= Viscous torque in an accretion disk
{title2=$\mathcal G$}
The <viscous torque in an accretion disk> is the rate of outward <angular momentum> transport by an internal shear stress. For an axisymmetric <thin disk> rotating with <angular velocity> $\Omega(r)$, a local Newtonian <viscous stress> has $T_{r\phi}=\rho\nu r\,d\Omega/dr$. Integrating that stress over a cylinder and multiplying by the lever arm gives
$$
\mathcal G=-2\pi\bar\nu\Sigma r^3\frac{d\Omega}{dr},
\qquad
\Sigma=\int\rho\,dz,\qquad
\bar\nu=\frac{1}{\Sigma}\int\rho\nu\,dz.
$$
Here $\bar\nu$ is the density-weighted mean <kinematic viscosity>, with units of <length> squared per <time>. The vertically integrated quantity is $\bar\nu\Sigma$. For decreasing $\Omega$, the sign convention makes $\mathcal G>0$, describing outward <angular momentum> transport. More general internal stresses can be represented by an effective $\bar\nu$ through the same integrated torque relation.<Dynamics of Astrophysical Discs by Gordon I. Ogilvie>[Lecture 4, sections 4.1–4.2, cylinder torque and mean effective kinematic viscosity]{ref}
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