A study of shear banding in polymer solutions

Abstract

In a recent letter [M. Cromer, M. C. Villet, G. H. Fredrickson, and L. G. Leal, “Shear banding in polymer solutions,” Phys. Fluids 25, 051703 (2013)], we showed the existence of a steady shear-banded velocity profile for a model polymer solution with an underlying monotonic constitutive curve. The driving mechanism is the coupling of the polymer stress to an inhomogeneous concentration profile. To further understand this phenomenon, in this paper we investigate the underlying linear instability as well as probe the model parameters and their effect on transient and steady state solutions. The linear stability analysis of the steady, base homogeneous model shows that, in opposition to diffusion, the polymer concentration moves up stress gradients in a shear flow creating a critical balance such that, for a range of parameters, an instability occurs that drives the system away from homogeneity. The simulation of the full nonlinear equations in planar one-dimensional shear reveals a window within which the linear instability manifests itself as a shear-banded flow. Unlike the case for a nonmonotonic constitutive curve for which two bands are predicted, there is no apparent selection process for a monotonic curve that sets the number of bands in planar shear. Thus, we find the possibility of greater than two bands, the number of which is determined by the ratio of the polymer correlation length to the channel width. In addition to steady shear banding, transient phenomena are also probed revealing a complicated band transition (i.e., number of bands changing in time) as well as elastic recoil in a Taylor-Couette cell, each of which have been observed in experiment. Finally, as we showed in our letter, a nonlinear subcritical instability exists resulting in multiple steady states depending upon the wall ramp speed. Here, we show that this phenomenon can occur for realistic parameter values, in particular those obtained for a particular polymer solution that has shown this multiple steady state behavior experimentally.