Advection-dominated accretion: global transonic solutions
  We obtained global transonic solutions representing optically thin advection-dominated accretion flow by solving the full set of differential equations describing such systems. We found that far from the sonic point self-similar solutions are an excellent approximation of the global flow structure if the accretion rate is well below the maximum value above which no optically thin solutions exist. 
Xingming Chen, Marek A. Abramowicz, Jean-Pierre Lasota