593:
2692:. R. M. Alexander found that animals of different sizes and masses travelling at different speeds, but with the same Froude number, consistently exhibit similar gaits. This study found that animals typically switch from an amble to a symmetric running gait (e.g., a trot or pace) around a Froude number of 1.0. A preference for asymmetric gaits (e.g., a canter, transverse gallop, rotary gallop, bound, or pronk) was observed at Froude numbers between 2.0 and 3.0.
206:
1441:
332:
588:{\displaystyle \rho ^{*}\equiv {\frac {\rho }{\rho _{0}}},\quad u^{*}\equiv {\frac {u}{u_{0}}},\quad r^{*}\equiv {\frac {r}{r_{0}}},\quad t^{*}\equiv {\frac {u_{0}}{r_{0}}}t,\quad \nabla ^{*}\equiv r_{0}\nabla ,\quad \mathbf {g} ^{*}\equiv {\frac {\mathbf {g} }{g_{0}}},\quad {\boldsymbol {\sigma }}^{*}\equiv {\frac {\boldsymbol {\sigma }}{p_{0}}},}
1001:
1413:
1884:
So, these flows are associated with the elevation of the topographic slopes that induce the gravity potential energy together with the pressure potential energy during the flow. Therefore, the classical Froude number should include this additional effect. For such a situation, Froude number needs to
1848:
on dynamically sensitive structures such as suspension bridges it is sometimes necessary to simulate the combined effect of the vibrating mass of the structure with the fluctuating force of the wind. In such cases, the Froude number should be respected. Similarly, when simulating hot smoke plumes
854:
2714:
One can easily see the line of "critical" flow in a kitchen or bathroom sink. Leave it unplugged and let the faucet run. Near the place where the stream of water hits the sink, the flow is supercritical. It 'hugs' the surface and moves quickly. On the outer edge of the flow pattern the flow is
2252:
The Froude number finds also a similar application in powder mixers. It will indeed be used to determine in which mixing regime the blender is working. If Fr<1, the particles are just stirred, but if Fr>1, centrifugal forces applied to the powder overcome gravity and the bed of particles
2518:
201:
introduced first the ratio of the flow velocity to the square root of the gravity acceleration times the flow depth. When the ratio was less than unity, the flow behaved like a fluvial motion (i.e., subcritical flow), and like a torrential flow motion when the ratio was greater than unity.
1280:
1169:
2683:
If total leg length is used as the characteristic length, then the theoretical maximum speed of walking has a Froude number of 1.0 since any higher value would result in takeoff and the foot missing the ground. The typical transition speed from bipedal walking to
895:
1285:
172:
the Froude number is not frequently considered since usually the equations are considered in the high Froude limit of negligible external field, leading to homogeneous equations that preserve the mathematical aspects. For example, homogeneous
758:
2679:
2419:
1968:
270:
2107:
are the pressure potential and gravity potential energies, respectively. In the classical definition of the shallow-water or granular flow Froude number, the potential energy associated with the surface elevation,
1202:
1758:
2416:, where the center of mass goes through a circular arc centered at the foot. The Froude number is the ratio of the centripetal force around the center of motion, the foot, and the weight of the animal walking:
1611:
3501:
1072:
738:
662:
2392:
2718:
The Froude number has been used to study trends in animal locomotion in order to better understand why animals use different gait patterns as well as to form hypotheses about the gaits of extinct species.
2715:
subcritical. This flow is thicker and moves more slowly. The boundary between the two areas is called a "hydraulic jump". The jump starts where the flow is just critical and Froude number is equal to 1.0.
2250:
3253:
Jikar, P. C.; Dhokey, N. B.; Shinde, S. S. (2021). "Numerical
Modeling Simulation and Experimental Study of Dynamic Particle Bed Counter Current Reactor and Its Effect on SolidâGas Reduction Reaction".
2315:
1503:
1810:
1067:
135:
1703:
2172:
is unbounded even though the kinetic energy is bounded. So, formally considering the additional contribution due to the gravitational potential energy, the singularity in Fr is removed.
3494:
2570:
2558:
may be chosen to suit the study at hand. For instance, some studies have used the vertical distance of the hip joint from the ground, while others have used total leg length.
996:{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+\nabla \cdot \left({\frac {1}{2}}\mathbf {u} \otimes \mathbf {u} \right)={\frac {1}{\mathrm {Fr} ^{2}}}\mathbf {g} }
1888:
229:
had put forward the concept much earlier in 1852 for testing ships and propellers but Froude was unaware of it. Speedâlength ratio was originally defined by Froude in his
217:(below). A sequence of 3, 6, and 12 (shown in the picture) foot scale models were constructed by Froude and used in towing trials to establish resistance and scaling laws.
3846:
3487:
2412:
The Froude number may be used to study trends in animal gait patterns. In analyses of the dynamics of legged locomotion, a walking limb is often modeled as an inverted
238:
2054:
1408:{\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} {\frac {\nabla p}{\rho }}={\frac {1}{\mathrm {Re} }}\nabla ^{2}u+{\frac {1}{\mathrm {Fr} ^{2}}}{\hat {g}}}
329:, need to be defined. These should be chosen such that the dimensionless variables are all of order one. The following dimensionless variables are thus obtained:
1708:
849:{\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} {\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}={\frac {1}{\mathrm {Fr} ^{2}}}\mathbf {g} }
1554:
671:
600:
2513:{\displaystyle \mathrm {Fr} ={\frac {\text{centripetal force}}{\text{gravitational force}}}={\frac {\;{\frac {mv^{2}}{l}}\;}{mg}}={\frac {v^{2}}{gl}}}
1849:
combined with natural wind, Froude number scaling is necessary to maintain the correct balance between buoyancy forces and the momentum of the wind.
3510:
3417:
Vaughan, Christopher L.; O'Malley, Mark J. (2005). "Froude and the contribution of naval architecture to our understanding of bipedal locomotion".
291:
The term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did. In France, it is sometimes called
2327:
2209:
1455:
2122:, is not considered. The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term
3457:
3407:
3307:
3215:
2900:
2272:
1769:
1039:
97:
2262:
2206:
is the impeller radius (in engineering the diameter is much more frequently employed), the Froude number then takes the following form:
3474:
1547:
In the case of planing craft, where the waterline length is too speed-dependent to be meaningful, the Froude number is best defined as
2056:
is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum;
1173:
Free Euler equations are conservative. The limit of high Froude numbers (low external field) is thus notable and can be studied with
225:, who used a series of scale models to measure the resistance each model offered when towed at a given speed. The naval constructor
1665:
2754:
2128:
emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows
1027:
185:
the Froude number is a significant figure used to determine the resistance of a partially submerged object moving through water.
