Squeeze resistance of eccentric nested spheres ($\mathcal{R}=\frac{6\pi \mu a^4}{{\Delta }^3{\alpha }^3}[\frac{2\alpha }{1-{\alpha }^2}+\log \frac{1-\alpha }{1+\alpha }]$)

For an inner sphere of radius $a$, outer radius $a+\Delta$, and signed offset $\alpha \Delta$ along a fixed axis, the leading lubrication pressure generated by offset speed $V=\Delta \dot{\alpha }$ is $p=3\mu a^{2}V/[\alpha {\Delta }^3(1-\alpha \cos \theta {)}^2]+\text{constant}$. Its force on the inner sphere is $-\mathcal{R}V$ along that axis. Remove the spatially constant divergent term before taking $\alpha \to 0$; the resistance then tends to $8\pi \mu a^4/{\Delta }^3$. As $\alpha \to 1$, it is $6\pi \mu a^4/[{\Delta }^3(1-\alpha )]$ to leading order.