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I think this is the best place to post this, as I see a couple of other fluid questions in here. For some reason, everyone always defines stream function as only applying to 2D flow.

The total flow is represented by [tex]\vec{U}[/tex], where [tex]\vec{U}[/tex] is a 3 dimensional vector with components u1, u2, and u3, or whatever your favorite notation is.

If this flow is perfectly non-divergent, then the stream function is defined as the function [tex]\Psi[/tex], such that: [tex]\vec{U}[/tex] = curl([tex]\Psi[/tex]).

Since the curl is defined in 3 dimensions, I see no reason that a 3 dimensional stream function cannot exist. Why do we always pretend that it doesn't?