= Solution
Inside the <innermost stable circular orbit>, nearly circular motion is unstable and gas enters the <plunging region of a black-hole accretion disk>. Its inflow time becomes shorter than the time on which internal stress can communicate <angular momentum> back to the disk, motivating the <zero-torque inner boundary condition> at $r_{\rm ISCO}$.
In a <steady state>, the given diffusion equation implies
$$
r^{1/2}\frac d{dr}\left(r^{1/2}\bar\nu\Sigma\right)=C.
$$
The constant mass supply fixes $C=\dot M/(6\pi)$, and a second integration gives
$$
\bar\nu\Sigma=\frac{\dot M}{3\pi}
\left[1-C_0r^{-1/2}\right].
$$
Zero torque means $\bar\nu\Sigma=0$ at $r=r_{\rm ISCO}$, so $C_0=r_{\rm ISCO}^{1/2}$. Thus the <Keplerian accretion disk> relation is
$$
\boxed{\bar\nu\Sigma=\dot M f(r)},
\qquad
\boxed{f(r)=\frac1{3\pi}
\left[1-\left(\frac{r_{\rm ISCO}}r\right)^{1/2}\right]}.
$$
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