= Solution
Choose the axial origin at the <sphere>'s centre. Reflection $x\mapsto-x$ leaves the straight tube, the spherical geometry and the transverse displacement of the centre unchanged. It reverses the prescribed axial <Hagen-Poiseuille flow>, while it leaves a hypothetical transverse particle velocity unchanged. On the other hand, <kinematic reversibility of Stokes flow> says that reversing the imposed flow reverses every particle velocity, including that transverse component. The <force-free> and <torque-free> constraints are preserved by both operations. Uniqueness of the coupled <Stokes flow> and rigid-body motion therefore makes the transverse velocity equal to its own negative. It must vanish.
Consequently \b[$\dot c=0$] as the particle is advected. This is <no lateral migration of a single sphere in a uniform tube in Stokes flow>. The argument uses the fore-aft symmetry of a rigid sphere and a uniform tube; it does not extend automatically to inertial migration or deformable particles.
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