Physics: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, [math]\displaystyle{ \nabla \times \mathbf{u} =0 \,, }[/math] where u denotes the flow velocity. As a result, u can be represented as the gradient of a scalar function Φ: [math]\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} \,. }[/math]
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. [math]\displaystyle{ \nabla ^2 \Phi - \frac{1}{c^2} \frac{ \partial^2 \Phi }{ \partial t ^2 } = 0 }[/math] 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 [math]\displaystyle{ p = -\rho \frac{\partial\Phi}{\partial t} \,. }[/math]
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]
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