174:
2180:
In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is
2722:
In addition particle bed behavior can be quantified by Froude number (Fr) in order to establish the optimum operating window.
1275:{\displaystyle {\boldsymbol {\sigma }}=p\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathsf {T}}\right)}
3656:
1164:{\displaystyle {\frac {D\mathbf {u} }{Dt}}+\mathrm {Eu} {\frac {\nabla p}{\rho }}={\frac {1}{\mathrm {Fr} ^{2}}}{\hat {g}}}
2396:
The densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the
1638:
flow velocity, averaged over the cross-section perpendicular to the flow direction. The wave velocity, termed celerity
1186:
3856:
3382:
3183:
2400:
which is more commonly encountered when considering stratified shear layers. For example, the leading edge of a
1885:
be re-defined. The extended Froude number is defined as the ratio between the kinetic and the potential energy:
3733:
3716:
2880:
2743:
316:
597:
Substitution of these inverse relations in the Euler momentum equations, and definition of the Froude number:
3851:
3711:
3611:
3245:
2701:
2197:
1997:
1862:
1541:
665:
2917:
3189:
Essai sur la solution numerique de quelques problemes relatifs au mouvement permanent des eaux courantes
158:
3571:
3566:
3291:
2968:
2780:
2748:
75:
3475:
https://web.archive.org/web/20070927085042/http://www.qub.ac.uk/waves/fastferry/reference/MCA457.pdf
3194:
An essay on the numerical solution to some problems relative to the steady movement of running water
3636:
1174:
878:
741:
31:
3606:
3546:
3023:"Powder Mixing - Powder Mixers Design - Ribbon blender, Paddle mixer, Drum blender, Froude Number"
3820:
3351:
3279:
3142:
2992:
2708:
2162:
are both of order unity. If the mass is shallow with a virtually bed-parallel free-surface, then
1825:
1424:
1015:
1007:
182:
2539:
2168:
can be disregarded. In this situation, if the gravity potential is not taken into account, then
1551:
and the reference length is taken as the cubic root of the volumetric displacement of the hull:
205:
1448:
In marine hydrodynamic applications, the Froude number is usually referenced with the notation
871:(corresponding to negligible stress) general Cauchy momentum equation becomes an inhomogeneous
3775:
3740:
3521:
3453:
3434:
3403:
3369:
3303:
3271:
3227:"Development of the BĂ©langer Equation and Backwater Equation by Jean-Baptiste BĂ©langer (1828)"
3226:
3211:
3171:
2984:
2937:
2896:
2397:
194:
2674:{\displaystyle \mathrm {Fr} ={\frac {v^{2}}{gl}}={\frac {(lf)^{2}}{gl}}={\frac {lf^{2}}{g}}.}
3825:
3701:
3696:
3681:
3646:
3556:
3426:
3359:
3343:
3263:
3241:
3187:
3163:
3134:
2976:
2929:
2888:
1845:
1817:
1518:
873:
178:
51:
3479:
2711:
or subcritical) depends upon whether the Froude number is greater than or less than unity.
2032:
3790:
3770:
3728:
3723:
3551:
2760:
2401:
2187:
1440:
1420:
2972:
3800:
3780:
3765:
3760:
3706:
3691:
3666:
3661:
3651:
3631:
3621:
3586:
3531:
3364:
3331:
3201:
2980:
2933:
2734: â Vector field which is used to mathematically describe the motion of a continuum
1625:
1537:
1196:
307:
we start from the Cauchy momentum equation in its dimensionless (nondimensional) form.
303:
To show how the Froude number is linked to general continuum mechanics and not only to
226:
222:
169:
45:
1191:
Incompressible NavierâStokes momentum equation is a Cauchy momentum equation with the
3840:
3810:
3805:
3795:
3785:
3750:
3745:
3686:
3626:
3596:
3591:
3561:
3526:
3283:
3146:
2892:
2757: â Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
2731:
1963:{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {\beta h+s_{g}\left(x_{d}-x\right)}}},}
304:
150:
142:
3815:
3755:
3671:
3641:
3601:
3576:
3536:
3430:
3167:
2784:
1881:
take place on inclined slopes which then merge into gentle and flat run-out zones.
3317:
Normand, J.A. (1888). "On the
Fineness of vessels in relation to size and speed".
17:
3447:
3397:
3205:
3154:
Alexander, RM (1989). "Optimization and gaits in the locomotion of vertebrates".
3676:
3581:
3541:
1878:
1011:
165:
3267:
3138:
2737:
1192:
1033:
83:
3275:
2988:
2941:
3299:
3022:
1874:
1858:
79:
3438:
3373:
3347:
3125:
Alexander, R. McN. (1984). "The Gaits of
Bipedal and Quadrupedal Animals".
1536:
in some notations. It is an important parameter with respect to the ship's
265:{\displaystyle {\text{speedâlength ratio}}={\frac {u}{\sqrt {\text{LWL}}}}}
27:
Dimensionless number; ratio of a fluid's flow inertia to the external field
3175:
2253:
becomes fluidized, at least in some part of the blender, promoting mixing
3046:
3044:
2549:
2413:
3355:
3332:"Estimating dinosaur maximum running speeds using evolutionary robotics"
2996:
2956:
2685:
1635:
1621:
321:
In order to make the equations dimensionless, a characteristic length r
87:
1753:{\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g{\dfrac {A}{B}}}}}.}
221:
Quantifying resistance of floating objects is generally credited to
2561:
The Froude number may also be calculated from the stride frequency
1606:{\displaystyle \mathrm {Fn} _{V}={\frac {u}{\sqrt {g{\sqrt{V}}}}}.}
733:{\displaystyle \mathrm {Eu} ={\frac {p_{0}}{\rho _{0}u_{0}^{2}}},}
657:{\displaystyle \mathrm {Fr} ={\frac {u_{0}}{\sqrt {g_{0}r_{0}}}},}
204:
3483:
1444:
Wave pattern versus speed, illustrating various Froude numbers.
1032:
Euler momentum equation is a Cauchy momentum equation with the
2763: â Ratio of inertial to viscous forces acting on a liquid
3080:
3078:
63:
2387:{\displaystyle g'=g{\frac {\rho _{1}-\rho _{2}}{\rho _{1}}}}
2245:{\displaystyle \mathrm {Fr} =\omega {\sqrt {\frac {r}{g}}}.}
1644:, is equal to the square root of gravitational acceleration
1865:
of terrestrial animals, including antelope and dinosaurs.
