Predictions of polymer migration in a dilute solution between rotating eccentric cylinders

Our recent continuum theory for stress-gradient-induced migration of polymers in confined solutions, including the depletion from the solid boundaries [Hajizadeh, E., and R. G. Larson, Soft Matter, 13, 5942–5949 (2017)], is applied to a two-dimensional rotational shearing flow in the gap between eccentric cylinders. Analytical results for the steady-state distribution of polymer dumbbells in the limit of dilute polymer solution c/c∗≪1 (c∗ is the chain overlap concentration) and in the absence of hydrodynamic interactions are obtained. The effects of eccentricity e and of three perturbation variables, namely, Weissenberg number Wi, gradient number Gd(which defines the level of polymer chain confinement), and Peclet number Pe on the polymer concentration pattern, are investigated. The stress-gradient-induced migration results in polymer migration toward the inner cylinder, while wall-depletion-induced migration results in near-zero polymer concentration close to flow boundaries, which couples to a stress-gradient-induced migration effect. In the presence of wall-depletion, we obtain first order concentration variation proportional to Wi. However, in the absence of wall-depletion, there is no first order contribution and, therefore, the lowest-order concentration variation is proportional to Wi2. An upper limit of Wi=1.6 exists, beyond which the numerical solution demands an excessive under-relaxation to converge. In addition, for a high degree of polymer chain confinement, i.e., for Gd greater than 0.5, the continuum theory fails to be accurate and mesoscopic simulations that track individual polymer molecules are needed.