Abstract
The flow of viscoelastic fluids may, under certain conditions, exhibit shear-banding characteristics that result from their susceptibility to unusual flow instabilities. In this work, we explore both the existing shear banding mechanisms in the literature, namely; constitutive instabilities and flow-induced inhomogeneities. Shear banding due to constitutive instabilities is modelled via either the Johnson–Segalman or the Giesekus constitutive models. Shear banding due to flow-induced inhomogeneities is modelled via the Rolie–Poly constitutive model. The Rolie–Poly constitutive equation is especially chosen because it expresses, precisely, the shear rheometry of polymer solutions for a large number of strain rates. For the Rolie–Poly approach, we use the two-fluid model wherein the stress dynamics are coupled with concentration equations. We follow a computational analysis approach via an efficient and versatile numerical algorithm. The numerical algorithm is based on the Finite Volume Method (FVM) and it is implemented in the open-source software package, OpenFOAM. The efficiency of our numerical algorithms is enhanced via two possible stabilization techniques, namely; the Log-Conformation Reformulation (LCR) and the Discrete Elastic Viscous Stress Splitting (DEVSS) methodologies. We demonstrate that our stabilized numerical algorithms accurately simulate these complex (shear banded) flows of complex (viscoelastic) fluids. Verification of the shear-banding results via both the Giesekus and Johnson-Segalman models show good agreement with existing literature using the DEVSS technique. A comparison of the Rolie–Poly two-fluid model results with existing literature for the concentration and velocity profiles is also in good agreement.
Highlights
Shear banding in flow of polymer solutions has been largely investigated in simple shear geometries.Such flows may, under certain conditions of, say, the prevailing shear rates, develop localized bands in response to the development of flow instabilities
The variables are initialized; the (Log-Conformation Reformulation (LCR) stabilized) stress equations are solved for the conformation tensor Q; the (DEVSS stabilized) momentum equations are solved for the intermediate velocity field; the pressure equation is solved and the under-relaxation applied, with an under-relaxation factor of 0.3; both pressure and velocity are corrected via the SIMPLE algorithm; the volume fraction is obtained; and, the solutions at the time level, tnew = told + δt, are obtained and the processes is repeated from step 2 until either the predetermined final time is reached or certain predetermined conditions are met
Shear banding has been observed in the shear flow of polymer solutions as well as polymer melts
Summary
Shear banding in flow of polymer solutions has been largely investigated in simple shear geometries. In addition to constitutive instabilities via non-monotonic stress–strain models, an alternative mechanism to explain shear banding in flows of polymer solutions is flow-induced concentration fluctuations. The premise of this concept is the Helfand–Fredrickson formalism. We employ finite volume methodologies, on the OpenFOAM software platform, in order to investigate shear banding in the transient shear flow of polymer solutions as modelled via the two-fluid Rolie-Poly constitutive equation. One may reasonably assume that the shear banding processes in the shear flow of such polymer melts should, expectedly be largely driven by non-monotonic stress-strain profiles and, modelled as constitutive instabilities. Our robust and versatile algorithm is the first to be tested in investigating the various shear banding characteristics in this way
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