Stream function
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In fluid dynamics, the stream function is defined for two-dimensional flows. The difference between the values of the stream function at any two points gives the volumetric flow rate (or flux) through a line connecting the the two points.
Note that since streamlines are tangent to the flow, the value of the stream function must be the same 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 streamfunction may be defined for any flow of dimensions greater then 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 fluid flow.
Two dimensional streamfunction
The streamfunction <math>\psi<math> for a two dimensional flow is defined such that the flow velocity can be expressed as:
- <math>
u= - \frac{\partial\psi}{\partial y},\qquad v=\frac{\partial\psi}{\partial x} <math>
Where <math>u<math> and <math>v<math> are the velocities in the <math>x<math> and <math>y<math> directions, respectively.
- <math>
u = \frac{\partial x}{\partial t},\qquad v = \frac{\partial y}{\partial t} <math>
This forumlation of the streamfunction satisfies the two dimensional continuity equation:
- <math>
\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 <math>