 Solution

In a steady axisymmetric outflow, mass conservation makes $2\pi r\Sigma (r)v_r(r)$ independent of $r$. The steady radial dust outflow therefore hasIt has an integrable $(r-a_p{)}^{-1/2}$ pile-up just outside the source ring, where the radial speed starts from zero, and approaches $Sigma\propto r^{-1}$ at large radius.