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Fluid mechanics

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Branch of physics
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Continuum mechanics
J=−Ddφdx{\displaystyle J=-D{\frac {d\varphi }{dx}}}

Fluid mechanics is the branch ofphysics concerned with themechanics offluids (liquids,gases, andplasmas) and theforces on them.[1]: 3 Originally applied to water (hydromechanics), it found applications in a wide range of disciplines, includingmechanical,aerospace,civil,chemical, andbiomedical engineering, as well asgeophysics,oceanography,meteorology,astrophysics, andbiology.

It can be divided intofluid statics, the study of various fluids at rest; andfluid dynamics, the study of the effect of forces on fluid motion.[1]: 3 It is a branch ofcontinuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from amacroscopic viewpoint rather than frommicroscopic.

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 bynumerical methods, typically using computers. A modern discipline, calledcomputational fluid dynamics (CFD), is devoted to this approach.[2]Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.

Etymology

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The first use of the phrase "fluid mechanics" is attributed to A. H. Jameson in his work, An Introduction to Fluid Mechanics (1937).[3]

History

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Main article:History of fluid mechanics
For a chronological guide, seeTimeline of fluid and continuum mechanics.

Historical Development of Fluid Mechanics

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The study of fluid mechanics goes back at least to the days ofancient Greece, whenArchimedes investigated fluid statics andbuoyancy and formulated his famous law known now as theArchimedes' principle, which was published in his workOn Floating Bodies—generally considered to be the first major work on fluid mechanics. Iranian scholarAbu Rayhan Biruni and laterAl-Khazini appliedexperimentalscientific methods to fluid mechanics.[4] Rapid advancement in fluid mechanics began withLeonardo da Vinci (observations and experiments),Evangelista Torricelli (invented thebarometer),Isaac Newton (investigatedviscosity) andBlaise Pascal (researchedhydrostatics, formulatedPascal's law), and was continued byDaniel Bernoulli with the introduction of mathematical fluid dynamics inHydrodynamica (1739).

Advancements in Fluid Flow Analysis

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Inviscid flow was further analyzed by various mathematicians (Jean le Rond d'Alembert,Joseph Louis Lagrange,Pierre-Simon Laplace,Siméon Denis Poisson) and viscous flow was explored by a multitude ofengineers includingJean Léonard Marie Poiseuille andGotthilf Hagen. Further mathematical justification was provided byClaude-Louis Navier andGeorge Gabriel Stokes in theNavier–Stokes equations, andboundary layers were investigated (Ludwig Prandtl,Theodore von Kármán), while various scientists such asOsborne Reynolds,Andrey Kolmogorov, andGeoffrey Ingram Taylor advanced the understanding of fluid viscosity andturbulence.

Main branches

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Fluid statics

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Main article:Fluid statics

Fluid statics orhydrostatics is the branch of fluid mechanics that studiesfluids at rest. It embraces the study of the conditions under which fluids are at rest instableequilibrium; and is contrasted withfluid dynamics, the study of fluids in motion. Hydrostatics offers physical explanations for many phenomena of everyday life, such as whyatmospheric pressure changes withaltitude, why wood andoil float on water, and why the surface of water is always level whatever the shape of its container. Hydrostatics is fundamental tohydraulics, theengineering of equipment for storing, transporting and usingfluids. It is also relevant to some aspects ofgeophysics andastrophysics (for example, in understandingplate tectonics and anomalies in theEarth's gravitational field), tometeorology, tomedicine (in the context ofblood pressure), and many other fields.

Fluid dynamics

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Main article:Fluid dynamics

Fluid dynamics is a subdiscipline of fluid mechanics that deals withfluid flow—the science of liquids and gases in motion.[5] Fluid dynamics offers a systematic structure—which underlies thesepractical disciplines—that embraces empirical and semi-empirical laws derived fromflow measurement and used to solve practical problems. The solution to afluid dynamics problem typically involves calculating various properties of the fluid, such asvelocity,pressure,density, andtemperature, as functions of space and time. It has several subdisciplines itself, includingaerodynamics[6][7][8][9] (the study of air and other gases in motion) andhydrodynamics[10][11] (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculatingforces andmovements onaircraft, determining themass flow rate ofpetroleum through pipelines, predicting evolvingweather patterns, understandingnebulae ininterstellar space and modelingexplosions. Some fluid-dynamical principles are used intraffic engineering and crowd dynamics.

