Each and every engineering student finds himself confronted with the Biot-Savart law at one point or another of their undergraduate studies whether it is related to fluid mechanics or classical electromagnetics.
The Biot-Savart carries along the qualitative idea that knowing the curl of a vector field at one point allows us to infer something about the vector field itself at another point.
As attractive as the idea is, it’s often misleading as it frequently leads to confusion concerning cause and effect.
Moreover, the fact that Navier-Stokes equations may be straightforwardly transformed from velocity to vorticity formulation and the use of potential flow related models to create obstructions to the flow strengthens the Biot-Savart frequently inferred view that vorticity induces velocity.
Well, here lies the fallacy. In the absence of a gravitational or electromagnetic body forces there is no action at a distance in ordinary fluid flows.
Casting the equations in one form or another and appealing to the Bio-Savart law as a calculus relation between a vector field and its curl does not mean a vortex at point A can cause a velocity at a remote point B.
In conclusion, although the claim that a mathematical relation as the Biot-Savart allows us to infer both quantitative and qualitative information about the velocity field at a distant point is true, in fluid mechanics it does not represent the physics and such a direct cause and effect relation is somewhat misleading (as opposed to its counterpart analogy in classical electromagnetics).