2527:
1333:. Under the continuum assumption, macroscopic (observed/measurable) properties such as density, pressure, temperature, and bulk velocity are taken to be well-defined at "infinitesimal" volume elements—small in comparison to the characteristic length scale of the system, but large in comparison to molecular length scale. Fluid properties can vary continuously from one volume element to another and are average values of the molecular properties. The continuum hypothesis can lead to inaccurate results in applications like supersonic speed flows, or molecular flows on nano scale. Those problems for which the continuum hypothesis fails can be solved using
1238:
2489:
this is the flow far from solid surfaces. In many cases, the viscous effects are concentrated near the solid boundaries (such as in boundary layers) while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid (ideal flow). When the viscosity is neglected, the term containing the viscous stress tensor
2456:
2488:
In some applications, another rough broad division among fluids is made: ideal and non-ideal fluids. An ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of
1513:
2277:
2051:
1411:
2451:{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}-{\frac {2}{3}}\delta _{ij}\nabla \cdot \mathbf {v} \right)+\kappa \delta _{ij}\nabla \cdot \mathbf {v} }
1669:
For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media (this is related to the
Beavers and Joseph condition). Further, it is useful at low
1349:, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, but molecular approach (statistical mechanics) can be applied to find the fluid motion for larger Knudsen numbers.
1748:). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property—for example, most fluids with long molecular chains can react in a non-Newtonian manner.
1820:
1592:, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called
1915:
1253:
The assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. Fundamentally, every fluid mechanical system is assumed to obey:
1939:
91:
1721:. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the
750:
Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems are partly or wholly unsolved and are best addressed by
1744:(although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip
2509:
1403:
2087:
1519:
These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes in
2479:
2265:
2206:
2147:
2117:
1658:
at the solid. In some cases, the mathematics of a fluid mechanical system can be treated by assuming that the fluid outside of boundary layers is inviscid, and then
1636:
2840:
Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer
Science & Business Media.
2235:
2176:
1846:
2580:
1872:
1571:
2485:, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic.
1548:
1508:{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}\mathbf {u} }
1584:. In practical terms, only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which the
3747:
3517:
1729:). Important fluids, like water as well as most gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.
640:
2784:
3836:
1772:
2831:
Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge
University Press.
3092:
777:
2885:
Anderson, D., Tannehill, J. C., & Pletcher, R. H. (2016). Computational fluid mechanics and heat transfer. Taylor & Francis.
3172:
762:, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
1659:
984:
float on water, and why the surface of water is always level whatever the shape of its container. Hydrostatics is fundamental to
3715:
3017:
2999:
2959:
2941:
2794:
2637:
2595:
2046:{\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}
739:, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a
1879:
972:, the study of fluids in motion. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why
3041:
1642:, one that facilitates mathematical treatment. In fact, purely inviscid flows are only known to be realized in the case of
1122:
17:
3060:
2904:
2662:
3510:
2981:
2767:
2695:
Bertin, J. J., & Smith, M. L. (1998). Aerodynamics for engineers (Vol. 5). Upper Saddle River, NJ: Prentice Hall.
1736:
can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding,
633:
1725:
of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare
3160:
2512:
1283:
1246:
2876:
Wesseling, P. (2009). Principles of computational fluid dynamics (Vol. 29). Springer
Science & Business Media.
2526:
884:
2605:
2585:
47:
2481:
is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed a
1047:—the science of liquids and gases in motion. Fluid dynamics offers a systematic structure—which underlies these
3663:
2686:
Batchelor, C. K., & Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge
University Press.
606:
3315:
3195:
3085:
2819:
307:
144:
1762:
The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the
1290:
of the mass contained in that volume is equal to the rate at which mass is passing through the surface from
1229:; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.
3503:
3140:
2565:
1593:
755:
626:
347:
233:
1085:(the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating
3800:
3150:
1930:
1358:
900:
302:
211:
94:
2849:
Anderson, J. D., & Wendt, J. (1995). Computational fluid dynamics (Vol. 206). New York: McGraw-Hill.
