Density-weighted viscosity of a disk ($\bar{\nu }={\Sigma }^{-1}\int \rho \nu dz$)

If angular velocity is independent of height, integrating the viscous stress $\rho \nu r{\Omega }^{\prime }$ gives $\Sigma \bar{\nu }r{\Omega }^{\prime }$. Thus the surface density weighted average, rather than an unweighted height average, enters vertically averaged viscous disk equations.