The constraint can be satisfied by using a vector potential for (c.f., .
For two dimensional flow, need only have a non-zero -component, e.g., .
Note that . Thus and has the components
[check that is incompressible: ]
The scalar is called streamfunction since a contour const. is aligned with the velocity - i.e. it is a streamline. See Figure (6.1).
In 2D, .
Thus , and
given , we can solve (6.11) for , then use (6.9) to
Consequently, equation (6.8) reduces, in 2D to
in a given fluid element remains constant, since the convective derivative is zero, as illustrated in Figure (6.2).
Example 6.1 Rankine Vortex.
Use coordinates, rather than : , still. The geometry of the Rankine Vortex is illustrated in Figure (6.3) and is defined to be
since is axisymmetric we look for solutions of the form
Dividing by , followed by integration we find
since must be regular at
Now find using (6.16):
(This is similar to ``solid body'' rotation with angular velocity .)
is continuous at , i.e.,
Hence the full solution for is and (see Figure (6.4))