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.^[1]

Suppose a smooth vector field $\mathbf {u}$ in a simple connected region represents the flow velocity of a fluid at each point. This flow field is said to be irrotational when $\nabla \times \mathbf {u} =0\,.$ If the flow field is irrotational, then it can be also 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 $u$ . Velocity potentials are unique up to a constant and a function solely of the temporal variable. So if $\phi (x,y,z)$ is a velocity potential, then $\phi (x,y,z)+f(t)+C$ generates the same flow field as $\phi (x,y,z)$ .

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,^[2] it is often desirable to work with the acoustic wave equation of the velocity potential $\phi$ 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 $\phi$ 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.^{[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]}

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Velocity potential

Usage in acoustics

See also

Notes