Power-law coating on a rotating cylinder ($Q_{\max }=\frac{n+1}{2n+1}{\left(\frac{k}{\rho g}\right)}^{1/(n+1)}(\Omega a{)}^{(2n+1)/(n+1)}$)

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 $Uh-\mathrm{sgn}(\cos \theta )\frac{n}{2n+1}(\rho g|\cos \theta |/k{)}^{1/n}{h}^{(2n+1)/n}$. On the uphill side this function has a finite maximum. The tightest restriction is at $\cos \theta =1$, where differentiation gives $h_{\ast }=[k{U}^n/(\rho g){]}^{1/(n+1)}$ and the displayed global steady-flux bound. This is the leading gravity-driven model without higher-order pressure-gradient or capillary corrections.