Stream function

The stream function is a scalar field variable which is constant on each streamline. It exists only in two-dimensional and axisymmetric flows.

On a streamline in two-dimensional flow

$d\psi = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy = 0$

The equation of a streamline in two-dimensions is

$v dx - u dy = 0$

Comparing the two equations, we have

$u = - \frac{\partial \psi}{\partial y}$
$v = \frac{\partial \psi}{\partial x}$

Conversely, the stream function at any point $P$ can be obtained from the velocity field by a line integral

$\psi(P) = \psi(P_o) + \int_{P_o}^P [ v(x,y,t) dx - u(x,y,t) dy ]$

where $P_o$ is some reference point and one can assume $\psi(P_o) = 0$ since the stream function is determined only upto a constant.

If the flow is incompressible, then the continuity equation is identically satisfied

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = -\frac{\partial^2 \psi}{\partial x \partial y} + \frac{\partial^2 \psi}{\partial y \partial x} = 0$