Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / b / i

The steady disk should end at the innermost stable circular orbit, $r_{\mathrm{in}}=3r_S$. Inside it, matter plunges inward on an orbital timescale and cannot maintain the nearly circular shear flow assumed by the viscous-disk equations. If the plunging region communicates negligible stress back across the ISCO, the natural boundary condition is the zero-torque inner boundary condition