|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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'''.
| | Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her full name. He is an order clerk and it's something free psychic readings ([http://checkmates.co.za/index.php?do=/profile-56347/info/ checkmates.co.za]) he truly enjoy. One of the things she loves most is canoeing and she's been performing it for quite a whilst. My wife and I reside in Mississippi but now I'm considering other choices.<br><br>Stop by my site :: love [http://medialab.zendesk.com/entries/54181460-Will-You-Often-End-Up-Bored-Try-One-Of-These-Hobby-Ideas- online psychics] ([http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going www.zavodpm.ru]) |
| | |
| ==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]]
| |
Alyson Meagher is the title her parents gave her but she doesn't like when individuals use her full name. He is an order clerk and it's something free psychic readings (checkmates.co.za) he truly enjoy. One of the things she loves most is canoeing and she's been performing it for quite a whilst. My wife and I reside in Mississippi but now I'm considering other choices.
Stop by my site :: love online psychics (www.zavodpm.ru)