Lubrication resistance integrals for a nearly occluding sphere ($I_n=\int h^{-n}dz$)

A sphere almost filling a tube has local parabolic lubrication gap $h\simeq a[ϵ+(z/a{)}^2]/2$. The resistance integrals are $I_1=2\pi {ϵ}^{-1/2}$, $I_2=(2\pi /a){ϵ}^{-3/2}$ and $I_3=(3\pi /a^2){ϵ}^{-5/2}$, obtained with $z=a\sqrt{ϵ}\xi$ and $\int (1+{\xi }^2{)}^{-n}d\xi$. Moving-boundary lubrication flux weights shear-driven and pressure-driven transport by different powers of the gap, explaining the three distinct divergence rates.

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.

A sphere of radius $R$ settling in a stationary cylindrical tube of radius $R+d$, with $d\ll R$ and zero net through-flow, has leading speedThe neck gap is $H(z)=d+z^2/(2R)$. In the sphere frame, leading bypass flux per circumferential length is $q\simeq UR/2$; wall-motion transport is smaller by $d/R$. The pressure-driven flux through a parabolic lubrication gap gives $\Delta p=9\pi \mu UR\sqrt{2R}/(4d^{5/2})$. Pressure dominates the sphere drag, $F\simeq \pi R^2\Delta p$, and balancing $F$ against $4\pi R^3\Delta \rho g/3$ proves the speed. Tube end conditions allowing appreciable net flow change the bypass balance and can give other drag regimes.

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

Take a fluid control volume spanning a nearly occluding sphere. The pressure difference contributes $\pi a^2\Delta P$ upwards, while the outer-wall shear in the sphere frame contributes $2\pi a\mu \int u_y(h)dz$. Since sphere-frame flux in a tube gives $u_y(h)=4U/h-6q/h^2$, the drag on the sphere is $F=\pi a^2\Delta P+2\pi a\mu (4U{I}_1-6q{I}_2)$. Using the wall shear avoids integrating traction over the strongly curved sphere directly. The remaining tube pressure drop is fixed by Hagen-Poiseuille flow.

Use $F^{\ast }=F/(6\pi \mu aU)$ and $L^{\ast }=4L/(3a)$. The control-volume drag formula for a sphere in a tube gives the displayed uniform leading drag. If $L^{\ast }\ll {ϵ}^{-1/2}$, $F^{\ast }\sim (4\pi /3){ϵ}^{-1/2}$ and local annular shear dominates. If ${ϵ}^{-1/2}\ll L^{\ast }\ll {ϵ}^{-5/2}$, $F^{\ast }\sim L^{\ast }$ and the Hagen-Poiseuille flow resistance of the long tube dominates. If $L^{\ast }\gg {ϵ}^{-5/2}$, $F^{\ast }\sim 3\pi {ϵ}^{-5/2}$ and pressure-driven bypass through the narrow gap dominates. The laboratory flow is piston-like in the first two limits and negligible in the third.

Use radius-deficit $ϵ$, $F^{\ast }=F/(6\pi \mu aU)$ and $\delta =3a/(4L)$. The control-volume drag formula for a sphere in a tube and the radius-deficit normalization of sphere-tube lubrication integrals give three leading resistances: $c_3{ϵ}^{-5/2}$ for $\delta \ll {ϵ}^{5/2}$, $1/\delta$ for ${ϵ}^{5/2}\ll \delta \ll {ϵ}^{1/2}$, and $2\pi \sqrt{2}/(3\sqrt{ϵ})$ for $\delta \gg {ϵ}^{1/2}$. They correspond to gap bypass with almost no tube throughflow, pressure resistance of tube throughflow, and local annular resistance, respectively.

Two identical well-separated spheres falling at the same speed share one laboratory volume flux, rather than producing twice the single-sphere piston flux. In the middle of the three drag regimes for a nearly occluding sphere, the same overall Hagen-Poiseuille flow pressure drop is divided equally between them, halving each sphere's drag. In the local-shear and local-gap-pressure regimes each sphere retains its own leading resistance, so the drag per sphere is unchanged. This conclusion assumes nonoverlapping lubrication regions and sufficient separation from the tube ends.