= Solution
Reflect the configuration in a plane perpendicular to the tube axis. The geometry and radial displacement are unchanged, whereas the imposed pressure gradient and every axial velocity reverse. By <kinematic reversibility of Stokes flow>, a radial migration velocity would also have to reverse; the spatial reflection leaves that component unchanged. <Uniqueness of Stokes flow> therefore forces it to vanish. Reflection in the meridional plane through the two axes similarly excludes azimuthal drift, so $c$ remains constant.
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