }{\sqrt {2}}}\right)+\varepsilon ^{2}\Phi (\chi ,\eta )+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots }
3782:
must be obtained as part of the solution. The above set of equations may resemble normal boundary-layer equations, however it has an elliptic character since the pressure gradient term now is non-local, i.e., the pressure gradient at a location
2233:{\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\frac {\zeta }{\varepsilon }}-{\sqrt {2}}\beta -{\frac {\varepsilon }{\pi }}\int _{-\infty }^{+\infty }A'(\chi )\leftd{\hat {\chi }}+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots }
49:(1969) and V. Y. Neiland (1969) and by A. F. Messiter (1970). Stewartson and Messiter considered the separated flow near the trailing edge of a flat plate, whereas Neiland studied the case of a shock impinging on a boundary layer.
40:
and others were the first to realize that to explain boundary layer separation, different scales other than the classical boundary-layer scales need to be introduced. These scales were first introduced independently by
4114:
Jobe, C. E., & Burggraf, O. R. (1974). The numerical solution of the asymptotic equations of trailing edge flow. Proceedings of the Royal
Society of London. A. Mathematical and Physical Sciences, 340(1620),
1003:
1594:{\displaystyle {\text{Outer wake}}:{\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+{\frac {\mu x^{1/3}}{\lambda }}f_{B}'\left({\frac {\eta }{\sqrt {2}}}\right)+\cdots }
1380:
1282:
439:
3592:
803:
2612:
19:
is a theory that describes a three-layered boundary-layer structure when sufficiently large disturbances are present in the boundary layer. This theory is able to successfully explain the phenomenon of
2916:
2864:
2987:
223:
367:{\displaystyle {\begin{aligned}{\text{Lower deck}}:&\quad y\sim Re^{-5/8}\\{\text{Middle deck}}:&\quad y\sim Re^{-4/8}\\{\text{Upper deck}}:&\quad y\sim Re^{-3/8}.\end{aligned}}}
743:
700:
795:
2754:
1984:
1409:
958:
164:
2815:
2588:
1654:
211:
3722:
2267:
1954:
1220:
1628:
584:
2974:
2945:
1442:
991:
534:
2788:
3780:
3751:
1930:
1897:
3801:
656:
616:
113:
636:
494:
474:
90:
70:
1176:{\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+{\frac {x}{\sqrt {2}}}\left+\cdots }
3688:{\displaystyle \chi \to -\infty :\,\,\,A\to 0,\quad {\text{and}}\quad \chi \to +\infty :\,\,\,A\to {\frac {\mu }{\lambda }}\chi ^{1/3}}
921:{\displaystyle \chi =x/\varepsilon ^{3},\quad \xi =y/\varepsilon ^{5},\quad \eta =y/\varepsilon ^{4},\quad \zeta =y/\varepsilon ^{3}.}
385:
2711:{\displaystyle {\frac {\psi }{\varepsilon ^{4}}}=\varepsilon ^{2}\Psi (\chi ,\xi )+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots }
1936:, to be determined from the upper and lower deck problems. Note that the correction to the Blasius stream function is of the order
2869:
2594:. One may notice that the pressure function and the derivative of the displacement function (aka transverse velocity) forms a
2820:
1287:
25:
24:, but it has found applications in many other flow setups as well, including the scaling of the lower-branch instability (
2269:. Furthermore, the upper deck problem also provides the relation between the displacement and the pressure function as
705:
665:
4105:
Sobey, I. J. (2000). Introduction to interactive boundary layer theory (Vol. 3). Oxford Texts in
Applied and En.
1225:
748:
4130:
2591:
994:
29:
1444:), the solution should approach the asymptotic behaviour of the Goldstein's near wake, which is given by
2724:
1959:
1385:
934:
122:
3803:
depends on other locations as well. Because of this, these equations are sometimes referred to as the
2793:
3959:
3880:
2562:
1633:
169:
3701:
3371:
3841:
2246:
3807:
equations. The numerical solution of these equations were obtained by Jobe and
Burggraf in 1974.
1939:
1189:
3975:
3896:
1607:
539:
2950:
2921:
1414:
963:
499:
2759:
3756:
3727:
2595:
1906:
1873:
3950:(1953). "On boundary layers and upstream influence II. Supersonic flows without separation".
4073:
4044:
4013:
3967:
3927:
3888:
3853:
46:
45:
and E. A. MĂĽller in 1953. The triple-layer structure itself was independently discovered by
3786:
797:. The coordinate for horizontal interaction and for the three decks can then be defined by
641:
589:
3947:
3816:
659:
116:
42:
33:
21:
3871:
Smith, F. T. (1979). "On the non-parallel flow stability of the
Blasius boundary layer".
3963:
3952:
Proceedings of the Royal
Society of London. Series A. Mathematical and Physical Sciences
3884:
95:
3916:"On the stability of two-dimensional parallel flows. III. Stability in a viscous fluid"
3821:
621:
479:
459:
445:
75:
55:
4064:
Messiter, A. F. (1970). "Boundary-layer flow near the trailing edge of a flat plate".
4124:
3900:
3979:
586:
of the coordinate system be located at the trailing edge of the plate. Further let
4004:
4002:
Stewartson, K. (1969). "On the flow near the trailing edge of a flat plate II".
4017:
3857:
37:
2756:
will satisfy a boundary-layer type equations driven by the pressure gradient
380:
The interaction zone identified by
Lighthill in the streamwise direction is
3971:
3892:
4091:
92:
are the streamwise and transverse coordinate with respect to the wall and
3932:
3915:
4049:
4032:
4077:
745:. For shortness of notation, let us introduce the small parameter
4033:"Theory of laminar boundary layer separation in supersonic flow"
993:), the solution should approach the asymptotic behaviour of the
496:; then the only parameter in the problem is the Reynolds number
434:{\displaystyle {\text{Interaction zone}}:\quad x\sim Re^{-3/8}.}
2817:
generated by the middle deck. It is convenient to introduce
3498:
3992:
E. A. Müller (1953) Dissertation, University of Göttingen.
1656:. The Goldstein's inner wake solution is not needed here.
