= Steady viscous disk supplied by a radial source
{title2=$\mathcal F(r)=2\pi\int_r^{r_2}r'S(r')\,dr'$}
For a <Keplerian disk> with evolution $\partial_t\Sigma=(2\pi r)^{-1}\partial_r\mathcal F+S$ and $\mathcal F=6\pi r^{1/2}\partial_r(r^{1/2}\bar\nu\Sigma)$, take zero viscous <torque> at $r_1>0$ and zero flux outside the source region at $r_2>r_1$. The <steady state> gives $\mathcal F'=-2\pi rS$, and its outer condition gives the displayed flux. Integrating once more from the <zero-torque inner boundary condition> yields
$$
\bar\nu(r)\Sigma(r)=\frac{1}{3\sqrt r}\int_{r_1}^{r}\frac{du}{\sqrt u}\int_u^{r_2}vS(v)\,dv.
$$
The equation determines the viscosity-density product, so a thermodynamic viscosity closure can be imposed afterward. For nonnegative integrable $S$, it is nonnegative and the inner <accretion rate> is exactly the total supplied mass per unit time.
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