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 $\Phi+a\;$ 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 $b\;$. 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.

See also

• Hamiltonian fluid mechanics
• Potential flow