1620:
1341:
1615:{\displaystyle \overbrace {\underbrace {\partial {\bf {v}} \over {\partial t}} _{\begin{smallmatrix}{\text{Local}}\\{\text{Change}}\end{smallmatrix}}+\underbrace {{\bf {v}}\cdot \nabla {\bf {v}}} _{\text{Advection}}} ^{\text{Inertial Acceleration}}=-\underbrace {{1 \over {\rho }}\nabla p} _{\begin{smallmatrix}{\text{Pressure}}\\{\text{Gradient}}\end{smallmatrix}}+\underbrace {\nu \Delta {\bf {v}}} _{\text{Diffusion}}-\underbrace {\nabla \Phi } _{\text{Gravity}}+\underbrace {\bf {F}} _{\begin{smallmatrix}{\text{External}}\\{\text{Forces}}\end{smallmatrix}}}
55:
2267:
3321:
1906:
2071:
3189:
3105:
1730:
206:
has a minor contribution, but does not play a significant enough role in most circumstances to be a governing factor. Due to the presence of a free surface, gravity is generally the most significant driver of open-channel flow; therefore, the ratio of inertial to gravity forces is the most important
180:
The discharge of a steady flow is non-uniform along a channel. This happens when water enters and/or leaves the channel along the course of flow. An example of flow entering a channel would be a road side gutter. An example of flow leaving a channel would be an irrigation channel. This flow can be
2918:
1074:
2595:
2951:
3646:
2735:
1255:
2262:{\displaystyle {\partial u \over {\partial t}}+u{\partial u \over {\partial x}}+g{\partial \zeta \over {\partial x}}=F_{x}\implies {\partial u \over {\partial t}}+u{\partial u \over {\partial x}}+g{\partial \eta \over {\partial x}}-gS=F_{x}}
2794:
3316:{\displaystyle E=\underbrace {{1 \over {2}}\rho \|{\bf {v}}\|^{2}} _{\begin{smallmatrix}{\text{Kinetic}}\\{\text{Energy}}\end{smallmatrix}}+\underbrace {\rho \Phi } _{\begin{smallmatrix}{\text{Potential}}\\{\text{Energy}}\end{smallmatrix}}}
4048:
784:
933:
1901:{\displaystyle {\begin{aligned}{\partial u \over {\partial t}}+u{\partial u \over {\partial x}}&=-{1 \over {\rho }}{\partial p \over {\partial x}}+F_{x}\\-{1 \over {\rho }}{\partial p \over {\partial z}}-g&=0\end{aligned}}}
548:
1324:
938:
3415:
446:
70:
Open-channel flow can be classified and described in various ways based on the change in flow depth with respect to time and space. The fundamental types of flow dealt with in open-channel hydraulics are:
3160:
112:
The depth of flow is the same at every section of the channel. Uniform flow can be steady or unsteady, depending on whether or not the depth changes with time, (although unsteady uniform flow is rare).
47:. These two types of flow are similar in many ways but differ in one important respect: open-channel flow has a free surface, whereas pipe flow does not, resulting in flow dominated by gravity but not
2477:
2374:
2649:
3547:
1735:
122:
The depth of flow changes along the length of the channel. Varied flow technically may be either steady or unsteady. Varied flow can be further classified as either rapidly or gradually-varied:
251:
3542:
2644:
2019:
1164:
664:
3100:{\displaystyle {\partial \over {\partial t}}\left({1 \over {2}}\|{\bf {v}}\|^{2}\right)+{\bf {v}}\cdot \nabla \left({1 \over {2}}\|{\bf {v}}\|^{2}+{p \over {\rho }}+\Phi \right)=0}
2469:
692:
789:
1159:
601:
3678:
1941:
1113:
2317:
3511:
2785:
485:
2946:
383:
1721:
2755:
1688:
2039:
3713:
3447:
3341:
2394:
2066:
568:
3346:
1664:
3534:
3467:
3180:
2418:
1640:
687:
628:
315:
291:
271:
181:
described using the continuity equation for continuous unsteady flow requires the consideration of the time effect and includes a time element as a variable.
2913:{\displaystyle {\partial {\bf {v}} \over {\partial t}}+\omega \times {\bf {v}}=-\nabla \left({1 \over {2}}\|{\bf {v}}\|^{2}+{p \over {\rho }}+\Phi \right)}
166:
of the channel under consideration. This is often the case with a steady flow. This flow is considered continuous and therefore can be described using the
1263:
355:
for quantities that are useful in open-channel flow: mass, momentum, and energy. The governing equations result from considering the dynamics of the
2322:
388:
346:
3848:
337:. However, it is generally acceptable to assume that the Reynolds number is sufficiently large so that viscous forces may be neglected.
214:
3907:
1069:{\displaystyle {d \over {dt}}\int _{x}\left(\int _{A}dA\right)dx=-\int _{x}{\partial \over {\partial x}}\left(\int _{A}u\;dA\right)dx}
130:
The depth changes abruptly over a comparatively short distance. Rapidly varied flow is known as a local phenomenon. Examples are the
2590:{\displaystyle {\partial u \over {\partial t}}+u{\partial u \over {\partial x}}+g{\partial \eta \over {\partial x}}+g(S_{f}-S)=0}
85:
The depth of flow does not change over time, or if it can be assumed to be constant during the time interval under consideration.
3113:
3991:
1335:
4096:
3641:{\displaystyle {\begin{aligned}h&=e+{p \over {\rho g}}\\&={u^{2} \over {2g}}+z+{p \over {\gamma }}\end{aligned}}}
4030:
3890:
4282:
4017:
4000:
3970:
3917:
2612:
1946:
3789:
334:
3877:
689:
can change with both time and space in the channel. If we start from the integral form of the continuity equation:
3417:
Assuming that the energy density is time-independent and the flow is one-dimensional leads to the simplification:
4328:
4287:
633:
2730:{\displaystyle {\bf {v}}\cdot \nabla {\bf {v}}=\omega \times {\bf {v}}+{1 \over {2}}\nabla \|{\bf {v}}\|^{2}}
3752:
3420:
4165:
4155:
3833:
2427:
786:
it is possible to decompose the volume integral into a cross-section and length, which leads to the form:
449:
1250:{\displaystyle \int _{x}{\partial A \over {\partial t}}\;dx=-\int _{x}{\partial Q \over {\partial x}}dx}
4313:
4292:
318:
1121:
4170:
4160:
577:
3654:
1914:
4150:
4089:
3470:
1079:
667:
2272:
4318:
4241:
1724:
58:
4068:
3479:
2764:
4261:
4226:
2927:
364:
1697:
4206:
4185:
4134:
3108:
2791:. This leads to a form of the momentum equation, ignoring the external forces term, given by:
2740:
1673:
294:
2024:
779:{\displaystyle {d \over {dt}}\int _{V}\rho \;dV=-\int _{V}\nabla \cdot (\rho {\bf {v}})\;dV}
3853:
3691:
3326:
2379:
2044:
1909:
1116:
553:
455:
To simplify the final form of the equations, it is acceptable to make several assumptions:
3957:
2319:. To account for shear stress along the channel banks, we may define the force term to be:
1649:
928:{\displaystyle {d \over {dt}}\int _{x}\left(\int _{A}\rho \;dA\right)dx=-\int _{x}\leftdx}
8:
4323:
4201:
4082:
2471:, a way of quantifying friction losses, leads to the final form of the momentum equation:
1667:
479:
460:
452:, these components correspond to the flow velocity in the x, y, and z axes respectively.
