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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'''.
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| ==Definition==
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| The flow velocity '''''u''''' of a fluid is a vector field
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| :<math> \mathbf{u}=\mathbf{u}(\mathbf{x},t)</math>
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| which gives the [[velocity]] of an ''[[fluid parcel|element of fluid]]'' at a position <math>\mathbf{x}\,</math> and time <math> t\, </math>.
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| 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>
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| :<math>q = || \mathbf{u} ||</math>
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| and is a scalar field.
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| ==Uses==
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| 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:
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| ===Steady flow===
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| {{Main|Steady flow}}
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| The flow of a fluid is said to be ''steady'' if <math> \mathbf{u}</math> does not vary with time. That is if
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| :<math> \frac{\partial \mathbf{u}}{\partial t}=0.</math>
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| ===Incompressible flow===
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| {{Main|Incompressible flow}}
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| If a fluid is incompressible the [[divergence]] of <math>\mathbf{u}</math> is zero:
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| :<math> \nabla\cdot\mathbf{u}=0.</math>
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| That is, if <math>\mathbf{u}</math> is a [[solenoidal vector field]].
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| ===Irrotational flow===
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| {{main|Irrotational flow}}
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| A flow is ''irrotational'' if the [[Curl (mathematics)|curl]] of <math>\mathbf{u}</math> is zero:
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| :<math> \nabla\times\mathbf{u}=0. </math>
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| That is, if <math>\mathbf{u}</math> is an [[irrotational vector field]].
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| 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>
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| ===Vorticity===
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| {{Main| Vorticity}}
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| The ''vorticity'', <math>\omega</math>, of a flow can be defined in terms of its flow velocity by
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| :<math> \omega=\nabla\times\mathbf{u}.</math>
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| Thus in irrotational flow the vorticity is zero.
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| ==The velocity potential== | |
| {{main|Potential flow}}
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| If an irrotational flow occupies a [[simply-connected]] fluid region then there exists a [[scalar field]] <math> \phi </math> such that
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| :<math> \mathbf{u}=\nabla\mathbf{\phi} </math> | |
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| The scalar field <math>\phi</math> is called the [[velocity potential]] for the flow. (See [[Irrotational vector field]].)
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| ==Notes and references==
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| {{reflist}}
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| ==Further reading==
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| * {{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}}
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| * {{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}}
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| [[Category:Fluid dynamics]]
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| [[Category:Continuum mechanics]]
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| [[Category:Vector calculus]]
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