Simulations of accretion flows crossing the last stable orbit
  We use three dimensional magnetohydrodynamic simulations, in a pseudo-Newtonian potential, to study geometrically thin accretion disc flows crossing the marginally stable circular orbit around black holes. We concentrate on vertically unstratified and isothermal disk models, but also consider a model that includes stratification. In all cases, we find that the sonic point lies just inside the last stable orbit, with a modest increase in the importance of magnetic field energy, relative to the thermal energy, observed interior to this radius. The time-averaged gradient of the specific angular momentum of the flow, (dl/dr), is close to zero within the marginally stable orbit, despite the presence of large fluctuations and continuing magnetic stress in the plunging region. The result that the specific angular momentum is constant within the last stable orbit is in general agreement with traditional disk models computed using a zero-torque boundary condition at the last stable orbit. 
Philip J. Armitage, Christopher S. Reynolds, James Chiang