2335:
1250:
1005:
2111:
1478:
1245:{\displaystyle {\begin{aligned}{\boldsymbol {\omega }}=\nabla \times \mathbf {v} =\left({\dfrac {\partial v_{z}}{\partial y}}-{\dfrac {\partial v_{y}}{\partial z}},{\dfrac {\partial v_{x}}{\partial z}}-{\dfrac {\partial v_{z}}{\partial x}},{\dfrac {\partial v_{y}}{\partial x}}-{\dfrac {\partial v_{x}}{\partial y}}\right)\,.\end{aligned}}}
2609:
A rotating-vane vorticity meter was invented by
Russian hydraulic engineer A. Ya. Milovich (1874â1958). In 1913 he proposed a cork with four blades attached as a device qualitatively showing the magnitude of the vertical projection of the vorticity and demonstrated a motion-picture photography of the
2664:
is the vorticity relative to the Earth induced by the air velocity field. This air velocity field is often modeled as a two-dimensional flow parallel to the ground, so that the relative vorticity vector is generally scalar rotation quantity perpendicular to the ground. Vorticity is positive when â
2639:
may be approximated by assuming that each spanwise segment of the wing has a semi-infinite trailing vortex behind it. It is then possible to solve for the strength of the vortices using the criterion that there be no flow induced through the surface of the wing. This procedure is called the vortex
2328:
Another way to visualize vorticity is to imagine that, instantaneously, a tiny part of the continuum becomes solid and the rest of the flow disappears. If that tiny new solid particle is rotating, rather than just moving with the flow, then there is vorticity in the flow. In the figure below, the
2159:, where most particles rotate about some straight axis, with speed inversely proportional to their distances to that axis. A small parcel of continuum that does not straddle the axis will be rotated in one sense but sheared in the opposite sense, in such a way that their mean angular velocity
1922:
1353:
1849:
2490:
2361:), the vorticity field can be modeled by a collection of discrete vortices, the vorticity being negligible everywhere except in small regions of space surrounding the axes of the vortices. This is true in the case of two-dimensional
2152:, all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.
2613:
Rotating-vane vorticity meters are commonly shown in educational films on continuum mechanics (famous examples include the NCFMF's "Vorticity" and "Fundamental
Principles of Flow" by Iowa Institute of Hydraulic Research).
2560:
2856:
773:
1642:
2106:{\displaystyle {\boldsymbol {\Omega }}={\frac {1}{2}}\left\quad {\text{or}}\quad \Omega _{ij}={\frac {1}{2}}\left({\frac {\partial v_{i}}{\partial x_{j}}}-{\frac {\partial v_{i}}{\partial x_{i}}}\right).}
1473:{\displaystyle {\begin{aligned}{\boldsymbol {\omega }}=\nabla \times \mathbf {v} =\left({\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)\mathbf {e} _{z}\,.\end{aligned}}}
905:
1358:
1010:
1739:
2192:
2185:
2178:
2307:
2300:
2293:
2279:
2272:
2265:
80:
1731:
1709:
1664:
967:
933:
855:
822:
701:
1914:
2577:) is the integral of the vorticity across a cross-section of the tube, and is the same everywhere along the tube (because vorticity has zero divergence). It is a consequence of
1885:
2573:
is the surface in the continuum formed by all vortex lines passing through a given (reducible) closed curve in the continuum. The 'strength' of a vortex tube (also called
1526:
993:
730:
1345:
1572:
2402:
796:
2791:
1549:
2879:
1684:
1596:
1316:
1296:
1276:
662:
motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along with the
3286:
2376:
Vorticity is useful for understanding how ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a
2701:). The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the vertical direction, but the potential vorticity is
2585:) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence.
2380:
of vorticity away from the vortex cores into the general flow field; this flow is accounted for by a diffusion term in the vorticity transport equation.
3727:
2498:
2799:
738:
3251:
1255:
In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
3014:
629:
3256:"Professor Milovich's float", as Joukovsky refers this vorticity meter to, is schematically shown in figure on page 196 of Collected works.
2596:. This phenomenon occurs in the formation of a bathtub vortex in outflowing water, and the build-up of a tornado by rising air currents.
3629:
1604:
2665:
looking down onto the Earth's surface â the wind turns counterclockwise. In the northern hemisphere, positive vorticity is called
2141:
2680:
is computed from the air velocity relative to an inertial frame, and therefore includes a term due to the Earth's rotation, the
3720:
3670:
3661:
3652:
2155:
Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal
674:
and provides a convenient framework for understanding a variety of complex flow phenomena, such as the formation and motion of
3684:
3869:
3423:
3404:
876:
3180:
2717:
of air masses in the atmosphere over the timescale of a few days, particularly when viewed on levels of constant entropy.
1844:{\displaystyle \Omega _{ij}=\varepsilon _{ijk}\omega _{k},\qquad \omega _{i}={\frac {1}{2}}\varepsilon _{ijk}\Omega _{jk}}
4176:
2754:(GCMs), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a
3889:
3713:
3618:
3585:
3571:
3554:
3537:
3520:
3481:
3464:
3442:
3383:
3147:
3088:
3060:
622:
3488:
4037:
3283:
2955:
2582:
36:
4084:
3268:
595:
2396:
is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation
3916:
3884:
3831:
3371:
3076:
2898:
2744:
2721:
2253:
1714:
1692:
1647:
950:
916:
838:
805:
684:
296:
133:
4166:
3218:
2960:
2645:
2641:
1495:
1484:
615:
336:
222:
2592:
of the square of its magnitude) can be intensified when a vortex line is extended â a phenomenon known as
4145:
2970:
2649:
2351:
1894:
825:
291:
200:
83:
1857:
4079:
3836:
3302:
Scheeler, Martin W.; van Rees, Wim M.; Kedia, Hridesh; Kleckner, Dustin; Irvine, William T. M. (2017).
2903:
2702:
2329:
left subfigure demonstrates no vorticity, and the right subfigure demonstrates existence of vorticity.
947:
Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by
4059:
3993:
3821:
2751:
207:
3926:
3778:
3608:
2149:
502:
497:
286:
279:
112:
3140:
Modern
Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics
3011:
1509:
976:
713:
4171:
4127:
4049:
3497:
3223:
2762:
2578:
1689:
Since vorticity is a axial vector, it can be associated with a second-order antisymmetric tensor
832:
565:
560:
229:
2485:{\displaystyle {\frac {dx}{\omega _{x}}}={\frac {dy}{\omega _{y}}}={\frac {dz}{\omega _{z}}}\,,}
1321:
3981:
3941:
3894:
3879:
3861:
117:
2743:) over a limited amount of time (a few days). In the 1950s, the first successful programs for
1554:
4028:
3921:
3911:
3874:
2694:
2563:
2132:
The vorticity may be nonzero even when all particles are flowing along straight and parallel
1318:-axis, and therefore can be expressed as a scalar field multiplied by a constant unit vector
996:
781:
540:
158:
2365:(i.e. two-dimensional zero viscosity flow), in which case the flowfield can be modeled as a
2357:
In many real flows where the viscosity can be neglected (more precisely, in flows with high
4107:
4074:
3976:
3971:
3936:
3315:
2729:
910:
378:
195:
175:
163:
107:
2767:
8:
4069:
4054:
3599:
2982:
2755:
2740:
2713:
flow predominates in the atmosphere, the potential vorticity is useful as an approximate
2689:
643:
580:
428:
321:
27:
3319:
3240:. Vol. 4. Moscow; Leningrad. 1937. pp. 193â216, 231â233 (abstract in English).
