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Abstract
Recent numerical studies on pressure-drops in contraction flows have introduced a variety of constitutive models to compare and contrast the competing influences of extensional vis-cosity, normal stress and shear-thinning. Early work on pressure-drops employed the constant viscosity Oldroyd-B and Upper Convected Max- well (UCM) models to represent the behavior of so-called Boger fluids in axisymmetric contrac-tion flows, in (unsuccessful) attempts to predict the very large enhancements that were ob-served experimentally. In more recent studies, other constitutive models have been employed to interpret observed behavior and some pro-gress has been made, although finding a (re-spectable) model to describe observed contrac-tion-flow behavior, even qualitatively, has been frustratingly difficult. With this in mind, the present study discusses the ability of a well- known FENE type model (the so-called FENE- CR model) to describe observed behavior. For various reasons, an axisymmetric (4:1:4) con-traction/expansion geometry, with rounded corners, is singled out for special attention, and a new hybrid finite element/volume algo-rithm is utilized to conduct the modeling, which reflects an incremental pressure-correction time-stepping structure. New to this algo-rithmic formulation are techniques in time discretization, discrete treatment of pressure terms, and compatible stress/velocity-gradient representation. We shall argue that the current simulations for the FENE-CR model have re-sulted in a major improvement in the sort-for agreement between theory and experiment in this important bench-mark problem.
Highlights
In the early days of Computational Rheology, the socalled Upper-Convected Maxwell (UCM) and Oldroyd-B models were strongly favored. This was partly due to the fact that they were assumed to be the ‘bottom-line’ of acceptable mathematical simplicity, whilst being able to mimic the complex rheometrical behavior for a class of dilute polymer solutions known as Boger fluids, which became popular in the late 1970s. (A Boger fluid has a reasonably constant shear viscosity, a high extensional viscosity as the extensional strain rate increases, and a high first normal stress difference, which has a quadratic dependence on shear rate at low to moderate shear rates, before becoming weaker than quadratic at higher shear rates.)
It is well known that simulations for the UCM and Oldroyd B models failed to predict the large increases in the so-called Couette correction (or equivalently the “extra pressure difference”) in axisymmetric contraction flows
It must be emphasized that such a restriction is not relevant to the Oldroyd B model and is certainly not appropriate for Boger fluids, where we have argued that β=0.9 is more realistic
Summary
In the early days of Computational Rheology, the socalled Upper-Convected Maxwell (UCM) and Oldroyd-B models were strongly favored. It is well known that simulations for the UCM and Oldroyd B models failed to predict the large increases in the so-called Couette correction (or equivalently the “extra pressure difference” (epd)) in axisymmetric contraction flows. Nigen and Walters [4] found significant differences in pressure-drop between Boger and Newtonian liquids with the same shear viscosity for axisymmetric contraction flow. The increase in the Couette correction for Boger fluids flowing in various axisymmetric contractions can be very high These experimental observations clearly presented theoretical (computational) rheologists with several significant challenges, some of which have already been resolved (see for example, Phillips and Williams [6]; Aboubacar et al [7]; Walters and Webster [8]; Alves et al [9]). We attempt to relate the observed behavior of highlyelastic Boger fluids in complex flows to their rheometrical behavior
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