View source for Velocity potential

{{more citations needed|date=May 2014}}
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>

It is used in [[continuum mechanics]], when a continuum occupies a [[simply-connected]] region and is [[irrotational]]. In such a case,
<math display="block">\nabla \times \mathbf{u} =0 \,,</math>
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}}:
<math display="block"> \mathbf{u} = \nabla \Phi\ =
\frac{\partial \Phi}{\partial x} \mathbf{i} +
\frac{\partial \Phi}{\partial y} \mathbf{j} +
\frac{\partial \Phi}{\partial z} \mathbf{k} \,.</math>

{{math|Φ}} is known as a '''velocity potential''' for {{math|'''u'''}}.

A velocity potential is not unique. If {{math|Φ}} is a velocity potential, then {{math|Φ + ''a''(''t'')}} is also a velocity potential for {{math|'''u'''}}, where {{math|''a''(''t'')}} is a scalar function of time and can be constant.  In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.

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]].

Unlike a [[stream function]], a velocity potential can exist in three-dimensional flow.

==Usage in acoustics==

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'''}}. 
<math display="block"> \nabla ^2 \Phi - \frac{1}{c^2} \frac{ \partial^2 \Phi }{ \partial t ^2 } = 0 </math>
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
<math display="block"> p = -\rho \frac{\partial\Phi}{\partial t} \,.</math>

==See also==
*[[Vorticity]]
*[[Hamiltonian fluid mechanics]]
*[[Potential flow]]
*[[Potential flow around a circular cylinder]]

==Notes==
{{Reflist}}

==External links==
* [http://prj.dimanov.com/ Joukowski Transform Interactive WebApp]

{{DEFAULTSORT:Velocity Potential}}
[[Category:Continuum mechanics]]
[[Category:Physical quantities]]


{{Fluiddynamics-stub}}
[[Category:Potentials]]