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,
![{\displaystyle \nabla \times \mathbf {u} =0\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b545c763e7286ef8cfcf5d798481b8fc51c9acf)
where
u denotes the flow velocity. As a result,
u can be represented as the gradient of a scalar function
Φ:
![{\displaystyle \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} \,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8af2537cc5826d4b27719627054042e0e8bfa81a)
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
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.
![{\displaystyle \nabla ^{2}\Phi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c97881c686956f5da18eafcf256e8695235ca601)
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
![{\displaystyle p=-\rho {\frac {\partial \Phi }{\partial t}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a16894b6f93369721ca3665d7fb81979c07eefd)
See also
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]
External links
- Joukowski Transform Interactive WebApp