Ringleb flow

$q$ = magnitude of velocity
$\theta$ = angle between velocity and $x$-axis
$\psi$ = stream function

$$\psi = \frac{1}{q} \sin\theta$$

Take

$$k = \frac{1}{\psi}$$

to be constant as a streamline.

Streamline equations:

$$\begin{eqnarray*} x(q) &=& \frac{1}{2 \rho(q) } \left( \frac{1}{q^2} - \frac{2}{... ...\ y(q) &=& \pm \frac{1}{kq\rho(q)} \sqrt{ 1 - \frac{q^2}{k^2} } \end{eqnarray*}$$

where

$$\begin{eqnarray*} J(q) &=& \frac{1}{c} + \frac{1}{3c^3} + \frac{1}{5c^5} - \frac... ...t{ 1 - \frac{\gamma-1}{2} q^2 } \\ \rho(q) &=& c^{2/(\gamma-1)} \end{eqnarray*}$$