Velocity potential

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} A velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.

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 $\Phi \;$:

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

$\Phi \;$ 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 as

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

See also

• Vorticity
• Hamiltonian fluid mechanics
• Potential flow