I don't know details of your problem, however you should consider that if the problem is three-dimensional, you have a single potential scalar function (v= grad phi ) but a vector (3 scalar functions) for the "stream function". Actually, in 3D you can no longet call them stream function...
You can also solve stream function in 2D. Try and see! But In 3D things become not easier for stream function than for the potential. Check out Boundary Element Methods in Fluid Mechnaics books.