White-Metzner Model Research Articles

Polymer flooding characteristics in modified Hele–Shaw cells

In this study, numerical simulations are employed to investigate the displacement flow of oil by viscoelastic polymer solutions inside modified Hele–Shaw cells (HSCs). The flooding characteristics within converging and diverging geometrical configurations and various flow injection rates are explored by utilizing the White–Metzner model to capture the viscoelastic behavior of hydrolyzed polyacrylamide solutions at different concentrations. The simulations are conducted in OpenFOAM using the volume of fluid method to model the multiphase interface. Results demonstrate that higher elasticity numbers (El) and lower modified capillary numbers (Ca′) enhance the oil recovery to over 75% in terms of swept volume due to improved flow stability and reduced finger branching. Additionally, the diverging HSCs positively influence the flow patterns and facilitate an extra increase in sweeping efficiency (SE) by more than 12% via favorably alleviating the stability threshold in terms of Ca′–El relationship. In contrast, the opposite holds for the converging channel, decreasing SE by 16% and narrowing down the stability region. The findings indicate that adjusting polymer concentration and cell configuration can mitigate the adverse effects of finger instability, thereby improving the overall success of oil displacement processes. Moreover, a supervised machine learning study was carried out to predict the sweep efficiency based on the flow dimensionless numbers, where the multi-layered artificial neural network was exhibited as a high-accuracy model with an R2 value of 94.46%.

Sharp corner singularity of the White–Metzner model

The stress singularity is determined using matched asymptotics for complete flow of a White–Metzner (WM) fluid around a re-entrant corner. The model is considered in the absence of a solvent viscosity, with power-law forms for the relaxation time and polymer viscosity. In this form, the model shares the same stress singularity as the upper convected Maxwell (UCM) model, but its wall boundary layers may be thinner or thicker than those for UCM depending upon the relative difference in the power-law exponents. If the exponent for the relaxation time is greater than that for the polymer viscosity, the boundary layer is narrower, whilst it is thicker if the polymer viscosity exponent exceeds that of the relaxation time. When the exponents are the same, the WM boundary layer thickness is the same size as that for UCM. A self-similar solution is derived for the stress and velocity fields and matched to both upstream and downstream boundary layers. Restrictions on the sizes of the power-law exponents are also given for validity of this solution.

Quantification of viscous and damage dissipation of bituminous binder and mastic using White-Metzner model

ABSTRACT Bituminous pavement layers are composite in nature, consisting of binder, filler, and aggregates. Understanding the mechanical behaviour of bituminous layers demands investigation of their different material scales, such as the binder, mastic, and mixture scales. While substantial investigation has been carried out on binder and mixture related to fatigue damage, not much research is carried out on mastic. Of the methods used to characterise fatigue damage, energy dissipation is widely used; however, it is complicated due to the presence of other modes of dissipation, such as viscous dissipation. In this study, an attempt is made to separate viscous dissipation and dissipation due to damage for both binder and mastic using a nonlinear viscoelastic fluid model. For this purpose, six types of materials were tested at temperatures of 20 ∘ C and 25 ∘ C, 10 Hz frequency, and five different strain levels spanning the linear and non-linear regimes, for 20,000 cycles. Material parameters for the non-linear White-Metzner model incorporated with the Williamson model were determined for an initial cycle and using these parameters, the viscous component of energy dissipation was determined. Dissipation due to damage was estimated by separating viscous dissipation from total dissipation at every 2000th cycle and based on the evolution of viscous and damage dissipation across cycles, three trends were observed. Based on the trends observed, the influence of strain amplitude, temperature and modifiers on the fatigue resistance of the materials was ascertained.

Role of viscoelastic fluid rheology in miscible viscous fingering

Classical viscous fingering patterns in flow displacement can be modulated by incorporating polymers into either of the two fluids. The polymeric solutions are viscoelastic fluids exhibiting both shear-thinning as well as elastic behavior. The present numerical study examines the dynamics of viscous fingering in miscible flow displacement with either displacing or displaced fluid being treated as polymeric fluid. The nonlinear rheology of the polymeric fluid is described using the White–Metzner model. The role of fluid rheology in the transient evolution of concentration field and fingering patterns defined using quantities such as mixing length, relative contact area, finger width, and the number of fingers, is investigated by performing numerical simulations. The correlation between fingering structure and rheological properties is established. The study shows that incorporation of polymer into the fluid alters the viscous fingering attributed to the three main factors: viscosity modification, shear-thinning behavior, and fluid elasticity. Primarily, incorporating polymers into the displaced (displacing) fluid leads to suppressed (enhanced) fingering patterns, attributed to modification of the viscosity contrast between two fluids, the governing parameter for viscous fingering. Interestingly, when the viscosity contrast at a gap-averaged shear-rate is held fixed, the shear-thinning behavior promotes the longitudinal growth of fingers irrespective of the fluid to which polymers are added. Moreover, the fluid elasticity ends to inhibit the growth of fingers leading to a more efficient flow displacement regardless of the flow arrangement. Thus, the study suggests that in addition to the viscosity at the gap-averaged shear-rate, the nature of the entire flow curve representing nonlinear rheology plays an important role in fingering dynamics. The mechanism for the destabilization due to shear-thinning behavior is explained by analyzing the vorticity structures, local shear-rate, and local viscosity distribution in the flow domain. The normal stress distribution is analyzed to comprehend the stabilization attributed to the fluid elasticity. Overall, the interplay between the viscosity contrast and the complete rheological description of the fluid governs the fingering dynamics.

A constitutive equation for fiber suspensions in viscoelastic media

A viscosity overshoot of fibers filled in a polymer melt under a shear flow is much tougher to predict via the existing constitutive equations of suspension rheology in a viscous media, owing to the effect of fiber orientation on the viscoelastic behavior. The WMT-X (White–Metzner model eXtended by Tseng) viscoelastic fluid model coupled with the typical Dinh–Armstrong fiber suspension model, known as the suspended WMT-X model, is proposed herein. The primary procedure involves verifying the lower viscosity of the completely aligned suspension compared to that of the randomly oriented suspension. In addition, the viscosity overshoot depends on the off-diagonal orientation tensor component in the flow-gradient plane. As a validation, the numerical predictions of transient shear viscosity are in good agreement with the related experimental data.

A constitutive analysis of stress overshoot for polymer melts under startup shear flow

Predicting a transient stress overshoot for polymer melts under startup shear flow is challenging. In recent, the classical White–Metzner (WM) constitutive equation of nonlinear viscoelastic fluids was potentially extended. For viscoelastic material functions, the minus ratio of the second normal stress difference to the first normal stress difference (−N2/N1) is important in characterizing a fluid's elasticity related to molecular structures and molecular weight distribution. Using the extended WM model to analyze a dramatic change in stress overshoot with respect to the −N2/N1 ratio at high Weissenberg numbers would be significant. As a validation, numerical predictions of shear stress growth coefficient at different shear rates are in good agreement with experimental data.

A revisitation of White−Metzner viscoelastic fluids

The famous White–Metzner (WM) constitutive equation expresses a relatively simple nonlinear viscoelastic fluid of polymer melts. However, such a differential stress model, substantial with strong hyperbolic and singular problems, has hitherto always obtained unsatisfactory simulations of corner vortex in a typical contraction flow, especially for high Weissenberg numbers. A modified WM model useful for viscoelastic fluid computations is, therefore, proposed herein. As a validation, this model primarily fits the first normal stress difference for characterizing a fluid's elasticity, as well as shear viscosity and extensional viscosity. It is significant to discuss the vortex formation and growth, with the predicted vortex sizes in good agreement with the experimental data.

Numerical and experimental investigation of dough kneading in a three-dimensional spiral kneader

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.

