Stream surfaces in two-dimensional and three-dimensional divergence-free flows

It is demonstrated that two-dimensional flows and three-dimensional flows have fundamentally different influences on stream surfaces. An element that generates a two-dimensional flow only can affect the angle of intersection between two stream surfaces locally; an element that generates a three-dimensional flow may affect this;angle regionally. One consequence is that stream functions for three-dimensional flows may not in general be chosen such that their basis vectors are everywhere mutually orthogonal. Another consequence is that the deformation of stream surfaces produced by a three-dimensional element does not approach that produced by a two-dimensional element at large distances from the element; a two-dimensional model cannot reproduce stream surfaces that occur in a three-dimensional flow.