Continuum Mechanics

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Fluid mechanics is a subdiscipline ofcontinuum mechanics, as illustrated in the following table:

Continuum mechanics
The study of the physics of continuous materials		Solid mechanics
The study of the physics of continuous materials with a defined rest shape.		Elasticity
Describes materials that return to their rest shape after appliedstresses are removed.
Plasticity
Describes materials that permanently deform after a sufficient applied stress.		Rheology
The study of materials with both solid and fluid characteristics.
Fluid mechanics
The study of the physics of continuous materials which deform when subjected to a force.		Non-Newtonian fluid
Do not undergo strain rates proportional to the applied shear stress.
Newtonian fluids undergo strain rates proportional to the applied shear stress.

In a mechanical view, a fluid is a substance that does not supportshear stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.

Assumptions

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Balance for some integrated fluid quantity in acontrol volume enclosed by acontrol surface.

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:

For example, the assumption that mass is conserved means that for any fixedcontrol volume (for example, a spherical volume)—enclosed by acontrol surface—therate of change of the mass contained in that volume is equal to the rate at which mass is passing through the surface fromoutside toinside, minus the rate at which mass is passing frominside tooutside. This can be expressed as anequation in integral form over the control volume.[12]: 74 

Continuum Assumption

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Thecontinuum assumption is an idealization ofcontinuum mechanics under which fluids can be treated ascontinuous, even though, on a microscopic scale, they are composed ofmolecules. 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.[13] Those problems for which the continuum hypothesis fails can be solved usingstatistical mechanics. To determine whether or not the continuum hypothesis applies, theKnudsen number, defined as the ratio of the molecularmean free path to the characteristic lengthscale, 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.

Navier–Stokes equations

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Main article:Navier–Stokes equations

TheNavier–Stokes equations (named afterClaude-Louis Navier andGeorge Gabriel Stokes) aredifferential equations that describe the force balance at a given point within a fluid. For anincompressible fluid with vector velocity fieldu{\displaystyle \mathbf {u} }, the Navier–Stokes equations are[14][15][16][17]

∂u∂t+(u⋅∇)u=−1ρ∇p+ν∇2u{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\nu \nabla ^{2}\mathbf {u} }.

These differential equations are the analogues for deformable materials to Newton's equations of motion for particles – the Navier–Stokes equations describe changes inmomentum (force) in response topressurep{\displaystyle p} and viscosity, parameterized by thekinematic viscosityν{\displaystyle \nu }. Occasionally,body forces, such as the gravitational force or Lorentz force are added to the equations.

Solutions of the Navier–Stokes equations for a given physical problem must be sought with the help ofcalculus. In practical terms, only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which theReynolds number is small. For more complex cases, especially those involvingturbulence, 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 calledcomputational fluid dynamics.[18][19][20][21][22]

Inviscid and viscous fluids

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Aninviscid fluid has noviscosity,ν=0{\displaystyle \nu =0}. In practice, an inviscid flow is anidealization, one that facilitates mathematical treatment. In fact, purely inviscid flows are only known to be realized in the case ofsuperfluidity. Otherwise, fluids are generallyviscous, a property that is often most important within aboundary layer near a solid surface,[23] where the flow must match onto theno-slip condition 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 thenmatching its solution onto that for a thinlaminar boundary layer.

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 lowsubsonic speeds to assume that gas isincompressible—that is, the density of the gas does not change even though the speed andstatic pressure change.

Newtonian vs Non-Newtonian fluids

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ANewtonian fluid (named afterIsaac Newton) is defined to be afluid whoseshear stress is linearly proportional to thevelocitygradient in the directionperpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, itcontinues to flow. 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 thedrag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Comparefriction). Important fluids, like water as well as most gasses, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth.[12]: 145 

By contrast, stirring anon-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time—this behavior is seen in materials such as pudding,oobleck, orsand (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-drippaints). 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.[12]: 145 

Equations for a Newtonian fluid

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Main article:Newtonian fluid

The constant of proportionality between the viscous stress tensor and the velocity gradient is known as theviscosity. A simple equation to describe incompressible Newtonian fluid behavior is

τ=−μdudn{\displaystyle \tau =-\mu {\frac {\mathrm {d} u}{\mathrm {d} n}}}

where

τ{\displaystyle \tau } is the shear stress exerted by the fluid ("drag"),
μ{\displaystyle \mu } is the fluid viscosity—a constant of proportionality, and
dudn{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} n}}} is the velocity gradient perpendicular to the direction of shear.

