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 W i, gradient number G d (which defines the level of polymer chain confinement), and Péclet number P e 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 W i. However, in the absence of wall-depletion, there is no first order contribution and, therefore, the lowest-order concentration variation is proportional to W i 2. An upper limit of W i = 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 G d greater than 0.5, the continuum theory fails to be accurate and mesoscopic simulations that track individual polymer molecules are needed.