1531:
3841:
3349:
3245:
2987:
2965:
2918:
2864:
2681:
2347:
1888:
1669:
1581:
1488:
1301:
1281:
1261:
970:
704:
600:
234:
190:
185:
2121:
In a mass of continuum that is rotating like a rigid body, the vorticity is twice the
4181:
4122:
4018:
3906:
3748:
3697:
3581:
3567:
3550:
3533:
3516:
3477:
3460:
3438:
3419:
3400:
3379:
3341:
3333:
3143:
3084:
3056:
2933:
2666:
2593:
1504:
217:
168:
3705:
3353:
935:
is always perpendicular to the plane of the flow, and can therefore be considered a
3681:
3525:
3452:
3323:
2943:
2938:
2122:
858:
555:
530:
443:
418:
413:
368:
2652:, lift per unit of span is the product of circulation, airspeed, and air density.
3773:
3688:
3636:
3394:
3290:
3272:
3184:
3177:
3018:
2881:. In atmospheric science, helicity of the air motion is important in forecasting
2693:
is absolute vorticity divided by the vertical spacing between levels of constant
2589:
2358:
2125:
vector of that rotation. This is the case, for example, in the central core of a
866:
545:
469:
433:
383:
314:
303:
248:
150:
3612:". Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma.
4064:
2366:
2362:
2126:
862:
799:
667:
663:
550:
408:
373:
274:
180:
16:
Pseudovector field describing the local rotation of a continuum near some point
2644:. The strengths of the vortices are then summed to find the total approximate
1711:(the so-called vorticity or rotation tensor), which is said to be the dual of
4160:
3846:
3559:
3337:
3030:
2737:
2555:{\displaystyle {\boldsymbol {\omega }}=(\omega _{x},\omega _{y},\omega _{z})}
2370:
2137:
1492:
708:
590:
423:
3433:
Guyon, Etienne; Hulin, Jean-Pierre; Petit, Luc; Mitescu, Catalin D. (2001).
3328:
3303:
3051:
Guyon, Etienne; Hulin, Jean-Pierre; Petit, Luc; Mitescu, Catalin D. (2001).
2334:
4023:
3998:
3931:
3808:
3542:
3532:". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994.
3345:
3235:
2913:
2851:{\displaystyle H(t)=\int _{V}\mathbf {v} \cdot {\boldsymbol {\omega }}\,dV}
2632:
2628:
2145:
1733:. The relation between the two quantities, in index notation, are given by
936:
768:{\displaystyle {\boldsymbol {\omega }}\equiv \nabla \times \mathbf {v} \,,}
655:
651:
575:
570:
535:
267:
2191:
2184:
2177:
4112:
3951:
3851:
3644:
2928:
2725:
2714:
2670:
2636:
1599:
675:
585:
488:
2306:
2299:
2292:
2278:
2271:
2264:
4117:
4089:
4003:
3798:
3783:
3763:
3135:
2992:
1487:(line integral of the velocity) along a closed path by the (classical)
870:
507:
403:
1891:. The vorticity tensor is simply the antisymmetric part of the tensor
3959:
3768:
3753:
3033:(2015), "Fluid Dynamics", in Nicholas J. Higham; et al. (eds.),
2908:
2882:
2710:
2706:
2377:
1575:
479:
474:
308:
3265:
4137:
4099:
4013:
3758:
3304:"Complete measurement of helicity and its dynamics in vortex tubes"
2133:
1258:
In a two-dimensional flow where the velocity is independent of the
659:
458:
363:
343:
329:
3633:". School of the Environment, University of Leeds. September 2001.
4008:
3609:
A Primer on
Vorticity for Application in Supercells and Tornadoes
2886:
2698:
2346:
The evolution of the vorticity field in time is described by the
212:
3671:
Kelvin's vorticity theorem for incompressible or barotropic flow
2610:
float's motion on the water surface in a model of a river bend.
3199:(1967), Section 2.6, Cambridge University Press ISBN 0521098173
2923:
2156:
1637:{\displaystyle {\boldsymbol {\omega }}\cdot (\mathbf {n} \,dA)}
824:
could be determined by marking parts of a continuum in a small
671:
353:
3816:
3301:
3284:
Films by Hunter Rouse â IIHR â Hydroscience & Engineering
2733:
2286:
Relative velocities (magnified) around the highlighted point
257:
3653:
The vorticity equation: incompressible and barotropic fluids
3416:
Introduction to
Theoretical and Computational Fluid Dynamics
2252:
is the distance to the center of the vortex and â indicates
3793:
2673:; the nomenclature is reversed in the Southern Hemisphere.
2588:
In a three-dimensional flow, vorticity (as measured by the
1298:-component, the vorticity vector is always parallel to the
3826:
942:
393:
3692:". (includes a collection of FORTRAN vorticity program)
3221:(1914). "On the motion of water at a turn of a river".
900:{\displaystyle \nabla \cdot {\boldsymbol {\omega }}=0.}
3432:
3050:
3735:
3513:
Elementary
Account Of Vorticity And Related Equations
2867:
2802:
2770:
2501:
2405:
1925:
1897:
1860:
1742:
1717:
1695:
1672:
1650:
1607:
1584:
1557:
1534:
1512:
1356:
1324:
1304:
1284:
1264:
1201:
1169:
1137:
1105:
1073:
1041:
1008:
979:
953:
919:
879:
841:
808:
784:
741:
716:
687:
39:
2724:
is the simplest way for forecasting the movement of
3580:", 3rd ed. Academic Press, Orlando, Florida. 1985.
2750:In modern numerical weather forecasting models and
869:. By its own definition, the vorticity vector is a
3055:. Oxford University Press. pp. 105, 268â310.
2873:
2850:
2785:
2554:
2484:
2258:Absolute velocities around the highlighted point:
2105:
1908:
1879:
1843:
1725:
1703:
1678:
1658:
1636:
1590:
1566:
1543:
1520:
1472:
1339:
1310:
1290:
1270:
1244:
987:
961:
927:
899:
849:
816:
790:
767:
724:
695:
74:
3515:". Cambridge University Press. January 30, 2005.
3023:
4158:
2604:
2383:
3492:"' The Weather Channel Interactive, Inc.. 2004.
3162:
3134:
3392:
3217:
3037:, Princeton University Press, pp. 467â476
3035:The Princeton Companion to Applied Mathematics
3721:
828:of the point in question, and watching their
623:
3870:Convective available potential energy (CAPE)
3295:
3266:National Committee for Fluid Mechanics Films
1483:The vorticity is also related to the flow's
861:vector of those particles relative to their
3628:ENVI 2210 â Atmosphere and Ocean Dynamics,
3142:. Princeton University Press. p. 741.