For a Newtonian fluid, the viscosity, by definition, depends only ontemperature, not on the forces acting upon it. If the fluid isincompressible the equation governing the viscous stress (inCartesian coordinates) is

τij=μ(∂vi∂xj+∂vj∂xi){\displaystyle \tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)}

where

τij{\displaystyle \tau _{ij}} is the shear stress on theith{\displaystyle i^{th}} face of a fluid element in thejth{\displaystyle j^{th}} direction
vi{\displaystyle v_{i}} is the velocity in theith{\displaystyle i^{th}} direction
xj{\displaystyle x_{j}} is thejth{\displaystyle j^{th}} direction coordinate.

If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is

τij=μ(∂vi∂xj+∂vj∂xi−23δij∇⋅v)+κδij∇⋅v{\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} }

whereκ{\displaystyle \kappa } is the second viscosity coefficient (or bulk viscosity). If a fluid does not obey this relation, it is termed anon-Newtonian fluid, of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic.

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 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 tensorτ{\displaystyle \mathbf {\tau } } in the Navier–Stokes equation vanishes. The equation reduced in this form is called theEuler equation.

See also

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References

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  1. ^abWhite, Frank M. (2011).Fluid Mechanics (7th ed.). McGraw-Hill.ISBN 978-0-07-352934-9.
  2. ^Tu, Jiyuan; Yeoh, Guan Heng; Liu, Chaoqun (Nov 21, 2012).Computational Fluid Dynamics: A Practical Approach. Butterworth-Heinemann.ISBN 978-0080982434.
  3. ^"OED- oxford English Dictionary fluid mechanics online entry".
  4. ^Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642,
  5. ^Batchelor, C. K., & Batchelor, G. K. (2000). An introduction to fluid dynamics. Cambridge University Press.
  6. ^Bertin, J. J., & Smith, M. L. (1998). Aerodynamics for engineers (Vol. 5). Upper Saddle River, NJ: Prentice Hall.
  7. ^Anderson Jr, J. D. (2010). Fundamentals of aerodynamics. Tata McGraw-Hill Education.
  8. ^Houghton, E. L., & Carpenter, P. W. (2003). Aerodynamics for engineering students. Elsevier.
  9. ^Milne-Thomson, L. M. (1973). Theoretical aerodynamics. Courier Corporation.
  10. ^Milne-Thomson, L. M. (1996). Theoretical hydrodynamics. Courier Corporation.
  11. ^Birkhoff, G. (2015). Hydrodynamics. Princeton University Press.
  12. ^abcBatchelor, George K. (1967).An Introduction to Fluid Dynamics. Cambridge University Press. p. 74.ISBN 0-521-66396-2.
  13. ^Greenkorn, Robert (3 October 2018).Momentum, Heat, and Mass Transfer Fundamentals. CRC Press. p. 18.ISBN 978-1-4822-9297-8.
  14. ^Constantin, P., & Foias, C. (1988). Navier-stokes equations. University of Chicago Press.
  15. ^Temam, R. (2001). Navier-Stokes equations: theory and numerical analysis (Vol. 343).American Mathematical Society.
  16. ^Foias, C., Manley, O., Rosa, R., & Temam, R. (2001). Navier-Stokes equations and turbulence (Vol. 83). Cambridge University Press.
  17. ^Girault, V., & Raviart, P. A. (2012). Finite element methods for Navier-Stokes equations: theory and algorithms (Vol. 5). Springer Science & Business Media.
  18. ^Anderson, J. D., & Wendt, J. (1995). Computational fluid dynamics (Vol. 206). New York: McGraw-Hill.
  19. ^Chung, T. J. (2010). Computational fluid dynamics. Cambridge University Press.
  20. ^Blazek, J. (2015). Computational fluid dynamics: principles and applications. Butterworth-Heinemann.
  21. ^Wesseling, P. (2009). Principles of computational fluid dynamics (Vol. 29). Springer Science & Business Media.
  22. ^Anderson, D., Tannehill, J. C., & Pletcher, R. H. (2016). Computational fluid mechanics and heat transfer. Taylor & Francis.
  23. ^Kundu, Pijush K.; Cohen, Ira M.; Dowling, David R. (27 March 2015). "10".Fluid Mechanics (6th ed.). Academic Press.ISBN 978-0124059351.

Further reading

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External links

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Fluid mechanics at Wikipedia'ssister projects
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