 Solution

The sink speed scales as $U\sim A/R^2$, so the Reynolds number is $Re\sim UR/\nu =A/(\nu R)$. The approximation applies in the high-Reynolds-number region $R\ll A/\nu$. Radial inertia scales as $UdU/dR\sim A^2/R^5$, so the radial pressure gradient has the same $R^{-5}$ dependence.