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is known as a velocity potential for .
A velocity potential is not unique. If is a constant, or a function solely of the temporal variable, then is also a velocity potential for . Conversely, if is a velocity potential for then for some constant, or a function solely of the temporal variable . In other words, velocity potentials are unique up to a constant, , or a function solely of the temporal variable.
If a velocity potential satisfies Laplace equation, the flow is incompressible ; one can check this statement by, for instance, developing and using, thanks to the Clairaut-Schwarz's theorem, the commutation between the gradient and the laplacian operators.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.
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it is defined as scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction.