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In this numerical study, an original approach to simulate non-isothermal viscoelastic fluid flows at high Weissenberg numbers is presented. Stable computations over a wide range of Weissenberg numbers are assured by using the root conformation approach in a finite volume framework on general unstructured meshes. The numerical stabilization framework is extended to consider thermo-rheological properties in Oldroyd-B type viscoelastic fluids. The temperature dependence of the viscoelastic fluid is modeled with the time-temperature superposition principle. Both Arrhenius and WLF shift factors can be chosen, depending on the flow characteristics. The internal energy balance takes into account both energy and entropy elasticity. Partitioning is achieved by a constant split factor. An analytical solution of the balance equations in planar channel flow is derived to verify the results of the main field variables and to estimate the numerical error. The more complex entry flow of a polyisobutylene-based polymer solution in an axisymmetric 4:1 contraction is studied and compared to experimental data from the literature. We demonstrate the stability of the method in the experimentally relevant range of high Weissenberg numbers. The results at different imposed wall temperatures, as well as Weissenberg numbers, are found to be in good agreement with experimental data. Furthermore, the division between energy and entropy elasticity is investigated in detail with regard to the experimental setup., Comment: 25 pages, 13 figures, 2 tables

In this work, the novel "Tensor Diffusion" approach for simulating viscoelastic fluids is proposed, which is based on the idea, that the extra-stress tensor in the momentum equation of the flow model is replaced by a product of the strain-rate tensor and a tensor-valued viscosity. At least for simple flows, this approach offers the possibility to reduce the full nonlinear viscoelastic model to a generalized "Tensor Stokes" problem, avoiding the need of considering a separate stress tensor in the solution process. Besides fully developed channel flows, the “Tensor Diffusion” approach is evaluated as well in the context of general two-dimensional flow configurations, which are simulated by a suitable four-field formulation of the viscoelastic model respecting the "Tensor Diffusion". However, substituting the extra-stress tensor by the "Tensor Diffusion" in the complete flow model is desired for general two-dimensional flows as well, again to be able to reduce the full nonlinear viscoelastic model to a Stokes-like problem., Ergebnisberichte des Instituts für Angewandte Mathematik;617

This study is concerned with finite element/volume modelling of contraction–expansion axisymmetric pipe flows for thixotropic and non-thixotropic viscoelastic models. To obtain solutions at high Weissenberg numbers (Wi) under a general differential form W i τ p ∇ = 2 ( 1 − β ) D − f τ p , both thixotropic Bautista–Manero micellar and non-thixotropic EPTT f-functionals have been investigated. Here, three key modifications have been implemented: first, that of convoluting EPTT and micellar Bautista–Manero f-functionals, either in a multiplicative (Conv*) or additive (Conv+) form; second, by adopting f-functionals in absolute form (ABS-f-correction); and third, by imposing pure uniaxial-extension velocity-gradient components at the pure-stretch flow-centreline (VGR-correction). With this combination of strategies, highly non-linear solutions have been obtained to impressively high Wi [=O(5000+)]. This capability permits analysis of industrial applications, typically displaying non-linear features such as thixotropy, yield stress and shear banding. The scope of applications covers enhanced oil-recovery, industrial processing of plastics and foods, as well as in biological and microfluidic flows. The impact of rheological properties across convoluted models (moderate-hardening, shear-thinning) has been observed through steady-state solutions and their excess pressure-drop (epd) production, stress, f-functional field structure, and vortex dynamics. Three phases of vortex-behaviour have been observed with rise in elasticity, along with upstream–downstream Moffatt vortices and plateauing epd-behaviour at high-Wi levels. Moreover, enhancement of positive-definiteness in stress has improved high-Wi solution attenuation.

This hybrid finite element/volume study is concerned with the modelling of worm-like micellar systems, employing a new micellar thixotropic constitutive model with viscoelasticity within network-structure construction–destruction kinetics. The work focuses on steady-state solutions for axisymmetric, rounded-corner, 4:1:4 contraction–expansion flows. This has importance in industrial and healthcare applications such as in enhanced oil-reservoir recovery. Material functions for the micellar models (time-dependent, thixotropic) have been fitted to match two different extensional configurations of the exponential Phan-Thien/Tanner (PTT) model (rubber network-based, non-thixotropic). This covers mild and strong-hardening response, and re solvent fraction, highly-polymeric (β = 1/9) and solvent-dominated (β = 0.9) fluids. Solution results are described through normalised Excess Pressure Drop (EPD), vortex intensity and stream function, stress (N1 and N2), and f-functional data. EPD predictions with the new micellar models prove to be consistent (at low rates, some rising) with Newtonian results, contrary to the base-reference modified Bautista–Manero (MBM) results. Markedly different vortex intensity trends are found in comparing micellar and EPTT solutions, which correspond with N2 − N1 and f data. In order to address the highly-elastic regime for thixotropic materials, a convoluted approach between EPPT and micellar models has been proposed. Here, numerically stable solutions are reported for impressively large We up to 300 and new vortex structures are revealed.

A time-stepping finite element method is used to predict the viscoelastic stresses that arise in a tube-tooling wire-coating problem. The polymer melt HDPE is modelled by a multi-mode Phan-Thien/Tanner constitutive equation. Different flow geometries are considered to address optimisation of the process with respect to minimising the stress induced within the coating produced. The influence of the die itself and the various modes are considered. Relaxation times range for a three-mode model from 10 −2 to 10 2 s and for a seven-mode model from 10 −3 to 10 3 s. Typical Weissenberg numbers may range up to 10 4 . Three modes are sufficient to adequately describe the flow, and shorter/narrower draw-down regions are identified as being preferable. Once an adequate land length has been gathered, that has relaxed the flow stresses prior to draw-down, the actual details of die design are found to be inconsequential to the induced stresses in the delivered coatings.