= Power-law coating on a rotating cylinder
{title2=$Q_{\max}=\frac{n+1}{2n+1}\left(\frac{k}{\rho g}\right)^{1/(n+1)}(\Omega a)^{(2n+1)/(n+1)}$}
For a thin, shear-free coating on a horizontal cylinder, <lubrication theory> balances tangential gravity against the transverse <shear stress> gradient. With angle measured counterclockwise from the rightward horizontal and wall speed $U=\Omega a>0$, the flux is $Uh-\operatorname{sgn}(\cos\theta)\frac{n}{2n+1}(\rho g|\cos\theta|/k)^{1/n}h^{(2n+1)/n}$. On the uphill side this function has a finite maximum. The tightest restriction is at $\cos\theta=1$, where differentiation gives $h_*=[kU^n/(\rho g)]^{1/(n+1)}$ and the displayed global steady-flux bound. This is the leading gravity-driven model without higher-order pressure-gradient or capillary corrections.
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