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For a smooth passing encounter of a force-driven sphere with a fixed sphere, isotropy and reflection symmetry imply $\mathbf{U}=[m_{\perp }(R)I+(m_{\parallel }(R)-m_{\perp }(R))\mathbf{nn}]\mathbf{F}$. Along a force axis $X$, $U_X$ is even and the transverse velocity is odd in $X$. Uniqueness then makes the trajectory fore-aft symmetric and restores the incoming impact parameter downstream. This consequence of kinematic reversibility of Stokes flow remains valid with lubrication resistance, provided the particles remain smooth, noncontacting and subject only to the stated forces. Surface roughness and contact can invalidate these assumptions.

A planar freezing front can push a spherical particle steadily when repulsive sphere-plane dispersion force balances lubrication resistance. The thin parabolic gap gives hydraulic drag $6\pi \mu V{a}^2/d$, and balancing it against $Aa/(6d^2)$ gives the displayed speed. This balance assumes a nonretarded interaction, small gap and negligible inertial and other particle forces.

Let $\delta \ll a$ be the minimum gap as the spheres approach, and $w=-\dot{\delta }>0$ their normal relative approach speed. For smooth equal-radius spheres the gap has local form $\delta +O(s^2/a)$, so the thin-gap region has lateral scale $\ell \sim (a\delta {)}^{1/2}$. Incompressibility drives a radial escape speed of order $w\ell /\delta$. The lubrication pressure therefore scales asThe available gravitational forcing remains finite. Thus lubrication resistance gives $w=O(W\delta /(\mu a^2))$: the normal approach becomes proportionally slower as the gap closes. In particular, for a bounded coefficient $C$, $\dot{\delta }\ge -C\delta$, so $\delta (t)\ge \delta (t_1)e^{-C(t-t_1)}>0$ at every finite time. Equivalently the estimated time to reach zero contains ${\int }_0^{{\delta }_1}d\delta /\delta$, which diverges. This is how lubrication prevents finite-time collision of smooth spheres.

The small nonzero initial lateral offset breaks the exactly collinear configuration; the spheres can deflect sideways and pass with a positive gap rather than touch. The condition $R>2a$ leaves space for a side-by-side passage. Smooth surfaces, continuum hydrodynamics and the no-slip boundary condition are essential to the noncontact conclusion; the argument is not a model of surface roughness or molecular contact.

The mobility correction from a fixed distant sphere givesTaking their ratio and keeping the first nonzero transverse correction yieldsto the stated leading order. Set $b=Y_{\infty }\gg a$. The deflection is $O(a^2/b)$, so replacing $Y$ by $b$ on the right introduces only higher-order errors. Integrating from $X=-\infty$ givesThe maximum occurs at $X=0$, and the deflection and spin in a distant sphere encounter areFor the rotation, use $dt=dX/V$ and $Y=b$ to leading order in the angular velocity from part (b). Its signed angle about the positive $z$ axis isThus the rotation is clockwise when viewed from positive $z$, with the magnitude of the displayed leading term.

The deflection tends back to zero downstream: $Y(+\infty )=Y_{\infty }$. More generally, kinematic reversibility of Stokes flow combined with reflection in the plane $X=0$ makes a passing trajectory fore-aft symmetric. One can see this without using the distant-sphere approximation: the relevant translational hydrodynamic mobility matrix has the form $m_{\perp }(R)I+[m_{\parallel }(R)-m_{\perp }(R)]\mathbf{nn}$. Hence $\dot{X}$ is even in $X$ and $\dot{Y}$ is odd in $X$. Uniqueness of the trajectory through $X=0$ then gives $Y(X)=Y(-X)$.

For $Y_{\infty }=a/100$, the numerical deflection and spin approximations above are invalid: the encounter enters a narrow gap and requires lubrication theory. However, the same return to the incoming offset holds for an ideal passing encounter of perfectly smooth spheres in Stokes flow. Lubrication resistance prevents finite-time contact under a bounded force, and does not itself destroy kinematic reversibility of Stokes flow. Contact, surface roughness or nonhydrodynamic forces could change that conclusion; they are additional physics, not part of the ideal model. This distinction is the fore-aft symmetry of a sedimenting-sphere encounter.