167:
1257:
Finally, this leads to the continuity equation for incompressible, 1D open-channel flow:
4231:
4221:
3519:
3452:
3165:
2403:
1625:
672:
613:
300:
276:
256:
48:
4008:
1727:. By invoking the high Reynolds number and 1D flow assumptions, we have the equations:
466:
The
Reynolds number is sufficiently large such that viscous diffusion can be neglected
4216:
4041:
4036:
4013:
3996:
3966:
3913:
3886:
163:
3727:
that was ignored by discounting the external forces term in the momentum equation.
2421:
1691:
543:{\displaystyle {\partial \rho \over {\partial t}}+\nabla \cdot (\rho {\bf {v}})=0}
352:
3828:
321:. Depending on the effect of viscosity relative to inertia, as represented by the
4246:
4236:
4211:
4179:
4124:
3843:
3811:
3681:
3474:
1319:{\displaystyle {\partial A \over {\partial t}}+{\partial Q \over {\partial x}}=0}
322:
203:
20:
3934:
3290:
3246:
1589:
1491:
1381:
4266:
4251:
3838:
3757:
3514:
3183:
2788:
608:
330:
135:
131:
40:
4063:
4058:
4053:
1334:
The momentum equation for open-channel flow may be found by starting from the
4307:
4256:
3806:
3410:{\displaystyle {\partial E \over {\partial t}}+{\bf {v}}\cdot \nabla (E+p)=0}
356:
208:
3779:
2397:
359:
326:
36:
4129:
3716:
2921:
441:{\displaystyle {\bf {v}}={\begin{pmatrix}u&v&w\end{pmatrix}}^{T}}
4119:
4105:
3794:
3762:
3742:
3724:
935:
Under the assumption of incompressible, 1D flow, this equation becomes:
607:
operator. Under the assumption of incompressible flow, with a constant
604:
24:
3816:
3784:
3767:
3685:
2758:
195:
44:
54:
3720:
1643:
3737:
571:
199:
4012:. Water Science and Technology Library. New York, NY: Springer.
194:
The behavior of open-channel flow is governed by the effects of
2606:
32:
4074:
3155:{\displaystyle {\bf {v}}\cdot (\omega \times {\bf {v}})=0}
2369:{\displaystyle F_{x}=-{1 \over {\rho }}{\tau \over {R}}}
3684:. However, realistic systems require the addition of a
463:(this is not a good assumption for rapidly-varied flow)
351:
It is possible to formulate equations describing three
207:
dimensionless parameter. The parameter is known as the
482:, describing the conservation of mass, takes the form:
408:
3694:
3657:
3545:
3522:
3482:
3473:. Of particular interest in open-channel flow is the
3455:
3423:
3349:
3329:
3192:
3168:
3116:
2954:
2930:
2797:
2767:
2743:
2652:
2615:
2480:
2430:
2406:
2382:
2325:
2275:
2074:
2047:
2027:
2021:
is the difference between the free surface elevation
1949:
1917:
1733:
1700:
1676:
1652:
1628:
1344:
1266:
1167:
1124:
1082:
941:
792:
695:
675:
636:
616:
580:
556:
488:
391:
367:
303:
279:
259:
217:
2609:
equation, note that the advective acceleration term
3707:
3672:
3640:
3528:
3505:
3461:
3441:
3409:
3335:
3315:
3174:
3154:
3099:
2940:
2912:
2779:
2749:
2729:
2638:
2589:
2463:
2412:
2388:
2368:
2311:
2261:
2060:
2033:
2013:
1935:
1900:
1715:
1682:
1658:
1634:
1614:
1318:
1249:
1153:
1107:
1068:
927:
778:
681:
658:
622:
595:
562:
542:
440:
377:
309:
285:
265:
246:{\displaystyle {\text{Fr}}={U \over {\sqrt {gD}}}}
245:
4305:
4037:Derivation of the Equations of Open Channel Flow
3995:. IAHR Monograph. Rotterdam, NL: A.A. Balkema.
3936:Basic Hydraulic Principles of Open-Channel Flow
3933:Jobson, Harvey E.; Froehlich, David C. (1988).
3932:
3906:Battjes, Jurjen A.; Labeur, Robert Jan (2017).
3343:is time-independent, we arrive at the equation:
2639:{\displaystyle {\bf {v}}\cdot \nabla {\bf {v}}}
2014:{\displaystyle \eta (t,x)=\zeta (t,x)-z_{b}(x)}
4090:
4009:Numerical Modeling in Open Channel Hydraulics
3912:. Cambridge, UK: Cambridge University Press.
3905:
2068:. Substitution into the first equation gives:
469:The flow is one-dimensional across the x-axis
43:. The other type of flow within a conduit is
3229:
3218:
3056:
3045:
2998:
2987:
2875:
2864:
2774:
2768:
2718:
2707:
347:Computational methods for free surface flow
4097:
4083:
2164:
2160:
1198:
1144:
1048:
907:
836:
769:
724:
630:, this equation has the simple expression
65:
162:The discharge is constant throughout the
4042:Surface Profiles for Steady Channel Flow
3965:. New York, NY: McGraw-Hill. p. 2.
3469:being a constant; this is equivalent to
659:{\displaystyle \nabla \cdot {\bf {v}}=0}
95:The depth of flow does change with time.
53:
3989:Nezu, Iehisa; Nakagawa, Hiroji (1993).
3107:This equation was arrived at using the
148:The depth changes over a long distance.
4306:
1336:incompressible Navier-Stokes equations
473:
4078:
3955:
3942:. Reston, VA: U.S. Geological Survey.
3885:. Caldwell, NJ: The Blackburn Press.
