Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields

Numerical simulation is used to investigate the flow of polymer solutions around a periodic, linear array of cylinders by using three constitutive equations derived from kinetic theory of dilute polymer solutions: the Giesekus model; the finitely extensible, nonlinear elastic dumbbell model with Peterlin's approximation (FENE-P); and the FENE dumbbell model of Chilcott–Rallison (CR). In the Giesekus model, intramolecular forces are described by a Hookean spring, whereas a finitely extensible spring whose modulus is given by the Warner approximation is used in both the FENE-P and CR models. Hydro dynamic drag on the beads is taken to be anisotropic for the Giesekus model and isotropic for the other two models. The CR and FENE-P models differ subtly in their approximate treatment of the nonlinear force law. The three models exhibit very similar rheological behavior in viscometric flow and steady elongational flow, with the notable exception that the viscosity for the CR model is shear-rate independent. Finite element simulations are performed by using two different formulations: the elastic-viscous split-stress gradient (EVSS-G) method and a new variant of this formulation, the discrete EVSS-G (DEVSS-G) formulation, in which the elliptic stabilization term is added only to the discrete version of the momentum equation, and the constitutive equation is solved directly in terms of the polymer contribution to the stress tensor. Calculations are performed for all models up to a Weissenberg number We, where the configuration tensor 〈QQ〉 loses positive definiteness. However, by locally refining the mesh in the gap region, the positive definiteness of 〈QQ〉 is recovered. The flow and stress fields predicted by the three constitutive equations are qualitatively similar. A `birefringent strand' of highly stretched polymer molecules, which appears to emanate from the rear stagnation point in the cylinder, strengthens as We is increased. Not surprisingly, the molecular extension computed for the Giesekus model is considerably larger than that of the two FENE spring models. The drag force on the cylinders differs for the FENE-P and CR models, because of the difference in the shear-thinning viscosity resulting from the different approximations used in these models.