Review of the entry flow problem: Experimental and numerical

This paper is concerned with a review of both experimental and numerical studies of axisymmetric and planar entry flows which have been considered as test problems for the numerical simulation of viscoelastic fluids. The test of the method is usually based upon whether the numerical model predicts vortices in the entry corners. However, it is not clear as to whether one should observe vortices for all viscoelastic fluids. Polyacrylamide solutions and Boger fluids exhibit vortices in axisymmetric flow and the size of the vortex does increase with fluid elasticity. However, the vortex is nearly suppressed in planar entry flow. On the other hand, not all polymer melts are found to exhibit vortices in either axisymmetric or planar entry flow. It is our belief that the origin of vortices is not related to the elasticity based on shear flow propertes but to the behavior of the transient extensional viscosity. Certain polymer melts such as low density polyethylene exhibit vortices in both planar and axisymmetric flow along with unbounded stress growth at the start up of extensional flow. It is believed that the constitutive equations used in the numerical simulation must reflect this extensional behavior if vortices are to be predicted. A review of the numerical simulations concerned with entry flow shows that there is considerable doubt about the accuracy of the predictions for most of the studies. Even for those where the numerical solution is thought to be accurate, the magnitude of the stream function associated with the vortices is usually very low. None of the differential models used to date predicts strain hardening extensional viscosity, but those which are thought to predict vortices do rise more rapidly to the steady-state extensional viscosity values with time. It is recommended that the search of test fluids be widened beyond polymer solutions as there may already exist a number of polymer melts which behave similarly to the predictions of existing constitutive equations.     

Two-dimensional viscoelastic flows: experimentation and modeling

Extensive experimental data on the birefringence in converging and diverging flows of a polymeric melt have been obtained. The birefringence and pressure drop measurements were carried out in working cells of planar geometry having different contraction angles and contraction ratios. For investigation of diverging or abrupt expansion flow, the direction of flow in the cells was reversed. The theoretical predictions are based upon the Leonov constitutive equation and a finite element scheme with streamwise integration.In contrast to Newtonian and second-order fluids, viscoelastic fluids at high shear rates show significant differences in pressure drop and birefringence (i.e. stresses) in converging and diverging flows. For a constant flow rate, the pressure drop is higher and the birefringence smaller in diverging flows than in converging flows. This difference increases with increasing flow rate. Further, for the same contraction ratio but different contraction angles, the birefringence maximum increases considerably with contraction angle. In addition, an increase in contraction ratio has the same effect.The viscoelastic constitutive equation of Leonov has been shown to describe all the above viscoelastic effects observed in the experiments. In general, a reasonable agreement between theory and experiment has been obtained, which shows the usefulness of the Leonov model in describing actual flows.     

Numerical simulation studies of the planar entry flow of polymer melts

This paper is concerned with the numerical simulation of planar entry flow using a penalty finite element method and the comparison of predictions with flow visualization and birefringence data for two polymer melts. The Phan-Thien Tanner (PTT) model was fit to the steady state shear and extensional viscosity data and the transient extensional viscosity data of both polystyrene and low-density polyethylene (LDPE) melts to obtain the parameters λ, ξ, and ϵ in this model. Agreement was found between the flow visualization and birefringence data and the predictions of streamlines and stress. With some modification of the constitutive equation, the vortex growth and intensity observed for LDPE could be predicted by the use of the PTT model and the material parameters fit to the rheological properties. Likewise, the flow behavior of polystyrene, in which only small vortices with no growth were observed, was also predicted. Furthermore, it was found that the size and intensity of the vortex could be affected by the parameter ϵ in the PTT model which controls the predictions of the extensional viscosity. Based on these results it seems that accurate simulation of entry flow behavior requires the use of a constitutive equation which is capable of giving realistic preciction's of a fluid's extentional flow properties.     

聚合物注射成型流动残余应力的数值分析

建立了可压缩黏弹性聚合物熔体在薄壁型腔中充模/保压过程中非等温、非稳态流动 的数学模型,用数值方法实现了注射成型过程中流动应力和取向建立及松弛过程的模拟,研 究了熔体温度、模具温度和注射速率等工艺条件对分子冻结取向的影响,取得了与实验相符 的结果.     

Lubricated optical rheometer for the study of two-dimensional complex flows of polymer melts

We describe a novel optical cross-slot channel rheometer generating two-dimensional and isothermal complex flows of polymer melts. This is made possible by lubricating the channel front and back viewing windows. Flow-induced birefringence and particle tracking velocimetry are reviewed and used to investigate the cross-slot flow of a low density polyethylene melt involving mixed shear and planar extensional deformations. This new device solves the issue of end effects in flow birefringence experiments where no variations of the optical properties along the light path are expected. It greatly facilitates the interpretation of stress field data by providing reliable measurements of the polymer melt extinction angle χ$\chi$ and retardation δ$\delta$, with a spatial resolution of one tenth of a millimeter. At the same time, it offers an enhanced temperature control and an increased optical accuracy due to an improved laser beam shaping. The capabilities and performances of this unique type of lubricated rheometer are discussed in detail and compared with previous approaches.     

