= Solution
Vertically integrating the <continuity equation> and imposing a steady <axisymmetric flow> gives
$$
\frac1R\frac d{dR}(R\Sigma u_R)=0.
$$
Thus the inward-positive <mass accretion rate> is
$$
\boxed{\dot m=-2\pi R\Sigma u_R=\text{constant}}.
$$
The zero-torque condition at $R_{\rm ISCO}$ gives the standard <Shakura--Sunyaev thin disk> relation
$$
\nu\Sigma=\frac{\dot m}{3\pi}
\left[1-\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right].
$$
Combining this relation with mass conservation, or substituting it into the stated drift equation, yields
$$
\boxed{u_R=-\frac{3\nu}{2R}
\left[1-\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right]^{-1}}.
$$
Far outside the inner edge this reduces to $u_R\simeq-3\nu/(2R)$.
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