Define the terms velocity potential and stream functions.
Answers
Streamlines – lines with a constant value of the stream function – for the incompressible potential flow around a circular cylinder in a uniform onflow.
The stream function is defined for incompressible (divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocitycomponents can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781.[1] The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.[2]
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 flow velocity vector of the flow, the value of the stream function must be constant along a streamline. The usefulness of the stream function lies in the fact that the flow velocity components in the x- and y- directions at a given point are given by the partial derivativesof 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.
For two-dimensional potential flow, streamlines are perpendicular to equipotentiallines. Taken together with the velocity potential, the stream function may be used to derive a complex potential. In other words, the stream function accounts for the solenoidalpart of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.
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