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