1498:{\displaystyle \mathrm {Fn} _{L}={\frac {u}{\sqrt {gL}}},}
3065:
3063:
3061:
3059:
2740: â Force which acts throughout the volume of a body
1511:
is the relative flow velocity between the sea and ship,
3336:
Proceedings of the Royal
Society B: Biological Sciences
863:(corresponding to negligible external field) are named
3399:
3330:
Sellers, William Irvin; Manning, Phillip Lars (2007).
2310:{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g'h}}}}
1805:{\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {gd}}}.}
1527:
is the length of the ship at the water line level, or
2573:
2422:
2330:
2275:
2212:
2035:
1891:
1772:
1733:
1711:
1668:
1557:
1458:
1288:
1205:
1075:
1062:{\displaystyle {\boldsymbol {\sigma }}=p\mathbf {I} }
1042:
898:
761:
674:
603:
335:
241:
130:{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {gL}}}}
100:
57:
3210:(2nd ed.). ButterworthâHeinemann. p. 650.
66:
60:
3319:
3050:
54:
2673:
2512:
2386:
2309:
2244:
2048:
1962:
1804:
1760:For rectangular cross-sections with uniform depth
1752:
1697:
1605:
1497:
1407:
1274:
1187:NavierâStokes equations § Incompressible flow
1163:
1061:
995:
848:
732:
656:
587:
264:
129:
3108:
2881:"Chapter 2. Body Support, Scaling, and Allometry"
2404:moves with a front Froude number of about unity.
3207:Hydraulics of Open Channel Flow: An Introduction
2918:"Allometry of the limbs of antelopes (Bovidae)"
859:Cauchy-type equations in the high Froude limit
86:(the latter in many applications simply due to
3127:The International Journal of Robotics Research
3084:
2707:In free-surface flow, the nature of the flow (
1181:Incompressible NavierâStokes momentum equation
740:the equations are finally expressed (with the
3495:
8:
2887:. Harvard University Press. pp. 26â37.
2704:between bodies of various sizes and shapes.
1698:{\displaystyle c={\sqrt {g{\frac {A}{B}}}},}
867:. On the other hand, in the low Euler limit
164:The Froude number has some analogy with the
1857:The Froude number has also been applied in
3502:
3488:
3480:
3196:] (in French). Paris: Carilian-Goeury.
2473:
2450:
1766:, the Froude number can be simplified to:
1705:so the Froude number in shallow water is:
3363:
3096:
3069:
3009:
2700:The Froude number is used to compare the
2656:
2646:
2626:
2610:
2591:
2585:
2574:
2572:
2494:
2488:
2461:
2451:
2447:
2434:
2423:
2421:
2376:
2365:
2352:
2345:
2329:
2287:
2276:
2274:
2227:
2213:
2211:
2040:
2034:
1937:
1922:
1903:
1892:
1890:
1784:
1773:
1771:
1732:
1723:
1712:
1710:
1680:
1675:
1667:
1590:
1585:
1576:
1567:
1559:
1556:
1477:
1468:
1460:
1457:
1394:
1393:
1385:
1377:
1371:
1359:
1345:
1340:
1322:
1314:
1295:
1289:
1287:
1282:in nondimensional convective form it is:
1260:
1259:
1250:
1236:
1217:
1206:
1204:
1199:being the stress constitutive relations:
1150:
1149:
1141:
1133:
1127:
1109:
1101:
1082:
1076:
1074:
1054:
1043:
1041:
988:
980:
972:
966:
953:
945:
935:
905:
899:
897:
841:
833:
825:
819:
811:
795:
787:
768:
762:
760:
718:
713:
703:
692:
686:
675:
673:
642:
632:
621:
615:
604:
602:
574:
565:
556:
551:
538:
529:
527:
518:
513:
499:
486:
467:
457:
451:
442:
426:
417:
408:
392:
383:
374:
358:
349:
340:
334:
250:
242:
240:
112:
101:
99:
3847:Dimensionless numbers of fluid mechanics
3511:Dimensionless numbers in fluid mechanics
3383:"Chapter 6 Incompressible Inviscid Flow"
1540:, or resistance, especially in terms of
1439:
1036:being the stress constitutive relation:
198:
3246:10.1061/(ASCE)0733-9429(2009)135:3(159)
2842:
2830:
2818:
2772:
1207:
1044:
812:
567:
552:
2866:
2196:is the impeller frequency (usually in
2029:is the channel downslope position and
1261:
1069:in nondimensional Lagrangian form is:
2806:
2751: â Partial differential equation
90:). The Froude number is based on the
7:
3256:Mining, Metallurgy & Exploration
2879:Alexander, R. McNeill (2013-10-01).
2854:
2794:
2779:Merriam Webster Online (for brother
1423:. Free NavierâStokes equations are
2981:10.1038/scientificamerican0491-130
2934:10.1111/j.1469-7998.1977.tb04177.x
2578:
2575:
2427:
2424:
2280:
2277:
2217:
2214:
1896:
1893:
1777:
1774:
1716:
1713:
1563:
1560:
1464:
1461:
1381:
1378:
1356:
1349:
1346:
1325:
1318:
1315:
1247:
1233:
1137:
1134:
1112:
1105:
1102:
976:
973:
924:
912:
902:
829:
826:
805:
791:
788:
679:
676:
608:
605:
505:
483:
105:
102:
25:
3452:(4th ed.). WCB/McGraw-Hill.
1620:For shallow water waves, such as
325:, and a characteristic velocity u
233:in 1868 in dimensional terms as:
3234:Journal of Hydraulic Engineering
2893:10.4159/harvard.9780674184404.c2
2885:Functional Vertebrate Morphology
2755:Euler equations (fluid dynamics)
2261:When used in the context of the
1656:, divided by free-surface width
1296:
1251:
1237:
1218:
1083:
1055:
1028:Euler equations (fluid dynamics)
989:
954:
946:
906:
889:nondimensional conservation form
842:
769:
530:
514:
50:
3109:Jikar, Dhokey & Shinde 2021
1873:Geophysical mass flows such as
744:and now omitting the indexes):
549:
511:
481:
437:
403:
369:
299:Definition and main application
3431:10.1016/j.gaitpost.2004.01.011
3168:10.1152/physrev.1989.69.4.1199
2955:Alexander, R. McNeill (1991).
2623:
2613:
2532:is the characteristic length,
1628:, the characteristic velocity
1399:
1256:
1244:
1155:
1006:This is an inhomogeneous pure
752:nondimensional convective form
1:
1824:the flow is characterised as
1650:, times cross-sectional area
3298:. Cambridge, Massachusetts:
2552:. The characteristic length
78:defined as the ratio of the
3396:Takahashi, Tamotsu (2007).
3051:Vaughan & O'Malley 2005
2916:Alexander, R. McN. (1977).
