The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines. In most cases, the stream function is the imaginary part of the complex potential, while the potential function is the real part.
Considering the particular case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate (or volumetric flux) through a line connecting the two points.
Since streamlines are tangent 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, hence would not be a streamline.
The usefulness of the stream function lies in the fact that the velocity components in the x- and y- directions at a given point are given by the partial derivatives of the stream function at that point. A stream function may be defined for any flow of dimensions greater than or equal to two, however the two dimensional case is generally the easiest to visualize and derive.
Taken together with the velocity potential, the stream function may be used to derive a complex potential for a potential flow. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.
Famous quotes containing the words stream and/or function:
“When cats run home and light is come,
And dew is cold upon the ground,
And the far-off stream is dumb,
And the whirring sail goes round,
And the whirring sail goes round;
Alone and warming his five wits,
The white owl in the belfry sits.”
—Alfred Tennyson (18091892)
“The fact remains that the human being in early childhood learns to consider one or the other aspect of bodily function as evil, shameful, or unsafe. There is not a culture which does not use a combination of these devils to develop, by way of counterpoint, its own style of faith, pride, certainty, and initiative.”
—Erik H. Erikson (19041994)