3951:
3949:
3875:
1329:
16:Type of liquid flow within a conduit
2464:{\displaystyle S_{f}=\tau /\rho gR}
666:. However, it is possible that the
13:
3983:
3383:
3361:
3353:
3330:
3279:
3083:
3025:
2961:
2957:
2902:
2844:
2813:
2801:
2704:
2663:
2626:
2600:
2544:
2536:
2518:
2510:
2492:
2484:
2228:
2220:
2202:
2194:
2176:
2168:
2138:
2130:
2112:
2104:
2086:
2078:
1869:
1861:
1817:
1809:
1775:
1767:
1749:
1741:
1701:
1677:
1556:
1553:
1524:
1477:
1421:
1366:
1354:
1301:
1293:
1278:
1270:
1232:
1224:
1189:
1181:
1021:
1017:
885:
747:
637:
581:
512:
500:
492:
14:
4340:
4024:
3946:
3289:
3245:
1588:
1490:
1380:
297:scale for a channel's depth, and
189:
3992:Turbulence in Open-Channel Flows
3375:
3223:
3138:
3119:
3050:
3017:
2992:
2933:
2869:
2833:
2806:
2712:
2684:
2668:
2655:
2631:
2618:
1578:
1529:
1426:
1413:
1359:
1154:{\displaystyle Q=\int _{A}u\;dA}
899:
761:
645:
526:
394:
370:
3513:, which is used to compute the
596:{\displaystyle \nabla \cdot ()}
4104:
3926:
3909:Unsteady Flow in Open Channels
3899:
3869:
3673:{\displaystyle \gamma =\rho g}
3398:
3386:
3143:
3127:
2578:
2559:
2424:. Defining the friction slope
2161:
2008:
2002:
1986:
1974:
1965:
1953:
1936:{\displaystyle p=\rho g\zeta }
1908:The second equation implies a
904:
891:
766:
753:
590:
587:
531:
518:
340:
1:
4069:Simulation of Turbulent Flows
4006:Syzmkiewicz, Romuald (2010).
3862:
1161:, the equation is reduced to:
1108:{\displaystyle \int _{A}dA=A}
35:flow within a conduit with a
3753:Computational fluid dynamics
2948:with this equation leads to:
2312:{\displaystyle S=-dz_{b}/dx}
2269:where the channel bed slope
198:and gravity relative to the
7:
3730:
170:for continuous steady flow.
10:
4345:
4054:Open Channel Flow Concepts
3506:{\displaystyle e=E/\rho g}
2780:{\displaystyle \|\cdot \|}
1943:, where the channel depth
344:
319:gravitational acceleration
4275:
4194:
4171:Hydrological optimization
4161:Groundwater flow equation
4143:
4112:
4064:Open Channel Flow Example
4059:What is a Hydraulic Jump?
2941:{\displaystyle {\bf {v}}}
378:{\displaystyle {\bf {v}}}
325:, the flow can be either
3956:Sturm, Terry W. (2001).
1716:{\displaystyle \Phi =gz}
4166:Hazen–Williams equation
4156:Darcy–Weisbach equation
3959:Open Channel Hydraulics
3879:Open-Channel Hydraulics
3834:Darcy-Weisbach equation
2750:{\displaystyle \omega }
2041:and the channel bottom
1725:gravitational potential
1683:{\displaystyle \Delta }
66:Classifications of flow
59:Central Arizona Project
3849:Saint-Venant equations
3824:Other related articles
3715:to account for energy
3709:
3674:
3642:
3530:
3507:
3463:
3443:
3411:
3337:
3317:
3176:
3156:
3101:
2942:
2914:
2781:
2751:
2731:
2640:
2591:
2465:
2414:
2390:
2370:
2313:
2263:
2062:
2035:
2034:{\displaystyle \zeta }
2015:
1937:
1902:
1717:
1684:
1660:
1636:
1616:
1320:
1251:
1155:
1109:
1070:
929:
780:
683:
660:
624:
597:
564:
544:
442:
379:
311:
287:
273:is the mean velocity,
267:
247:
103:Space as the criterion
62:
4329:Hydraulic engineering
4186:Pipe network analysis
4151:Bernoulli's principle
4135:Hydraulic engineering
3876:Chow, Ven Te (2008).
3710:
3708:{\displaystyle h_{f}}
3675:
3643:
3531:
3508:
3471:Bernoulli's principle
3464:
3444:
3442:{\displaystyle E+p=C}
3412:
3338:
3336:{\displaystyle \Phi }
3318:
3177:
3157:
3109:scalar triple product
3102:
2943:
2915:
2782:
2752:
2732:
2646:may be decomposed as:
2641:
2592:
2466:
2415:
2391:
2389:{\displaystyle \tau }
2371:
2314:
2264:
2063:
2061:{\displaystyle z_{b}}
2036:
2016:
1938:
1903:
1718:
1685:
1661:
1637:
1617:
1451:Inertial Acceleration
1321:
1252:
1156:
1110:
1071:
930:
781:
684:
661:
625:
598:
565:
563:{\displaystyle \rho }
545:
450:Cartesian coordinates
443:
380:
345:Further information:
312:
295:characteristic length
288:
268:
248:
176:Spatially-varied flow
144:Gradually-varied flow
76:Time as the criterion
57:
3854:Standard step method
3692:
3655:
3543:
3520:
3480:
3453:
3421:
3347:
3327:
3190:
3166:
3114:
2952:
2928:
2795:
2765:
2741:
2650:
2613:
2478:
2428:
2404:
2380:
2323:
2273:
2072:
2045:
2025:
1947:
1915:
1910:hydrostatic pressure
1731:
1698:
1674:
1659:{\displaystyle \nu }
1650:
1626:
1342:
1264:
1165:
1122:
1117:volumetric flow rate
1080:
939:
790:
693:
673:
668:cross-sectional area
634:
614:
578:
554:
486:
389:
365:
301:
277:
257:
215:
211:, and is defined as:
202:forces of the flow.
