= Radius-deficit normalization of sphere-tube lubrication integrals
{title2=$I_1^*=\frac{\pi}{\sqrt2}\epsilon^{-1/2},\quad I_2^*=\frac{\pi}{2\sqrt2}\epsilon^{-3/2},\quad I_3^*=\frac{3\pi}{8\sqrt2}\epsilon^{-5/2}$}
When the tube radius is $a$ and the sphere radius is $a(1-\epsilon)$, the minimum gap is $a\epsilon$ and its leading neck profile is $h=a\epsilon+z^2/(2a)$. Substitution $z=a\sqrt{2\epsilon}\,t$ in $I_n^*=a^{n-1}\int h^{-n}dz$ gives the displayed coefficients. Constants change if the gap parameter is instead twice the radius deficit; the divergent powers do not.
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