2540:acceleration due to gravity
1976:is the mean flow velocity,
1519:acceleration due to gravity
877:(here we make explicit the
3873:
3381:Shih, Y.C. (Spring 2009),
3268:10.1007/s42461-021-00516-6
3139:10.1177/027836498400300205
3085:Sellers & Manning 2007
1998:earth pressure coefficient
1549:displacement Froude number
1184:
1025:
314:
3517:
2267:densimetric Froude number
2257:Densimetric Froude number
244:speed–length ratio
3446:White, Frank M. (1999).
3225:Chanson, Hubert (2009).
2744:Cauchy momentum equation
2324:is the reduced gravity:
2263:Boussinesq approximation
748:Cauchy momentum equation
317:Cauchy momentum equation
311:Cauchy momentum equation
3184:Belanger, Jean Baptiste
1832:the flow is denoted as
1022:Euler momentum equation
3348:10.1098/rspb.2007.0846
2702:wave making resistance
2675:
2514:
2388:
2311:
2246:
2050:
1964:
1869:Extended Froude number
1806:
1754:
1699:
1607:
1542:wave making resistance
1499:
1445:
1409:
1276:
1165:
1063:
997:
850:
734:
658:
589:
295:after Frederic Reech.
266:
218:
179:conservation equations
131:
3292:Newman, John Nicholas
3262:. Springer: 139â152.
3156:Physiological Reviews
2676:
2515:
2408:Walking Froude number
2389:
2312:
2247:
2051:
2049:{\displaystyle x_{d}}
1965:
1816:the flow is called a
1807:
1755:
1700:
1608:
1517:is in particular the
1500:
1443:
1410:
1277:
1166:
1064:
998:
851:
735:
659:
590:
287:= length of waterline
267:
208:
159:characteristic length
132:
94:which he defined as:
3296:Marine hydrodynamics
2781:James Anthony Froude
2571:
2420:
2328:
2273:
2210:
2033:
1889:
1770:
1709:
1666:
1555:
1456:
1427:(non conservative).
1286:
1203:
1073:
1040:
896:
759:
672:
601:
333:
239:
98:
84:external force field
76:dimensionless number
3053:, pp. 350â362.
2973:1991SciAm.264d.130A
2961:Scientific American
2957:"How Dinosaurs Ran"
2833:, pp. 257â261.
2821:, pp. 159â163.
2440:gravitational force
1616:Shallow water waves
1452:and is defined as:
1175:perturbation theory
879:material derivative
742:material derivative
723:
293:ReechâFroude number
32:continuum mechanics
3857:Naval architecture
3657:KeuleganâCarpenter
3419:Gait & Posture
2922:Journal of Zoology
2671:
2510:
2384:
2307:
2242:
2046:
1960:
1826:supercritical flow
1802:
1750:
1742:
1695:
1603:
1495:
1446:
1436:Ship hydrodynamics
1431:Other applications
1405:
1272:
1161:
1059:
1016:diffusion equation
1008:advection equation
993:
846:
730:
709:
654:
585:
262:
219:
195:open channel flows
183:naval architecture
127:
92:speedâlength ratio
18:Speedâlength ratio
3834:
3833:
3459:978-0-07-116848-9
3409:978-0-203-94628-2
3309:978-0-262-14026-3
3217:978-0-7506-5978-9
3027:powderprocess.net
2902:978-0-674-18440-4
2749:Burgers' equation
2666:
2641:
2605:
2508:
2483:
2471:
2442:
2441:
2438:
2437:centripetal force
2398:Richardson number
2382:
2305:
2304:
2237:
2236:
1955:
1954:
1844:When considering
1797:
1796:
1745:
1744:
1741:
1690:
1688:
1598:
1597:
1595:
1490:
1489:
1402:
1391:
1353:
1335:
1309:
1158:
1147:
1122:
1096:
1010:, as much as the
986:
943:
919:
839:
803:
782:
725:
649:
648:
580:
544:
473:
432:
398:
364:
260:
259:
258:
245:
231:Law of Comparison
168:. In theoretical
125:
124:
16:(Redirected from
3864:
3504:
3497:
3490:
3481:
3463:
3442:
3413:
3392:
3387:
3377:
3367:
3342:(1626): 2711â6.
3326:
3313:
3287:
3249:
3231:
3221:
3197:
3179:
3150:
3112:
3106:
3100:
3094:
3088:
3082:
3073:
3067:
3054:
3048:
3039:
3038:
3036:
3034:
3019:
3013:
3007:
3001:
3000:
2952:
2946:
2945:
2913:
2907:
2906:
2876:
2870:
2864:
2858:
2852:
2846:
2845:, p. xxvii.
2840:
2834:
2828:
2822:
2816:
2810:
2804:
2798:
2792:
2786:
2777:
2691:
2680:
2678:
2677:
2672:
2667:
2662:
2661:
2660:
2647:
2642:
2640:
2632:
2631:
2630:
2611:
2606:
2604:
2596:
2595:
2586:
2581:
2566:
2557:
2547:
2537:
2531:
2525:
2519:
2517:
2516:
2511:
2509:
2507:
2499:
2498:
2489:
2484:
2482:
2474:
2472:
2467:
2466:
2465:
2452:
2448:
2443:
2439:
2436:
2435:
2430:
2393:
2391:
2390:
2385:
2383:
2381:
2380:
2371:
2370:
2369:
2357:
2356:
2346:
2338:
2323:
2316:
2314:
2313:
2308:
2306:
2300:
2292:
2288:
2283:
2251:
2249:
2248:
2243:
2238:
2229:
2228:
2220:
2205:
2195:
2185:
2171:
2167:
2161:
2140:
2134:
2127:
2121:
2120:
2119:
2106:
2086:
2085:
2073:
2068:
2067:
2055:
2053:
2052:
2047:
2045:
2044:
2028:
2022:
2005:
1995:
1989:
1975:
1969:
1967:
1966:
1961:
1956:
1953:
1949:
1942:
1941:
1927:
1926:
1908:
1904:
1899:
1861:to studying the
1840:Wind engineering
1831:
1823:
1818:subcritical flow
1815:
1811:
1809:
1808:
1803:
1798:
1789:
1785:
1780:
1765:
1759:
1757:
1756:
1751:
1746:
1743:
1734:
1728:
1724:
1719:
1704:
1702:
1701:
1696:
1691:
1689:
1681:
1676:
1661:
1655:
1649:
1643:
1633:
1612:
1610:
1609:
1604:
1599:
1596:
1594:
1586:
1581:
1577:
1572:
1571:
1566:
1535:
1526:
1516:
1510:
1504:
1502:
1501:
1496:
1491:
1482:
1478:
1473:
1472:
1467:
1451:
1418:
1414:
1412:
1411:
1406:
1404:
1403:
1395:
1392:
1390:
1389:
1384:
1372:
1364:
1363:
1354:
1352:
1341:
1336:
1331:
1323:
1321:
1310:
1308:
1300:
1299:
1290:
1281:
1279:
1278:
1273:
1271:
1267:
1266:
1265:
1264:
1254:
1240:
1221:
1210:
1170:
1168:
1167:
1162:
1160:
1159:
1151:
1148:
1146:
1145:
1140:
1128:
1123:
1118:
1110:
1108:
1097:
1095:
1087:
1086:
1077:
1068:
1066:
1065:
1060:
1058:
1047:
1002:
1000:
999:
994:
992:
987:
985:
984:
979:
967:
962:
958:
957:
949:
944:
936:
920:
918:
910:
909:
900:
885:Burgers equation
874:Burgers equation
870:
862:
855:
853:
852:
847:
845:
840:
838:
837:
832:
820:
815:
804:
796:
794:
783:
781:
773:
772:
763:
739:
737:
736:
731:
726:
724:
722:
717:
708:
707:
697:
696:
687:
682:
663:
661:
660:
655:
650:
647:
646:
637:
636:
627:
626:
625:
616:
611:
594:
592:
591:
586:
581:
579:
578:
566:
561:
560:
555:
545:
543:
542:
533:
528:
523:
522:
517:
504:
503:
491:
490:
474:
472:
471:
462:
461:
452:
447:
446:
433:
431:
430:
418:
413:
412:
399:
397:
396:
384:
379:
378:
365:
363:
362:
350:
345:
344:
286:
280:
271:
269:
268:
263:
261:
256:
255:
251:
246:
243:
156:
148:
140:
136:
134:
133:
128:
126:
117:
113:
108:
73:
72:
69:
68:
65:
62:
59:
56:
43:
21:
3872:
3871:
3867:
3866:
3865:
3863:
3862:
3861:
3837:
3836:
3835:
3830:
3513:
3508:
3471:
3466:
3460:
3449:Fluid mechanics
3445:
3416:
3410:
3395:
3390:Fluid Mechanics
3385:
3380:
3329:
3316:
3310:
3290:
3252:
3229:
3224:
3218:
3202:Chanson, Hubert
3200:
3182:
3162:(4): 1199â227.
3153:
3124:
3120:
3115:
3107:
3103:
3095:
3091:
3083:
3076:
3068:
3057:
3049:
3042:
3032:
3030:
3021:
3020:
3016:
3008:
3004:
2954:
2953:
2949:
2915:
2914:
2910:
2903:
2878:
2877:
2873:
2865:
2861:
2853:
2849:
2841:
2837:
2829:
2825:
2817:
2813:
2805:
2801:
2793:
2789:
2778:
2774:
2770:
2761:Reynolds number
2728:
2698:
2689:
2652:
2648:
2633:
2622:
2612:
2597:
2587:
2569:
2568:
2562:
2553:
2543:
2533:
2527:
2521:
2500:
2490:
2475:
2457:
2453:
2449:
2418:
2417:
2410:
2402:gravity current
2372:
2361:
2348:
2347:
2331:
2326:
2325:
2318:
2293:
2271:
2270:
2259:
2208:
2207:
2201:
2191:
2188:circular motion
2181:
2178:
2169:
2163:
2154:
2147:
2142:
2136:
2129:
2123:
2118:
2115:
2114:
2113:
2109:
2099:
2092:
2084:
2081:
2080:
2079:
2075:
2066:
2063:
2062:
2061:
2057:
2036:
2031:
2030:
2024:
2012:
2007:
2006:is the slope),
2001:
1991:
1977:
1971:
1933:
1932:
1928:
1918:
1887:
1886:
1871:
1855:
1842:
1829:
1821:
1813:
1768:
1767:
1761:
1707:
1706:
1664:
1663:
1657:
1651:
1645:
1639:
1629:
1626:hydraulic jumps
1618:
1558:
1553:
1552:
1534:
1528:
1522:
1512:
1506:
1459:
1454:
1453:
1449:
1438:
1433:
1421:Reynolds number
1416:
1376:
1355:
1324:
1301:
1291:
1284:
1283:
1255:
1232:
1228:
1201:
1200:
1189:
1183:
1132:
1111:
1088:
1078:
1071:
1070:
1038:
1037:
1030:
1024:
1012:Stokes equation
1004:
971:
934:
930:
911:
901:
894:
893:
868:
860:
857:
824:
774:
764:
757:
756:
699:
698:
688:
670:
669:
638:
628:
617:
599:
598:
570:
550:
534:
512:
495:
482:
463:
453:
438:
422:
404:
388:
370:
354:
336:
331:
330:
328:
324:
319:
313:
301:
284:
276:
237:
236:
191:
175:Euler equations
154:
146:
138:
96:
95:
53:
49:
39:
28:
23:
22:
15:
12:
11:
5:
3870:
3868:
3860:
3859:
3854:
3852:Fluid dynamics
3849:
3839:
3838:
3832:
3831:
3829:
3828:
3823:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3773:
3768:
3763:
3758:
3753:
3748:
3743:
3738:
3737:
3736:
3726:
3721:
3720:
3719:
3714:
3704:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3554:
3549:
3544:
3539:
3534:
3529:
3524:
3518:
3515:
3514:
3509:
3507:
3506:
3499:
3492:
3484:
3478:
3477:
3470:
3469:External links
3467:
3465:
3464:
3458:
3443:
3414:
3408:
3393:
3378:
3327:
3314:
3308:
3288:
3250:
3222:
3216:
3198:
3180:
3151:
3121:
3119:
3116:
3114:
3113:
3101:
3097:Alexander 1989
3089:
3074:
3070:Alexander 1984
3055:
3040:
3014:
3010:Takahashi 2007
3002:
2967:(4): 130â137.
2947:
2928:(1): 125â146.
2908:
2901:
2871:
2859:
2847:
2835:
2823:
2811:
2809:, p. 294.