3046:
3044:
3012:Lecture Notes from University of Washington
2761:Related to the concept of vorticity is the
835:as they move along the flow. The vorticity
75:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
3728:
3714:
3566:". Van Nostrand Reinhold, New York. 1977.
3413:
3250:: CS1 maint: location missing publisher (
3178:Introduction to Astrophysical Gas Dynamics
2861:where the integral is over a given volume
2140:(that is, if the flow speed varies across
630:
616:
3641:Astronomy 202: Astrophysical Gas Dynamics
3451:
3327:
3097:
2976:
2841:
2478:
1624:
1462:
1234:
761:
3662:Interpretation of the vorticity equation
3041:
3370:
3083:. Oxford University Press. p. 10.
3075:
3029:
2837:
2655:
2503:
1726:{\displaystyle {\boldsymbol {\omega }}}
1719:
1704:{\displaystyle {\boldsymbol {\Omega }}}
1659:{\displaystyle {\boldsymbol {\omega }}}
1652:
1609:
1362:
1014:
962:{\displaystyle {\boldsymbol {\omega }}}
955:
928:{\displaystyle {\boldsymbol {\omega }}}
921:
887:
850:{\displaystyle {\boldsymbol {\omega }}}
843:
817:{\displaystyle {\boldsymbol {\omega }}}
810:
743:
696:{\displaystyle {\boldsymbol {\omega }}}
689:
4159:
3698:Mesoscale Compressible Community (MC2)
3393:Landau, L. D.; Lifshitz, E.M. (1987).
943:Mathematical definition and properties
3709:
3549:". Cambridge University Press; 2002.
3476:, Pitman Publishing Limited, London
3156:
2617:
995:describing the continuum motion. In
3832:Convective condensation level (CCL)
3578:Mathematical Methods for Physicists
2599:
1909:{\displaystyle \nabla \mathbf {v} }
13:
4038:Equivalent potential temperature (
3623:". Foundations of Fluid Mechanics.
3505:
2079:
2064:
2042:
2027:
1994:
1963:
1949:
1898:
1880:{\displaystyle \varepsilon _{ijk}}
1829:
1744:
1666:is the vorticity at the center of
1561:
1436:
1421:
1406:
1391:
1369:
1219:
1204:
1187:
1172:
1155:
1140:
1123:
1108:
1091:
1076:
1059:
1044:
1021:
880:
785:
750:
14:
4193:
3890:Conditional symmetric instability
3736:Meteorological data and variables
3702:". (Potential vorticity analysis)
3591:
3547:Vorticity and Incompressible Flow
3457:An Introduction to Fluid Dynamics
3197:An Introduction to Fluid Dynamics
2949:
2648:about the wing. According to the
666:. It is an important quantity in
3837:Lifting condensation level (LCL)
3125:Guyon, et al (2001), pp. 289â290
2829:
2350:, which can be derived from the
2333:
2305:
2298:
2291:
2277:
2270:
2263:
2190:
2183:
2176:
1967:
1953:
1927:
1902:
1697:
1620:
1514:
1452:
1376:
1028:
981:
757:
718:
3822:Cloud condensation nuclei (CCN)
3364:
3277:
3259:
3211:
3202:
3189:
3171:
1992:
1986:
1788:
4085:Wet-bulb potential temperature
3927:Level of free convection (LFC)
3459:, Cambridge University Press,
3138:; Blandford, Roger D. (2017).
3128:
3119:
3110:
3069:
3005:
2812:
2806:
2780:
2774:
2622:
2549:
2510:
1972:
1960:
1631:
1616:
1331:
681:Mathematically, the vorticity
1:
4128:Pressure-gradient force (PGF)
4050:Sea surface temperature (SST)
3885:Convective momentum transport
2998:
2899:Barotropic vorticity equation
2745:numerical weather forecasting
2722:barotropic vorticity equation
2605:Rotating-vane vorticity meter
2384:Vortex lines and vortex tubes
2248:is the velocity of the flow,
3942:Bulk Richardson number (BRN)
3619:Vorticity Transport Theorems
3603:". Scienceworld.wolfram.com.
2669:, and negative vorticity is
2642:computational fluid dynamics
2583:Kelvin's circulation theorem
2341:
2148:within a pipe with constant
1521:{\displaystyle \mathbf {n} }
988:{\displaystyle \mathbf {v} }
865:, oriented according to the
725:{\displaystyle \mathbf {v} }
7:
4146:Maximum potential intensity
3912:Free convective layer (FCL)
3875:Convective inhibition (CIN)
3700:Real-Time Model Predictions
3617:NavierâStokes Equations --
3437:. Oxford University Press.
3418:. Oxford University Press.
3378:. Oxford University Press.
2892:
2562:is the vorticity vector in
2167:
2116:
10:
4198:
4080:Wet-bulb globe temperature
3937:Maximum parcel level (MPL)
3606:Doswell III, Charles A., "
3399:(2nd ed.). Elsevier.
2752:general circulation models
2304:
2297:
2290:
2276:
2269:
2262:
2189:
2182:
2175:
2161:about their center of mass
1340:{\displaystyle {\hat {z}}}
4177:Meteorological quantities
4136:
4098:
4060:Thermodynamic temperature
3994:Forest fire weather index
3950:
3860:
3807:
3741:
3643:". Astronomy Department,
3501:". Integrated Publishing.
3376:Elementary Fluid Dynamics
3271:October 21, 2016, at the
3081:Elementary Fluid Dynamics
3017:October 16, 2015, at the
2285:
2243:
2212:Parallel flow with shear
2170:
1887:is the three-dimensional
658:that describes the local
3982:Equivalent temperature (
3895:Convective temperature (
3779:Surface weather analysis
3530:Vorticity and Turbulence
2747:utilized that equation.
1567:{\displaystyle d\Gamma }
857:would be twice the mean
134:ClausiusâDuhem (entropy)
84:Fick's laws of diffusion
4029:Potential temperature (
3774:Surface solar radiation
3564:Physical Fluid Dynamics
3545:, Andrea L. Bertozzi, "
3414:Pozrikidis, C. (2011).
3329:10.1126/science.aam6897
3289:April 21, 2016, at the
3224:Matematicheskii Sbornik
2971:KuttaâJoukowski theorem
2695:(potential) temperature
2650:KuttaâJoukowski theorem
2352:NavierâStokes equations
2144:). For example, in the
1278:-coordinate and has no
791:{\displaystyle \nabla }
292:NavierâStokes equations
230:Material failure theory
4019:Relative humidity (RH)
3907:Equilibrium level (EL)
3880:Convective instability
3435:Physical Hydrodynamics
3208:Batchelor, section 5.2
3183:June 14, 2011, at the
3053:Physical Hydrodynamics
2885:and the potential for
2875:
2852:
2787:
2556:
2486:
2199:Rigid-body-like vortex
2107:
1910:
1881:
1845:
1727:
1705:
1680:
1660:
1638:
1592:
1568:
1545:
1522:
1474:
1341:
1312:
1292:
1272:
1246:
989:
973:of the velocity field
963:
929:
901:
851:
818:
792:
769:
726:
697:
76:
3597:Weisstein, Eric W., "
3472:Clancy, L.J. (1975),
3163:Kundu P and Cohen I.
