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Viscous torque in protostellar discs
Protostellar discs are sticky. Each ring exerts a viscous torque on its adjacent rings which acts to:
- transfer angular momentum along velocity gradients via random particle motion
- allow mass transfer inwards
- heat the gas through dissipation, converting gravitational energy into heat and reradiating it
Derivation
Consider two adjacent annuli of width \(\lambda\). The sheer velocity at a point \(r\) with respect to a point \(R\) is given by: \[u(r) = r(\Omega(r) - \Omega(R))\] Now lets consider the sheer velocities with respect to a point P at two points A and B represented by \(+\) and \(-\) in the following equation. \[u(R \pm \frac{\lambda}{2}) = (R \pm \frac{\lambda}{2})[\Omega(R \pm \frac{\lambda}{2}) - \Omega(R)]\] \[= \pm \frac{\lambda}{2}(R\pm \frac{\lambda}{2})\frac{d\Omega}{dr}\]
Chaotic motions carry particles across adjacent annuli. These motions have a length scale \(\lambda\), which could either be the mean free path
or the size of the largest eddies; and a mean velocity \(\tilde u\), which could either be the sound speed or the turnover speed of the largest eddies.
Mass crosses the boundary at an equal rate in and out, so the mass flow per unit arc length is \(H\rho \tilde u\) (kg/m/s).
The surface density, \(\Sigma = H \rho\), is commonly used which gives a mass flow per unit arc length of \(H \tilde u\)
The net angular momentum flux is therefore:
\[\Sigma \tilde u [(ru_\phi)_B - (ru_\phi)_A]\]
Which is also a torque per unit arc length which we can expand out such that torque per unit arc length exerted on the *outer ring by the inner ring* is
\[\frac{\delta Q(R)}{\delta l} = \Sigma \tilde u [(R - \frac{\lambda}{2})(-\frac{\lambda}{2})(R -\frac{\lambda}{2})\Omega' - (R + \frac{\lambda}{2})(+\frac{\lambda}{2})(R +\frac{\lambda}{2})\Omega']\]
\[\approx -\Sigma \tilde u \lambda R^2 \Omega'\]
Therefore the total torque exerted on the *inner ring by the outer ring* is
\[2\pi R \nu \Sigma R^2 \Omega'\]
Where \(\nu = \lambda \tilde u\)