2799:
2787:
2771:
2769:
2766:
2765:
2764:
2758:
2752:
2746:
2741:
2735:
2727:
2724:
2697:
2694:
2670:
2665:
2659:
2655:
2651:
2645:
2639:
2636:
2629:
2625:
2621:
2618:
2615:
2609:
2603:
2600:
2594:
2590:
2584:
2580:
2577:
2506:
2503:
2497:
2493:
2487:
2481:
2478:
2470:
2464:
2460:
2456:
2446:
2433:
2429:
2426:
2409:
2406:
2379:
2375:
2368:
2364:
2360:
2355:
2351:
2344:
2341:
2337:
2334:
2303:
2299:
2296:
2291:
2286:
2282:
2279:
2269:is defined as
2258:
2255:
2241:
2235:
2232:
2226:
2223:
2219:
2216:
2177:
2174:
2152:
2145:
2116:
2097:
2090:
2082:
2064:
2043:
2039:
2010:
1959:
1952:
1948:
1945:
1940:
1936:
1931:
1925:
1921:
1917:
1914:
1911:
1907:
1902:
1898:
1895:
1870:
1867:
1854:
1851:
1841:
1838:
1820:, further for
1801:
1795:
1792:
1788:
1783:
1779:
1776:
1749:
1740:
1737:
1731:
1727:
1722:
1718:
1715:
1694:
1687:
1684:
1679:
1674:
1671:
1617:
1614:
1602:
1593:
1589:
1584:
1580:
1575:
1570:
1565:
1562:
1532:
1494:
1488:
1485:
1481:
1476:
1471:
1466:
1463:
1437:
1434:
1432:
1429:
1401:
1398:
1388:
1383:
1380:
1375:
1370:
1367:
1362:
1358:
1351:
1348:
1344:
1339:
1334:
1330:
1327:
1320:
1317:
1313:
1307:
1304:
1298:
1294:
1270:
1263:
1258:
1253:
1249:
1246:
1243:
1239:
1235:
1231:
1227:
1224:
1220:
1216:
1213:
1209:
1182:
1179:
1157:
1154:
1144:
1139:
1136:
1131:
1126:
1121:
1117:
1114:
1107:
1104:
1100:
1094:
1091:
1085:
1081:
1057:
1053:
1050:
1046:
1023:
1020:
991:
983:
978:
975:
970:
965:
961:
956:
952:
948:
942:
939:
933:
929:
926:
923:
917:
914:
908:
904:
883:
865:free equations
844:
836:
831:
828:
823:
818:
814:
810:
807:
802:
799:
793:
790:
786:
780:
777:
771:
767:
746:
729:
721:
716:
712:
706:
702:
695:
691:
685:
681:
678:
653:
645:
641:
635:
631:
624:
620:
614:
610:
607:
584:
577:
573:
569:
564:
559:
554:
548:
541:
537:
532:
526:
521:
516:
510:
507:
502:
498:
494:
489:
485:
480:
477:
470:
466:
460:
456:
450:
445:
441:
436:
429:
425:
421:
416:
411:
407:
402:
395:
391:
387:
382:
377:
373:
368:
361:
357:
353:
348:
343:
339:
326:
322:
312:
309:
300:
297:
289:
288:
282:
254:
249:
227:Frederic Reech
223:William Froude
190:
187:
181:. However, in
170:fluid dynamics
153:(in m/s), and
123:
120:
116:
111:
107:
104:
46:William Froude
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3869:
3858:
3855:
3853:
3850:
3848:
3845:
3844:
3842:
3827:
3824:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3782:
3779:
3777:
3774:
3772:
3769:
3767:
3764:
3762:
3759:
3757:
3754:
3752:
3749:
3747:
3744:
3742:
3739:
3735:
3732:
3731:
3730:
3727:
3725:
3722:
3718:
3715:
3713:
3710:
3709:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3567:Chandrasekhar
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3543:
3540:
3538:
3535:
3533:
3530:
3528:
3525:
3523:
3520:
3519:
3516:
3512:
3505:
3500:
3498:
3493:
3491:
3486:
3485:
3482:
3476:
3473:
3472:
3468:
3461:
3455:
3451:
3450:
3444:
3440:
3436:
3432:
3428:
3425:(3): 350â62.
3424:
3420:
3415:
3411:
3405:
3402:. CRC Press.
3401:
3400:
3394:
3391:
3384:
3379:
3375:
3371:
3366:
3361:
3357:
3353:
3349:
3345:
3341:
3337:
3333:
3328:
3324:
3320:
3315:
3311:
3305:
3301:
3297:
3293:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3261:
3257:
3251:
3247:
3243:
3240:(3): 159â63.
3239:
3235:
3228:
3223:
3219:
3213:
3209:
3208:
3203:
3199:
3195:
3191:
3190:
3185:
3181:
3177:
3173:
3169:
3165:
3161:
3157:
3152:
3148:
3144:
3140:
3136:
3132:
3128:
3123:
3122:
3117:
3110:
3105:
3102:
3098:
3093:
3090:
3086:
3081:
3079:
3075:
3071:
3066:
3064:
3062:
3060:
3056:
3052:
3047:
3045:
3041:
3028:
3024:
3018:
3015:
3011:
3006:
3003:
2998:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2958:
2951:
2948:
2943:
2939:
2935:
2931:
2927:
2923:
2919:
2912:
2909:
2904:
2898:
2894:
2890:
2886:
2882:
2875:
2872:
2869:, p. 28.
