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Neglect tube-end effects and use the localized flow with no imposed throughflow, as implied by the sphere being the only cause of motion. Extend the fluid velocity through the sphere by its rigid-body velocity. This extended field is continuous across the surface by the no-slip boundary condition and divergence-free both inside and outside. The divergence theorem therefore makes its total flux through a complete tube section independent of height. In sections far from the sphere the fluid is quiescent, so this total flux is zero.

At height $z$ cutting the sphere, let $A_s(z)$ be the area of its circular slice and take $z$ upward. The translation contributes $V_z{A}_s$. The vertical component of $\mathbf{\Omega }\times (\mathbf{x}-{\mathbf{x}}_c)$ integrates to zero over that circle because its first moments about the centre vanish. Hence the zero total flux for localized tube Stokes flow giveswhere $V_z<0$ for a falling sphere. Thus the net fluid flux is upwards in every section intersecting the sphere's interior, even though some fluid locally moves downwards. At a tangential section $A_s=0$ and the flux vanishes. The zero-total-flux condition is important: an imposed throughflow or appreciable open-tube end effects would change this argument.

When both spheres have the same effective downward weight, the second sphere adds a force-driven disturbance instead of being merely a force-free rigid inclusion. The power now contains both $W{V}_{1z}$ and $W{V}_{2z}$, so the one-forced-sphere bound from part (iv) no longer applies.

For $a\ll R$, place the spheres in a tandem configuration with $a\ll s\ll R$. Locally the flow around the second sphere is dominated by its downward Stokeslet. Its axial incident velocity at the first sphere is downward, so the first sphere falls faster to leading order. By contrast, the zero total flux for localized tube Stokes flow requires compensating upward flow away from the downward-moving particle core in sections cutting the sphere. There are therefore positions in the return flow where the disturbance acts upwards.

One can choose the second sphere in such an upward-flow region generated by the first. The Lorentz reciprocal theorem makes the axial cross coefficients of the hydrodynamic mobility matrix equal, so the downward force on the second sphere then produces an upward contribution to the first sphere's velocity. With small well-separated spheres, this force-driven interaction dominates the higher-order self-resistance change from simply inserting the other sphere. In mobility notation, with downward positive,and the cross coefficient can have either sign in the tube. ThusThe distinction is the sign of the incident flow, not whether adding a second geometrical obstacle lowers a fixed-velocity dissipation.