456:
Let the length scales be normalized with the plate length
2911:{\displaystyle {\hat {v}}=-\partial \Psi /\partial \chi }
2859:{\displaystyle {\hat {u}}=\partial \Psi /\partial \xi }
1375:{\displaystyle f_{B}(0)=f_{B}'(0)=f_{B}'(\infty )-1=0}
3842:"On the high Reynolds number theory of laminar flows"
3789:
3759:
3730:
3704:
3595:
3350:
2985:
2953:
2924:
2872:
2823:
2796:
2762:
2727:
2615:
2565:
2278:
2249:
2002:
1993:
In the upper deck, the solution is found to given by
1962:
1942:
1909:
1876:
1673:
1636:
1610:
1453:
1417:
1388:
1290:
1228:
1192:
1006:
966:
937:
806:
751:
708:
668:
644:
624:
592:
542:
502:
482:
462:
388:
221:
172:
125:
98:
78:
58:
1956:, although the pressure perturbation is only order
476:and the velocity scale by the free-stream velocity
3795:
3774:
3745:
3716:
3687:
3580:
3330:
2968:
2939:
2910:
2858:
2809:
2782:
2748:
2710:
2582:
2548:
2261:
2232:
1978:
1948:
1924:
1891:
1859:
1648:
1622:
1593:
1436:
1403:
1374:
1276:
1214:
1175:
985:
952:
920:
789:
737:
694:
650:
630:
610:
578:
528:
488:
468:
433:
366:
205:
158:
107:
84:
64:
3341:These equations are subjected to the conditions
1664:The solution in the middle deck is found to be
8:
2606:In the lower deck, the solution is given by
738:{\displaystyle v=-\partial \psi /\partial x}
618:be the non-dimensional velocity components,
695:{\displaystyle u=\partial \psi /\partial y}
452:Flow near the trailing edge of a flat plate
638:be the non-dimensional pressure field and
4048:
3931:
3788:
3758:
3729:
3703:
3675:
3671:
3657:
3650:
3649:
3648:
3627:
3613:
3612:
3611:
3594:
3566:
3536:
3522:
3521:
3503:
3482:
3481:
3461:
3460:
3435:
3434:
3428:
3411:
3410:
3390:
3389:
3375:
3374:
3366:
3365:
3364:
3363:
3349:
3312:
3292:
3291:
3285:
3278:
3264:
3263:
3246:
3245:
3223:
3222:
3208:
3199:
3191:
3177:
3163:
3131:
3130:
3124:
3113:
3112:
3087:
3086:
3080:
3069:
3068:
3029:
3028:
3022:
2997:
2996:
2990:
2986:
2984:
2955:
2954:
2952:
2926:
2925:
2923:
2897:
2874:
2873:
2871:
2845:
2825:
2824:
2822:
2801:
2795:
2769:
2761:
2726:
2684:
2640:
2625:
2616:
2614:
2566:
2564:
2532:
2531:
2514:
2513:
2491:
2490:
2481:
2472:
2464:
2450:
2436:
2407:
2395:
2394:
2377:
2376:
2354:
2353:
2339:
2330:
2322:
2308:
2294:
2277:
2248:
2206:
2175:
2174:
2156:
2132:
2131:
2122:
2106:
2071:
2063:
2049:
2036:
2023:
2012:
2003:
2001:
1967:
1961:
1941:
1908:
1875:
1833:
1789:
1766:
1753:
1715:
1705:
1694:
1683:
1674:
1672:
1635:
1609:
1569:
1556:
1536:
1532:
1522:
1503:
1493:
1482:
1471:
1462:
1454:
1452:
1428:
1416:
1387:
1342:
1317:
1295:
1289:
1259:
1249:
1233:
1227:
1222:is the Blasisus function which satisfies
1197:
1191:
1146:
1133:
1117:
1098:
1088:
1067:
1048:
1038:
1027:
1016:
1007:
1005:
977:
965:
936:
909:
900:
881:
872:
853:
844:
825:
816:
805:
777:
773:
761:
750:
724:
707:
681:
667:
643:
623:
591:
541:
518:
501:
481:
461:
418:
411:
389:
387:
347:
340:
316:
302:
295:
271:
257:
250:
226:
222:
220:
193:
189:
171:
146:
139:
124:
97:
77:
57:
3832:
1277:{\displaystyle f_{B}'''+f_{B}f_{B}''=0}
790:{\displaystyle \varepsilon =1/Re^{1/8}}
119:, the boundary layer thickness is then
213:. Then the thickness of each deck is
7:
4066:SIAM Journal on Applied Mathematics
166:. The boundary layer coordinate is
3642:
3605:
3515:
3448:
3431:
3305:
3282:
3203:
3195:
3170:
3164:
3144:
3127:
3100:
3083:
3042:
3025:
3010:
2993:
2902:
2894:
2891:
2850:
2842:
2839:
2728:
2646:
2573:
2567:
2476:
2468:
2443:
2437:
2334:
2326:
2301:
2295:
2075:
2067:
1795:
1398:
1354:
947:
729:
718:
686:
675:
14:
2749:{\displaystyle \Psi (\chi ,\xi )}
1979:{\displaystyle \varepsilon ^{2}.}
1404:{\displaystyle \chi \to +\infty }
953:{\displaystyle \chi \to -\infty }
159:{\displaystyle \delta =Re^{-1/2}}
2810:{\displaystyle \varepsilon ^{2}}
3632:
3626:
3508:
3502:
2790:and the slip-velocity of order
2673:
2583:{\displaystyle \mathrm {p.v.} }
2412:
2406:
2195:
1822:
1649:{\displaystyle \lambda =0.8789}
890:
862:
834:
397:
326:
281:
236:
206:{\displaystyle \eta =yRe^{1/2}}
3769:
3763:
3740:
3734:
3717:{\displaystyle \alpha =1.6552}
3654:
3636:
3617:
3599:
3554:
3548:
3533:
3527:
3512:
3466:
3440:
3395:
3380:
3297:
3269:
3251:
3234:
3228:
3219:
3136:
3118:
3092:
3074:
3034:
3002:
2960:
2931:
2879:
2830:
2743:
2731:
2699:
2693:
2661:
2649:
2537:
2519:
2502:
2496:
2487:
2427:
2421:
2400:
2382:
2365:
2359:
2350:
2288:
2282:
2221:
2215:
2180:
2137:
2094:
2088:
1919:
1913:
1886:
1880:
1848:
1842:
1810:
1798:
1746:
1740:
1421:
1392:
1357:
1351:
1332:
1326:
1307:
1301:
1209:
1203:
970:
941:
605:
593:
573:
561:
555:
543:
1:
2262:{\displaystyle \beta =1.2168}
4093:Verh. III. Int. Math. Kongr.
