Horizontal rotation in a nearly touching spherical shell ($G\sim (16\pi /5)(\mu \Omega a^4/\Delta )\log (1/ϵ)$)

For $ϵ=1-\alpha \ll 1$, the near-pole gap is $h=ϵ\Delta +r^2/(2R)$ with $R\simeq a^2/\Delta$, and horizontal rotation gives local sliding speed $V\simeq a\Omega$. The annulus $a\sqrt{ϵ}\ll r\ll a$ has $h\sim \Delta r^2/a^2$. Its shear dissipation is at least $\mu V^2\int dA/h\sim 2\pi \mu V^{2}R\log (1/ϵ)$, so a uniformly bounded torque of order $\mu \Omega a^4/\Delta$ is impossible. The exact local Reynolds lubrication equation solution $p=-(6\mu V/5)x/h^2$ gives the displayed logarithmic torque and a dominant horizontal holding-force magnitude $(12\pi /5)(\mu \Omega a^4/{\Delta }^2)\log (1/ϵ)$. The smallest patch and fixed-angle outer region each contribute only a bounded amount; the intermediate annulus supplies the logarithm. A fully flooded film is assumed.