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| {{Unreferenced|date=March 2009}}
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| A '''velocity potential''' is used in [[fluid dynamics]], when a fluid occupies a [[simply-connected]] region and is [[irrotational]]. In such a case,
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| : <math>\nabla \times \mathbf{u} =0,</math>
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| where <math> \mathbf{u} </math> denotes the [[flow velocity]] of the fluid. As a result, <math> \mathbf{u} </math> can be represented as the [[gradient]] of a [[scalar field|scalar]] function <math>\Phi\;</math>:
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| :<math> \mathbf{u} = \nabla \Phi\;</math>,
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| <math>\Phi\;</math> is known as a '''velocity potential''' for <math>\mathbf{u}</math>.
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| A velocity potential is not unique. If <math>a\;</math> is a constant, or a function solely of the temporal variable, then <math>\Phi+a(t)\;</math> is also a velocity potential for <math>\mathbf{u}\;</math>. Conversely, if <math>\Psi\;</math> is a velocity potential for <math>\mathbf{u}\;</math> then <math>\Psi=\Phi+b\;</math> for some constant, or a function solely of the temporal variable <math>b(t)\;</math>. In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.
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| If a velocity potential satisfies [[Laplace equation]], the [[potential flow|flow]] is [[incompressible flow|incompressible]] ; one can check this statement by, for instance, developing <math>\nabla \times (\nabla \times \mathbf{u}) </math> and using, thanks to the [[Symmetry of second derivatives|Clairaut-Schwarz's theorem]], the commutation between the gradient and the laplacian operators.
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| Unlike a [[stream function]], a velocity potential can exist in three-dimensional flow.
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| It is defined as scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction.
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| ==See also== | |
| *[[Hamiltonian fluid mechanics]]
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| *[[Potential flow]]
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| {{DEFAULTSORT:Velocity Potential}}
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| [[Category:Fluid dynamics]]
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| [[Category:Equations of fluid dynamics]]
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| {{Fluiddynamics-stub}}
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