= Eccentricity increases narrow-gap bypass mobility
{title2=$U(e)/U(0)=(2\pi)^{-1}\int_0^{2\pi}(1+(e/d)\cos\phi)^{5/2}d\phi$}
For an off-axis nearly occluding <sphere> with radial offset $e<d$, the local minimum gap is $d(\phi)=d+e\cos\phi$ to leading order. Each circumferential strip has pressure-driven conductance proportional to $d(\phi)^{5/2}$, and the strips act in parallel. Thus its speed at fixed load is multiplied relative to the concentric value by
$$
\frac1{2\pi}\int_0^{2\pi}(1+\epsilon\cos\phi)^{5/2}d\phi,\qquad\epsilon=e/d.
$$
<Convexity> and the zero average of cosine make this greater than one for nonzero $\epsilon$. It increases strictly with $\epsilon$: pair the positive and negative cosine contributions to the derivative, obtaining factors $c[(1+\epsilon c)^{3/2}-(1-\epsilon c)^{3/2}]>0$ for $c>0$. Consequently an off-axis <sphere> falls faster in this narrow-gap, no-through-flow regime.
Back to article page