Linear instability of shear thinning pressure driven channel flow

We study theoretically pressure driven planar channel flow of shear thinning viscoelastic fluids. Combining linear stability analysis and full nonlinear simulation, we study the instability of an initially one-dimensional base state to the growth of two-dimensional perturbations with wavevector in the flow direction. We do so within three widely used constitutive models: the microscopically motivated Rolie-Poly model, and the phenomenological Johnson-Segalman and White-Metzner models. In each model, we find instability when the degree of shear thinning exceeds some level characterised by the logarithmic slope of the flow curve at its shallowest point, n=dlogΣ/dlogγ˙|min. Specifically, we find instability for n < n*, with n* ≈ 0.21, 0.11 and 0.30 in the Rolie-Poly, Johnson-Segalman and White-Metzner models respectively. Within each model, we show that the critical pressure drop for the onset of instability obeys a criterion expressed in terms of this degree of shear thinning, n, together with the derivative of the first normal stress with respect to shear stress. Both shear thinning and rapid variations in first normal stress across the channel are therefore key ingredients driving the instability. In the Rolie-Poly and Johnson-Segalman models, the underlying mechanism appears to involve the destabilisation of a quasi-interface that exists in each half of the channel, across which the normal stress varies rapidly. (The flow is not however shear banded in any parameter regime that we consider.) In the White-Metzner model, no such quasi-interface exists, but the criterion for instability nonetheless appears to follow the same form as in the Rolie-Poly and Johnson-Segalman models. This presents an outstanding puzzle concerning any possibly generic nature of the instability mechanism. We finally make some brief comments on the Giesekus model, which is rather different in its predictions from the other three.

A numerical approximation with WLS/SUPG algorithm for solving White–Metzner viscoelastic flows

In this paper, we consider a numerical method for the White–Metzner model of viscoelastic fluid flows by a combination of the weighted least-squares (WLS) method and the streamline upwind/Petrov–Galerkin (SUPG) method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes-like problem and a linearized constitutive equation using WLS and SUPG, respectively. The elastic viscous split stress formulation (EVSS) introduced in [27] is used for the discretization of the constitutive equation. An a priori error estimate for the WLS/SUPG method is derived and numerical results supporting the estimate are presented. We encounter no limit of the Weissenberg number for all values of ε satisfying 0<ε<1, a dimensionless material parameter, in our calculations.

From elastic deformation to flow in tempered chocolate

We characterize the yield behavior and flow curve of a commercially available, molten, tempered milk chocolate with a fat content of 30%, from rheological measurements performed with a rotational rheometer fitted with a four-bladed vane spindle in a large-gap setup. We show that the yield behavior can be captured by the viscoelastic White–Metzner model, which indicates that elastic deformation precedes flow as the shear stress is increased. Following an initial fluidization cycle, the shear modulus of the tempered chocolate increases by a factor of 60 to G=4.4×104 Pa. We find that the Bingham model provides the best fit to our extensive flow curve data, where the yield stress is τyield=36.1±0.6 Pa and the Newtonian plateau viscosity is μB=8.7±0.02 Pa s.

Simulation of collagen solution flow in rectangular capillary

The viscoelastic properties of foods and polymers can be evaluated from flow of the material in capillary with specified dimension and shape. The extrusion rheometer equipped by capillary with rectangular cross-section was used for determination of the rheological behaviour of water collagen solution. The measurements of the axial profiles in longitudinal direction of the total stresses at capillary wall were performed for various shear rates. The linear viscoelastic model of Oldroyd B type: White-Metzner model was used for simulation of fluid flow in OpenFOAM software package. The simulations describe the effect of relaxation time on wall total stress in convergent-divergent capillary.

Application of the LS-STAG Immersed Boundary/Cut-Cell Method to Viscoelastic Flow Computations

AbstractThis paper presents the extension of a well-established Immersed Boundary (IB)/cut-cell method, the LS-STAG method (Y. Cheny &amp; O. Botella, J. Comput. Phys. Vol. 229, 1043-1076, 2010), to viscoelastic flow computations in complex geometries. We recall that for Newtonian flows, the LS-STAG method is based on the finite-volume method on staggered grids, where the IB boundary is represented by its level-set function. The discretization in the cut-cells is achieved by requiring that global conservation properties equations be satisfied at the discrete level, resulting in a stable and accurate method and, thanks to the level-set representation of the IB boundary, at low computational costs.In the present work, we consider a general viscoelastic tensorial equation whose particular cases recover well-known constitutive laws such as the Oldroyd-B, White-Metzner and Giesekus models. Based on the LS-STAG discretization of the Newtonian stresses in the cut-cells, we have achieved a compatible velocity-pressure-stress discretization that prevents spurious oscillations of the stress tensor. Applications to popular benchmarks for viscoelastic fluids are presented: the four-to-one abrupt planar contraction flows with sharp and rounded re-entrant corners, for which experimental and numerical results are available. The results show that the LS-STAG method demonstrates an accuracy and robustness comparable to body-fitted methods.

Evaluation of Thermally Induced Degradation of Branched Polypropylene by Using Rheology and Different Constitutive Equations.

