Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 1 / c / Solution

At leading order, $X_1=Vt$, $d{X}_1/dt=V$, $Y_1=0$, and the much smaller motion of sphere 2 may be neglected when evaluating the separation:The $y$ component of the leading $a/R$ term in the mobility from part b isThusIntegrating from the initial position $X_1=0$ to infinity gives the hydrodynamic displacement of a force-free sphere

The leading horizontal velocity isConsequently $X_2\sim (3a/2)\log (X_1/Y_0)$ andlogarithmically. The sphere is carried arbitrarily far downstream even though its transverse displacement approaches a finite limit.