Why must stream function be 2d?

[Weather Freak]

Hi Folks,

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 $$\vec{U}$$, where $$\vec{U}$$ 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 $$\Psi$$, such that: $$\vec{U}$$ = curl($$\Psi$$).

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?

 

[arildno]

Alternatively, we call that "stream function" the vector potential of the velocity field.
And, it sure exists, but that is irrelevant!

The reason why we don't bother about the vector potential in 3-D fluid mechanics is that no simplification whatsoever is gained by doing so!

In the 2-D case, our stream function is a scalar function, and we therefore reduce our dimensions by one (from the two unknown components of the velocity field).

This will very often simplify our calculations, even though introduction of the stream function will introduce higher order derivatives than by keeping our velocity field as the primary unknown.

In the general 3-D case, however, no such simplification occurs by introducing the vector potential.
Instead, we merely increase the order of derivatives by one by switching from the 3-D velocity field as our primary unknown to the 3-D vector potential.

Thus, we choose not to use it.