2868:
2863:
2860:
2856:
2851:
2848:
2844:
2839:
2836:
2832:
2827:
2824:
2820:
2815:
2812:
2808:
2803:
2800:
2796:
2791:
2788:
2785:
2782:
2776:
2773:
2767:
2762:
2759:
2756:
2753:
2750:
2747:
2745:
2742:
2739:
2736:
2733:
2732:Flow velocity
2730:
2729:
2725:
2723:
2720:
2716:
2712:
2710:
2709:supercritical
2705:
2703:
2695:
2693:
2687:
2681:
2668:
2663:
2657:
2653:
2649:
2643:
2637:
2634:
2627:
2619:
2616:
2607:
2601:
2598:
2592:
2588:
2582:
2565:
2559:
2556:
2551:
2546:
2541:
2536:
2530:
2526:is the mass,
2524:
2504:
2501:
2495:
2491:
2485:
2479:
2476:
2468:
2462:
2458:
2454:
2444:
2431:
2415:
2407:
2405:
2403:
2399:
2394:
2377:
2373:
2366:
2362:
2358:
2353:
2349:
2342:
2339:
2335:
2332:
2321:
2301:
2297:
2294:
2289:
2284:
2268:
2264:
2256:
2254:
2239:
2233:
2230:
2224:
2221:
2204:
2199:
2194:
2189:
2184:
2176:Stirred tanks
2175:
2173:
2166:
2159:
2155:
2148:
2139:
2132:
2126:
2112:
2104:
2100:
2093:
2078:
2072:
2060:
2041:
2037:
2027:
2021:
2017:
2013:
2004:
1999:
1994:
1988:
1984:
1980:
1974:
1957:
1950:
1946:
1943:
1938:
1934:
1929:
1923:
1919:
1915:
1912:
1909:
1905:
1900:
1882:
1880:
1876:
1868:
1866:
1864:
1860:
1852:
1850:
1847:
1839:
1837:
1835:
1834:critical flow
1827:
1819:
1799:
1793:
1790:
1786:
1781:
1764:
1747:
1738:
1735:
1729:
1725:
1720:
1692:
1685:
1682:
1677:
1672:
1669:
1660:
1654:
1648:
1642:
1637:
1632:
1627:
1623:
1615:
1613:
1600:
1591:
1587:
1582:
1578:
1573:
1568:
1550:
1545:
1543:
1539:
1531:
1525:
1520:
1515:
1509:
1492:
1486:
1483:
1479:
1474:
1469:
1442:
1435:
1430:
1428:
1426:
1422:
1396:
1386:
1373:
1368:
1365:
1360:
1342:
1337:
1332:
1328:
1311:
1305:
1302:
1292:
1268:
1241:
1229:
1225:
1222:
1214:
1211:
1198:
1194:
1188:
1180:
1178:
1176:
1171:
1152:
1142:
1129:
1124:
1119:
1115:
1098:
1092:
1089:
1079:
1051:
1048:
1035:
1029:
1021:
1019:
1017:
1013:
1009:
1003:
981:
968:
963:
959:
950:
940:
937:
931:
927:
921:
915:
890:
886:
882:
880:
876:
875:
866:
856:
834:
821:
816:
808:
800:
797:
784:
778:
775:
765:
753:
749:
745:
743:
727:
719:
714:
710:
704:
700:
693:
689:
683:
667:
651:
643:
639:
633:
629:
622:
618:
612:
595:
582:
575:
571:
562:
557:
546:
539:
535:
524:
519:
508:
500:
496:
492:
487:
478:
475:
468:
464:
458:
454:
448:
443:
439:
434:
427:
423:
419:
414:
409:
405:
400:
393:
389:
385:
380:
375:
371:
366:
359:
355:
351:
346:
341:
337:
318:
310:
308:
306:
305:hydrodynamics
298:
296:
294:
283:
279:
275:
274:
273:
252:
247:
234:
232:
228:
224:
216:
212:
209:The hulls of
207:
203:
200:
199:Belanger 1828
196:
188:
186:
184:
180:
176:
171:
167:
162:
160:
152:
151:gravity field
149:is the local
144:
143:flow velocity
141:is the local
121:
118:
114:
109:
93:
89:
85:
81:
77:
71:
47:
42:
37:
36:Froude number
33:
19:
3616:
3448:
3422:
3418:
3398:
3389:
3339:
3335:
3322:
3318:
3295:
3259:
3255:
3237:
3233:
3206:
3193:
3188:
3159:
3155:
3133:(2): 49â59.
3130:
3126:
3104:
3092:
3031:. Retrieved
3026:
3017:
3012:, p. 6.
3005:
2964:
2960:
2950:
2925:
2921:
2911:
2884:
2874:
2862:
2850:
2843:Chanson 2004
2838:
2831:Normand 1888
2826:
2819:Chanson 2009
2814:
2802:
2797:, p. 7.
2790:
2775:
2721:
2717:
2713:
2706:
2699:
2688:occurs with
2682:
2567:as follows:
2563:
2560:
2554:
2544:
2534:
2528:
2522:
2411:
2395:
2319:
2266:
2260:
2202:
2192:
2182:
2179:
2164:
2157:
2150:
2143:
2137:
2130:
2124:
2110:
2102:
2095:
2088:
2076:
2070:
2058:
2025:
2019:
2015:
2008:
2002:
1992:
1986:
1982:
1978:
1972:
1883:
1879:debris flows
1872:
1856:
1846:wind effects
1843:
1833:
1762:
1658:
1652:
1646:
1640:
1630:
1619:
1548:
1546:
1529:
1523:
1513:
1507:
1447:
1197:Stokes's law
1190:
1172:
1031:
1005:
892:
888:
884:
872:
864:
858:
755:
751:
747:
666:Euler number
596:
320:
302:
292:
290:
281:= flow speed
277:
235:
230:
220:
214:
213:(above) and
210:
192:
163:
91:
80:flow inertia
40:
35:
29:
3821:Weissenberg
2867:Newman 1977
1425:dissipative
166:Mach number
3841:Categories
3741:Richardson
3522:Archimedes
3325:: 257â261.
3118:References
2807:White 1999
2738:Body force
1875:avalanches
1863:locomotion
1193:Pascal law
1185:See also:
1034:Pascal law
1026:See also:
1014:is a pure
315:See also:
145:(in m/s),
3826:Womersley
3717:turbulent
3697:Ohnesorge
3682:Marangoni
3647:Iribarren
3572:Damköhler
3557:Capillary
3300:MIT Press
3284:244507908
3276:2524-3462
3147:120138903
2989:0036-8733
2942:0952-8369
2855:Shih 2009
2795:Shih 2009
2374:ρ
2363:ρ
2359:−
2350:ρ
2225:ω
2190:), where
1944:−
1910:β
1859:allometry
1853:Allometry
1822:Fr > 1
1814:Fr < 1
1400:^
1357:∇
1333:ρ
1326:∇
1248:∇
1234:∇
1226:μ
1208:σ
1156:^
1120:ρ
1113:∇
1045:σ
951:⊗
928:⋅
925:∇
913:∂
903:∂
813:σ
809:⋅
806:∇
801:ρ
701:ρ
568:σ
563:≡
558:∗
553:σ
525:≡
520:∗
506:∇
493:≡
488:∗
484:∇
449:≡
444:∗
415:≡
410:∗
381:≡
376:∗
356:ρ
352:ρ
347:≡
342:∗
338:ρ
161:(in m).
3801:Suratman
3791:Strouhal
3771:Sherwood
3734:magnetic
3729:Reynolds
3724:Rayleigh
3712:magnetic
3552:Brinkman
3439:15760752
3374:17711833
3356:25249388
3294:(1977).
3204:(2004).
3186:(1828).
