(2) into the above yields:

$$R^{3} \Omega^{2} \Sigma < 2 \left( \frac{d \tau}{dR} \right),$$

as the condition for the onset of radial flow in the disk.

We can now introduce a model for the torques in terms of the rate of shear and the effective viscosity $\nu$ (e.g. Von Wiezsäcker 1943, Pringle 1981) as:

$$\tau=2 \pi R^{3} \nu \Sigma \frac{d\Omega}{dR}.$$

Taking $\Omega^{2} = GM/R^{3}$ to substitute the above equation into condition (5) gives

$$(GM)^{1/2} \Sigma < -6 \pi \frac{d}{dR} \left(R^{1/2} \nu \Sigma \right).$$

To proceed further we can take for example, $\nu=$constant and a model for the disk surface density profile of the form:

$$\Sigma=\Sigma_{0} \left(\frac{R}{R_{0}} \right)^{-n},$$

of the type often used in models of accretion disks when fitted to observations (e.g. Hughes et al. 2009). Use of the two above forms for $\nu$ and $\Sigma(R)$ reduces condition(7) to:

$$6 \pi \nu (n-1/2) > (G M R)^{1/2}.$$

The above condition will always be met in any accretion disk with $n>1/2$, interior to a transition radius $R_{T}$ given by:

$$R_{T} = \frac{[6 \pi \nu (n-1/2)]^{1/2}}{GM}.$$

It is interesting that directly observed accretion disks have spectra which when modelled typically yield $n \sim 1$, in general $0<n<1.5$, e.g. Hartmann et al. (1998), 