= Solution
Assume an axisymmetric <thin disk> in the fixed <Newtonian potential of a point mass>, negligible disk self-gravity, and slow evolution compared with an orbital period. Radial pressure and radial inertia are small compared with gravity and centrifugal acceleration, so $u_\phi=r\Omega(r)$ with $\Omega=(GM/r^3)^{1/2}$, independent of height and time. Assume no mass loss or torque through the upper and lower disk faces. Define the <surface density of a disk>, mass-weighted radial velocity and integrated shear stress by
$$
\Sigma=\int\rho\,dz,\qquad \Sigma\bar u_r=\int\rho u_r\,dz,\qquad W=\int T_{r\phi}\,dz.
$$
Vertical integration of the <continuity equation> gives
$$
\partial_t\Sigma+\frac1r\partial_r(r\Sigma\bar u_r)=0.
$$
The azimuthal momentum equation, including the geometric term $u_ru_\phi/r$, gives <conservation of angular momentum> in terms of $j=r^2\Omega$:
$$
\partial_t(\Sigma j)+\frac1r\partial_r(r\Sigma\bar u_rj)=\frac1r\partial_r(r^2W).
$$
Subtracting $j$ times mass conservation eliminates the time derivative and leaves $\Sigma\bar u_rj'=r^{-1}(r^2W)'$.
Define the effective <density-weighted viscosity of a disk> through the actual stress:
$$
\boxed{W=\bar\nu\Sigma r\Omega',\qquad
\bar\nu=\frac{\int T_{r\phi}\,dz}{\Sigma r\Omega'}}.
$$
If the local viscous stress is $T_{r\phi}=\rho\nu r\Omega'$, this reduces to $\bar\nu=\Sigma^{-1}\int\rho\nu\,dz$. The definition can also describe an effective turbulent transport stress, provided the same shear closure is appropriate. Since $\Omega'<0$, outward transport corresponds to $W<0$ and positive $\bar\nu$.
For <Keplerian rotation>, $j=\sqrt{GMr}$ and $r^2W=-(3/2)\sqrt{GM}\,r^{1/2}\bar\nu\Sigma$. Therefore
$$
\boxed{\bar u_r=-\frac3{\Sigma r^{1/2}}\partial_r(r^{1/2}\bar\nu\Sigma)}.
$$
Substitution into mass conservation yields the <Keplerian viscous diffusion equation>
$$
\boxed{\partial_t\Sigma=\frac3r\partial_r\left[r^{1/2}\partial_r(r^{1/2}\bar\nu\Sigma)\right]}.
$$
The absence of surface fluxes, fixed rotation law and leading thin-disk approximation are essential: a wind, appreciable radial inertia or evolving gravitational potential would alter this reduction.
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