3862:
3631:
3604:
1307:
864:
771:
759:
2492:
1580:
Solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of
218:
2867:
Blazek, J. (2015). Computational fluid dynamics: principles and applications. Butterworth-Heinemann.
3857:
3737:
3392:
3375:
3122:
3078:
3032:
2590:
2555:
2271:
If the fluid is not incompressible the general form for the viscous stress in a
Newtonian fluid is
1267:
912:
796:
513:
508:
297:
290:
123:
1386:
3370:
1737:
965:
576:
571:
240:
2713:
Houghton, E. L., & Carpenter, P. W. (2003). Aerodynamics for engineering students. Elsevier.
2062:
876:
3821:
3683:
3270:
1376:
1262:
962:
924:
825:
701:
685:
128:
3826:
3795:
3658:
3579:
2809:
Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of
Chicago Press.
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2464:
1372:
1334:
1048:
896:
868:
689:
551:
169:
2240:
2181:
2122:
2092:
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to the plane of shear. This definition means regardless of the forces acting on a fluid, it
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3569:
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2213:
2154:
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872:
697:
389:
206:
186:
174:
118:
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1857:
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8:
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1368:
1326:
1322:
1205:
1143:
1134:
892:
736:
591:
439:
332:
38:
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2511:
in the Navier–Stokes equation vanishes. The equation reduced in this form is called the
1201:
The study of the physics of continuous materials which deform when subjected to a force.
3816:
3693:
3688:
3641:
3479:
3265:
3205:
2600:
1533:
1114:
801:
611:
245:
201:
196:
27:
Branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas)
3831:
3780:
3725:
3705:
3591:
3434:
3399:
3180:
3013:
2995:
2977:
2973:
2955:
2937:
2900:
2818:
Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343).
2790:
2763:
2658:
2633:
2550:
1655:
920:
821:
817:
806:
805:—generally considered to be the first major work on fluid mechanics. Iranian scholar
751:
693:
228:
179:
2704:
Anderson Jr, J. D. (2010). Fundamentals of aerodynamics. Tata McGraw-Hill
Education.
961:
at rest. It embraces the study of the conditions under which fluids are at rest in
3770:
3710:
3636:
3484:
3355:
3340:
3305:
3215:
2929:
2755:
1167:
1090:
1052:
1009:
916:
853:
677:
566:
541:
454:
429:
424:
379:
3360:
1075:, as functions of space and time. It has several subdisciplines itself, including
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3700:
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3429:
3387:
3382:
3210:
3064:
3045:
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1216:
1152:
1059:
problem typically involves calculating various properties of the fluid, such as
1005:
904:
849:
556:
480:
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394:
325:
314:
161:
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3720:
3675:
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3419:
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3325:
3320:
3310:
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3280:
3245:
3190:
3155:
3132:
2575:
2570:
2532:
1849:
1722:
1671:
1651:
1577:, such as the gravitational force or Lorentz force are added to the equations.
1342:
1338:
1279:
1272:
1242:
1098:
1056:
1039:
1033:
1021:
969:
908:
888:
784:
732:
561:
419:
384:
285:
191:
2858:
Chung, T. J. (2010). Computational fluid dynamics. Cambridge
University Press.
1181:
Describes materials that permanently deform after a sufficient applied stress.
3851:
3730:
3469:
3464:
3454:
3444:
3409:
3404:
3345:
3285:
3275:
3255:
3250:
3220:
3185:
3109:
3057:
2933:
1714:
1643:
1346:
950:
945:
841:
728:
601:
434:
3495:
2731:
Milne-Thomson, L. M. (1996). Theoretical hydrodynamics. Courier
Corporation.
1278:
For example, the assumption that mass is conserved means that for any fixed
3755:
3474:
3414:
3330:
3300:
3260:
3235:
3145:
3051:
2722:
Milne-Thomson, L. M. (1973). Theoretical aerodynamics. Courier Corporation.
2545:
1766:. A simple equation to describe incompressible Newtonian fluid behavior is
1703:
1695:
1678:—that is, the density of the gas does not change even though the speed and
1663:
1226:
1157:
The study of the physics of continuous materials with a defined rest shape.
