= Horizontal rotation in a nearly touching spherical shell
{title2=$G\sim(16\pi/5)(\mu\Omega a^4/\Delta)\log(1/\epsilon)$}
For $\epsilon=1-\alpha\ll1$, the near-pole gap is $h=\epsilon\Delta+r^2/(2R)$ with $R\simeq a^2/\Delta$, and horizontal rotation gives local sliding speed $V\simeq a\Omega$. The annulus $a\sqrt\epsilon\ll r\ll a$ has $h\sim\Delta r^2/a^2$. Its shear dissipation is at least $\mu V^2\int dA/h\sim2\pi\mu V^2R\log(1/\epsilon)$, 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 $(12\pi/5)(\mu\Omega a^4/\Delta^2)\log(1/\epsilon)$. 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.
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