Stream function

Remarks

If a stream function exists, then $\partial u / \partial x + \partial v / \partial y = \psi_{yx} - \psi_{xy} = 0$, so the flow is incompressible. Conversely, on a simply connected domain every divergence-free field has a stream function, unique up to an additive constant. Example: the uniform flow $\mathbf{u} = (U, 0)$ has the stream function $\psi = U y$; check: $\psi_y = U$ and $-\psi_x = 0$. Non-example: the field $\mathbf{u} = (x, y)$ has divergence $2$, so it has no stream function. Along a streamline the function $\psi$ is constant, since $\mathbf{u} \cdot \nabla \psi = \psi_y \psi_x - \psi_x \psi_y = 0$.