1077:
1001:
845:
833:
717:
709:
586:
581:
546:
278:
2895:
Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (27 March 2015). "10".
3335:
3240:
3200:
1922:
1639:
1072:
1013:
989:
713:
596:
499:
1379:
that describe the force balance at a given point within a fluid. For an
3775:
3765:
3117:
1589:
1574:
1287:
1118:
997:
985:
928:
814:
810:
788:
735:, the study of the effect of forces on fluid motion. It is a branch of
705:
684:
on them. It has applications in a wide range of disciplines, including
518:
414:
3626:
1921:
For a Newtonian fluid, the viscosity, by definition, depends only on
1763:
1609:
1337:. To determine whether or not the continuum hypothesis applies, the
1330:
1210:
Do not undergo strain rates proportional to the applied shear stress.
1102:
880:
837:
829:
661:
490:
485:
319:
820:
to fluid mechanics. Rapid advancement in fluid mechanics began with
1726:
1710:
1707:
1581:
1528:
1520:
1225:
In a mechanical view, a fluid is a substance that does not support
1185:
1094:
1064:
1060:
1017:
977:
792:
469:
374:
354:
340:
3070:
1166:
Describes materials that return to their rest shape after applied
1917:
is the velocity gradient perpendicular to the direction of shear.
1815:{\displaystyle \tau =-\mu {\frac {\mathrm {d} u}{\mathrm {d} n}}}
1190:
The study of materials with both solid and fluid characteristics.
1106:
1068:
721:
657:
223:
2677:
Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642,
2740:
Birkhoff, G. (2015). Hydrodynamics. Princeton University Press.
1219:
undergo strain rates proportional to the applied shear stress.
1110:
993:
783:
The study of fluid mechanics goes back at least to the days of
669:
364:
1051:—that embraces empirical and semi-empirical laws derived from
863:
Inviscid flow was further analyzed by various mathematicians (
1745:
1699:
1588:
is small. For more complex cases, especially those involving
1524:
1086:
958:
681:
665:
268:
1329:, even though, on a microscopic scale, they are composed of
1741:
1237:
2653:
Tu, Jiyuan; Yeoh, Guan Heng; Liu, Chaoqun (Nov 21, 2012).
754:, typically using computers. A modern discipline, called
1874:
is the fluid viscosity—a constant of proportionality, and
1654:
near a solid surface, where the flow must match onto the
1055:
and used to solve practical problems. The solution to a
981:
673:
404:
2581:
Different types of boundary conditions in fluid dynamics
1685:
856:
with the introduction of mathematical fluid dynamics in
1910:{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} n}}}
1128:
1043:
is a subdiscipline of fluid mechanics that deals with
2495:
2467:
2280:
2243:
2216:
2184:
2157:
2125:
2095:
2065:
1942:
1882:
1860:
1834:
1775:
1618:
1559:
1536:
1414:
1389:
891:. Further mathematical justification was provided by
50:
2522:
1925:, not on the forces acting upon it. If the fluid is
1650:, a property that is often most important within a
927:advanced the understanding of fluid viscosity and
3054:– the Computational Fluid Dynamics reference wiki.
2655:Computational Fluid Dynamics: A Practical Approach
2503:
2473:
2450:
2259:
2229:
2200:
2170:
2141:
2111:
2081:
2045:
1909:
1866:
1840:
1814:
1751:
1630:
1565:
1542:
1507:
1397:
879:) and viscous flow was explored by a multitude of
85:
2894:
1081:(the study of air and other gases in motion) and
992:of equipment for storing, transporting and using
3849:
2989:
2750:
2748:
2746:
1282:(for example, a spherical volume)—enclosed by a
1241:Balance for some integrated fluid quantity in a
1148:The study of the physics of continuous materials
2926:Fluid Mechanics (A short course for physicists)
1298:, minus the rate at which mass is passing from
1121:. Some fluid-dynamical principles are used in
795:and formulated his famous law known now as the
2786:Momentum, Heat, and Mass Transfer Fundamentals
1929:the equation governing the viscous stress (in
1599:
957:is the branch of fluid mechanics that studies
3525:
3511:
3086:
2776:
2743:
634:
2623:
2621:
1848:is the shear stress exerted by the fluid ("
86:{\displaystyle J=-D{\frac {d\varphi }{dx}}}
3518:
3504:
3093:
3079:
3012:, CRC Press (Taylor & Francis group),
2949:
2762:. Cambridge University Press. p. 74.