In this work, virgin as well as thermally degraded branched polypropylenes were investigated by using rotational and Sentmanat extensional rheometers, gel permeation chromatography and different constitutive equations. Based on the obtained experimental data and theoretical analysis, it has been found that even if both chain scission and branching takes place during thermal degradation of the tested polypropylene, the melt strength (quantified via the level of extensional strain hardening) can increase at short degradation times. It was found that constitutive equations such as Generalized Newtonian law, modified White-Metzner model, Yao and Extended Yao models have the capability to describe and interpret the measured steady-state rheological data of the virgin as well as thermally degraded branched polypropylenes. Specific attention has been paid to understanding molecular changes during thermal degradation of branched polypropylene by using physical parameters of utilized constitutive equations.

Measurements and modeling of temperature-strain rate dependent uniaxial and planar extensional viscosities for branched LDPE polymer melt

Abstract In this work, novel rectangle and circular orifice (zero-length) dies have been utilized for temperature-strain rate dependent planar and uniaxial extensional viscosity measurements for the LDPE polymer melt by using standard twin bore capillary rheometer and Cogswell model and the capability of five different constitutive equations (novel generalized Newtonian model, original Yao model, extended Yao model, modified White-Metzner model, modified Leonov model) to describe the measured experimental data has been tested. It has been shown that chain branching causes the strain hardening occurrence in both uniaxial and planar extensional viscosities and its maximum is shifted to the higher strain rates if the temperature is increased. The level of uniaxial extensional strain hardening for the branched LDPE sample has been found to be higher in comparison with the planar extensional viscosity within wide range of temperatures.

The stabilizing effect of shear thinning on the onset of purely elastic instabilities in serpentine microflows.

We determine both experimentally and numerically the onset of elastic flow instabilities in viscoelastic polymer solutions with different levels of shear thinning. Previous experiments realized in microfluidic serpentine channels using dilute polymeric solutions showed that the onset of elastic instabilities strongly depends on the channel curvature. The scaling dependence is well captured by the general instability scaling criterion proposed by Pakdel and McKinley [Phys. Rev. Lett., 1996, 76, 2459:1-4]. We determine here the influence of fluid shear thinning on the onset of such purely-elastic flow instabilities. By testing a set of polyethylene oxide solutions of high molecular weight at different polymer concentrations in microfluidic serpentine channels we observe that shear thinning has a stabilizing effect on the microfluidic flow. Three-dimensional numerical simulations performed using the White-Metzner model predict similar trends, which are not captured by a simple scaling analysis using the Pakdel-McKinley criterion.

Review of Non-Newtonian Mathematical Models for Rheological Characteristics of Viscoelastic Composites

This study presents an overview of viscoelastic characteristics of biocomposites derived of natural-fibre-reinforced thermoplastic polymers and predictive models has been presented in order to understand their rheological behavior. Various constitutive equations are reviewed for a better understanding of their applicability to polymer melt in determining the viscosity. The models to be investigated are the Giesekus-Leonov model, the Upper Convected Maxwell (UCM) model, the White-Metzner model, K-BKZ model, the Oldroyd-B model, and the Phan-Thien-Tanner models. The aforementioned models are the most powerful for predicting the rheological behavior of hybrid and green viscoelastic materials in the presence of high shear rate and in all dimensions. The Phan-Thien Tanner model, the Oldroyd-B model, and the Giesekus model can be used in various modes to fit the relaxation modulus accurately and to predict the shear thinning as well as shear thickening characteristics. The Phan-Thien Tanner, K-BKZ, Upper convected Maxwell, Oldroyd-B, and Giesekus models predicted the steady shear viscosity and the transient first normal stress co-efficient better than the White-Metzner model for green-fibre-reinforced thermoplastic composites.

