Logarithmic lubrication resistance of a sphere near a wall

A sphere of radius $a$ at gap $aϵ$ translates at $U{\mathbf{e}}_x$ and rotates at $\omega {\mathbf{e}}_y$ above a stationary wall in a right-handed coordinate system. At leading order its gap is $h=aϵ+r^2/(2a)$. In its translating frame the lower and upper tangential velocities are $-U{\mathbf{e}}_x$ and $-a\omega {\mathbf{e}}_x$. The Reynolds lubrication equation givesIntegrating the upper-boundary traction, including pressure on the sloping surface, and its moment about the sphere centre gives the sphere-on-fluid force and torqueThe off-diagonal symmetry is required by the Lorentz reciprocal theorem for Stokes flow. A torque-free sphere therefore has $a\omega =U/4$ and leading drag $3\pi \mu aU\log (1/ϵ)$. Compare the rigid-wall terms of Bertin et al., equations (4.4)–(4.5); those forces act on the sphere and have the opposite sign.

For a fixed outer radius $b\ll a$ and $h=aϵ+r^2/(2a)$,Indeed, for $a\sqrt{ϵ}\ll r\ll b$ the integrand is $(2a{)}^n/r$; integrating from a scale proportional to $a\sqrt{ϵ}$ to $b$ produces half of $\log (1/ϵ)$. The inner region contributes a bounded term after rescaling $r=a\sqrt{ϵ}s$. In particular, $\int h^{-1}dxdy$ and $\int x^2/(a{h}^2)dxdy$ have the same leading coefficient $2\pi a$.