The stream function and two-dimensional incompressible flow

In two dimensions the condition \(u_x + v_y = 0\) can be solved once and for all. Choose any smooth function \(\psi(x,y,t)\) and set \[ u = \frac{\partial\psi}{\partial y}, \qquad v = -\frac{\partial\psi}{\partial x}. \] Then \(u_x + v_y = \psi_{yx} - \psi_{xy} = 0\) automatically. Conversely, on a simply connected region every divergence-free field arises this way, because the condition says the one-form \(-v\,dx + u\,dy\) is closed, hence exact, and \(\psi\) is its potential. The function \(\psi\) is the stream function (introduced by Lagrange in 1781).

Its level curves are the streamlines. Along a curve where \(\psi\) is constant, \(\mathbf{u}\cdot\nabla\psi = \psi_y\psi_x - \psi_x\psi_y = 0\), so the velocity is tangent to the level set. Better still, the difference of \(\psi\) between two points is the volume flux across any curve joining them: \[ \int_A^B (u\,dy - v\,dx) = \int_A^B (\psi_y\,dy + \psi_x\,dx) = \psi(B) - \psi(A). \] Closely spaced streamlines therefore mean fast flow, since the same flux squeezes through a narrower gap.

The vorticity of a 2D flow is the scalar \(\omega = v_x - u_y\). In terms of the stream function, \[ \omega = -\psi_{xx} - \psi_{yy} = -\Delta\psi. \] So if you know the vorticity you recover the flow by solving a Poisson equation, \(\Delta\psi = -\omega\). This is the starting point of the vorticity-stream function formulation that makes 2D flow so much more tractable than 3D.

A few examples. \(\psi = Uy\) is a uniform stream \((U,0)\). \(\psi = xy\) gives \(\mathbf{u} = (x,-y)\), flow into a corner or towards a wall, with hyperbolic streamlines. \(\psi = -\frac{\Gamma}{2\pi}\ln r\) is a point vortex: the speed is \(\Gamma/(2\pi r)\) around circles, and \(\Delta\psi = 0\) away from the origin, so the flow is irrotational everywhere except at the centre. On a region with a hole, such as the outside of a cylinder, \(\psi\) can take different values on different boundaries, and the difference is the flux between them.

Picture it: a contour map of \(\psi\) is a map of the flow; water runs along the contours, and faster where they bunch together. Think it: in the language of forms, \(\mathbf{u}\) is divergence-free exactly when a certain one-form is closed, and the stream function is the statement that closed forms are exact on simply connected domains.