Let $\varepsilon=H/R\ll1$. Vertical hydrostatic balance gives $c_s\sim H\Omega_K$, whereas leading radial force balance gives $u_\phi\sim R\Omega_K$. Hence
$$
\frac{u_\phi}{c_s}\sim\frac RH=\varepsilon^{-1}\gg1.
$$
Using part (a) and $\nu=\alpha c_sH$ far from the inner edge,
$$
\frac{|u_R|}{c_s}
\simeq\frac32\frac{\nu}{Rc_s}
=\frac32\alpha\frac HR
=O(\alpha\varepsilon)\ll1.
$$
Thus <subsonic radial drift in a thin disk> coexists with highly supersonic orbital motion. Gas follows nearly circular, pressure-coherent orbits and loses angular momentum only slowly, completing of order $(\alpha\varepsilon^2)^{-1}$ revolutions during its viscous inflow.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 347 / 2 / c / Solution

In a steady thin alpha disk, $|u_R|/c_s$ is of order $\alpha H/R\ll1$, whereas $u_\phi/c_s$ is of order $R/H\gg1$. Matter therefore completes many nearly circular orbits during its slow viscous drift inward.


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 Incoming links: Subsonic radial drift in a thin disk