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 $u$ denotes the flow velocity. As a result, $u$ can be represented as the gradient of a scalar function $Φ$ :

\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} \,.

$Φ$ is known as a velocity potential for $u$ .

A velocity potential is not unique. If $a$ is a constant, or a function solely of the temporal variable, then $Φ + a (t)$ is also a velocity potential for $u$ . Conversely, if $Ψ$ is a velocity potential for $u$ then $Ψ = Φ + b$ 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 $\nabla \times (\nabla \times u)$ 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,^[2] it is often desirable to work with the acoustic wave equation of the velocity potential $Φ$ instead of pressure $p$ and/or particle velocity $u$ .

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

Solving the wave equation for either $p$ field or $u$ field does not necessarily provide a simple answer for the other field. On the other hand, when $Φ$ is solved for, not only is $u$ found as given above, but $p$ is also easily found – from the (linearised) Bernoulli equation for irrotational and unsteady flow – as

p=-\rho {\frac {\partial \Phi }{\partial t}}\,.

Notes

↑ Anderson, John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 978-0521669559.
↑ Pierce, A. D. (1994). Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America. ISBN 978-0883186121.

Velocity potential

Usage in acoustics

Notes

See also