3775:Types of fluid flow
3536:that is defined as:
1668:kinematic viscosity
480:continuity equation
474:Continuity equation
168:continuity equation
126:Rapidly-varied flow
3705:
3670:
3638:
3636:
3526:
3503:
3459:
3439:
3407:
3333:
3313:
3312:
3310:
3309:
3286:
3268:
3266:
3265:
3242:
3172:
3152:
3097:
2938:
2910:
2777:
2747:
2727:
2636:
2587:
2461:
2410:
2386:
2366:
2309:
2259:
2058:
2031:
2011:
1933:
1898:
1896:
1713:
1680:
1656:
1632:
1612:
1611:
1609:
1608:
1585:
1570:
1563:
1545:
1538:
1513:
1511:
1510:
1487:
1442:
1435:
1403:
1401:
1400:
1377:
1316:
1247:
1151:
1105:
1066:
925:
776:
679:
656:
620:
593:
560:
540:
438:
426:
375:
307:
283:
263:
243:
63:
49:hydraulic pressure
4314:Civil engineering
4301:
4300:
4176:Open-channel flow
4049:Open-Channel Flow
3790:Transitional flow
3632:
3611:
3579:
3529:{\displaystyle h}
3462:{\displaystyle C}
3368:
3305:
3296:
3274:
3272:
3261:
3252:
3213:
3201:
3199:
3175:{\displaystyle E}
3078:
3043:
2985:
2968:
2897:
2862:
2820:
2702:
2551:
2525:
2499:
2413:{\displaystyle R}
2364:
2352:
2235:
2209:
2183:
2145:
2119:
2093:
1876:
1856:
1824:
1804:
1782:
1756:
1635:{\displaystyle p}
1604:
1595:
1576:
1574:
1568:
1551:
1549:
1543:
1519:
1517:
1506:
1497:
1475:
1463:
1461:
1454:
1452:
1447:
1440:
1409:
1407:
1396:
1387:
1373:
1351:
1349:
1330:Momentum equation
1308:
1285:
1239:
1196:
1115:and defining the
1028:
955:
806:
709:
682:{\displaystyle A}
623:{\displaystyle V}
507:
353:conservation laws
310:{\displaystyle g}
286:{\displaystyle D}
266:{\displaystyle U}
241:
239:
221:
29:open-channel flow
4336:
4099:
4092:
4085:
4076:
4075:
3977:
3976:
3964:
3953:
3944:
3943:
3941:
3930:
3924:
3923:
3903:
3897:
3896:
3884:
3873:
3802:Fluid properties
3714:
3712:
3711:
3706:
3704:
3703:
3679:
3677:
3676:
3671:
3647:
3645:
3644:
3639:
3637:
3633:
3631:
3623:
3612:
3610:
3602:
3601:
3592:
3584:
3580:
3578:
3567:
3535:
3533:
3532:
3527:
3512:
3510:
3509:
3504:
3496:
3468:
3466:
3465:
3460:
3448:
3446:
3445:
3440:
3416:
3414:
3413:
3408:
3379:
3378:
3369:
3367:
3359:
3351:
3342:
3340:
3339:
3334:
3322:
3320:
3319:
3314:
3311:
3306:
3303:
3297:
3294:
3287:
3282:
3267:
3262:
3259:
3253:
3250:
3243:
3238:
3237:
3236:
3227:
3226:
3214:
3212:
3204:
3181:
3179:
3178:
3173:
3161:
3159:
3158:
3153:
3142:
3141:
3123:
3122:
3106:
3104:
3103:
3098:
3090:
3086:
3079:
3077:
3069:
3064:
3063:
3054:
3053:
3044:
3042:
3034:
3021:
3020:
3011:
3007:
3006:
3005:
2996:
2995:
2986:
2984:
2976:
2969:
2967:
2956:
2947:
2945:
2944:
2939:
2937:
2936:
2919:
2917:
2916:
2911:
2909:
2905:
2898:
2896:
2888:
2883:
2882:
2873:
2872:
2863:
2861:
2853:
2837:
2836:
2821:
2819:
2811:
2810:
2809:
2799:
2786:
2784:
2783:
2778:
2761:of the flow and
2756:
2754:
2753:
2748:
2736:
2734:
2733:
2728:
2726:
2725:
2716:
2715:
2703:
2701:
2693:
2688:
2687:
2672:
2671:
2659:
2658:
2645:
2643:
2642:
2637:
2635:
2634:
2622:
2621:
2596:
2594:
2593:
2588:
2571:
2570:
2552:
2550:
2542:
2534:
2526:
2524:
2516:
2508:
2500:
2498:
2490:
2482:
2470:
2468:
2467:
2462:
2451:
2440:
2439:
2422:hydraulic radius
2419:
2417:
2416:
2411:
2395:
2393:
2392:
2387:
2375:
2373:
2372:
2367:
2365:
2363:
2355:
2353:
2351:
2343:
2335:
2334:
2318:
2316:
2315:
2310:
2302:
2297:
2296:
2268:
2266:
2265:
2260:
2258:
2257:
2236:
2234:
2226:
2218:
2210:
2208:
2200:
2192:
2184:
2182:
2174:
2166:
2159:
2158:
2146:
2144:
2136:
2128:
2120:
2118:
2110:
2102:
2094:
2092:
2084:
2076:
2067:
2065:
2064:
2059:
2057:
2056:
2040:
2038:
2037:
2032:
2020:
2018:
2017:
2012:
2001:
2000:
1942:
1940:
1939:
1934:
1907:
1905:
1904:
1899:
1897:
1877:
1875:
1867:
1859:
1857:
1855:
1847:
1838:
1837:
1825:
1823:
1815:
1807:
1805:
1803:
1795:
1783:
1781:
1773:
1765:
1757:
1755:
1747:
1739:
1722:
1720:
1719:
1714:
1692:Laplace operator
1689:
1687:
1686:
1681:
1665:
1663:
1662:
1657:
1641:
1639:
1638:
1633:
1621:
1619:
1618:
1613:
1610:
1605:
1602:
1596:
1593:
1586:
1581:
1569:
1566:
1564:
1559:
1544:
1541:
1539:
1534:
1533:
1532:
1512:
1507:
1504:
1498:
1495:
1488:
1483:
1476:
1474:
1466:
1453:
1450:
1448:
1443:
1441:
1438:
1436:
1431:
1430:
1429:
1417:
1416:
1402:
1397:
1394:
1388:
1385:
1378:
1372:
1364:
1363:
1362:
1352:
1347:
1345:
1325:
1323:
1322:
1317:
1309:
1307:
1299:
1291:
1286:
1284:
1276:
1268:
1256:
1254:
1253:
1248:
1240:
1238:
1230:
1222:
1220:
1219:
1197:
1195:
1187:
1179:
1177:
1176:
1160:
1158:
1157:
1152:
1140:
1139:
1114:
1112:
1111:
1106:
1092:
1091:
1075:
1073:
1072:
1067:
1059:
1055:
1044:
1043:
1029:
1027:
1016:
1014:
1013:
992:
988:
981:
980:
966:
965:
956:
954:
943:
934:
932:
931:
926:
918:
914:
903:
902:
884:
883:
869:
868:
847:
843:
832:
831:
817:
816:
807:
805:
794:
785:
783:
782:
777:
765:
764:
746:
745:
720:
719:
710:
708:
697:
688:
686:
685:
680:
665:
663:
662:
657:
649:
648:
629:
627:
626:
621:
602:
600:
599:
594:
569:
567:
566:
561:
549:
547:
546:
541:
530:
529:
508:
506:
498:
490:
447:
445:
444:
439:
437:
436:
431:
430:
398:
397:
385:with components
384:
382:
381:
376:
374:
373:
316:
314:
313:
308:
292:
290:
289:
284:
272:
270:
269:
264:
252:
250:
249:
244:
242:
240:
232:
227:
222:
219:
4344:
4343:
4339:
4338:
4337:
4335:
4334:
4333:
4304:
4303:
4302:
4297:
4276:Public networks
4271:
4190:
4180:Manning formula
4139:
4125:Hydraulic fluid
4108:
4103:
4071:(p. 26-38)
4033:lecture notes:
4027:
3986:
3984:Further reading
3981:
3980:
3973:
3962:
3954:
3947:
3939:
3931:
3927:
3920:
3904:
3900:
3893:
3882:
3874:
3870:
3865:
3860:
3844:Manning formula
3812:Reynolds number
3748:Fields of study
3733:
3699:
3695:
3693:
3690:
3689:
3682:specific weight
3656:
3653:
3652:
3649:
3635:
3634:
3627:
3622:
3603:
3597:
3593:
3591:
3582:
3581:
3571:
3566:
3553:
3546:
3544:
3541:
3540:
3521:
3518:
3517:
3492:
3481:
3478:
3477:
3475:specific energy
3454:
3451:
3450:
3422:
3419:
3418:
3374:
3373:
3360:
3352:
3350:
3348:
3345:
3344:
3328:
3325:
3324:
3308:
3307:
3302:
3299:
3298:
3293:
3288:
3275:
3273:
3264:
3263:
3258:
3255:
3254:
3249:
3244:
3232:
3228:
3222:
3221:
3208:
3203:
3202:
3200:
3191:
3188:
3187:
3167:
3164:
3163:
3137:
3136:
3118:
3117:
3115:
3112:
3111:
3073:
3068:
3059:
3055:
3049:
3048:
3038:
3033:
3032:
3028:
3016:
3015:
3001:
2997:
2991:
2990:
2980:
2975:
2974:
2970:
2960:
2955:
2953:
2950:
2949:
2932:
2931:
2929:
2926:
2925:
2892:
2887:
2878:
2874:
2868:
2867:
2857:
2852:
2851:
2847:
2832:
2831:
2812:
2805:
2804:
2800:
2798:
2796:
2793:
2792:
2766:
2763:
2762:
2742:
2739:
2738:
2721:
2717:
2711:
2710:
2697:
2692:
2683:
2682:
2667:
2666:
2654:
2653:
2651:
2648:
2647:
2630:
2629:
2617:
2616:
2614:
2611:
2610:
2603:
2601:Energy equation
2598:
2566:
2562:
2543:
2535:
2533:
2517:
2509:
2507:
2491:
2483:
2481:
2479:
2476:
2475:
2447:
2435:
2431:
2429:
2426:
2425:
2405:
2402:
2401:
2381:
2378:
2377:
2359:
2354:
2347:
2342:
2330:
2326:
2324:
2321:
2320:
2298:
2292:
2288:
2274:
2271:
2270:
2253:
2249:
2227:
2219:
2217:
2201:
2193:
2191:
2175:
2167:
2165:
2154:
2150:
2137:
2129:
2127:
2111:
2103:
2101:
2085:
2077:
2075:
2073:
2070:
2069:
2052:
2048:
2046:
2043:
2042:
2026:
2023:
2022:
1996:
1992:
1948:
1945:
1944:
1916:
1913:
1912:
1895:
1894:
1884:
1868:
1860:
1858:
1851:
1846:
1840:
1839:
1833:
1829:
1816:
1808:
1806:
1799:
1794:
1784:
1774:
1766:
1764:
1748:
1740:
1738:
1734:
1732:
1729:
1728:
1699:
1696:
1695:
1675:
1672:
1671:
1651:
1648:
1647:
1627:
1624:
1623:
1607:
1606:
1601:
1598:
1597:
1592:
1587:
1577:
1575:
1565:
1552:
1550:
1540:
1528:
1527:
1520:
1518:
1509:
1508:
1503:
1500:
1499:
1494:
1489:
1470:
1465:
1464:
1462:
1449:
1437:
1425:
1424:
1412:
1411:
1410:
1408:
1399:
1398:
1393:
1390:
1389:
1384:
1379:
1365:
1358:
1357:
1353:
1350:
1348:
1346:
1343:
1340:
1339:
1332:
1327:
1300:
1292:
1290:
1277:
1269:
1267:
1265:
1262:
1261:
1231:
1223:
1221:
1215:
1211:
1188:
1180:
1178:
1172:
1168:
1166:
1163:
1162:
1135:
1131:
1123:
1120:
1119:
1087:
1083:
1081:
1078:
1077:
1076:By noting that
1039:
1035:
1034:
1030:
1020:
1015:
1009:
1005:
976:
972:
971:
967:
961:
957:
947:
942:
940:
937:
936:
898:
897:
879:
875:
874:
870:
864:
860:
827:
823:
822:
818:
812:
808:
798:
793:
791:
788:
787:
760:
759:
741:
737:
715:
711:
701:
696:
694:
691:
690:
674:
671:
670:
644:
643:
635:
632:
631:
615:
612:
611:
579:
576:
575:
555:
552:
551:
525:
524:
499:
491:
489:
487:
484:
483:
476:
432:
425:
424:
419:
414:
404:
403:
402:
393:
392:
390:
387:
386:
369:
368:
366:
363:
362:
349:
343:
323:Reynolds number
302:
299:
298:
278:
275:
274:
258:
255:
254:
231:
226:
218:
216:
213:
212:
204:Surface tension
192:
158:Continuous flow
68:
21:fluid mechanics
17:
12:
11:
5:
4342:
4332:
4331:
4326:
4321:
4319:Fluid dynamics
4316:
4299:
4298:
4296:
4295:
4290:
4285:
4279:
4277:
4273:
4272:
4270:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4198:
4196:
4192:
4191:
4189:
4188:
4183:
4173:
4168:
4163:
4158:
4153:
4147:
4145:
4141:
4140:
4138:
4137:
4132:
4127:
4122:
4116:
4114:
4110:
4109:
4102:
4101:
4094:
4087:
4079:
4073:
4072:
4066:
4061:
4056:
4051:
4046:
4045:
4044:
4039:
4026:
4025:External links
4023:
4022:
4021:
4004:
3985:
3982:
3979:
3978:
3971:
3945:
3925:
3918:
3898:
3892:978-1932846188
3891:
3867:
3866:
3864:
3861:
3859:
3858:
3857:
3856:
3851:
3846:
3841:
3839:Hydraulic jump
3836:
3831:
3821:
3820:
3819:
3814:
3809:
3799:
3798:
3797:
3795:Turbulent flow
3792:
3787:
3782:
3772:
3771:
3770:
3765:
3760:
3758:Fluid dynamics
3755:
3745:
3740:
3734:
3732:
3729:
3702:
3698:
3669:
3666:
3663:
3660:
3630:
3626:
3621:
3618:
3615:
3609:
3606:
3600:
3596:
3590:
3587:
3585:
3583:
3577:
3574:
3570:
3565:
3562:
3559:
3556:
3554:
3552:
3549:
3548:
3538:
3525:
3515:hydraulic head
3502:
3499:
3495:
3491:
3488:
3485:
3458:
3438:
3435:
3432:
3429:
3426:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3377:
3372:
3366:
3363:
3358:
3355:
3332:
3301:
3300:
3292:
3291:
3285:
3281:
3278:
3271:
3257:
3256:
3248:
3247:
3241:
3235:
3231:
3225:
3220:
3217:
3211:
3207:
3198:
3195:
3184:energy density
3171:
3151:
3148:
3145:
3140:
3135:
3132:
3129:
3126:
3121:
3096:
3093:
3089:
3085:
3082:
3076:
3072:
3067:
3062:
3058:
3052:
3047:
3041:
3037:
3031:
3027:
3024:
3019:
3014:
3010:
3004:
3000:
2994:
2989:
2983:
2979:
2973:
2966:
2963:
2959:
2935:
2908:
2904:
2901:
2895:
2891:
2886:
2881:
2877:
2871:
2866:
2860:
2856:
2850:
2846:
2843:
2840:
2835:
2830:
2827:
2824:
2818:
2815:
2808:
2803:
2789:Euclidean norm
2776:
2773:
2770:
2746:
2724:
2720:
2714:
2709:
2706:
2700:
2696:
2691:
2686:
2681:
2678:
2675:
2670:
2665:
2662:
2657:
2633:
2628:
2625:
2620:
2602:
2599:
2586:
2583:
2580:
2577:
2574:
2569:
2565:
2561:
2558:
2555:
2549:
2546:
2541:
2538:
2532:
2529:
2523:
2520:
2515:
2512:
2506:
2503:
2497:
2494:
2489:
2486:
2473:
2460:
2457:
2454:
2450:
2446:
2443:
2438:
2434:
2409:
2385:
2362:
2358:
2350:
2346:
2341:
2338:
2333:
2329:
2308:
2305:
2301:
2295:
2291:
2287:
2284:
2281:
2278:
2256:
2252:
2248:
2245:
2242:
2239:
2233:
2230:
2225:
2222:
2216:
2213:
2207:
2204:
2199:
2196:
2190:
2187:
2181:
2178:
2173:
2170:
2163:
2157:
2153:
2149:
2143:
2140:
2135:
2132:
2126:
2123:
2117:
2114:
2109:
2106:
2100:
2097:
2091:
2088:
2083:
2080:
2055:
2051:
2030:
2010:
2007:
2004:
1999:
1995:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1932:
1929:
1926:
1923:
1920:
1893:
1890:
1887:
1885:
1883:
1880:
1874:
1871:
1866:
1863:
1854:
1850:
1845:
1842:
1841:
1836:
1832:
1828:
1822:
1819:
1814:
1811:
1802:
1798:
1793:
1790:
1787:
1785:
1780:
1777:
1772:
1769:
1763:
1760:
1754:
1751:
1746:
1743:
1737:
1736:
1712:
1709:
1706:
1703:
1679:
1655:
1631:
1600:
1599:
1591:
1590:
1584:
1580:
1573:
1562:
1558:
1555:
1548:
1537:
1531:
1526:
1523:
1516:
1502:
1501:
1493:
1492:
1486:
1482:
1479:
1473:
1469:
1460:
1457:
1446:
1434:
1428:
1423:
1420:
1415:
1406:
1392:
1391:
1383:
1382:
1376:
1371:
1368:
1361:
1356:
1331:
1328:
1315:
1312:
1306:
1303:
1298:
1295:
1289:
1283:
1280:
1275:
1272:
1259:
1246:
1243:
1237:
1234:
1229:
1226:
1218:
1214:
1210:
1207:
1204:
1201:
1194:
1191:
1186:
1183:
1175:
1171:
1150:
1147:
1143:
1138:
1134:
1130:
1127:
1104:
1101:
1098:
1095:
1090:
1086:
1065:
1062:
1058:
1054:
1051:
1047:
1042:
1038:
1033:
1026:
1023:
1019:
1012:
1008:
1004:
1001:
998:
995:
991:
987:
984:
979:
975:
970:
964:
960:
953:
950:
946:
924:
921:
917:
913:
910:
906:
901:
896:
893:
890:
887:
882:
878:
873:
867:
863:
859:
856:
853:
850:
846:
842:
839:
835:
830:
826:
821:
815:
811:
804:
801:
797:
775:
772:
768:
763:
758:
755:
752:
749:
744:
740:
736:
733:
730:
727:
723:
718:
714:
707:
704:
700:
678:
655:
652:
647:
642:
639:
619:
609:control volume
592:
589:
586:
583:
559:
539:
536:
533:
528:
523:
520:
517:
514:
511:
505:
502:
497:
494:
475:
472:
471:
470:
467:
464:
461:incompressible
435:
429:
423:
420:
418:
415:
413:
410:
409:
407:
401:
396:
372:
342:
339:
306:
282:
262:
238:
235:
230:
225:
191:
190:States of flow
188:
187:
186:
185:
184:
183:
182:
173:
172:
171:
155:
154:
153:
152:
151:
150:
149:
141:
140:
139:
136:hydraulic drop
132:hydraulic jump
115:
114:
113:
100:
99:
98:
97:
96:
88:
87:
86:
67:
64:
15:
9:
6:
4:
3:
2:
4341:
4330:
4327:
4325:
4322:
4320:
4317:
4315:
4312:
4311:
4309:
4294:
4291:
4289:
4286:
4284:
4281:
4280:
4278:
4274:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4242:Power network
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4199:
4197:
4193:
4187:
4184:
4181:
4177:
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4152:
4149:
4148:
4146:
4142:
4136:
4133:
4131:
4128:
4126:
4123:
4121:
4118:
4117:
4115:
4111:
4107:
4100:
4095:
4093:
4088:
4086:
4081:
4080:
4077:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4043:
4040:
4038:
4035:
4034:
4032:
4029:
4028:
4019:
4018:9789048136735
4015:
4011:
4010:
4005:
4002:
4001:9789054101185
3998:
3994:
3993:
3988:
3987:
3974:
3972:9780073397870
3968:
3961:
3960:
3952:
3950:
3938:
3937:
3929:
3921:
3919:9781316576878
3915:
3911:
3910:
3902:
3894:
3888:
3881:
3880:
3872:
3868:
3855:
3852:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3829:Chézy formula
3827:
3826:
3825:
3822:
3818:
3815:
3813:
3810:
3808:
3807:Froude number
3805:
3804:
3803:
3800:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3777:
3776:
3773:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3750:
3749:
3746:
3744:
3741:
3739:
3736:
3735:
3728:
3726:
3722:
3718:
3700:
3696:
3687:
3683:
3667:
3664:
3661:
3658:
3648:
3628:
3624:
3619:
3616:
3613:
3607:
3604:
3598:
3594:
3588:
3586:
3575:
3572:
3568:
3563:
3560:
3557:
3555:
3550:
3537:
3523:
3516:
3500:
3497:
3493:
3489:
3486:
3483:
3476:
3472:
3456:
3436:
3433:
3430:
3427:
3424:
3404:
3401:
3395:
3392:
3389:
3380:
3370:
3364:
3356:
3283:
3276:
3269:
3239:
3233:
3215:
3209:
3205:
3196:
3193:
3185:
3169:
3149:
3146:
3133:
3130:
3124:
3110:
3094:
3091:
3087:
3080:
3074:
3070:
3065:
3060:
3039:
3035:
3029:
3022:
3012:
3008:
3002:
2981:
2977:
2971:
2964:
2923:
2906:
2899:
2893:
2889:
2884:
2879:
2858:
2854:
2848:
2841:
2838:
2828:
2825:
2822:
2816:
2790:
2771:
2760:
2744:
2722:
2698:
2694:
2689:
2679:
2676:
2673:
2660:
2623:
2608:
2605:To derive an
2597:
2584:
2581:
2575:
2572:
2567:
2563:
2556:
2553:
2547:
2539:
2530:
2527:
2521:
2513:
2504:
2501:
2495:
2487:
2472:
2458:
2455:
2452:
2448:
2444:
2441:
2436:
2432:
2423:
2407:
2399:
2383:
2360:
2356:
2348:
2344:
2339:
2336:
2331:
2327:
2306:
2303:
2299:
2293:
2289:
2285:
2282:
2279:
2276:
2254:
2250:
2246:
2243:
2240:
2237:
2231:
2223:
2214:
2211:
2205:
2197:
2188:
2185:
2179:
2171:
2155:
2151:
2147:
2141:
2133:
2124:
2121:
2115:
2107:
2098:
2095:
2089:
2081:
2053:
2049:
2028:
2005:
1997:
1993:
1989:
1983:
1980:
1977:
1971:
1968:
1962:
1959:
1956:
1950:
1930:
1927:
1924:
1921:
1918:
1911:
1891:
1888:
1886:
1881:
1878:
1872:
1864:
1852:
1848:
1843:
1834:
1830:
1826:
1820:
1812:
1800:
1796:
1791:
1788:
1786:
1778:
1770:
1761:
1758:
1752:
1744:
1726:
1710:
1707:
1704:
1693:
1669:
1653:
1645:
1629:
1582:
1571:
1560:
1546:
1535:
1521:
1514:
1484:
1480:
1471:
1467:
1458:
1455:
1444:
1432:
1418:
1404:
1374:
1369:
1337:
1326:
1313:
1310:
1304:
1296:
1287:
1281:
1273:
1258:
1244:
1241:
1235:
1227:
1216:
1212:
1208:
1205:
1202:
1199:
1192:
1184:
1173:
1169:
1148:
1145:
1141:
1136:
1132:
1128:
1125:
1118:
1102:
1099:
1096:
1093:
1088:
1084:
1063:
1060:
1056:
1052:
1049:
1045:
1040:
1036:
1031:
1024:
1010:
1006:
1002:
999:
996:
993:
989:
985:
982:
977:
973:
968:
962:
958:
951:
948:
944:
922:
919:
915:
911:
908:
894:
888:
880:
876:
871:
865:
861:
857:
854:
851:
848:
844:
840:
837:
833:
828:
824:
819:
813:
809:
802:
799:
795:
773:
770:
756:
750:
742:
738:
734:
731:
728:
725:
721:
716:
712:
705:
702:
698:
676:
669:
653:
650:
640:
617:
610:
606:
584:
573:
570:is the fluid
557:
537:
534:
521:
515:
509:
503:
495:
481:
468:
465:
462:
458:
457:
456:
453:
451:
433:
427:
421:
416:
411:
405:
399:
361:
358:
357:flow velocity
354:
348:
338:
336:
332:
328:
324:
320:
304:
296:
280:
260:
236:
233:
228:
223:
210:
209:Froude number
205:
201:
197:
179:
178:
177:
174:
169:
165:
161:
160:
159:
156:
147:
146:
145:
142:
137:
133:
129:
128:
127:
124:
123:
121:
120:
119:
116:
111:
110:
109:
106:
105:
104:
101:
94:
93:
92:
91:Unsteady flow
89:
84:
83:
82:
79:
78:
77:
74:
73:
72:
60:
56:
52:
50:
46:
42:
39:, known as a
38:
34:
31:is a type of
30:
26:
22:
4262:Rescue tools
4227:Drive system
4195:Technologies
4175:
4007:
3990:
3958:
3935:
3928:
3908:
3901:
3878:
3871:
3823:
3801:
3780:Laminar flow
3774:
3747:
3650:
3539:
3323:Noting that
2604:
2474:
2398:shear stress
1333:
1260:
478:The general
477:
459:The flow is
454:
360:vector field
350:
335:transitional
193:
175:
157:
143:
125:
117:
108:Uniform flow
107:
102:
90:
80:
75:
69:
37:free surface
28:
18:
4207:Accumulator
4130:Fluid power
3717:dissipation
2922:dot product
2920:Taking the
341:Formulation
118:Varied flow
81:Steady flow
4324:Hydraulics
4308:Categories
4293:Manchester
4120:Hydraulics
4106:Hydraulics
3863:References
3763:Hydraulics
3743:Streamflow
3725:turbulence
3680:being the
3182:to be the
3162:. Define
605:divergence
25:hydraulics
4283:Liverpool
4202:Machinery
3817:Viscosity
3785:Pipe flow
3768:Hydrology
3686:head loss
3665:ρ
3659:γ
3629:γ
3573:ρ
3498:ρ
3384:∇
3381:⋅
3362:∂
3354:∂
3331:Φ
3295:Potential
3284:⏟
3280:Φ
3277:ρ
3240:⏟
3230:‖
3219:‖
3216:ρ
3134:×
3131:ω
3125:⋅
3084:Φ
3075:ρ
3057:‖
3046:‖
3026:∇
3023:⋅
2999:‖
2988:‖
2962:∂
2958:∂
2903:Φ
2894:ρ
2876:‖
2865:‖
2845:∇
2842:−
2829:×
2826:ω
2814:∂
2802:∂
2775:‖
2772:⋅
2769:‖
2759:vorticity
2745:ω
2719:‖
2708:‖
2705:∇
2680:×
2677:ω
2664:∇
2661:⋅
2627:∇
2624:⋅
2573:−
2545:∂
2540:η
2537:∂
2519:∂
2511:∂
2493:∂
2485:∂
2453:ρ
2445:τ
2384:τ
2357:τ
2349:ρ
2340:−
2283:−
2238:−
2229:∂
2224:η
2221:∂
2203:∂
2195:∂
2177:∂
2169:∂
2162:⟹
2139:∂
2134:ζ
2131:∂
2113:∂
2105:∂
2087:∂
2079:∂
2029:ζ
1990:−
1972:ζ
1951:η
1931:ζ
1925:ρ
1879:−
1870:∂
1862:∂
1853:ρ
1844:−
1818:∂
1810:∂
1801:ρ
1792:−
1776:∂
1768:∂
1750:∂
1742:∂
1702:Φ
1678:Δ
1654:ν
1583:⏟
1561:⏟
1557:Φ
1554:∇
1547:−
1542:Diffusion
1536:⏟
1525:Δ
1522:ν
1485:⏟
1478:∇
1472:ρ
1459:−
1445:⏞
1439:Advection
1433:⏟
1422:∇
1419:⋅
1375:⏟
1367:∂
1355:∂
1302:∂
1294:∂
1279:∂
1271:∂
1233:∂
1225:∂
1213:∫
1209:−
1190:∂
1182:∂
1170:∫
1133:∫
1085:∫
1037:∫
1022:∂
1018:∂
1007:∫
1003:−
974:∫
959:∫
895:ρ
889:⋅
886:∇
877:∫
862:∫
858:−
834:ρ
825:∫
810:∫
757:ρ
751:⋅
748:∇
739:∫
735:−
722:ρ
713:∫
641:⋅
638:∇
585:⋅
582:∇
558:ρ
522:ρ
516:⋅
513:∇
501:∂
496:ρ
493:∂
331:turbulent
196:viscosity
45:pipe flow
4232:Manifold
4222:Cylinder
4144:Modeling
4113:Concepts
3731:See also
3721:friction
1644:pressure
1594:External
1505:Gradient
1496:Pressure
200:inertial
134:and the
61:channel.