2997:24936872
2726:See also
2690:Fr â 0.5
2550:velocity
2414:pendulum
2336:′
2298:′
2135:, while
1622:tsunamis
664:and the
44:, after
3781:Stanton
3776:Shields
3766:Scruton
3761:Schmidt
3707:Prandtl
3692:Nusselt
3667:Laplace
3662:Knudsen
3652:Kapitza
3637:Görtler
3632:Grashof
3622:Galilei
3587:Deborah
3532:Bagnold
3365:2279215
3176:2678167
2969:Bibcode
2686:running
2548:is the
2538:is the
1996:is the
1828:. When
1636:average
1634:is the
1419:is the
272:where:
189:Origins
88:gravity
82:to the
74:) is a
3811:Ursell
3806:Taylor
3796:Stuart
3786:Stokes
3751:Rossby
3746:Roshko
3702:PĂ©clet
3687:Morton
3627:Graetz
3617:Froude
3607:Eötvös
3597:Eckert
3592:Dukhin
3562:Cauchy
3527:Atwood
3456:
3437:
3406:
3372:
3362:
3354:
3306:
3282:
3274:
3214:
3174:
3145:
3033:31 May
2995:
2987:
2940:
2899:
2520:where
2317:where
2200:) and
1970:where
1830:Fr â 1
1521:, and
1505:where
1415:where
869:Eu â 0
861:Fr â â
137:where
34:, the
3816:Weber
3756:Rouse
3672:Lewis
3642:Hagen
3612:Euler
3602:Ekman
3577:Darcy
3537:Bejan
3386:(PDF)
3352:JSTOR
3280:S2CID
3230:(PDF)
3192:[
3143:S2CID
3029:. n.d
2993:JSTOR
2768:Notes
2696:Usage
215:raven
157:is a
3677:Mach
3582:Dean
3547:Bond
3542:Biot
3454:ISBN
3435:PMID
3404:ISBN
3370:PMID
3304:ISBN
3272:ISSN
3212:ISBN
3172:PMID
3035:2019
2985:ISSN
2938:ISSN
2897:ISBN
2542:and
2265:the
2141:and
2074:and
2018:sin
1985:cos
1877:and
1812:For
1624:and
1538:drag
1195:and
211:swan
177:are
3427:doi
3360:PMC
3344:doi
3340:274
3264:doi
3242:doi
3238:135
3164:doi
3135:doi
2977:doi
2965:264
2930:doi
2926:183
2889:doi
2198:rpm
2133:âȘ 1
2117:pot
2083:pot
2065:pot
1990:, (
881:):
285:LWL
257:LWL
193:In
30:In
3843::
3433:.
3423:21
3421:.
3388:,
3368:.
3358:.
3350:.
3338:.
3334:.
3323:29
3321:.
3302:.
3278:.
3270:.
3260:39
3258:.
3236:.
3232:.
3170:.
3160:69
3158:.
3141:.
3129:.
3077:^
3058:^
3043:^
3025:.
2991:.
2983:.
2975:.
2963:.
2959:.
2936:.
2924:.
2920:.
2895:.
2883:.
2783:)
2183:Ïr
2170:Fr
2165:ÎČh
2156:â
2131:ÎČh
2125:ÎČh
2101:â
2087:=
2071:ÎČh
2069:=
2023:,
2014:=
2000:,
1983:gK
1981:=
1836:.
1662::
1544:.
1533:wl
1450:Fn
1417:Re
1177:.
1018:.
891:)
754:)
668::
197:,
64:uË
48:,
41:Fr
3503:e
3496:t
3489:v
3462:.
3441:.
3429::
3412:.
3376:.
3346::
3312:.
3286:.
3266::
3248:.
3244::
3220:.
3178:.
3166::
3149:.
3137::
3131:3
3111:.
3099:.
3087:.
3072:.
3037:.
2999:.
2979::
2971::
2944:.
2932::
2905:.
2891::
2857:.
2669:.
2664:g
2658:2
2654:f
2650:l
2644:=
2638:l
2635:g
2628:2
2624:)
2620:f
2617:l
2614:(
2608:=
2602:l
2599:g
2593:2
2589:v
2583:=
2579:r
2576:F
2564:f
2555:l
2545:v
2535:g
2529:l
2523:m
2505:l
2502:g
2496:2
2492:v
2486:=
2480:g
2477:m
2469:l
2463:2
2459:v
2455:m
2445:=
2432:=
2428:r
2425:F
2378:1
2367:2
2354:1
2343:g
2340:=
2333:g
2322:âČ
2320:g
2302:h
2295:g
2290:u
2285:=
2281:r
2278:F
2240:.
2234:g
2231:r
2222:=
2218:r
2215:F
2203:r
2193:Ï
2186:(
2160:)
2158:x
2153:d
2151:x
2149:(
2146:g
2144:s
2138:u
2111:E
2105:)
2103:x
2098:d
2096:x
2094:(
2091:g
2089:s
2077:E
2059:E
2042:d
2038:x
2026:x
2020:ζ
2016:g
2011:g
2009:s
2003:ζ
1993:K
1987:ζ
1979:ÎČ
1973:u
1958:,
1951:)
1947:x
1939:d
1935:x
1930:(
1924:g
1920:s
1916:+
1913:h
1906:u
1901:=
1897:r
1894:F
1800:.
1794:d
1791:g
1787:U
1782:=
1778:r
1775:F
1763:d
1748:.
1739:B
1736:A
1730:g
1726:U
1721:=
1717:r
1714:F
1693:,
1686:B
1683:A
1678:g
1673:=
1670:c
1659:B
1653:A
1647:g
1641:c
1631:U
1601:.
1592:3
1588:V
1583:g
1579:u
1574:=
1569:V
1564:n
1561:F
1530:L
1524:L
1514:g
1508:u
1493:,
1487:L
1484:g
1480:u
1475:=
1470:L
1465:n
1462:F
1397:g
1387:2
1382:r
1379:F
1374:1
1369:+
1366:u
1361:2
1350:e
1347:R
1343:1
1338:=
1329:p
1319:u
1316:E
1312:+
1306:t
1303:D
1297:u
1293:D
1269:)
1262:T
1257:)
1252:u
1245:(
1242:+
1238:u
1230:(
1223:+
1219:I
1215:p
1212:=
1153:g
1143:2
1138:r
1135:F
1130:1
1125:=
1116:p
1106:u
1103:E
1099:+
1093:t
1090:D
1084:u
1080:D
1056:I
1052:p
1049:=
990:g
982:2
977:r
974:F
969:1
964:=
960:)
955:u
947:u
941:2
938:1
932:(
922:+
916:t
907:u
887:(
843:g
835:2
830:r
827:F
822:1
817:=
798:1
792:u
789:E
785:+
779:t
776:D
770:u
766:D
750:(
728:,
720:2
715:0
711:u
705:0
694:0
690:p
684:=
680:u
677:E
652:,
644:0
640:r
634:0
630:g
623:0
619:u
613:=
609:r
606:F
583:,
576:0
572:p
547:,
540:0
536:g
531:g
515:g
509:,
501:0
497:r
479:,
476:t
469:0
465:r
459:0
455:u
440:t
435:,
428:0
424:r
420:r
406:r
401:,
394:0
390:u
386:u
372:u
367:,
360:0
327:0
323:0
278:u
253:u
248:=
155:L
147:g
139:u
122:L
119:g
115:u
110:=
106:r
103:F
70:/
67:d
61:r
58:f
55:Ë
52:/
38:(
20:)
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