Getting the speed of a fluid streamline I have the following streamline function, $$(r^2 - \frac{a^3}{r})sin^2\theta = b$$ how do I get the speed of this fluid at each point on the streamfunction? Incompressible, this is the streamline for very large r. It is derived from the streamfunction in spherical polars for a fluid perturbed by a sphere, $$\Psi(r,\theta) = -\frac{1}{2}U(r^2 - \frac{a^3}{r})sin^2\theta$$

The velocity of a fluid in terms of the streamfunction is $u = \
abla\wedge(0,0,\psi)$. The speed is then the magnitude of this vector. In terms of spherical polar co-ordinates:

$$u =\bigg(\frac1r\frac{\partial\psi}{\partial\theta},-\frac{\partial\psi}{\partial r},0\bigg)$$

Note that in general, the streamfunction contains a lot more information than the streamline equation. The streamline equation will give the curves in the plane that the streamlines flow across, but will not have information about the speed of the flow on an particular streamline.