3724:. The displacement function
1949:{\displaystyle \varepsilon }
1215:{\displaystyle f_{B}(\eta )}
444:classical boundary layer of
1623:{\displaystyle \mu =1.1321}
579:{\displaystyle (x,y)=(0,0)}
4147:
3805:interactive boundary-layer
2969:{\displaystyle {\hat {v}}}
2940:{\displaystyle {\hat {u}}}
1437:{\displaystyle x\to 0^{+}}
986:{\displaystyle x\to 0^{-}}
529:{\displaystyle Re=UL/\nu }
4018:10.1112/S0025579300004678
2783:{\displaystyle dP/d\chi }
22:boundary layer separation
4031:Neiland, V. Ya. (1969).
3948:Lighthill, Michael James
3775:{\displaystyle P(\chi )}
3746:{\displaystyle A(\chi )}
1925:{\displaystyle P(\chi )}
1892:{\displaystyle A(\chi )}
3858:10.1093/imamat/28.3.207
658:be the non-dimensional
3972:10.1098/rspa.1953.0075
3893:10.1098/rspa.1979.0041
3797:
3776:
3747:
3718:
3689:
3582:
3332:
2970:
2941:
2912:
2860:
2811:
2784:
2750:
2712:
2592:Cauchy principal value
2584:
2550:
2263:
2234:
1980:
1950:
1932:is referred to as the
1926:
1899:is referred to as the
1893:
1861:
1650:
1624:
1595:
1438:
1405:
1376:
1278:
1216:
1177:
987:
954:
922:
791:
739:
696:
652:
632:
612:
580:
530:
490:
470:
435:
368:
207:
160:
109:
86:
66:
3840:Smith, F. T. (1982).
3798:
3796:{\displaystyle \chi }
3777:
3748:
3719:
3690:
3583:
3333:
2971:
2942:
2913:
2861:
2812:
2785:
2751:
2713:
2585:
2551:
2264:
2235:
1981:
1951:
1927:
1901:displacement function
1894:
1862:
1651:
1625:
1596:
1439:
1406:
1377:
1279:
1217:
1178:
988:
955:
923:
792:
740:
697:
653:
651:{\displaystyle \psi }
633:
613:
611:{\displaystyle (u,v)}
581:
531:
491:
471:
436:
369:
208:
161:
110:
87:
67:
3787:
3757:
3728:
3702:
3593:
3348:
2983:
2951:
2922:
2870:
2821:
2794:
2760:
2725:
2613:
2563:
2276:
2247:
2000:
1960:
1940:
1907:
1874:
1671:
1634:
1608:
1451:
1415:
1386:
1288:
1226:
1190:
1004:
997:, which is given by
964:
935:
804:
749:
706:
666:
642:
622:
590:
540:
500:
480:
460:
386:
219:
170:
123:
96:
76:
56:
3964:1953RSPSA.217..478L
3914:Lin, C. C. (1946).
3885:1979RSPSA.366...91S
3207:
2480:
2338:
2079:
1761:
1564:
1350:
1325:
1267:
1241:
1141:
4050:10.1007/BF01094681
3873:Proc. R. Soc. Lond
3793:
3772:
3743:
3714:
3685:
3578:
3497:
3328:
3326:
3187:
2966:
2937:
2908:
2856:
2807:
2780:
2746:
2708:
2580:
2546:
2460:
2318:
2259:
2230:
2059:
1976:
1946:
1922:
1889:
1857:
1749:
1646:
1620:
1591:
1552:
1434:
1401:
1372:
1338:
1313:
1274:
1255:
1229:
1212:
1173:
1129:
983:
950:
918:
787:
735:
692:
648:
628:
608:
576:
526:
486:
466:
431:
364:
362:
203:
156:
108:{\displaystyle Re}
105:
82:
62:
17:Triple-deck theory
3958:(1131): 478–507.
3933:10.1090/qam/14894
3920:Quart. Appl. Math
3846:IMA J. Appl. Math
3665:
3630:
3573:
3572:
3530:
3506:
3469:
3455:
3443:
3398:
3383:
3319:
3300:
3272:
3258:
3254:
3231:
3185:
3151:
3139:
3121:
3107:
3095:
3077:
3049:
3037:
3017:
3005:
2963:
2934:
2882:
2833:
2631:
2596:Hilbert transform
2540:
2526:
2522:
2499:
2458:
2410:
2403:
2389:
2385:
2362:
2316:
2183:
2164:
2147:
2140:
2057:
2041:
2031:
2018:
1934:pressure function
1776:
1775:
1725:
1724:
1699:
1689:
1579:
1578:
1550:
1513:
1512:
1487:
1477:
1457:
1156:
1155:
1127:
1126:
1108:
1107:
1077:
1076:
1058:
1057:
1032:
1022:
631:{\displaystyle p}
536:. Let the origin
489:{\displaystyle U}
469:{\displaystyle L}
392:
319:
274:
229:
85:{\displaystyle y}
65:{\displaystyle x}
4138:
4116:
4112:
4106:
4103:
4097:
4096:
4088:
4082:
4081:
4061:
4055:
4054:
4052:
4028:
4022:
4021:
3999:
3993:
3990:
3984:
3983:
3944:
3938:
3937:
3935:
3911:
3905:
3904:
3879:(1724): 91–109.
