The stream function is defined for two-dimensional flows of various kinds and can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines and in most cases, the stream function is the imaginary part of the complex potential, while the potential function is the real part.
Considering the case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate or flux through a line connecting the two points.
Since streamlines are in tangential direction to the velocity vector of the flow, the value of the stream function must be constant along a streamline. If there were a flux across a line, it would necessarily not be tangent to the flow and thus would not be a streamline.
The usefulness of the stream function lies in the fact that the velocity components at a given point are given by the partial derivatives of the stream function at that point and a stream function may be defined for any flow of dimensions greater than two, however the two dimensional case is generally the easiest to visualize and derive.
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