# Velocity potential

A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.[1]

It 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, or a function solely of the temporal variable, then $\Phi+a(t)\;$ is also a velocity potential for $\mathbf{u}\;$. Conversely, if $\Psi\;$ is a velocity potential for $\mathbf{u}\;$ then $\Psi=\Phi+b\;$ for some constant, or a function solely of the temporal variable $b(t)\;$. In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.

If a velocity potential satisfies Laplace equation, the flow is incompressible ; one can check this statement by, for instance, developing $\nabla \times (\nabla \times \mathbf{u})$ and using, thanks to the Clairaut-Schwarz's theorem, the commutation between the gradient and the laplacian operators.

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

## Notes

1. ^ Anderson John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 0521669553.