Time-dependent evolution of quasi-spherical, self-gravitating accretion   flow
  A self-similar solution for time evolution of quasi-spherical, self-gravitating accretion flows is obtained under the assumption that the generated heat by viscosity is retained in the flow. The solutions are parameterized by the ratio of the mass of the accreting gas to the central object mass and the viscosity coefficient. While the density and the pressure are obtained simply by solving a set of ordinary differential equations, the radial and the rotational velocities are presented analytically. Profiles of the density and the rotational velocities show two distinct features. Low density outer accreting flow with relatively flat rotation velocity, surrounds an inner high density region. In the inner part, the rotational velocity increases from the center to a transition radius where separates the inner and outer portions. We show that the behaviour of the solutions in the inner region depends on the ratio of the heat capacities, $\gamma$, and the viscosity coefficient, $\alpha$. 
Mohsen Shadmehri