3868:
3862:
3861:
3837:
3802:
3800:
3799:
3794:
3781:
3779:
3778:
3773:
3752:
3750:
3749:
3744:
3723:
3721:
3720:
3715:
3694:
3692:
3691:
3686:
3684:
3683:
3679:
3666:
3658:
3631:
3628:
3587:
3585:
3584:
3579:
3574:
3571:
3570:
3558:
3557:
3537:
3532:
3531:
3523:
3507:
3504:
3501:
3500:
3471:
3470:
3462:
3456:
3454:
3446:
3445:
3444:
3436:
3429:
3400:
3399:
3391:
3385:
3384:
3376:
3337:
3335:
3334:
3329:
3327:
3320:
3318:
3317:
3316:
3303:
3302:
3301:
3293:
3290:
3289:
3279:
3274:
3273:
3265:
3259:
3257:
3256:
3255:
3247:
3237:
3233:
3232:
3224:
3218:
3209:
3206:
3198:
3186:
3178:
3176:
3152:
3150:
3142:
3141:
3140:
3132:
3125:
3123:
3122:
3114:
3108:
3106:
3098:
3097:
3096:
3088:
3081:
3079:
3078:
3070:
3050:
3048:
3040:
3039:
3038:
3030:
3023:
3018:
3016:
3008:
3007:
3006:
2998:
2991:
2975:
2973:
2972:
2967:
2965:
2964:
2956:
2946:
2944:
2943:
2938:
2936:
2935:
2927:
2917:
2915:
2914:
2909:
2901:
2884:
2883:
2875:
2865:
2863:
2862:
2857:
2849:
2835:
2834:
2826:
2816:
2814:
2813:
2808:
2806:
2805:
2789:
2787:
2786:
2781:
2773:
2755:
2753:
2752:
2747:
2717:
2715:
2714:
2709:
2689:
2688:
2645:
2644:
2632:
2630:
2629:
2617:
2589:
2587:
2586:
2581:
2579:
2555:
2553:
2552:
2547:
2542:
2541:
2533:
2527:
2525:
2524:
2523:
2515:
2505:
2501:
2500:
2492:
2482:
2479:
2471:
2459:
2451:
2449:
2420:
2411:
2408:
2405:
2404:
2396:
2390:
2388:
2387:
2386:
2378:
2368:
2364:
2363:
2355:
2349:
2340:
2337:
2329:
2317:
2309:
2307:
2268:
2266:
2265:
2260:
2239:
2237:
2236:
2231:
2211:
2210:
2185:
2184:
2176:
2170:
2166:
2165:
2157:
2152:
2148:
2143:
2142:
2141:
2133:
2123:
2114:
2113:
2087:
2078:
2070:
2058:
2050:
2042:
2037:
2032:
2024:
2019:
2017:
2016:
2004:
1985:
1983:
1982:
1977:
1972:
1971:
1955:
1953:
1952:
1947:
1931:
1929:
1928:
1923:
1898:
1896:
1895:
1890:
1866:
1864:
1863:
1858:
1838:
1837:
1794:
1793:
1781:
1777:
1771:
1767:
1757:
1730:
1726:
1720:
1716:
1710:
1709:
1700:
1695:
1690:
1688:
1687:
1675:
1655:
1653:
1652:
1647:
1629:
1627:
1626:
1621:
1600:
1598:
1597:
1592:
1584:
1580:
1574:
1570:
1560:
1551:
1546:
1545:
1544:
1540:
1523:
1518:
1514:
1508:
1504:
1498:
1497:
1488:
1483:
1478:
1476:
1475:
1463:
1458:
1455:
1443:
1441:
1440:
1435:
1433:
1432:
1410:
1408:
1407:
1402:
1381:
1379:
1378:
1373:
1346:
1321:
1300:
1299:
1283:
1281:
1280:
1275:
1263:
1254:
1253:
1237:
1221:
1219:
1218:
1213:
1202:
1201:
1182:
1180:
1179:
1174:
1166:
1162:
1161:
1157:
1151:
1147:
1137:
1128:
1122:
1118:
1113:
1109:
1103:
1099:
1093:
1092:
1078:
1072:
1068:
1063:
1059:
1053:
1049:
1043:
1042:
1033:
1028:
1023:
1021:
1020:
1008:
995:Blasius solution
992:
990:
989:
984:
982:
981:
959:
957:
956:
951:
927:
925:
924:
919:
914:
913:
904:
886:
885:
876:
858:
857:
848:
830:
829:
820:
796:
794:
793:
788:
786:
785:
781:
765:
744:
742:
741:
736:
728:
701:
699:
698:
693:
685:
657:
655:
654:
649:
637:
635:
634:
629:
617:
615:
614:
609:
585:
583:
582:
577:
535:
533:
532:
527:
522:
495:
493:
492:
487:
475:
473:
472:
467:
440:
438:
437:
432:
427:
426:
422:
393:
391:Interaction zone
390:
373:
371:
370:
365:
363:
356:
355:
351:
320:
317:
311:
310:
306:
275:
272:
266:
265:
261:
230:
227:
212:
210:
209:
204:
202:
201:
197:
165:
163:
162:
157:
155:
154:
150:
114:
112:
111:
106:
91:
89:
88:
83:
71:
69:
68:
63:
47:Keith Stewartson
4146:
4145:
4141:
4140:
4139:
4137:
4136:
4135:
4121:
4120:
4119:
4113:
4109:
4104:
4100:
4090:
4089:
4085:
4078:10.1137/0118020
4063:
4062:
4058:
4030:
4029:
4025:
4001:
4000:
3996:
3991:
3987:
3946:
3945:
3941:
3913:
3912:
3908:
3870:
3869:
3865:
3839:
3838:
3834:
3830:
3817:Flow separation
3813:
3785:
3784:
3755:
3754:
3726:
3725:
3700:
3699:
3667:
3591:
3590:
3562:
3538:
3496:
3495:
3447:
3430:
3425:
3424:
3367:
3346:
3345:
3325:
3324:
3308:
3304:
3281:
3280:
3238:
3211:
3210:
3153:
3143:
3126:
3099:
3082:
3065:
3064:
3051:
3041:
3024:
3009:
2992:
2981:
2980:
2949:
2948:
2920:
2919:
2868:
2867:
2819:
2818:
2797:
2792:
2791:
2758:
2757:
2723:
2722:
2680:
2636:
2621:
2611:
2610:
2604:
2561:
2560:
2506:
2483:
2413:
2369:
2342:
2341:
2274:
2273:
2245:
2244:
2202:
2124:
2118:
2102:
2101:
2097:
2080:
2008:
1998:
1997:
1991:
1963:
1958:
1957:
1938:
1937:
1905:
1904:
1872:
1871:
1829:
1785:
1762:
1711:
1701:
1679:
1669:
1668:
1662:
1632:
1631:
1606:
1605:
1565:
1528:
1524:
1499:
1489:
1467:
1449:
1448:
1424:
1413:
1412:
1384:
1383:
1291:
1286:
1285:
1245:
1224:
1223:
1193:
1188:
1187:
1142:
1094:
1084:
1083:
1079:
1044:
1034:
1012:
1002:
1001:
973:
962:
961:
933:
932:
905:
877:
849:
821:
802:
801:
769:
747:
746:
704:
703:
664:
663:
660:stream function
640:
639:
620:
619:
588:
587:
538:
537:
498:
497:
478:
477:
458:
457:
454:
407:
384:
383:
361:
360:
336:
324:
313:
312:
291:
279:
268:
267:
246:
234:
217:
216:
185:
168:
167:
135:
121:
120:
117:Reynolds number
94:
93:
74:
73:
54:
53:
43:James Lighthill
34:James Lighthill
12:
11:
5:
4144:
4142:
4134:
4133:
4131:Fluid dynamics
4123:
4122:
4118:
4117:
4107:
4098:
4083:
4072:(1): 241–257.