1352:
996:. It is also relevant to some aspects of
641:
627:
3010:Fluid Dynamics via Examples and Solutions
3007:
2923:
2782:
2754:
2652:
1137:, as illustrated in the following table.
778:Timeline of fluid and continuum mechanics
2954:(4th revised ed.), Academic Press,
2950:Kundu, Pijush K.; Cohen, Ira M. (2008),
2618:
1341:, defined as the ratio of the molecular
1236:
1105:through pipelines, predicting evolving
14:
3850:
3716:Atomic, molecular, and optical physics
3058:Educational Particle Image Velocimetry
2994:(8th ed.), Taylor & Francis,
2967:
1638:. In practice, an inviscid flow is an
1133:Fluid mechanics is a subdiscipline of
3499:
3074:
2627:
2596:Stochastic Eulerian Lagrangian method
1686:Newtonian versus non-Newtonian fluids
1325:under which fluids can be treated as
758:(CFD), is devoted to this approach.
2783:Greenkorn, Robert (3 October 2018).
1550:and viscosity, parameterized by the
915:), while various scientists such as
3100:
2990:Massey, B.; Ward-Smith, J. (2005),
1405:, the Navier–Stokes equations are
1129:Relationship to continuum mechanics
731:, the study of fluids at rest; and
24:
2917:
2437:
2402:
2360:
2345:
2323:
2308:
2022:
2007:
1985:
1970:
1897:
1887:
1802:
1792:
1662:its solution onto that for a thin
1646:. Otherwise, fluids are generally
1491:
1478:
1451:
1428:
1418:
799:, which was published in his work
25:
3874:
3026:
2760:An Introduction to Fluid Dynamics
1027:
3038:Annual Review of Fluid Mechanics
2899:(6th ed.). Academic Press.
2525:
2504:{\displaystyle \mathbf {\tau } }
2444:
2409:
1706:is linearly proportional to the
1501:
1458:
1444:
1422:
1391:
939:
934:
824:(observations and experiments),
3837:Timeline of physics discoveries
2970:Fundamental Mechanics of Fluids
2888:
2879:
2870:
2861:
2852:
2843:
2834:
2825:
2812:
2803:
2734:
2606:Smoothed-particle hydrodynamics
2119:face of a fluid element in the
1752:Equations for a Newtonian fluid
1004:(for example, in understanding
791:investigated fluid statics and
776:For a chronological guide, see
3067:– resources and demonstrations
2928:, Cambridge University Press,
2725:
2716:
2707:
2698:
2689:
2680:
2671:
2646:
1454:
1440:
1306:. This can be expressed as an
1232:
13:
1:
2820:American Mathematical Society
2632:(7th ed.). McGraw-Hill.
2611:
1674:speeds to assume that gas is
1345:to the characteristic length
885:Jean LĂ©onard Marie Poiseuille
3141:Computational fluid dynamics
2566:Computational fluid dynamics
1594:computational fluid dynamics
1398:{\displaystyle \mathbf {u} }
756:computational fluid dynamics
7:
3801:Quantum information science
2924:Falkovich, Gregory (2011),
2586:Fluid–structure interaction
2518:
2089:is the shear stress on the
1600:Inviscid and viscous fluids
1383:with vector velocity field
1010:Earth's gravitational field
743:viewpoint rather than from
10:
3879:
3632:Classical electromagnetism
3033:Free Fluid Mechanics books
3008:Nazarenko, Sergey (2014),
2082:{\displaystyle \tau _{ij}}
1755:
1356:
1031:
1024:), and many other fields.