Experimental and numerical analysis of performance of two fluted mixer designs

In this paper, the effect of the Maddock mixer design on its performance in the single screw extrusion of High-density Polyethylene (HDPE) melt has been investigated experimentally on a special extrusion line equipped with a barrel having several glass windows as well as theoretically using a three-dimensional finite element method simulation. The red–green–blue spectral analysis has been used for the experimental quantification of the mixing efficiency, whereas a generalised Newtonian model (with Carreau–Yasuda viscosity function) and a viscoelastic modified White–Metzner model have been utilised in flow simulations. Based on the performed experimental work, it has been found that the mixer with no undercut wiping flight (‘Closed’ mixer) has a faster mixing/purging but a similar final mixing performance as the mixer with undercut wiping flight (‘Open’ mixer). It has also been revealed that the ‘Open’ mixer creates an unwanted dead zone (in the form of the stagnation layer of material rotating between the mixer and the barrel) at which the polymer melt degradation may take place. It has been found that based on the stress, residence time and pathline calculation, it is possible to predict the experimentally observed tendency of the ‘Closed’ mixer for a faster mixing/purging in comparison with the ‘Open’ mixer as well as the presence of a stagnation layer in the ‘Open’ mixer. The performed theoretical parametric study indicates that the gradual opening of the wiping gap causes the pressure driven flow to become more dominant than the drag flow in shearing as well as in the wiping gap independently of the utilised rheological model. On the other hand, the predicted particle trajectories inside the mixer were found to be shorter for the viscoelastic model, tending to occur mainly in the middle of the channel and thus leaving the mixer a little bit faster in comparison with the purely viscous calculations. Finally, it has been revealed that for the highest tested wiping flight opening, the viscoelastic modified White–Metzner model predicts back flow over the wiping flight, whereas the purely viscous Carreau–Yasuda does not, which can be explained by the elongational viscosity, considered in the viscoelastic modified White–Metzner model. This suggests that the modified White–Metzner model should be preferred more than the purely viscous Carreau–Yasuda model in the mixing element die design optimisation.

“Experimentation and modeling for the apparent elongation viscosity of polymer melts with the White-Metzner model”

The measurement of the apparent elongation viscosity (ηe) of several polyolefin melts was conducted in this study by using the isothermal fiber-spinning method. The White–Metzner (W–M) model was used to analyze the spinning flow of the polymer melts and, thus, the elongation viscosity was predicted at elongation strain rates ranging from 0 to approximately 5 s−1. The values of the model parameters required in the W–M model were obtained by curve fitting the experimental data obtained from the shear measurements. The elongation viscosity predicted using the W–M model was in good agreement with the experimental results of fiber spinning. In addition, ηe could also be estimated directly from the measured shear viscosity (ηS) with a formulation using the W–M model; the subsequently obtained elongation viscosity and Trouton ratio (TR) were reasonable within a wide range of strain rates. Based on the experimental and theoretical results, the polyolefin with a high molecular weight was observed to have high elongation viscosity, and the polymer with a broad molecular weight distribution also possessed high ηe. The TR value of the commercial polypropylene (PP-1040) began to increase from 3 at a deformation rate of 0.1 s−1 and grew up asymptotically to 10, whereas the TR of high-density polyethylene (HDPE-606) remained nearly at 3 within the entire range of strain rates. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical investigations of polyester coextrusion instabilities

Interfacial instabilities occurring in the coextrusion process of molten polymers have been widely studied. The theoretical work based on the stability of 2D Poiseuille multilayer flows (invariant along the flow motion) have pointed out the convective nature of this instability (it is either amplified or damped in the flow direction). This behavior has been confirmed by experimental observations. As this instability is purely elastic (the Reynold number in coextrusion process is very small), it is necessary to introduce a viscoelastic constitutive equation for the polymers. Comparisons between experimental observations and theoretical approaches based on the Giesekus or White–Metzner models give good agreements for laboratory simple devices. However, the applicability of this approach for industrial coextrusion process remains an important challenge, because one has to deal with the influence of complex geometries, varying temperature and more realistic constitutive equations. As it is still very costly in computational time to perform 3D instationary computations for viscoelastic fluids, a methodology to analyze the appearance of interfacial instabilities in a complex geometry is proposed within this paper. First, a stationary 3D computation is performed in the real geometry for a single polymer. This computation allows to find zones in which the flow motion is essentially 2D, isotherm and with maximum values of the shear rate and, thus, the Weissenberg number. Instationary 2D finite element computations for two layer flows and a multi-mode differential viscoelastic model are then performed in these zones. Spatio-temporal analyses give information on the spatial behavior of interfacial disturbances. This methodology is applied to the coextrusion of two polyesters for which experimental observations are available. The simulations predict well the experimental results and show that the outlet zone can amplify low frequencies perturbations. Finally, the influence of processing parameters (temperatures, flow rates, etc.) and die geometries has been checked.