Velocity potential

Within the applied mathematical study of fluid dynamics and continuum mechanics, a velocity potential is a scalar potential used in potential flow theory. It was introduced by Joseph-Louis Lagrange in 1788.

Suppose a smooth vector field $𝐮$ in a simple connected region represents the flow velocity of a fluid at each point. This flow field is said to be irrotational when $\nabla \times 𝐮 = 0 .$ If the flow field is irrotational, then it can be also be represented as the gradient of a scalar function $ϕ$ : $𝐮 = \nabla ϕ = \frac{\partial ϕ}{\partial x} 𝐢 + \frac{\partial ϕ}{\partial y} 𝐣 + \frac{\partial ϕ}{\partial z} 𝐤 .$

$ϕ$ is known as a velocity potential for $u$ . Velocity potentials are unique up to a constant and a function solely of the temporal variable. So if $ϕ (x, y, z)$ is a velocity potential, then $ϕ (x, y, z) + f (t) + C$ generates the same flow field as $ϕ (x, y, z)$ .

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

In theoretical acoustics, it is often desirable to work with the acoustic wave equation of the velocity potential $ϕ$ instead of pressure $p$ and/or particle velocity $u$ . $\nabla^{2} ϕ - \frac{1}{c^{2}} \frac{\partial^{2} ϕ}{\partial t^{2}} = 0$ 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 $p = - ρ \frac{\partial ϕ}{\partial t} .$

Notes

↑ 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]}

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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]}

Velocity potential

Usage in acoustics

See also

Notes