| 11. | For an irrotational flow, the flow velocity can be described as the gradient of a velocity potential.
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| 12. | Taken together with the velocity potential, the stream function may be used to derive a complex potential.
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| 13. | An analogous result for the velocity potential of a fluid had been obtained some years previously by Leonhard Euler.
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| 14. | In the case of an incompressible flow the velocity potential satisfies Laplace's equation, and potential theory is applicable.
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| 15. | In other words, velocity potentials are unique up to a constant, or a function solely of the temporal variable.
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| 16. | The calculation of the velocity and the velocity potential at a given point is encapsulated inside the class definition of the vortex rings.
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| 17. | Now, for irrotational flow the velocity is the gradient of the velocity potential, and the local Mach number components are defined as:
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| 18. | Here \ varphi ( x, y, t ) is the velocity potential at the mean free-surface level z = 0.
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| 19. | If a velocity potential satisfies Laplace equation, the Clairaut-Schwarz's theorem, the commutation between the gradient and the laplacian operators.
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| 20. | If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero : \ Delta \ Phi = 0.
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