= Constant-pressure-ratio vertical disk model
Assume a <thin disk> has height-independent <opacity> $\kappa_{\rm op}$ and <radiation-to-gas pressure ratio> $\beta$, with vertical gravity $-\Omega_z^2z$. <Hydrostatic equilibrium> and <radiative diffusion> imply
$$
F_z=\frac\beta{1+\beta}\frac{c\Omega_z^2}{\kappa_{\rm op}}z.
$$
Balancing its derivative against viscous heating $\rho\nu q^2\Omega^2$ fixes the <dynamic viscosity>
$$
\rho\nu=\frac\beta{1+\beta}\frac{c}{q^2\kappa_{\rm op}}\frac{\Omega_z^2}{\Omega^2}.
$$
It is independent of height under these assumptions. For a <spherically symmetric potential>, $\Omega_z=\Omega$. Integrating $\rho\nu$ through the full surface-to-surface thickness $2H$ gives $\bar\nu\Sigma=2H\rho\nu$, which determines the thickness from the radial <accretion rate> solution.
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