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 $Φ$ is a velocity potential, then $Φ + a (t)$ is also a velocity potential for $u$ , where $a (t)$ is a scalar function of time and can be constant. In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.

The Laplacian of a velocity potential is equal to the divergence of the corresponding flow. Hence if a velocity potential satisfies Laplace equation, the flow is incompressible.

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

Usage in acoustics[edit]

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[edit]

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

External links[edit]

Joukowski Transform Interactive WebApp

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[1] Anderson, John (1998). A History of Aerodynamics. Cambridge University Press. ISBN 978-0521669559.^{[page needed]}

[2] Pierce, A. D. (1994). Acoustics: An Introduction to Its Physical Principles and Applications. Acoustical Society of America. ISBN 978-0883186121.^{[page needed]}

[1]

[2]

Velocity potential

Usage in acoustics[edit]

See also[edit]

Notes[edit]

External links[edit]