4217:Circuit
4031:Caltech
3738:HEC-RAS
3719:due to
3251:Kinetic
2787:is the
2757:is the
2420:is the
2396:is the
1723:is the
1690:is the
1666:is the
1642:is the
1567:Gravity
1338: :
603:is the
572:density
327:laminar
317:is the
293:is the
41:channel
4288:London
4016:
3999:
3969:
3916:
3889:
3304:Energy
3260:Energy
2737:where
2607:energy
2376:where
1694:, and
1622:where
1603:Forces
1395:Change
550:where
253:where
33:liquid
4247:Press
4237:Motor
4212:Brake
3963:(PDF)
3940:(PDF)
3883:(PDF)
3688:term
3651:with
3449:with
1386:Local
448:. In
333:, or
164:reach
4267:Seal
4252:Pump
4014:ISBN
3997:ISBN
3967:ISBN
3914:ISBN
3887:ISBN
3723:and
2400:and
574:and
23:and
4257:Ram
2924:of
19:In
4310::
3948:^
1670:,
1646:,
329:,
220:Fr
51:.
27:,
4182:)
4178:(
4098:e
4091:t
4084:v
4020:.
4003:.
3975:.
3922:.
3895:.
3701:f
3697:h
3668:g
3662:=
3625:p
3620:+
3617:z
3614:+
3608:g
3605:2
3599:2
3595:u
3589:=
3576:g
3569:p
3564:+
3561:e
3558:=
3551:h
3524:h
3501:g
3494:/
3490:E
3487:=
3484:e
3457:C
3437:C
3434:=
3431:p
3428:+
3425:E
3405:0
3402:=
3399:)
3396:p
3393:+
3390:E
3387:(
3376:v
3371:+
3365:t
3357:E
3270:+
3234:2
3224:v
3210:2
3206:1
3197:=
3194:E
3186::
3170:E
3150:0
3147:=
3144:)
3139:v
3128:(
3120:v
3095:0
3092:=
3088:)
3081:+
3071:p
3066:+
3061:2
3051:v
3040:2
3036:1
3030:(
3018:v
3013:+
3009:)
3003:2
2993:v
2982:2
2978:1
2972:(
2965:t
2934:v
2907:)
2900:+
2890:p
2885:+
2880:2
2870:v
2859:2
2855:1
2849:(
2839:=
2834:v
2823:+
2817:t
2807:v
2723:2
2713:v
2699:2
2695:1
2690:+
2685:v
2674:=
2669:v
2656:v
2632:v
2619:v
2585:0
2582:=
2579:)
2576:S
2568:f
2564:S
2560:(
2557:g
2554:+
2548:x
2531:g
2528:+
2522:x
2514:u
2505:u
2502:+
2496:t
2488:u
2459:R
2456:g
2449:/
2442:=
2437:f
2433:S
2408:R
2361:R
2345:1
2337:=
2332:x
2328:F
2307:x
2304:d
2300:/
2294:b
2290:z
2286:d
2280:=
2277:S
2255:x
2251:F
2247:=
2244:S
2241:g
2232:x
2215:g
2212:+
2206:x
2198:u
2189:u
2186:+
2180:t
2172:u
2156:x
2152:F
2148:=
2142:x
2125:g
2122:+
2116:x
2108:u
2099:u
2096:+
2090:t
2082:u
2054:b
2050:z
2009:)
2006:x
2003:(
1998:b
1994:z
1987:)
1984:x
1981:,
1978:t
1975:(
1969:=
1966:)
1963:x
1960:,
1957:t
1954:(
1928:g
1922:=
1919:p
1892:0
1889:=
1882:g
1873:z
1865:p
1849:1
1835:x
1831:F
1827:+
1821:x
1813:p
1797:1
1789:=
1779:x
1771:u
1762:u
1759:+
1753:t
1745:u
1711:z
1708:g
1705:=
1630:p
1579:F
1572:+
1530:v
1515:+
1481:p
1468:1
1456:=
1427:v
1414:v
1405:+
1370:t
1360:v
1314:0
1311:=
1305:x
1297:Q
1288:+
1282:t
1274:A
1245:x
1242:d
1236:x
1228:Q
1217:x
1206:=
1203:x
1200:d
1193:t
1185:A
1174:x
1149:A
1146:d
1142:u
1137:A
1129:=
1126:Q
1103:A
1100:=
1097:A
1094:d
1089:A
1064:x
1061:d
1057:)
1053:A
1050:d
1046:u
1041:A
1032:(
1025:x
1011:x
1000:=
997:x
994:d
990:)
986:A
983:d
978:A
969:(
963:x
952:t
949:d
945:d
923:x
920:d
916:]
912:A
909:d
905:)
900:v
892:(
881:A
872:[
866:x
855:=
852:x
849:d
845:)
841:A
838:d
829:A
820:(
814:x
803:t
800:d
796:d
774:V
771:d
767:)
762:v
754:(
743:V
732:=
729:V
726:d
717:V
706:t
703:d
699:d
677:A
654:0
651:=
646:v
618:V
591:)
588:(
538:0
535:=
532:)
527:v
519:(
510:+
504:t
434:T
428:)
422:w
417:v
412:u
406:(
400:=
395:v
371:v
305:g
281:D
261:U
237:D
234:g
229:U
224:=
138:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
