Although shear banding is a ubiquitous phenomenon observed in soft materials, the mechanisms that give rise to shear-band formation are not always the same. In this work, we develop a new two-fluid model for semi-dilute entangled polymer solutions using the generalized bracket approach of nonequilibrium thermodynamics. The model is based on the hypothesis that the direct coupling between polymer stress and concentration is the driving mechanism of steady shear-band formation. To obtain smooth banded profiles in the two-fluid framework, a new stress-di↵usive term is added to the time evolution equation for the conformation tensor. The advantage of the new model is that the di↵erential velocity is treated as a state variable. This allows a straightforward implementation of the additional boundary conditions arising from the derivative di↵usive terms with respect to this new state variable. To capture the overshoot of the shear stress during the start of a simple shear flow, we utilize a nonlinear Giesekus relaxation. Moreover, we include an additional relaxation term that resembles the term used in the Rouse linear entangled polymer model to account for convective constraint release and chain stretch to generate the upturn of the flow curve at large shear rates. Numerical calculations performed for cylindrical Couette flow confirm the independency of the solution from the deformation history and initial conditions. Furthermore, we find that stress-induced migration is the responsible di↵usive term for steady-state shear banding. Because of its simplicity, the new model is an ideal candidate for the use in the simulation of more complex flows.

We study shear banding in a planar 4:1 contraction flow using our recently developed two-fluid model for semidilute entangled polymer solutions derived from the generalized bracket approach of nonequilibrium thermodynamics. In our model, the differential velocity between the constituents of the solution allows for coupling between the viscoelastic stress and the polymer concentration. Stress-induced migration is assumed to be the triggering mechanism of shear banding. To solve the benchmark problem, we used the OpenFOAM software package with the viscoelastic solver RheoTool v.2.0. The convection terms are discretized using the high-resolution scheme CUBISTA, and the governing equations are solved using the SIMPLEC algorithm. To enter into the shear banding regime, the uniform velocity at the inlet was gradually increased. The velocity increases after the contraction due to the mass conservation; therefore, shear banding is first observed at the downstream. While the velocity profile in the upstream channel is still parabolic, the corresponding profile changes to plug-like after the contraction. In agreement with experimental data, we found that shear banding competes with flow recirculation. Finally, the profile of the polymer concentration shows a peak in the shear banding regime, which is closer to the center of the channel for larger inlet velocities. Nevertheless, the increase in the polymer concentration in the region of flow recirculation was significantly larger for the inlet velocities studied in this work. With our two-fluid finite-volume solver, localized shear bands in industrial applications can be simulated.

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This work reports on the first three-dimensional viscoelastic dough kneading simulation performed in a spiral kneader. Unstructured tetrahedral grids were generated using ICEM CFD 17.1. Viscoelastic volume-of-fluid simulations were performed using OpenFOAM v.4.0 in combination with the RheoTool package v.2.0. The White-Metzner model with a Bird-Carreau type of shear-rate dependency of the viscosity and relaxation time was utilized to describe the rheology of the dough matrix. We validated our numerical method by simulating the viscoelastic rod climbing benchmark problem in a cylindrical bowl. The temporal evolution of the dough surface was compared with screenshots obtained with a high-speed video camera during laboratory kneading. We found that the curvature of the free surface matches the experimental data well. With our numerical approach, we were able to predict the formation, extension, and breakup of dough pockets. The dough is convected around the inner stationary rod by the rotation of the outer cylindrical bowl, whereas the spiral arm located in between these two parts produces spiral flow patterns. Vertical mixing is not as good as radial mixing and may be enhanced by utilizing two spiral arms similar to hand kneading. Industrial kneading geometries and processes may be further optimized by performing such types of simulations.

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