3116:Acheson (1990), p. 15
2876:
2853:
2788:
2671:anticyclonic rotation
2581:(or equivalently, of
2564:Cartesian coordinates
2557:
2487:
2108:
1911:
1882:
1846:
1728:
1706:
1681:
1661:
1639:
1593:
1569:
1546:
1523:
1475:
1342:
1313:
1293:
1273:
1247:
997:Cartesian coordinates
990:
964:
930:
902:
852:
819:
793:
770:
727:
698:
287:Bernoulli's principle
280:Archimedes' principle
77:
4108:Atmospheric pressure
4075:Wet-bulb temperature
3977:Dry-bulb temperature
3972:Dew point depression
3526:Chorin, Alexandre J.
2977:Atmospheric sciences
2904:D'Alembert's paradox
2865:
2800:
2786:{\displaystyle H(t)}
2768:
2656:Atmospheric sciences
2635:distribution over a
2579:Helmholtz's theorems
2499:
2403:
1923:
1895:
1858:
1740:
1715:
1693:
1670:
1648:
1605:
1582:
1555:
1532:
1510:
1354:
1322:
1302:
1282:
1262:
1006:
977:
951:
917:
911:two-dimensional flow
877:
839:
806:
782:
739:
714:
685:
668:the dynamical theory
379:Cohesion (chemistry)
201:Infinitesimal strain
37:
4167:Continuum mechanics
4070:Virtual temperature
4055:Temperature anomaly
3749:Adiabatic processes
3320:2017Sci...357..487S
2983:Prognostic equation
2966:Vorticity equations
2756:prognostic equation
2741:geopotential height
2690:potential vorticity
2215:Irrotational vortex
2157:irrotational vortex
644:continuum mechanics
297:Poiseuille equation
28:Continuum mechanics
22:Part of a series on
3842:Precipitable water
3687:2004-06-22 at the
3626:Parker, Douglas, "
2988:Carl-Gustaf Rossby
2919:Velocity potential
2871:
2848:
2783:
2682:Coriolis parameter
2678:absolute vorticity
2662:relative vorticity
2552:
2482:
2348:vorticity equation
2103:
1906:
1889:Levi-Civita tensor
1877:
1841:
1723:
1701:
1676:
1656:
1634:
1588:
1564:
1551:, the circulation
1544:{\displaystyle dA}
1541:
1518:
1491:. Namely, for any
1470:
1468:
1337:
1308:
1288:
1268:
1242:
1240:
1227:
1195:
1163:
1131:
1099:
1067:
985:
959:
925:
897:
847:
814:
788:
765:
722:
693:
654:(or axial vector)
503:Magnetorheological
498:Electrorheological
235:Fracture mechanics
72:
4154:
4153:
4123:Pressure gradient
3932:Lifted index (LI)
3425:978-0-19-975207-2
3406:978-0-08-057073-0
3314:(6350): 487â491.
2934:Vortex stretching
2874:{\displaystyle V}
2667:cyclonic rotation
2618:Specific sciences
2594:vortex stretching
2476:
2451:
2426:
2324:
2323:
2093:
2056:
2017:
1990:
1942:
1810:
1679:{\displaystyle C}
1591:{\displaystyle C}
1443:
1413:
1334:
1311:{\displaystyle z}
1291:{\displaystyle z}
1271:{\displaystyle z}
1226:
1194:
1162:
1130:
1098:
1066:
969:, defined as the
640:
639:
515:
514:
449:
448:
218:Contact mechanics
141:
140:
70:
4189:
3730:
3723:
3716:
3707:
3706:
3637:Graham, James R.
3615:Cramer, M. S., "
3543:Majda, Andrew J.
3489:Weather Glossary
3469:
3453:Batchelor, G. K.
3448:
3429:
3410:
3389:
3358:
3357:
3331:
3299:
3293:
3281:
3275:
3263:
3257:
3255:
3249:
3241:
3233:. Reprinted in:
3232:
3215:
3209:
3206:
3200:
3195:G.K. Batchelor,
3193:
3187:
3175:
3169:
3168:
3160:
3154:
3153:
3132:
3126:
3123:
3117:
3114:
3108:
3101:
3095:
3094:
3073:
3067:
3066:
3048:
3039:
3038:
3027:
3021:
3009:
2944:Wingtip vortices
2939:Horseshoe vortex
2880:
2878:
2877:
2872:
2857:
2855:
2854:
2849:
2840:
2832:
2827:
2826:
2792:
2790:
2789:
2784:
2640:panel method of
2600:Vorticity meters
2561:
2559:
2558:
2553:
2548:
2547:
2535:
2534:
2522:
2521:
2506:
2491:
2489:
2488:
2483:
2477:
2475:
2474:
2465:
2457:
2452:
2450:
2449:
2440:
2432:
2427:
2425:
2424:
2415:
2407:
2337:
2309:
2302:
2295:
2281:
2274:
2267:
2251:
2247:
2239:
2238:
2236:
2235:
2230:
2227:
2209:
2194:
2187:
2180:
2168:
2123:angular velocity
2112:
2110:
2109:
2104:
2099:
2095:
2094:
2092:
2091:
2090:
2077:
2076:
2075:
2062:
2057:
2055:
2054:
2053:
2040:
2039:
2038:
2025:
2018:
2010:
2005:
2004:
1991:
1988:
1985:
1981:
1980:
1979:
1970:
1956:
1943:
1935:
1930:
1915:
1913:
1912:
1907:
1905:
1886:
1884:
1883:
1878:
1876:
1875:
1850:
1848:
1847:
1842:
1840:
1839:
1827:
1826:
1811:
1803:
1798:
1797:
1784:
1783:
1774:
1773:
1755:
1754:
1732:
1730:
1729:
1724:
1722:
1710:
1708:
1707:
1702:
1700:
1685:
1683:
1682:
1677:
1665:
1663:
1662:
1657:
1655:
1643:
1641:
1640:
1635:
1623:
1612:
1597:
1595:
1594:
1589:
1573:
1571:
1570:
1565:
1550:
1548:
1547:
1542:
1527:
1525:
1524:
1519:
1517:
1505:normal direction
1502:
1479:
1477:
1476:
1471:
1469:
1461:
1460:
1455:
1449:
1445:
1444:
1442:
1434:
1433:
1432:
1419:
1414:
1412:
1404:
1403:
1402:
1389:
1379:
1365:
1346:
1344:
1343:
1338:
1336:
1335:
1327:
1317:
1315:
1314:
1309:
1297:
1295:
1294:
1289:
1277:
1275:
1274:
1269:
1251:
1249:
1248:
1243:
1241:
1233:
1229:
1228:
1225:
1217:
1216:
1215:
1202:
1196:
1193:
1185:
1184:
1183:
1170:
1164:
1161:
1153:
1152:
1151:
1138:
1132:
1129:
1121:
1120:
1119:
1106:
1100:
1097:
1089:
1088:
1087:
1074:
1068:
1065:
1057:
1056:
1055:
1042:
1031:
1017:
994:
992:
991:
986:
984:
968:
966:
965:
960:
958:
934:
932:
931:
926:
924:
906:
904:
903:
898:
890:
859:angular velocity
856:
854:
853:
848:
846:
823:
821:
820:
815:
813:
802:. Conceptually,
797:
795:
794:
789:
774:
772:
771:
766:
760:
746:
731:
729:
728:
723:
721:
702:
700:
699:
694:
692:
632:
625:
618:
464:
463:
429:Gay-Lussac's law
419:Combined gas law
369:Capillary action
254:
253:
97:
96:
81:
79:
78:
73:
71:
69:
61:
53:
19:
18:
4197:
4196:
4192:
4191:
4190:
4188:
4187:
4186:
4157:
4156:
4155:
4150:
4132:
4094:
4044:
3988:
3966:
3946:
3901:
3856:
3803:
3737:
3734:
3689:Wayback Machine
3594:
3511:Ohkitani, K., "
3508:
3506:Further reading
3467:
3445:
3426:
3407:
3396:Fluid Mechanics
3386:
3367:
3362:
3361:
3300:
3296:
3291:Wayback Machine
3282:
3278:
3273:Wayback Machine
3264:
3260:
3243:
3242:
3237:Collected works
3234:
3216:
3212:
3207:
3203:
3194:
3190:
3185:Wayback Machine
3176:
3172:
3165:Fluid Mechanics
3161:
3157:
3150:
3133:
3129:
3124:
3120:
3115:
3111:
3102:
3098:
3091:
3074:
3070:
3063:
3049:
3042:
3028:
3024:
3019:Wayback Machine
3010:
3006:
3001:
2979:
2956:BiotâSavart law
2952:
2895:
2866:
2863:
2862:
2836:
2828:
2822:
2818:
2801:
2798:
2797:
2769:
2766:
2765:
2658:
2625:
2620:
2607:
2602:
2590:volume integral
2543:
2539:
2530:
2526:
2517:
2513:
2502:
2500:
2497:
2496:
2470:
2466:
2458:
2456:
2445:
2441:
2433:
2431:
2420:
2416:
2408:
2406:
2404:
2401:
2400:
2386:
2359:Reynolds number
2344:
2257:
2254:proportionality
2249:
2245:
2231:
2228:
2225:
2224:
2222:
2217:
2216:
2201:
2200:
2171:Example flows:
2119:
2086:
2082:
2078:
2071:
2067:
2063:
2061:
2049:
2045:
2041:
2034:
2030:
2026:
2024:
2023:
2019:
2009:
1997:
1993:
1987:
1975:
1971:
1966:
1952:
1948:
1944:
1934:
1926:
1924:
1921:
1920:
1901:
1896:
1893:
1892:
1865:
1861:
1859:
1856:
1855:
1832:
1828:
1816:
1812:
1802:
1793:
1789:
1779:
1775:
1763:
1759:
1747:
1743:
1741:
1738:
1737:
1718:
1716:
1713:
1712:
1696:
1694:
1691:
1690:
1671:
1668:
1667:
1651:
1649:
1646:
1645:
1619:
1608:
1606:
1603:
1602:
1583:
1580:
1579:
1556:
1553:
1552:
1533:
1530:
1529:
1513:
1511:
1508:
1507:
1498:
1496:surface element
1489:Stokes' theorem
1467:
1466:
1456:
1451:
1450:
1435:
1428:
1424:
1420:
1418:
1405:
1398:
1394:
1390:
1388:
1387:
1383:
1375:
1361:
1357:
1355:
1352:
1351:
1326:
1325:
1323:
1320:
1319:
1303:
1300:
1299:
1283:
1280:
1279:
1263:
1260:
1259:
1239:
1238:
1218:
1211:
1207:
1203:
1200:
1186:
1179:
1175:
1171:
1168:
1154:
1147:
1143:
1139:
1136:
1122:
1115:
1111:
1107:
1104:
1090:
1083:
1079:
1075:
1072:
1058:
1051:
1047:
1043:
1040:
1039:
1035:
1027:
1013:
1009:
1007:
1004:
1003:
980:
978:
975:
974:
954:
952:
949:
948:
945:
920:
918:
915:
914:
886:
878:
875:
874:
867:right-hand rule
842:
840:
837:
836:
809:
807:
804:
803:
783:
780:
779:
756:
742:
740:
737:
736:
717:
715:
712:
711:
688:
686:
683:
682:
636:
607:
606:
605:
525:
517:
516:
470:Viscoelasticity
461:
451:
450:
438:
388:
384:Surface tension
348:
251:
249:Fluid mechanics
241:
240:
239:
153:
151:Solid mechanics
143:
142:
94:
86:
62:
54:
52:
38:
35:
34:
17:
12:
11:
5:
4195:
4185:
4184:
4179:
4174:
4172:Fluid dynamics
4169:
4152:
4151:
4149:
4148:
4142:
4140:
4134:
4133:
4131:
4130:
4125:
4120:
4115:
4110:
4104:
4102:
4096:
4095:
4093:
4092:
4087:
4082:
4077:
4072:
4067:
4065:Vapor pressure
4062:
4057:
4052:
4047:
4042:
4035:
4026:
4021:
4016:
4011:
4006:
4001:
3996:
3991:
3986:
3979:
3974:
3969:
3964:
3956:
3954:
3948:
3947:
3945:
3944:
3939:
3934:
3929:
3924:
3919:
3914:
3909:
3904:
3899:
3892:
3887:
3882:
3877:
3872:
3866:
3864:
3858:
3857:
3855:
3854:
3849:
3844:
3839:
3834:
3829:
3824:
3819:
3813:
3811:
3805:
3804:
3802:
3801:
3796:
3791:
3786:
3781:
3776:
3771:
3766:
3761:
3756:
3751:
3745:
3743:
3739:
3738:
3733:
3732:
3725:
3718:
3710:
3704:
3703:
3693:
3682:Spherepack 3.1
3677:
3676:
3675:
3666:
3657:
3634:
3624:
3621:: Introduction
3613:
3604:
3593:
3592:External links
3590:
3589:
3588:
3574:
3560:Tritton, D. J.
3557:
3540:
3523:
3507:
3504:
3503:
3502:
3493:
3484:
3470:
3465:
3449:
3443:
3430:
3424:
3411:
3405:
3390:
3384:
3366:
3363:
3360:
3359:
3294:
3276:
3258:
3219:Joukovsky N.E.
3210:
3201:
3188:
3170:
3155:
3148:
3136:Thorne, Kip S.
