Lubrication resistance ($F\sim \mu a^{2}U/h$)

Lubrication theory predicts large viscous resistance when fluid must be squeezed through a narrow gap. For normal approach of smooth curved surfaces with curvature scale $a$ and minimum separation $h\ll a$, the resisting force scales as $\mu a^{2}U/h$. Under a bounded normal force this gives $|\dot{h}|=O(h)$, so ideal smooth surfaces do not make contact in finite time. Tangential and rolling motions have different resistance laws; the normal squeeze law should not be applied to them.

Rotation about the offset axis has azimuthal Couette flow and no pressure variation at leading lubrication order. Integrating the tangential traction gives torque $-{\mathcal{R}}_{\Omega }\Omega$ on the inner sphere. The concentric limit is $8\pi \mu a^4/(3\Delta )$ and the contact limit is finite, $4\pi \mu a^4/\Delta$, because the wall velocity vanishes at the closest pole.

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.

For smooth no-slip spheres of radius scale $a$ with minimum gap $\delta \ll a$, the lateral gap scale is $(a\delta {)}^{1/2}$. Squeeze-film lubrication theory gives pressure $p\sim \mu a(-\dot{\delta })/{\delta }^2$ and the displayed normal resistance. Bounded forcing therefore gives $-\dot{\delta }=O(\delta )$, so the gap cannot vanish in finite time. The conclusion uses smooth surfaces, no slip and continuum creeping flow; it is not a contact model for rough particles.