943:
775:
772:History of fluid mechanics
769:
765:
760:Particle image velocimetry
3809:
3746:
3674:
3590:
3562:
3534:
3171:
3131:
3108:
2789:. CRC Press. p. 18.
2657:. Butterworth-Heinemann.
1310:over the control volume.
1308:equation in integral form
1215:
1195:
1184:
1160:
1151:
1142:
968:; and is contrasted with
3738:Condensed matter physics
2934:10.1017/CBO9780511794353
2628:White, Frank M. (2011).
2591:Immersed boundary method
1732:By contrast, stirring a
1273:The continuum assumption
1268:Conservation of momentum
1109:patterns, understanding
852:), and was continued by
145:Clausius–Duhem (entropy)
95:Fick's laws of diffusion
3151:Navier–Stokes equations
2474:{\displaystyle \kappa }
2178:is the velocity in the
1365:Navier–Stokes equations
1359:Navier–Stokes equations
1353:Navier–Stokes equations
901:Navier–Stokes equations
865:Jean le Rond d'Alembert
727:It can be divided into
303:Navier–Stokes equations
241:Material failure theory
3822:Nobel Prize in Physics
3684:Relativistic mechanics
2968:Currie, I. G. (1974),
2505:
2475:
2452:
2261:
2260:{\displaystyle j^{th}}
2231:
2202:
2201:{\displaystyle i^{th}}
2172:
2143:
2142:{\displaystyle j^{th}}
2113:
2112:{\displaystyle i^{th}}
2083:
2047:
1911:
1868:
1842:
1816:
1632:
1631:{\displaystyle \nu =0}
1567:
1544:
1509:
1399:
1377:differential equations
1321:is an idealization of
1263:Conservation of energy
1250:
925:Geoffrey Ingram Taylor
826:Evangelista Torricelli
702:biomedical engineering
87:
3827:Philosophy of physics
3173:Dimensionless numbers
3123:Archimedes' principle
2561:Communicating vessels
2556:Bernoulli's principle
2506:
2476:
2453:
2267:direction coordinate.
2262:
2232:
2230:{\displaystyle x_{j}}
2203:
2173:
2171:{\displaystyle v_{i}}
2144:
2114:
2084:
2048:
1931:Cartesian coordinates
1912:
1869:
1843:
1841:{\displaystyle \tau }
1817:
1698:) is defined to be a
1633:
1568:
1545:
1510:
1400:
1373:George Gabriel Stokes
1335:statistical mechanics
1240:
1049:practical disciplines
1008:and anomalies in the
897:George Gabriel Stokes
869:Joseph Louis Lagrange
797:Archimedes' principle
298:Bernoulli's principle
291:Archimedes' principle
88:
3786:Mathematical physics
2756:Batchelor, George K.
2493:
2465:
2278:
2241:
2214:
2182:
2155:
2123:
2093:
2063:
1940:
1880:
1867:{\displaystyle \mu }
1858:
1832:
1773:
1616:
1566:{\displaystyle \nu }
1557:
1534:
1412:
1387:
1381:incompressible fluid
1317:continuum assumption
1258:Conservation of mass
1125:and crowd dynamics.
974:atmospheric pressure
877:Siméon Denis Poisson
873:Pierre-Simon Laplace
390:Cohesion (chemistry)
212:Infinitesimal strain
48:
18:Continuum assumption
3761:Atmospheric physics
3600:Classical mechanics
3528:branches of physics
2992:Mechanics of Fluids
2541:Transport phenomena
2483:non-Newtonian fluid
1734:non-Newtonian fluid
1552:kinematic viscosity
1369:Claude-Louis Navier
1323:continuum mechanics
1206:Non-Newtonian fluid
1144:Continuum mechanics
1135:continuum mechanics
1123:traffic engineering
1020:(in the context of
913:Theodore von Kármán
907:were investigated (
893:Claude-Louis Navier
737:continuum mechanics
660:concerned with the
308:Poiseuille equation
39:Continuum mechanics
33:Part of a series on
3817:History of physics
3316:Keulegan–Carpenter
3063:2017-08-03 at the
3044:2009-01-19 at the
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802:On Floating Bodies
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509:Electrorheological
246:Fracture mechanics
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3845:
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3832:Physics education
3781:Materials science
3748:Interdisciplinary
3706:Quantum mechanics
3493:
3492:
3019:978-1-43-988882-7
3001:978-0-415-36206-1
2974:McGraw-Hill, Inc.