3127:
3118:
3109:
3107:, Section 7.11
3103:Clancy, L.J.,
3096:
3089:
3068:
3061:
3040:
3022:
3003:
3002:
3000:
2997:
2996:
2995:
2990:
2985:
2978:
2975:
2974:
2973:
2968:
2963:
2958:
2951:
2950:Fluid dynamics
2948:
2947:
2946:
2941:
2936:
2931:
2926:
2921:
2916:
2911:
2906:
2901:
2894:
2891:
2870:
2859:
2858:
2847:
2844:
2839:
2835:
2831:
2825:
2821:
2817:
2814:
2811:
2808:
2805:
2782:
2779:
2776:
2773:
2728:(that is, the
2657:
2654:
2624:
2621:
2619:
2616:
2606:
2603:
2601:
2598:
2551:
2546:
2542:
2538:
2533:
2529:
2525:
2520:
2516:
2512:
2509:
2505:
2493:
2492:
2481:
2473:
2469:
2464:
2461:
2455:
2448:
2444:
2439:
2436:
2430:
2423:
2419:
2414:
2411:
2394:vorticity line
2385:
2382:
2367:complex-valued
2363:potential flow
2343:
2340:
2339:
2338:
2326:
2325:
2322:
2321:
2320:Vorticity = 0
2318:
2317:Vorticity â 0
2315:
2314:Vorticity â 0
2311:
2310:
2303:
2296:
2288:
2287:
2283:
2282:
2275:
2268:
2260:
2259:
2241:
2240:
2213:
2210:
2196:
2195:
2188:
2181:
2173:
2172:
2136:, if there is
2127:Rankine vortex
2118:
2115:
2114:
2113:
2102:
2098:
2089:
2085:
2081:
2074:
2070:
2066:
2060:
2052:
2048:
2044:
2037:
2033:
2029:
2022:
2016:
2013:
2008:
2003:
2000:
1996:
1984:
1978:
1974:
1969:
1965:
1962:
1959:
1955:
1951:
1947:
1941:
1938:
1933:
1929:
1904:
1900:
1874:
1871:
1868:
1864:
1852:
1851:
1838:
1835:
1831:
1825:
1822:
1819:
1815:
1809:
1806:
1801:
1796:
1792:
1787:
1782:
1778:
1772:
1769:
1766:
1762:
1758:
1753:
1750:
1746:
1721:
1699:
1675:
1654:
1633:
1630:
1627:
1622:
1618:
1615:
1611:
1587:
1563:
1560:
1540:
1537:
1516:
1481:
1480:
1465:
1459:
1454:
1448:
1441:
1438:
1431:
1427:
1423:
1417:
1411:
1408:
1401:
1397:
1393:
1386:
1382:
1378:
1374:
1371:
1368:
1364:
1360:
1359:
1333:
1330:
1307:
1287:
1267:
1253:
1252:
1237:
1232:
1224:
1221:
1214:
1210:
1206:
1199:
1192:
1189:
1182:
1178:
1174:
1167:
1160:
1157:
1150:
1146:
1142:
1135:
1128:
1125:
1118:
1114:
1110:
1103:
1096:
1093:
1086:
1082:
1078:
1071:
1064:
1061:
1054:
1050:
1046:
1038:
1034:
1030:
1026:
1023:
1020:
1016:
1012:
1011:
983:
957:
944:
941:
923:
896:
893:
889:
885:
882:
863:center of mass
845:
812:
800:nabla operator
787:
776:
775:
764:
759:
755:
752:
749:
745:
720:
691:
638:
637:
635:
634:
627:
620:
612:
609:
608:
604:
603:
598:
593:
588:
583:
578:
573:
568:
563:
558:
553:
548:
543:
538:
533:
527:
526:
523:
522:
519:
518:
513:
512:
511:
510:
505:
500:
492:
491:
485:
484:
483:
482:
477:
472:
462:
457:
456:
453:
452:
447:
446:
440:
439:
437:
436:
431:
426:
421:
416:
411:
406:
400:
397:
396:
390:
389:
387:
386:
381:
376:
374:Chromatography
371:
366:
360:
357:
356:
350:
349:
347:
346:
327:
326:
325:
306:
294:
289:
277:
264:
261:
260:
252:
247:
246:
243:
242:
238:
237:
232:
227:
226:
225:
215:
210:
205:
204:
203:
198:
188:
183:
178:
173:
172:
171:
161:
155:
154:
149:
148:
145:
144:
139:
138:
137:
136:
128:
127:
123:
122:
121:
120:
115:
110:
102:
101:
95:
92:
91:
88:
87:
82:
68:
65:
60:
57:
51:
48:
45:
42:
31:
30:
24:
23:
15:
9:
6:
4:
3:
2:
4194:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4164:
4162:
4147:
4144:
4143:
4141:
4139:
4135:
4129:
4126:
4124:
4121:
4119:
4118:Barotropicity
4116:
4114:
4111:
4109:
4106:
4105:
4103:
4101:
4097:
4091:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4061:
4058:
4056:
4053:
4051:
4048:
4046:
4041:
4036:
4034:
4032:
4027:
4025:
4022:
4020:
4017:
4015:
4012:
4010:
4007:
4005:
4002:
4000:
3997:
3995:
3992:
3990:
3985:
3980:
3978:
3975:
3973:
3970:
3968:
3963:
3958:
3957:
3955:
3953:
3949:
3943:
3940:
3938:
3935:
3933:
3930:
3928:
3925:
3923:
3920:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3898:
3893:
3891:
3888:
3886:
3883:
3881:
3878:
3876:
3873:
3871:
3868:
3867:
3865:
3863:
3859:
3853:
3850:
3848:
3847:Precipitation
3845:
3843:
3840:
3838:
3835:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3814:
3812:
3810:
3806:
3800:
3797:
3795:
3792:
3790:
3787:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3746:
3744:
3740:
3731:
3726:
3724:
3719:
3717:
3712:
3711:
3708:
3701:
3699:
3694:
3691:
3690:
3686:
3683:
3678:
3673:
3672:
3667:
3664:
3663:
3658:
3655:
3654:
3649:
3648:
3646:
3642:
3638:
3635:
3632:
3631:
3625:
3622:
3620:
3614:
3611:
3610:
3605:
3602:
3601:
3596:
3595:
3587:
3586:0-12-059820-5
3583:
3579:
3576:Arfken, G., "
3575:
3573:
3572:0-19-854493-6
3569:
3565:
3561:
3558:
3556:
3555:0-521-63948-4
3552:
3548:
3544:
3541:
3539:
3538:0-387-94197-5
3535:
3531:
3527:
3524:
3522:
3521:0-521-81984-9
3518:
3514:
3510:
3509:
3500:
3499:
3494:
3491:
3490:
3485:
3483:
3482:0-273-01120-0
3479:
3475:
3471:
3468:
3466:0-521-66396-2
3462:
3458:
3454:
3450:
3446:
3444:0-19-851746-7
3440:
3436:
3431:
3427:
3421:
3417:
3412:
3408:
3402:
3398:
3397:
3391:
3387:
3385:0-19-859679-0
3381:
3377:
3373:
3372:Acheson, D.J.
3369:
3368:
3355:
3351:
3347:
3343:
3339:
3335:
3330:
3325:
3321:
3317:
3313:
3309:
3305:
3298:
3292:
3288:
3285:
3280:
3274:
3270:
3267:
3262:
3253:
3247:
3239:
3238:
3230:
3226:
3225:
3220:
3214:
3205:
3198:
3192:
3186:
3182:
3179:
3174:
3166:
3159:
3151:
3149:9780691159027
3145:
3141:
3137:
3131:
3122:
3113:
3106:
3100:
3092:
3090:0-19-859679-0
3086:
3082:
3078:
3077:Acheson, D.J.
