= Vertically averaged viscous disk equations
{title2=$\partial_t\Sigma+r^{-1}\partial_r(r\Sigma v_r)=0$}
In an axisymmetric <thin disc> with no vertical mass or <torque> flux, integrate mass and <conservation of angular momentum> over height. With fixed height-independent $l=r^2\Omega$, the equations are $\partial_t\Sigma+r^{-1}\partial_r(r\Sigma v_r)=0$ and $\partial_t(\Sigma l)+r^{-1}\partial_r(r\Sigma v_rl-r^3\bar\nu\Sigma\Omega')=0$. Their difference gives $r\Sigma v_rl'=(r^3\bar\nu\Sigma\Omega')'$. The <density-weighted viscosity of a disk> supplies the integrated stress.
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