4056:
4037:Fluid Dynamics
4023:
4012:(1): 106–121.
3994:
3985:
3939:
3926:(4): 277–301.
3906:
3863:
3852:(3): 207–281.
3831:
3829:
3826:
3825:
3824:
3822:Boundary layer
3819:
3812:
3809:
3792:
3771:
3768:
3765:
3762:
3753:and therefore
3742:
3739:
3736:
3733:
3713:
3710:
3707:
3696:
3695:
3682:
3678:
3674:
3670:
3664:
3661:
3656:
3653:
3647:
3644:
3641:
3638:
3635:
3625:
3622:
3619:
3616:
3610:
3607:
3604:
3601:
3598:
3588:
3577:
3569:
3565:
3561:
3556:
3553:
3550:
3547:
3544:
3541:
3535:
3529:
3526:
3520:
3517:
3514:
3511:
3499:
3494:
3491:
3488:
3485:
3480:
3477:
3474:
3468:
3465:
3459:
3453:
3450:
3442:
3439:
3433:
3427:
3426:
3423:
3420:
3417:
3414:
3409:
3406:
3403:
3397:
3394:
3388:
3382:
3379:
3373:
3372:
3370:
3362:
3359:
3356:
3353:
3339:
3338:
3323:
3315:
3311:
3307:
3299:
3296:
3288:
3284:
3277:
3271:
3268:
3262:
3253:
3250:
3244:
3241:
3236:
3230:
3227:
3221:
3217:
3214:
3205:
3202:
3197:
3194:
3190:
3184:
3181:
3175:
3172:
3169:
3166:
3162:
3159:
3156:
3154:
3149:
3146:
3138:
3135:
3129:
3120:
3117:
3111:
3105:
3102:
3094:
3091:
3085:
3076:
3073:
3067:
3066:
3063:
3060:
3057:
3054:
3052:
3047:
3044:
3036:
3033:
3027:
3021:
3015:
3012:
3004:
3001:
2995:
2989:
2988:
2962:
2959:
2933:
2930:
2907:
2904:
2900:
2896:
2893:
2890:
2887:
2881:
2878:
2855:
2852:
2848:
2844:
2841:
2838:
2832:
2829:
2804:
2800:
2779:
2776:
2772:
2768:
2765:
2745:
2742:
2739:
2736:
2733:
2730:
2719:
2718:
2707:
2704:
2701:
2698:
2695:
2692:
2687:
2683:
2679:
2676:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2643:
2639:
2635:
2628:
2624:
2620:
2603:
2600:
2578:
2575:
2572:
2569:
2557:
2556:
2545:
2539:
2536:
2530:
2521:
2518:
2512:
2509:
2504:
2498:
2495:
2489:
2486:
2478:
2475:
2470:
2467:
2463:
2457:
2454:
2448:
2445:
2442:
2439:
2435:
2432:
2429:
2426:
2423:
2419:
2416:
2402:
2399:
2393:
2384:
2381:
2375:
2372:
2367:
2361:
2358:
2352:
2348:
2345:
2336:
2333:
2328:
2325:
2321:
2315:
2312:
2306:
2303:
2300:
2297:
2293:
2290:
2287:
2284:
2281:
2258:
2255:
2252:
2241:
2240:
2229:
2226:
2223:
2220:
2217:
2214:
2209:
2205:
2201:
2198:
2194:
2191:
2188:
2182:
2179:
2173:
2169:
2163:
2160:
2155:
2151:
2146:
2139:
2136:
2130:
2127:
2121:
2117:
2112:
2109:
2105:
2100:
2096:
2093:
2090:
2086:
2083:
2077:
2074:
2069:
2066:
2062:
2056:
2053:
2048:
2045:
2040:
2035:
2030:
2027:
2022:
2015:
2011:
2007:
1990:
1987:
1975:
1970:
1966:
1945:
1921:
1918:
1915:
1912:
1888:
1885:
1882:
1879:
1868:
1867:
1856:
1853:
1850:
1847:
1844:
1841:
1836:
1832:
1828:
1825:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1792:
1788:
1784:
1780:
1774:
1770:
1765:
1760:
1756:
1752:
1748:
1745:
1742:
1739:
1736:
1733:
1729:
1723:
1719:
1714:
1708:
1704:
1698:
1693:
1686:
1682:
1678:
1661:
1658:
1645:
1642:
1639:
1619:
1616:
1613:
1602:
1601:
1590:
1587:
1583:
1577:
1573:
1568:
1563:
1559:
1555:
1549:
1543:
1539:
1535:
1531:
1527:
1521:
1517:
1511:
1507:
1502:
1496:
1492:
1486:
1481:
1474:
1470:
1466:
1461:
1431:
1427:
1423:
1420:
1400:
1397:
1394:
1391:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1349:
1345:
1341:
1337:
1334:
1331:
1328:
1324:
1320:
1316:
1312:
1309:
1306:
1303:
1298:
1294:
1273:
1270:
1266:
1262:
1258:
1252:
1248:
1244:
1240:
1236:
1232:
1211:
1208:
1205:
1200:
1196:
1184:
1183:
1172:
1169:
1165:
1160:
1154:
1150:
1145:
1140:
1136:
1132:
1125:
1121:
1116:
1112:
1106:
1102:
1097:
1091:
1087:
1082:
1075:
1071:
1066:
1062:
1056:
1052:
1047:
1041:
1037:
1031:
1026:
1019:
1015:
1011:
980:
976:
972:
969:
949:
946:
943:
940:
929:
928:
917:
912:
908:
903:
899:
896:
893:
889:
884:
880:
875:
871:
868:
865:
861:
856:
852:
847:
843:
840:
837:
833:
828:
824:
819:
815:
812:
809:
784:
780:
776:
772:
768:
764:
760:
757:
754:
734:
731:
727:
723:
720:
717:
714:
711:
691:
688:
684:
680:
677:
674:
671:
647:
627:
607:
604:
601:
598:
595:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
525:
521:
517:
514:
511:
508:
505:
485:
465:
453:
450:
430:
425:
421:
417:
414:
410:
406:
403:
400:
396:
377:disturbances.