3072:
3064:
3062:0-19-851746-7
3058:
3054:
3047:
3045:
3036:
3032:
3031:Moffatt, H.K.
3026:
3020:
3016:
3013:
3008:
3004:
2994:
2991:
2989:
2986:
2984:
2981:
2980:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2953:
2945:
2942:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2896:
2890:
2888:
2884:
2868:
2845:
2842:
2833:
2823:
2819:
2815:
2809:
2803:
2796:
2795:
2794:
2793:, defined as
2777:
2771:
2764:
2759:
2757:
2753:
2748:
2746:
2742:
2739:
2735:
2731:
2727:
2723:
2718:
2716:
2712:
2708:
2704:
2700:
2696:
2692:
2691:
2685:
2683:
2679:
2674:
2672:
2668:
2663:
2653:
2651:
2647:
2643:
2638:
2634:
2630:
2615:
2611:
2597:
2595:
2591:
2586:
2584:
2580:
2576:
2572:
2567:
2565:
2544:
2540:
2536:
2531:
2527:
2523:
2518:
2514:
2507:
2479:
2471:
2467:
2462:
2459:
2453:
2446:
2442:
2437:
2434:
2428:
2421:
2417:
2412:
2409:
2399:
2398:
2397:
2395:
2391:
2381:
2379:
2374:
2372:
2371:complex plane
2369:field on the
2368:
2364:
2360:
2355:
2353:
2349:
2336:
2332:
2331:
2330:
2319:
2316:
2313:
2312:
2308:
2301:
2294:
2289:
2284:
2280:
2273:
2266:
2261:
2255:
2242:
2234:
2220:
2214:
2211:
2208:
2204:
2198:
2197:
2193:
2186:
2179:
2174:
2169:
2166:
2165:
2164:
2162:
2158:
2153:
2151:
2150:cross section
2147:
2143:
2139:
2135:
2130:
2128:
2124:
2100:
2096:
2087:
2083:
2072:
2068:
2058:
2050:
2046:
2035:
2031:
2020:
2014:
2011:
2006:
2001:
1998:
1982:
1976:
1957:
1945:
1939:
1936:
1931:
1919:
1918:
1917:
1890:
1872:
1869:
1866:
1862:
1836:
1833:
1823:
1820:
1817:
1813:
1807:
1804:
1799:
1794:
1790:
1785:
1780:
1776:
1770:
1767:
1764:
1760:
1756:
1751:
1748:
1736:
1735:
1734:
1687:
1673:
1628:
1625:
1613:
1601:
1585:
1577:
1558:
1538:
1535:
1506:
1501:
1497:
1494:
1493:infinitesimal
1490:
1486:
1463:
1457:
1446:
1439:
1429:
1425:
1415:
1409:
1399:
1395:
1384:
1380:
1372:
1366:
1350:
1349:
1348:
1328:
1305:
1285:
1265:
1256:
1235:
1230:
1222:
1212:
1208:
1197:
1190:
1180:
1176:
1165:
1158:
1148:
1144:
1133:
1126:
1116:
1112:
1101:
1094:
1084:
1080:
1069:
1062:
1052:
1048:
1036:
1032:
1024:
1018:
1002:
1001:
1000:
998:
972:
940:
938:
912:
907:
894:
891:
883:
872:
868:
864:
860:
834:
833:displacements
831:
827:
801:
762:
753:
747:
735:
734:
733:
710:
709:flow velocity
706:
679:
677:
673:
669:
665:
661:
657:
653:
649:
645:
633:
628:
626:
621:
619:
614:
613:
611:
610:
602:
599:
597:
594:
592:
589:
587:
584:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
547:
544:
542:
539:
537:
534:
532:
529:
528:
521:
520:
509:
506:
504:
501:
499:
496:
495:
494:
493:
490:
487:
486:
481:
478:
476:
473:
471:
468:
467:
466:
465:
460:
455:
454:
445:
442:
441:
435:
432:
430:
427:
425:
422:
420:
417:
415:
414:Charles's law
412:
410:
407:
405:
402:
401:
399:
398:
395:
392:
391:
385:
382:
380:
377:
375:
372:
370:
367:
365:
362:
361:
359:
358:
355:
352:
351:
345:
342:
338:
335:
331:
328:
323:
322:non-Newtonian
320:
316:
312:
311:
310:
307:
305:
302:
298:
295:
293:
290:
288:
285:
281:
278:
276:
273:
269:
266:
265:
263:
262:
259:
256:
255:
250:
245:
244:
236:
233:
231:
228:
224:
221:
220:
219:
216:
214:
211:
209:
208:Compatibility
206:
202:
199:
197:
196:Finite strain
194:
193:
192:
189:
187:
184:
182:
179:
177:
174:
170:
167:
166:
165:
162:
160:
157:
156:
152:
147:
146:
135:
132:
131:
130:
129:
125:
124:
119:
116:
114:
111:
109:
106:
105:
104:
103:
100:Conservations
99:
98:
90:
89:
85:
66:
63:
58:
55:
49:
46:
43:
40:
33:
32:
29:
26:
25:
21:
20:
4039:
4030:
4024:Mixing ratio
3999:Haines Index
3983:
3961:
3896:
3809:Condensation
3788:
3696:
3680:
3669:
3660:
3651:
3640:
3630:9: Vorticity
3627:
3616:
3607:
3598:
3577:
3563:
3546:
3529:
3512:
3496:
3487:
3474:Aerodynamics
3473:
3456:
3434:
3415:
3395:
3375:
3365:Bibliography
3311:
3307:
3297:
3279:
3261:
3236:
3228:
3222:
3213:
3204:
3196:
3191:
3173:
3164:
3158:
3139:
3130:
3121:
3112:
3105:Aerodynamics
3104:
3099:
3080:
3071:
3052:
3034:
3025:
3007:
2914:Palinstrophy
2860:
2760:
2749:
2736:of 500
2726:Rossby waves
2719:
2688:
2686:
2677:
2675:
2661:
2659:
2629:aerodynamics
2626:
2612:
2608:
2587:
2574:
2570:
2568:
2494:
2393:
2389:
2387:
2375:
2356:
2345:
2327:
2232:
2218:
2206:
2202:
2160:
2154:
2146:laminar flow
2131:
2120:
1853:
1688:
1499:
1482:
1257:
1254:
946:
937:scalar field
908:
873:field since
829:
826:neighborhood
777:
680:
676:vortex rings
652:pseudovector
647:
641:
489:Smart fluids
434:Graham's law
340:
333:
318:
304:Pascal's law
300:
283:
271:
126:Inequalities
4113:Baroclinity
3960:Dew point (
3952:Temperature
3852:Water vapor
3645:UC Berkeley
2961:Circulation
2929:Vortex tube
2646:circulation
2637:finite wing
2623:Aeronautics
2575:vortex flux
2571:vortex tube
2390:vortex line
2142:streamlines
1600:dot product
1485:circulation
508:Ferrofluids
409:Boyle's law
181:Hooke's law
159:Deformation
4161:Categories
4090:Wind chill
4004:Heat index
3862:Convection
3799:Wind shear
3784:Visibility
3764:Lapse rate
2999:References
2993:Hans Ertel
2889:activity.
