I am trying to plot the phase trajectories of a spherically symmetric fluid flow having the following equation of motion: $$\frac{dv}{dr}=\frac{2v}{r^2}\frac{r-1}{v^2-1}$$
Integrating the above equation gives the velocity profile of the flow expressed as $$\frac{v^2}{2}-\ln (v)=\ln(r^2)+\frac{2}{r}+K$$ where $K$ is the integration constant.
The phase trajectories can be obtained by plotting this equation to obtain plots like
In the above equations, $u/c_s=v$ and $r_B$ is set to unity.
The various curves are obtained according to the following conditions on the value of $K$:
- $K=-1.5$: The red curves marked 'accretion' and 'wind'.
- $K<-1.5$: The grey curves marked 'unphysical'.
- $K>-1.5$: The green curves marked 'subsonic' and 'supersonic'.
I am having trouble to understand how to plot the curves. Can someone help me to understand the algorithm for the problem?
I tried the online phase plane plotter to obtain the following plot:
