= Solution
Let $l(R)=R^2\Omega=\sqrt{GMR}$ be the Keplerian <specific angular momentum>. In a source-free steady interval, the sum of advected and viscously transported angular momentum is constant:
$$
\dot m,l-2\pi R^3\nu\Sigma\frac{d\Omega}{dR}=C.
$$
Inside the injection radius, $\dot m=\dot m_0$. The <zero-torque inner boundary condition> at $R_{\rm ISCO}$ sets $C=\dot m_0l_{\rm ISCO}$, and therefore
$$
\boxed{\nu\Sigma=\frac{\dot m_0}{3\pi}
\left[1-\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right],
\qquad R<R_0.}
$$
Outside $R_0$ there is no net mass flow in the stated steady distribution, but it must carry outward the angular momentum deposited by matter moving from $R_0$ to the ISCO. Its constant <viscous torque in an accretion disk> is therefore
$$
3\pi\nu\Sigma l=\dot m_0(l_0-l_{\rm ISCO}).
$$
Thus
$$
\boxed{\nu\Sigma=\frac{\dot m_0}{3\pi}
\left[\left(\frac{R_0}R\right)^{1/2}
-\left(\frac{R_{\rm ISCO}}R\right)^{1/2}\right],
\qquad R>R_0.}
$$
The two expressions agree at $R_0$; the jump in mass flux there is exactly the injected rate.
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