2883:supercells
1574:along the
871:solenoidal
561:Gay-Lussac
524:Scientists
424:Fick's law
404:Atmosphere
223:frictional
176:Plasticity
164:Elasticity
3789:Vorticity
3769:Lightning
3754:Advection
3600:Vorticity
3498:Vorticity
3455:(2000) ,
3338:0036-8075
3246:cite book
2909:Enstrophy
2838:ω
2834:⋅
2820:∫
2711:adiabatic
2709:flow. As
2707:adiabatic
2703:conserved
2541:ω
2528:ω
2515:ω
2504:ω
2468:ω
2443:ω
2418:ω
2378:diffusion
2342:Evolution
2163:is zero.
2134:pathlines
2080:∂
2065:∂
2059:−
2043:∂
2028:∂
1995:Ω
1964:∇
1958:−
1950:∇
1928:Ω
1899:∇
1863:ε
1830:Ω
1814:ε
1791:ω
1777:ω
1761:ε
1745:Ω
1720:ω
1698:Ω
1653:ω
1614:⋅
1610:ω
1576:perimeter
1562:Γ
1528:and area
1437:∂
1422:∂
1416:−
1407:∂
1392:∂
1373:×
1370:∇
1363:ω
1332:^
1220:∂
1205:∂
1198:−
1188:∂
1173:∂
1156:∂
1141:∂
1134:−
1124:∂
1109:∂
1092:∂
1077:∂
1070:−
1060:∂
1045:∂
1025:×
1022:∇
1015:ω
956:ω
922:ω
888:ω
884:⋅
881:∇
844:ω
811:ω
786:∇
754:×
751:∇
748:≡
744:ω
690:ω
648:vorticity
601:Truesdell
531:Bernoulli
480:Rheometer
475:Rheometry
315:Newtonian
309:Viscosity
59:φ
47:−
4182:Rotation
4138:Velocity
4100:Pressure
4014:Humidity
3917:Helicity
3759:Buoyancy
3685:Archived
3374:(1990).
3354:23287311
3346:28774926
3287:Archived
3269:Archived
3181:Archived
3079:(1990).
3015:Archived
2893:See also
2887:tornadic
2763:helicity
2117:Examples
1916:, i.e.,
830:relative
660:spinning
459:Rheology
364:Adhesion
344:Pressure
330:Buoyancy
275:Dynamics
113:Momentum
4009:Humidex
3922:K Index
3742:General
3316:Bibcode
3308:Science
2730:troughs
2699:entropy
2237:
2223:
1598:is the
798:is the
707:of the
703:is the
546:Charles
354:Liquids
268:Statics
213:Bending
3584:
3570:
3553:
3536:
3519:
3480:
3463:
3441:
3422:
3403:
3382:
3352:
3344:
3336:
3146:
3087:
3059:
2924:Vortex
2734:ridges
2715:tracer
2705:in an
2631:, the
2495:where
2244:where
1854:where
1644:where
778:where
672:fluids
596:Stokes
591:Pascal
581:Navier
576:Newton
566:Graham
541:Cauchy
444:Plasma
339:
337:Mixing
332:
317:
299:
282:
270:
258:Fluids
191:Strain
186:Stress
169:linear
118:Energy
3817:Cloud
3350:S2CID
2138:shear
1503:with
909:In a
656:field
650:is a
571:Hooke
551:Euler
536:Boyle
394:Gases
3794:Wind
3582:ISBN
3568:ISBN
3551:ISBN
3534:ISBN
3517:ISBN
3478:ISBN
3461:ISBN
3439:ISBN
3420:ISBN
3401:ISBN
3380:ISBN
3342:PMID
3334:ISSN
3252:link
3144:ISBN
3085:ISBN
3057:ISBN
2732:and
2720:The
2697:(or
2687:The
2676:The
2660:The
2633:lift
971:curl
705:curl
664:flow
586:Noll
556:Fick
108:Mass
93:Laws
3827:Fog
3639:, "
3562:, "
3528:, "
3324:doi
3312:357
2738:hPa
2627:In
2392:or
1578:of
670:of
642:In
4163::
3674:".
3665:".
3656:".
3647:.
3348:.
3340:.
3332:.
3322:.
3310:.
3306:.
3248:}}
3244:{{
3229:28
3227:.
3043:^
2758:.
2684:.
2569:A
2566:.
2388:A
2373:.
2354:.
2221:â
2205:â
2129:.
1989:or
1686:.
1347::
999::
939:.
913:,
895:0.
732::
678:.
646:,
4045:)
4043:e
4040:θ
4033:)
4031:θ
3989:)
3987:e
3984:T
3967:)
3965:d
3962:T
3902:)
3900:c
3897:T
3729:e
3722:t
3715:v
3695:"
3679:"
3668:"
3659:"
3650:"
3495:"
3486:"
3447:.
3428:.
3409:.
3388:.
3356:.
3326::
3318::
3254:)
3231:.
3167:.
3152:.
3093:.
3065:.
2869:V
2846:V
2843:d
2830:v
2824:V
2816:=
2813:)
2810:t
2807:(
2804:H
2781:)
2778:t
2775:(
2772:H
2550:)
2545:z
2537:,
2532:y
2524:,
2519:x
2511:(
2508:=
2480:,
2472:z
2463:z
2460:d
2454:=
2447:y
2438:y
2435:d
2429:=
2422:x
2413:x
2410:d
2256:.
2250:r
2246:v
2233:r
2229:/
2226:1
2219:v
2207:r
2203:v
2101:.
2097:)
2088:i
2084:x
2073:i
2069:v
2051:j
2047:x
2036:i
2032:v
2021:(
2015:2
2012:1
2007:=
2002:j
1999:i
1983:]
1977:T
1973:)
1968:v
1961:(
1954:v
1946:[
1940:2
1937:1
1932:=
1903:v
1873:k
1870:j
1867:i
1837:k
1834:j
1824:k
1821:j
1818:i
1808:2
1805:1
1800:=
1795:i
1786:,
1781:k
1771:k
1768:j
1765:i
1757:=
1752:j
1749:i
1674:C
1632:)
1629:A
1626:d
1621:n
1617:(
1586:C
1559:d
1539:A
1536:d
1515:n
1500:C
1464:.
1458:z
1453:e
1447:)
1440:y
1430:x
1426:v
1410:x
1400:y
1396:v
1385:(
1381:=
1377:v
1367:=
1329:z
1306:z
1286:z
1266:z
1236:.
1231:)
1223:y
1213:x
1209:v
1191:x
1181:y
1177:v
1166:,
1159:x
1149:z
1145:v
1127:z
1117:x
1113:v
1102:,
1095:z
1085:y
1081:v
1063:y
1053:z
1049:v
1037:(
1033:=
1029:v
1019:=
982:v
892:=
763:,
758:v
719:v
631:e
624:t
617:v
341:¡
334:¡
324:)
319:¡
313:(
301:¡
284:¡
272:¡
67:x
64:d
56:d
50:D
44:=
41:J
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.