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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.
Unlike a stream function, a velocity potential can exist in three-dimensional flow.