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In [[continuum mechanics]] the '''macroscopic velocity'''<ref>{{harvnb|Duderstadt|1979|p=218}}, {{harvnb|Freidberg|2008|p=225}}</ref>, also '''flow velocity''' in [[fluid dynamics]] or '''drift velocity''' in [[electromagnetism]], of a fluid is a [[vector field]] which is used to mathematically describe the motion of a fluid. The length of the flow velocity vector is the '''flow speed'''. | |||
==Definition== | |||
The flow velocity '''''u''''' of a fluid is a vector field | |||
:<math> \mathbf{u}=\mathbf{u}(\mathbf{x},t)</math> | |||
which gives the [[velocity]] of an ''[[fluid parcel|element of fluid]]'' at a position <math>\mathbf{x}\,</math> and time <math> t\, </math>. | |||
The flow speed ''q'' is the length of the flow velocity vector<ref>{{cite book| first1=R. | last1=Courant | author1-link=Richard Courant | first2=K.O. | last2=Friedrichs | author2-link=Kurt Otto Friedrichs | edition=5th | publisher=Springer | origyear=First published in 1948 | isbn=0387902325 | pages=24 | title=Supersonic Flow and Shock Waves | oclc=44071435 | publisher=Springer-Verlag New York Inc | year=1999 | series=Applied mathematical sciences}}</ref> | |||
:<math>q = || \mathbf{u} ||</math> | |||
and is a scalar field. | |||
==Uses== | |||
The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: | |||
===Steady flow=== | |||
{{Main|Steady flow}} | |||
The flow of a fluid is said to be ''steady'' if <math> \mathbf{u}</math> does not vary with time. That is if | |||
:<math> \frac{\partial \mathbf{u}}{\partial t}=0.</math> | |||
===Incompressible flow=== | |||
{{Main|Incompressible flow}} | |||
If a fluid is incompressible the [[divergence]] of <math>\mathbf{u}</math> is zero: | |||
:<math> \nabla\cdot\mathbf{u}=0.</math> | |||
That is, if <math>\mathbf{u}</math> is a [[solenoidal vector field]]. | |||
===Irrotational flow=== | |||
{{main|Irrotational flow}} | |||
A flow is ''irrotational'' if the [[Curl (mathematics)|curl]] of <math>\mathbf{u}</math> is zero: | |||
:<math> \nabla\times\mathbf{u}=0. </math> | |||
That is, if <math>\mathbf{u}</math> is an [[irrotational vector field]]. | |||
A flow in a [[simply-connected domain]] which is irrotational can be described as a [[potential flow]], through the use of a [[velocity potential]] <math>\Phi,</math> with <math>\mathbf{u}=\nabla\Phi.</math> If the flow is both irrotational and incompressible, the [[Laplacian]] of the velocity potential must be zero: <math>\Delta\Phi=0.</math> | |||
===Vorticity=== | |||
{{Main| Vorticity}} | |||
The ''vorticity'', <math>\omega</math>, of a flow can be defined in terms of its flow velocity by | |||
:<math> \omega=\nabla\times\mathbf{u}.</math> | |||
Thus in irrotational flow the vorticity is zero. | |||
==The velocity potential== | |||
{{main|Potential flow}} | |||
If an irrotational flow occupies a [[simply-connected]] fluid region then there exists a [[scalar field]] <math> \phi </math> such that | |||
:<math> \mathbf{u}=\nabla\mathbf{\phi} </math> | |||
The scalar field <math>\phi</math> is called the [[velocity potential]] for the flow. (See [[Irrotational vector field]].) | |||
==Notes and references== | |||
{{reflist}} | |||
==Further reading== | |||
* {{cite book |author=Duderstadt, James J., Martin, William R.| title= Transport theory | editor=Wiley-Interscience Publications | location= New York| year= 1979 | ed= | ISBN=978-0471044925|chapter=Chapter 4:The derivation of continuum description from trasport equations}} | |||
* {{cite book | author=Freidberg, Jeffrey P.|title=Plasma Physics and Fusion Energy|edition=1|editor=Cambridge University Press|location=Cambridge|year=2008| ISBN=978-0521733175|chapter=Chapter 10:A self-consistent two-fluid model}} | |||
[[Category:Fluid dynamics]] | |||
[[Category:Continuum mechanics]] | |||
[[Category:Vector calculus]] | |||
Revision as of 15:51, 14 January 2014
In continuum mechanics the macroscopic velocity[1], also flow velocity in fluid dynamics or drift velocity in electromagnetism, of a fluid is a vector field which is used to mathematically describe the motion of a fluid. The length of the flow velocity vector is the flow speed.
Definition
The flow velocity u of a fluid is a vector field
which gives the velocity of an element of fluid at a position and time .
The flow speed q is the length of the flow velocity vector[2]
and is a scalar field.
Uses
The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow:
Steady flow
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The flow of a fluid is said to be steady if does not vary with time. That is if
Incompressible flow
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If a fluid is incompressible the divergence of is zero:
That is, if is a solenoidal vector field.
Irrotational flow
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A flow is irrotational if the curl of is zero:
That is, if is an irrotational vector field.
A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential with If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero:
Vorticity
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The vorticity, , of a flow can be defined in terms of its flow velocity by
Thus in irrotational flow the vorticity is zero.
The velocity potential
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field such that
The scalar field is called the velocity potential for the flow. (See Irrotational vector field.)
Notes and references
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Further reading
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
- ↑ Template:Harvnb, Template:Harvnb
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534