2961:978-0-12-373735-9
2943:978-1-107-00575-4
2796:978-1-4822-9297-8
2639:978-0-07-352934-9
2551:Applied mechanics
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3184:
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3178:
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3139:
3138:
3136:
3134:
3130:
3124:
3121:
3119:
3116:
3115:
3113:
3111:
3110:Fluid statics
3107:
3103:
3096:
3091:
3089:
3084:
3082:
3077:
3076:
3073:
3066:
3062:
3059:
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2983:0-07-015000-1
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2792:
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2769:0-521-66396-2
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2440:
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2397:
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2252:
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2163:
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2104:
2101:
2097:
2074:
2071:
2067:
2059:
2058:
2057:
2039:
2030:
2026:
2015:
2011:
2001:
1993:
1989:
1978:
1974:
1963:
1959:
1956:
1951:
1948:
1944:
1936:
1935:
1934:
1932:
1928:
1924:
1901:
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1861:
1854:
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1828:
1827:
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1765:
1759:
1749:
1747:
1743:
1739:
1735:
1730:
1728:
1724:
1720:
1716:
1715:perpendicular
1712:
1709:
1705:
1701:
1697:
1694:(named after
1693:
1683:
1681:
1677:
1673:
1667:
1665:
1661:
1657:
1653:
1649:
1645:
1644:superfluidity
1641:
1625:
1622:
1619:
1611:
1607:
1597:
1595:
1591:
1587:
1583:
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1406:
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1374:
1370:
1367:(named after
1366:
1360:
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1126:
1124:
1120:
1117:and modeling
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1083:hydrodynamics
1080:
1079:
1074:
1070:
1066:
1062:
1058:
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995:
991:
987:
983:
979:
976:changes with
975:
971:
967:
964:
960:
956:
952:
951:Fluid statics
947:
946:Fluid statics
940:Fluid statics
935:Main branches
932:
930:
926:
922:
918:
914:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
870:
866:
861:
859:
858:Hydrodynamica
855:
851:
848:, formulated
847:
843:
842:Blaise Pascal
839:
835:
831:
827:
823:
819:
816:
812:
808:
804:
803:
798:
794:
790:
786:
779:
773:
763:
761:
757:
753:
748:
746:
742:
738:
734:
730:
729:fluid statics
725:
723:
719:
715:
711:
707:
704:, as well as
703:
699:
695:
691:
687:
683:
679:
675:
671:
667:
663:
659:
655:
644:
639:
637:
632:
630:
625:
624:
622:
621:
613:
610:
608:
605:
603:
600:
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595:
593:
590:
588:
585:
583:
580:
578:
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570:
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560:
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555:
553:
550:
548:
545:
543:
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531:
520:
517:
515:
512:
510:
507:
506:
505:
504:
501:
498:
497:
492:
489:
487:
484:
482:
479:
478:
477:
476:
471:
466:
465:
456:
453:
452:
446:
443:
441:
438:
436:
433:
431:
428:
426:
425:Charles's law
423:
421:
418:
416:
413:
412:
410:
409:
406:
403:
402:
396:
393:
391:
388:
386:
383:
381:
378:
376:
373:
372:
370:
369:
366:
363:
362:
356:
353:
349:
346:
342:
339:
334:
333:non-Newtonian
331:
327:
323:
322:
321:
318:
316:
313:
309:
306:
304:
301:
299:
296:
292:
289:
287:
284:
280:
277:
276:
274:
273:
270:
267:
266:
261:
256:
255:
247:
244:
242:
239:
235:
232:
231:
230:
227:
225:
222:
220:
219:Compatibility
217:
213:
210:
208:
207:Finite strain
205:
204:
203:
200:
198:
195:
193:
190:
188:
185:
181:
178:
177:
176:
173:
171:
168:
167:
163:
158:
157:
146:
143:
142:
141:
140:
136:
135:
130:
127:
125:
122:
120:
117:
116:
115:
114:
111:Conservations
110:
109:
101:
100:
96:
77:
74:
69:
66:
60:
57:
54:
51:
44:
43:
40:
37:
36:
32:
31:
19:
3756:Astrophysics
3570:Experimental
3146:Aerodynamics
3101:
3009:
2991:
2969:
2951:
2925:
2896:
2890:
2881:
2872:
2863:
2854:
2845:
2836:
2827:
2814:
2805:
2785:
2778:
2759:
2736:
2727:
2718:
2709:
2700:
2691:
2682:
2673:
2654:
2648:
2629:
2546:Aerodynamics
2487:
2460:
2270:
2055:
1920:
1824:
1761:
1731:
1718:
1704:shear stress
1696:Isaac Newton
1691:
1689:
1668:
1647:
1640:idealization
1605:
1603:
1579:
1518:
1364:
1362:
1314:
1312:
1303:
1299:
1295:
1291:
1277:
1252:
1227:shear stress
1224:
1196:
1170:are removed.
1132:
1082:
1078:aerodynamics
1076:
1044:
1038:
1037:
1002:astrophysics
955:hydrostatics
954:
949:
862:
857:
850:Pascal's law
846:hydrostatics
844:(researched
834:Isaac Newton
815:experimental
800:
782:
749:
744:
740:
726:
718:astrophysics
710:oceanography
653:
652:
500:Smart fluids
445:Graham's law
351:
344:
329:
315:Pascal's law
311:
294:
282:
259:
137:Inequalities
3659:Statistical
3575:Theoretical
3552:Engineering
3480:Weissenberg
1923:temperature
1575:body forces
1233:Assumptions
1073:temperature
1014:meteorology
990:engineering
966:equilibrium
745:microscopic
741:macroscopic
714:meteorology
519:Ferrofluids
420:Boyle's law
192:Hooke's law
170:Deformation
3852:Categories
3776:Geophysics
3766:Biophysics
3610:Analytical
3563:Approaches
3400:Richardson
3181:Archimedes
3118:Hydraulics
2612:References
1590:turbulence
1327:continuous
1177:Plasticity
1162:Elasticity
1119:explosions
1045:fluid flow
998:geophysics
986:hydraulics
929:turbulence
883:including
811:Al-Khazini
809:and later
789:Archimedes
706:geophysics
686:mechanical
680:) and the
572:Gay-Lussac
535:Scientists
435:Fick's law
415:Atmosphere
234:frictional
187:Plasticity
175:Elasticity
3726:Molecular
3627:Acoustics
3620:Continuum
3615:Celestial
3605:Newtonian
3592:Classical
3535:Divisions
3485:Womersley
3376:turbulent
3356:Ohnesorge
3341:Marangoni
3306:Iribarren
3231:Damköhler
3216:Capillary
2498:τ
2469:κ
2441:⋅
2438:∇
2426:δ
2422:κ
2406:⋅
2403:∇
2391:δ
2377:−
2361:∂
2346:∂
2324:∂
2309:∂
2298:μ
2283:τ
2208:direction
2149:direction
2068:τ
2023:∂
2008:∂
1986:∂
1971:∂
1960:μ
1945:τ
1862:μ
1836:τ
1786:μ
1783:−
1777:τ
1764:viscosity
1620:ν
1610:viscosity
1561:ν
1492:∇
1488:ν
1479:∇
1474:ρ
1466:−
1452:∇
1449:⋅
1429:∂
1419:∂
1331:molecules
1103:petroleum
1091:movements
881:engineers
838:viscosity
830:barometer
690:aerospace
662:mechanics
612:Truesdell
542:Bernoulli
491:Rheometer
486:Rheometry
326:Newtonian
320:Viscosity
70:φ
58:−
3460:Suratman
3450:Strouhal
3430:Sherwood
3393:magnetic
3388:Reynolds
3383:Rayleigh
3371:magnetic
3211:Brinkman
3061:Archived
3042:Archived
2758:(1967).
