Radius-deficit normalization of sphere-tube lubrication integrals ($I_1^{\ast }=\frac{\pi }{\sqrt{2}}{ϵ}^{-1/2},I_2^{\ast }=\frac{\pi }{2\sqrt{2}}{ϵ}^{-3/2},I_3^{\ast }=\frac{3\pi }{8\sqrt{2}}{ϵ}^{-5/2}$)

When the tube radius is $a$ and the sphere radius is $a(1-ϵ)$, the minimum gap is $aϵ$ and its leading neck profile is $h=aϵ+z^2/(2a)$. Substitution $z=a\sqrt{2ϵ}t$ in $I_n^{\ast }=a^{n-1}\int h^{-n}dz$ gives the displayed coefficients. Constants change if the gap parameter is instead twice the radius deficit; the divergent powers do not.