Falling spheres in polymeric solutions. Limiting Results of the Two-Fluid Theory of Migration

In this paper we derive a few analytical results for the case of polymeric solutions flowing past a spherical obstacle. The calculations are meant to ascertain the possible importance of flow-induced migration (i.e., of changes in concentration), making use of the, recent two-fluid theory of Doi and Milner. In order to obtain solutions in closed form, the problem is suitably linearised; in particular, we have considered both the case of Newtonian behaviour of the polymeric liquid flowing past the sphere, and that of a linearly elastic polymeric gel perturbed by a localised force. The results, though confirming migration effects, indicate that, in these linear limits at least, the concentration change decays quadratically with the reciprocal distance from the centre of the perturbation, i.e., no long ''wake'' is predicted. Another outcome of these calculations refers specifically to the viscous case. The classical Stokes result is recovered only as a singular solution of the two-fluid problem