2519:See also
1727:friction
1711:gradient
1708:velocity
1682:change.
1672:subsonic
1660:matching
1582:calculus
1529:pressure
1521:momentum
1186:Rheology
1168:stresses
1095:aircraft
1065:pressure
1061:velocity
1018:medicine
978:altitude
860:(1739).
813:applied
793:buoyancy
698:chemical
470:Rheology
375:Adhesion
355:Pressure
341:Buoyancy
286:Dynamics
124:Momentum
3810:Related
3694:General
3689:Special
3547:Applied
3440:Stanton
3435:Shields
3425:Scruton
3420:Schmidt
3366:Prandtl
3351:Nusselt
3326:Laplace
3321:Knudsen
3311:Kapitza
3296:Görtler
3291:Grashof
3281:Galilei
3246:Deborah
3191:Bagnold
3052:CFDWiki
2237:is the
1738:oobleck
1664:laminar
1648:viscous
1608:has no
1304:outside
1292:outside
1111:nebulae
1107:weather
1069:density
899:in the
787:, when
766:History
722:biology
678:plasmas
670:liquids
658:physics
557:Charles
365:Liquids
279:Statics
224:Bending
3721:Atomic
3676:Modern
3526:Major
3470:Ursell
3465:Taylor
3455:Stuart
3445:Stokes
3410:Rossby
3405:Roshko
3361:PĂ©clet
3346:Morton
3286:Graetz
3276:Froude
3266:Eötvös
3256:Eckert
3251:Dukhin
3221:Cauchy
3186:Atwood
3016:
2998:
2980:
2958:
2940:
2903:
2793:
2766:
2661:
2636:
2461:where
2056:where
1825:where
1746:paints
1702:whose
1375:) are
1300:inside
1296:inside
1087:forces
1071:, and
1012:), to
994:fluids
988:, the
963:stable
959:fluids
923:, and
903:, and
840:) and
720:, and
700:, and
682:forces
676:, and
666:fluids
607:Stokes
602:Pascal
592:Navier
587:Newton
577:Graham
552:Cauchy
455:Plasma
350:
348:Mixing
343:
328:
310:
293:
281:
269:Fluids
202:Strain
197:Stress
180:linear
129:Energy
3475:Weber
3415:Rouse
3331:Lewis
3301:Hagen
3271:Euler
3261:Ekman
3236:Darcy
3196:Bejan
1933:) is
1740:, or
1700:fluid
1525:force
1347:scale
1286:—the
1016:, to
694:civil
674:gases
582:Hooke
562:Euler
547:Boyle
405:Gases
3647:Wave
3542:Pure
3336:Mach
3241:Dean
3206:Bond
3201:Biot
3014:ISBN
2996:ISBN
2978:ISBN
2956:ISBN
2938:ISBN
2901:ISBN
2791:ISBN
2764:ISBN
2659:ISBN
2634:ISBN
1850:drag
1742:sand
1723:drag
1371:and
1363:The
1313:The
1089:and
1000:and
895:and
887:and
597:Noll
567:Fick
119:Mass
104:Laws
3642:Ray
2930:doi
1852:"),
1604:An
1302:to
1294:to
1113:in
1101:of
1093:on
982:oil
953:or
832:),
747:.
664:of
3854::
3040:.
2976:,
2972:,
2936:,
2745:^
2620:^
2515:.
1690:A
1612:,
1596:.
1067:,
1063:,
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911:,
875:,
871:,
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724:.
716:,
712:,
708:,
696:,
692:,
688:,
672:,
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2822:.
2799:.
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2302:(
2295:=
2290:j
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2072:i
2040:)
2031:i
2027:x
2016:j
2012:v
2002:+
1994:j
1990:x
1979:i
1975:v
1964:(
1957:=
1952:j
1949:i
1902:n
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1780:=
1626:0
1623:=
1538:p
1523:(
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324:(
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