Velocity potential

A velocity potential is used in fluid dynamics, when a fluid occupies a simply-connected region and is irrotational. In such a case,

$\nabla \times \mathbf{u} =0,$

where $\mathbf{u}$ denotes the flow velocity of the fluid. As a result, $\mathbf{u}$ can be represented as the gradient of a scalar function $\Phi\;$ :

$\mathbf{u} = \nabla \Phi\;$ ,

$\Phi\;$ is known as a velocity potential for $\mathbf{u}$ .

A velocity potential is not unique. If $a\;$ is a constant then $\Phi%2Ba\;$ is also a velocity potential for $\mathbf{u}\;$ . Conversely, if $\Psi\;$ is a velocity potential for $\mathbf{u}\;$ then $\Psi=\Phi%2Bb\;$ for some constant $b\;$ . In other words, velocity potentials are unique up to a constant.If value of Φ satisfies Laplace equation,it indicates case of fluid flow.

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

Velocity potential

See also