 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 $\dot{c}=0$ 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.