Are there thin accretion disks?
  Total and internal energy requirements set constraints on the allowable temperature on the disk. In a gas pressure dominated disk, they lead to a cooling rate that decreases with temperature. These findings are used to show that the flow is mildly supersonic, with the azimuthal Mach number moderately exceeding one in a large extent of the disk. Only in the narrow outermost region, of about 12% of the disk extent, the Mach number greatly exceeds one. It is also shown that, under most favorable conditions, radiation can, at the most, transport 20% of the heat produced, close to the inner radius. Far away, under the same conditions, this efficiency can rise to 100%. This result depends only on boundary conditions, being independent of the accretion rate and mass of the central object.The geometrical thin disk approximation is violated everywhere in the disk, except in the narrow region where, under the unlikely assumption of thermodynamical equilibrium or, at least, local equilibrium, the flow is indeed supersonic. 
Cesar Meirelles Filho, Celso Luiz Lima, and Hideaki Miyake