http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Velocity_potentialVelocity potential - Revision history2026-05-04T22:26:56ZRevision history for this page on the wikiMediaWiki 1.43.1http://debianws.lexgopc.com/wiki143/index.php?title=Velocity_potential&diff=1743738&oldid=previmported>Dolphin51: /* top */ Erased redundant words in accordance with MOS:NOTE2024-09-26T11:37:40Z<p><span class="autocomment">top: </span> Erased redundant words in accordance with MOS:NOTE</p>
<p><b>New page</b></p><div>{{more citations needed|date=May 2014}}<br />
A '''velocity potential''' is a [[scalar potential]] used in [[potential flow]] theory. It was introduced by [[Joseph-Louis Lagrange]] in 1788.<ref>{{cite book|last=Anderson|first=John|title=A History of Aerodynamics|year=1998|publisher=Cambridge University Press|isbn=978-0521669559}}{{page needed|date=December 2017}}</ref><br />
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It is used in [[continuum mechanics]], when a continuum occupies a [[simply-connected]] region and is [[irrotational]]. In such a case,<br />
<math display="block">\nabla \times \mathbf{u} =0 \,,</math><br />
where {{math|'''u'''}} denotes the [[flow velocity]]. As a result, {{math|'''u'''}} can be represented as the [[gradient]] of a [[scalar field|scalar]] function {{math|ϕ}}:<br />
<math display="block"> \mathbf{u} = \nabla \varphi\ =<br />
\frac{\partial \varphi}{\partial x} \mathbf{i} +<br />
\frac{\partial \varphi}{\partial y} \mathbf{j} +<br />
\frac{\partial \varphi}{\partial z} \mathbf{k} \,.</math><br />
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{{math|ϕ}} is known as a '''velocity potential''' for {{math|'''u'''}}.<br />
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A velocity potential is not unique. If {{math|ϕ}} is a velocity potential, then {{math|ϕ + ''f''(''t'')}} is also a velocity potential for {{math|'''u'''}}, where {{math|''f''(''t'')}} is a scalar function of time and can be constant. Velocity potentials are unique up to a constant, or a function solely of the temporal variable.<br />
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The [[Laplace operator|Laplacian]] of a velocity potential is equal to the [[divergence]] of the corresponding flow. Hence if a velocity potential satisfies [[Laplace equation]], the [[potential flow|flow]] is [[incompressible flow|incompressible]].<br />
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Unlike a [[stream function]], a velocity potential can exist in three-dimensional flow.<br />
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==Usage in acoustics==<br />
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In theoretical [[acoustics]],<ref>{{cite book|last=Pierce |first=A. D.|title=Acoustics: An Introduction to Its Physical Principles and Applications|year=1994|publisher=Acoustical Society of America|isbn=978-0883186121}}{{page needed|date=December 2017}}</ref> it is often desirable to work with the [[acoustic wave equation]] of the velocity potential {{math|ϕ}} instead of pressure {{mvar|p}} and/or [[particle velocity]] {{math|'''u'''}}. <br />
<math display="block"> \nabla ^2 \varphi - \frac{1}{c^2} \frac{ \partial^2 \varphi }{ \partial t ^2 } = 0 </math><br />
Solving the wave equation for either {{mvar|p}} field or {{math|'''u'''}} field does not necessarily provide a simple answer for the other field. On the other hand, when {{math|ϕ}} is solved for, not only is {{math|'''u'''}} found as given above, but {{mvar|p}} is also easily found—from the (linearised) [[Bernoulli's principle|Bernoulli equation]] for [[irrotational flow|irrotational]] and [[unsteady flow]]—as<br />
<math display="block"> p = -\rho \frac{\partial\varphi}{\partial t} \,.</math><br />
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==See also==<br />
*[[Vorticity]]<br />
*[[Hamiltonian fluid mechanics]]<br />
*[[Potential flow]]<br />
*[[Potential flow around a circular cylinder]]<br />
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==Notes==<br />
{{Reflist}}<br />
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==External links==<br />
* [http://prj.dimanov.com/ Joukowski Transform Interactive WebApp]<br />
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{{DEFAULTSORT:Velocity Potential}}<br />
[[Category:Continuum mechanics]]<br />
[[Category:Physical quantities]]<br />
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{{Fluiddynamics-stub}}<br />
[[Category:Potentials]]</div>imported>Dolphin51