359:
354:
350:
346:
343:
339:
335:
332:
329:
325:
323:
315:
314:
309:
305:
301:
298:
294:
290:
287:
284:
280:
278:
270:
269:
264:
260:
256:
253:
249:
245:
242:
239:
235:
233:
225:
224:
200:
196:
192:
188:
184:
181:
178:
175:
153:
149:
145:
142:
138:
134:
131:
128:
104:
101:
81:
61:
13:
10:
9:
6:
4:
3:
2:
4143:
4132:
4129:
4128:
4126:
4111:
4108:
4102:
4099:
4094:
4087:
4084:
4079:
4075:
4071:
4067:
4060:
4057:
4051:
4046:
4042:
4038:
4034:
4027:
4024:
4019:
4015:
4011:
4007:
4006:
3998:
3995:
3989:
3986:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3953:
3949:
3943:
3940:
3934:
3929:
3925:
3921:
3917:
3910:
3907:
3902:
3898:
3894:
3890:
3886:
3882:
3878:
3874:
3867:
3864:
3859:
3855:
3851:
3847:
3843:
3836:
3833:
3827:
3823:
3820:
3818:
3815:
3814:
3810:
3808:
3806:
3790:
3766:
3760:
3737:
3731:
3711:
3708:
3705:
3680:
3676:
3672:
3668:
3662:
3659:
3651:
3645:
3639:
3633:
3623:
3620:
3614:
3608:
3602:
3596:
3589:
3575:
3567:
3563:
3559:
3551:
3545:
3542:
3539:
3524:
3518:
3509:
3492:
3489:
3486:
3483:
3478:
3475:
3472:
3463:
3457:
3451:
3437:
3421:
3418:
3415:
3412:
3407:
3404:
3401:
3392:
3386:
3377:
3368:
3360:
3357:
3354:
3351:
3344:
3343:
3342:
3321:
3313:
3309:
3294:
3286:
3275:
3266:
3260:
3248:
3242:
3239:
3225:
3215:
3212:
3200:
3192:
3188:
3182:
3179:
3173:
3167:
3160:
3157:
3155:
3147:
3133:
3115:
3109:
3103:
3089:
3071:
3061:
3058:
3055:
3053:
3045:
3031:
3019:
3013:
2999:
2979:
2978:
2977:
2976:must satisfy
2957:
2928:
2905:
2898:
2888:
2885:
2876:
2853:
2846:
2836:
2827:
2802:
2798:
2777:
2774:
2770:
2766:
2763:
2740:
2737:
2734:
2705:
2702:
2696:
2690:
2685:
2681:
2677:
2674:
2670:
2667:
2664:
2658:
2655:
2652:
2641:
2637:
2633:
2626:
2622:
2618:
2609:
2608:
2607:
2601:
2599:
2597:
2593:
2576:
2570:
2543:
2534:
2528:
2516:
2510:
2507:
2493:
2484:
2473:
2465:
2461:
2455:
2452:
2446:
2440:
2433:
2430:
2424:
2417:
2414:
2397:
2391:
2379:
2373:
2370:
2356:
2346:
2343:
2331:
2323:
2319:
2313:
2310:
2304:
2298:
2291:
2285:
2279:
2272:
2271:
2270:
2256:
2253:
2250:
2227:
2224:
2218:
2212:
2207:
2203:
2199:
2196:
2192:
2189:
2186:
2177:
2171:
2167:
2161:
2158:
2153:
2149:
2144:
2134:
2128:
2125:
2119:
2115:
2110:
2107:
2103:
2098:
2091:
2084:
2081:
2072:
2064:
2060:
2054:
2051:
2046:
2043:
2038:
2033:
2028:
2025:
2020:
2013:
2009:
2005:
1996:
1995:
1994:
1988:
1986:
1973:
1968:
1964:
1943:
1935:
1916:
1910:
1902:
1883:
1877:
1854:
1851:
1845:
1839:
1834:
1830:
1826:
1823:
1819:
1816:
1813:
1807:
1804:
1801:
1790:
1786:
1782:
1778:
1772:
1768:
1763:
1758:
1754:
1750:
1743:
1737:
1734:
1731:
1727:
1721:
1717:
1712:
1706:
1702:
1696:
1691:
1684:
1680:
1676:
1667:
1666:
1665:
1659:
1657:
1643:
1640:
1637:
1617:
1614:
1611:
1588:
1585:
1581:
1575:
1571:
1566:
1561:
1557:
1553:
1547:
1541:
1537:
1533:
1529:
1525:
1519:
1515:
1509:
1505:
1500:
1494:
1490:
1484:
1479:
1472:
1468:
1464:
1459:
1447:
1446:
1445:
1429:
1425:
1418:
1395:
1389:
1369:
1366:
1363:
1360:
1347:
1343:
1339:
1335:
1329:
1322:
1318:
1314:
1310:
1304:
1296:
1292:
1284:subjected to
1271:
1268:
1264:
1260:
1256:
1250:
1246:
1242:
1238:
1234:
1230:
1206:
1198:
1194:
1170:
1167:
1163:
1158:
1152:
1148:
1143:
1138:
1134:
1130:
1123:
1119:
1114:
1110:
1104:
1100:
1095:
1089:
1085:
1080:
1073:
1069:
1064:
1060:
1054:
1050:
1045:
1039:
1035:
1029:
1024:
1017:
1013:
1009:
1000:
999:
998:
996:
978:
974:
967:
944:
938:
915:
910:
906:
901:
897:
894:
891:
887:
882:
878:
873:
869:
866:
863:
859:
854:
850:
845:
841:
838:
835:
831:
826:
822:
817:
813:
810:
807:
800:
799:
798:
782:
778:
774:
770:
766:
762:
758:
755:
752:
732:
725:
721:
715:
712:
709:
689:
682:
678:
672:
669:
661:
645:
625:
602:
599:
596:
570:
567:
564:
558:
552:
549:
546:
523:
519:
515:
512:
509:
506:
503:
483:
463:
451:
449:
447:
441:
428:
423:
419:
415:
412:
408:
404:
401:
398:
394:
381:
378:
374:
357:
352:
348:
344:
341:
337:
333:
330:
327:
321:
307:
303:
299:
296:
292:
288:
285:
282:
276:
262:
258:
254:
251:
247:
243:
240:
237:
231:
214:
198:
194:
190:
186:
182:
179:
176:
173:
151:
147:
143:
140:
136:
132:
129:
126:
118:
102:
99:
79:
59:
50:
48:
44:
39:
35:
31:
27:
23:
18:
4110:
4101:
4092:
4086:
4069:
4065:
4059:
4043:(4): 33–35.
4040:
4036:
4026:
4009:
4003:
3997:
3988:
3955:
3951:
3942:
3923:
3919:
3909:
3876:
3872:
3866:
3849:
3845:
3835:
3804:
3697:
3340:
2720:
2605:
2558:
2242:
1992:
1933:
1900:
1869:
1663:
1603:
1185:
930:
455:
442:
382:
379:
375:
215:
51:
30:Blasius flow
16:
15:
4005:Mathematika
2590:stands for
1660:Middle deck
273:Middle deck
4095:: 484–491.
3828:References
2602:Lower deck
1989:Upper deck
1456:Outer wake
662:such that
318:Upper deck
228:Lower deck
38:Lev Landau
3901:112228524
3791:χ
3767:χ
3738:χ
3706:α
3669:χ
3663:λ
3660:μ
3655:→
3643:∞
3637:→
3634:χ
3618:→
3606:∞
3603:−
3600:→
3597:χ
3564:α
3552:χ
3540:ξ
3534:→
3528:^
3516:∞
3513:→
3510:ξ
3484:χ
3467:^
3452:ξ
3449:∂
3441:^
3432:∂
3416:≤
3413:χ
3396:^
3381:^
3352:ξ
3310:ξ
3306:∂
3298:^
3283:∂
3270:^
3267:χ
3252:^
3249:χ
3243:−
3240:χ
3229:^
3226:χ
3204:∞
3196:∞
3193:−
3189:∫
3183:π
3161:−
3148:ξ
3145:∂
3137:^
3128:∂
3119:^
3104:χ
3101:∂
3093:^
3084:∂
3075:^
3046:ξ
3043:∂
3035:^
3026:∂
3014:χ
3011:∂
3003:^
2994:∂
2961:^
2932:^
2906:χ
2903:∂
2895:Ψ
2892:∂
2889:−
2880:^
2854:ξ
2851:∂
2843:Ψ
2840:∂
2831:^
2799:ε
2778:χ
2741:ξ
2735:χ
2729:Ψ
2706:⋯
2697:χ
2682:ε
2668:⋯
2659:ξ
2653:χ
2647:Ψ
2638:ε
2623:ε
2619:ψ
2559:in which
2538:^
2535:χ
2520:^
2517:χ
2511:−
2508:χ
2497:^
2494:χ
2477:∞
2469:∞
2466:−
2462:∫
2456:π
2434:−
2425:χ
2401:^
2398:χ
2383:^
2380:χ
2374:−
2371:χ
2360:^
2357:χ
2335:∞
2327:∞
2324:−
2320:∫
2314:π
2286:χ
2251:β
2228:⋯
2219:χ
2204:ε
2190:⋯
2181:^
2178:χ
2159:π
2145:ζ
2138:^
2135:χ
2129:−
2126:χ
2116:
2108:−
2092:χ
2076:∞
2068:∞
2065:−
2061:∫
2055:π
2052:ε
2047:−
2044:β
2034:−
2029:ε
2026:ζ
2010:ε
2006:ψ
1965:ε
1944:ε
1917:χ
1884:χ
1855:⋯
1846:χ
1831:ε
1817:⋯
1808:η
1802:χ
1796:Φ
1787:ε
1769:η
1744:χ
1735:ε
1718:η
1681:ε
1677:ψ
1638:λ
1612:μ
1589:⋯
1572:η
1548:λ
1526:μ
1506:η
1469:ε
1465:ψ
1422:→
1399:∞
1393:→
1390:χ
1361:−
1355:∞
1207:η
1171:⋯
1149:η
1120:η
1115:−
1101:η
1051:η
1014:ε
1010:ψ
979:−
971:→
948:∞
945:−
942:→
939:χ
907:ε
892:ζ
879:ε
864:η
851:ε
836:ξ
823:ε
808:χ
753:ε
730:∂
722:ψ
719:∂
716:−
687:∂
679:ψ
676:∂
646:ψ
524:ν
413:−
402:∼
342:−
331:∼
297:−
286:∼
252:−
241:∼
174:η
141:−
127:δ
28:) of the
4125:Category
3980:95497146
3811:See also
3216:″
2918:, where
2418:′
2347:′
2085:′
1759:′
1562:′
1348:′
1323:′
1265:″
1239:‴
1139:′
52:Suppose
4115:91-111.
3960:Bibcode
3881:Bibcode
446:Prandtl
115:be the
32:, etc.
3978:
3899:
3712:1.6552
3698:where
2721:where
2598:pair.
2257:1.2168
2243:where
1870:where
1644:0.8789
1618:1.1321
1604:where
1186:where
3976:S2CID
3897:S2CID
1382:. As
3487:>
2947:and
2866:and
1903:and
1630:and
1411:(or
960:(or
702:and
72:and
4074:doi
4045:doi
4014:doi
3968:doi
3956:217
3928:doi
3889:doi
3877:366
3854:doi
3629:and
3505:and
2409:and
2104:tan
931:As
26:T-S
4127::
4070:18
4068:.
4039:.
4035:.
4010:16
4008:.
3974:.
3966:.
3954:.
3922:.
3918:.
3895:.
3887:.
3875:.
3850:28
3848:.
3844:.
448:.
36:,
4080:.
4076::
4053:.
4047::
4041:4
4020:.
4016::
3982:.
3970::
3962::
3936:.
3930::
3924:3
3903:.
3891::
3883::
3860:.
3856::
3770:)
3764:(
3761:P
3741:)
3735:(
3732:A
3709:=
3681:3
3677:/
3673:1
3652:A
3646::
3640:+
3624:,
3621:0
3615:A
3609::
3576:,
3568:3
3560:2
3555:)
3549:(
3546:A
3543:+
3525:u
3519::
3493:,
3490:0
3479:,
3476:0
3473:=
3464:v
3458:=
3438:u
3422:,
3419:0
3408:,
3405:0
3402:=
3393:v
3387:=
3378:u
3369:{
3361::
3358:0
3355:=
3322:.
3314:2
3295:u
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3276:+
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3235:)
3220:(
3213:A
3201:+
3180:1
3174:.
3171:v
3168:.
3165:p
3158:=
3134:u
3116:v
3110:+
3090:u
3072:u
3062:,
3059:0
3056:=
3032:v
3020:+
3000:u
2958:v
2929:u
2899:/
2886:=
2877:v
2847:/
2837:=
2828:u
2803:2
2775:d
2771:/
2767:P
2764:d
2744:)
2738:,
2732:(
2703:+
2700:)
2694:(
2691:P
2686:2
2678:=
2675:p
2671:,
2665:+
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2650:(
2642:2
2634:=
2627:4
2577:.
2574:v
2571:.
2568:p
2544:.
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2488:(
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2438:p
2431:=
2428:)
2422:(
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2351:(
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2332:+
2311:1
2305:.
2302:v
2299:.
2296:p
2292:=
2289:)
2283:(
2280:P
2254:=
2225:+
2222:)
2216:(
2213:P
2208:2
2200:=
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2187:+
2172:d
2168:]
2162:2
2154:+
2150:)
2120:(
2111:1
2099:[
2095:)
2089:(
2082:A
2073:+
2039:2
2021:=
2014:4
1974:.
1969:2
1920:)
1914:(
1911:P
1887:)
1881:(
1878:A
1852:+
1849:)
1843:(
1840:P
1835:2
1827:=
1824:p
1820:,
1814:+
1811:)
1805:,
1799:(
1791:2
1783:+
1779:)
1773:2
1764:(
1755:B
1751:f
1747:)
1741:(
1738:A
1732:+
1728:)
1722:2
1713:(
1707:B
1703:f
1697:2
1692:=
1685:4
1641:=
1615:=
1586:+
1582:)
1576:2
1567:(
1558:B
1554:f
1542:3
1538:/
1534:1
1530:x
1520:+
1516:)
1510:2
1501:(
1495:B
1491:f
1485:2
1480:=
1473:4
1460::
1430:+
1426:0
1419:x
1396:+
1370:0
1367:=
1364:1
1358:)
1352:(
1344:B
1340:f
1336:=
1333:)
1330:0
1327:(
1319:B
1315:f
1311:=
1308:)
1305:0
1302:(
1297:B
1293:f
1272:0
1269:=
1261:B
1257:f
1251:B
1247:f
1243:+
1235:B
1231:f
1210:)
1204:(
1199:B
1195:f
1168:+
1164:]
1159:)
1153:2
1144:(
1135:B
1131:f
1124:2
1111:)
1105:2
1096:(
1090:B
1086:f
1081:[
1074:2
1070:x
1065:+
1061:)
1055:2
1046:(
1040:B
1036:f
1030:2
1025:=
1018:4
975:0
968:x
916:.
911:3
902:/
898:y
895:=
888:,
883:4
874:/
870:y
867:=
860:,
855:5
846:/
842:y
839:=
832:,
827:3
818:/
814:x
811:=
783:8
779:/
775:1
771:e
767:R
763:/
759:1
756:=
733:x
726:/
713:=
710:v
690:y
683:/
673:=
670:u
626:p
606:)
603:v
600:,
597:u
594:(
574:)
571:0
568:,
565:0
562:(
559:=
556:)
553:y
550:,
547:x
544:(
520:/
516:L
513:U
510:=
507:e
504:R
484:U
464:L
429:.
424:8
420:/
416:3
409:e
405:R
399:x
395::
358:.
353:8
349:/
345:3
338:e
334:R
328:y
322::
308:8
304:/
300:4
293:e
289:R
283:y
277::
263:8
259:/
255:5
248:e
244:R
238:y
232::
199:2
195:/
191:1
187:e
183:R
180:y
177:=
152:2
148:/
144:1
137:e
133:R
130:=
103:e
100:R
80:y
60:x
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