# 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 continuum mechanics, when a continuum occupies a simply-connected region and is irrotational. In such a case,

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

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

$\mathbf{u} = \nabla \Phi\ = \frac{\partial \Phi}{\partial x} \mathbf{i} + \frac{\partial \Phi}{\partial y} \mathbf{j} + \frac{\partial \Phi}{\partial z} \mathbf{k} .$

$\Phi\;$ 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.

## Usage in acoustics

In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential $\Phi\;$ instead of pressure $p\;$ and/or particle velocity $\mathbf{u}\;$.

$\nabla ^2 \Phi - {1 \over c^2} { \partial^2 \Phi \over \partial t ^2 } = 0$

Solving the wave equation for either $p\;$ field or $\mathbf{u}\;$ field doesn't necessarily provide a simple answer for the other field. On the other hand, when $\Phi\;$ is solved for, not only is $\mathbf{u}\;$ found as given above, but $p\;$ is also easily found – from the (linearised) Bernoulli equation for irrotational and unsteady flow – as

$p = -\rho {\partial \over \partial t}\Phi$.

## Notes

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