Three dimensional finite element analysis of the flow of polymer melts

The finite element simulation of a selection of two- and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two- and three-dimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels.     

Formation of fibrous crystals in flowing blends of polyethylene melts

Flow induced crystallization of high density polyethylene has been studied in a two-phase flow system using low density polyethylene as the carrier phase. Extensional stresses were generated under slow flow conditions by either of two methods: one involving flow past a stationary seed, the other involving a droplet deformation and bursting mechanism. In both cases, oriented, fibrillar crystallization of the high density phase was observed optically and correlated with calculations indicating the presence of flow-induced extensional gradients. Morphological, thermal, and birefringence data indicate that the crystalline fibers produced are oriented and superheatable, and consist of a multifibrillar substructure. For fibers produced by the droplet bursting process a semi-quantitative agreement was found between fiber melting point and birefringence based on a simplified analysis for the bursting induced extensional flow. These results demonstrate that two-phase flows of crystallizable systems are a convenient means for studying the phenomenon of flow induced crystallization in polymer melts.     

The melt processing of monodisperse and polydisperse polystyrene melts within a slit entry and exit flow

This paper presents experimental results and numerical simulations for mono, blended and polydisperse polstryrenes of different molecular weights flowing within a slit geometry. Flow experiments were carried out on small (less than 10 g) quantities of polymer using a multi-pass rheometer and flow-induced birefringence images were obtained for well-defined flow boundary conditions. Experimental flow birefringence observations illustrate the similarities and differences in the flow behaviour between monodisperse and polydisperse polystyrene. For the case of monodisperse polystyrene a transition from “near-Newtonian” stress patterns for low molecular weight polystyrenes, to a highly unstable flow at high molecular weight was observed. Both blending and polydispersity enabled stable flows to be achieved at high flowrates.Experimental flow birefringence results and some pressure difference predictions were compared with numerical predictions. Two different computational approaches were followed, one using a viscoelastic integral K-BKZ/Wagner model within the finite element method solver Polyflow, and the other using the tube theory-based Pom-Pom constitutive equation and Lagrangian-Eulerian code flowSolve. Both numerical methods were able to capture certain experimental observations reasonably well in the stable flow regime, but were not able to predict the onset of the experimentally observed flow instabilities.     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

A novel technique for conducting creep experiments in equibiaxial elongation

A recently developed rheological technique known as continuous lubricated squeezing flow (CLSF) is adapted to perform constant stress, or creep, experiments in equibiaxial elongation flows of polymer melts. By modifying the CLSF technique, which was developed for constant strain rate deformations, we demonstrate that the technique can also be used to generate constant stress flows. Measured steady state viscosities are compared to constant rate elongation results for polymer melts having different molecular characteristics. Linear polymers show strain softening and compare well in constant stress and constant strain rate deformations. The branched polymer shows strain hardening and a viscosity that is slightly higher in constant stress for low rates. Limitations of the current version of the CLSF technique for creep flows are also briefly discussed.     

Birefringence and stress growth in uniaxial extension of polymer solutions

Simultaneous measurements of extensional stresses and birefringence are rare, especially for polymer solutions. This paper reports such measurements using the filament stretch rheometer and a phase modulated birefringence system. Both the extensional viscosity and the birefringence increase monotonically with strain and reach a plateau. Estimates of this saturation value for birefringence, using Peterlin’s formula for birefringence of a fully extended polymer chain are in agreement with the experimental results. However, estimates of the saturation value of the extensional viscosity using Batchelor’s formula for suspensions of elongated fibres are much higher than observed. Reasons for the inability of the flow field to fully unravel the polymer chain are examined using published Brownian dynamics simulations. It is tentatively concluded that the polymer chain forms a folded structure. Such folded chains can exhibit saturation in birefringence even though the stress is less than that expected for a fully extended molecule.Simultaneous measurements of stress and birefringence during relaxation indicate that the birefringence decays much more slowly than the stress. The stress-birefringence data show a pronounced hysteresis as predicted by bead-rod models. The failure of the stress optic coefficient in strong flows is noted.Experiments were also performed wherein the strain was increased linearly with time, then held constant for a short period before being increased again. The response of the stress and birefringence in such experiments is dramatically different and can be traced to the different configurations obtained during stretching and relaxation. The results cast doubt on the appropriateness of pre-averaging the non-linear terms in constitutive equations.     

Characteristic relations of flow birefringence

The general problems of the flow-induced birefringence in liquids have been presented and discussed in Part 1 of this paper, in which some particular responses of an aqueous solution of NGS 1828 were presented, namely, the pertinent mechanical responses and the birefringence responses in transmitted radiation. This paper presents the optical responses of NGS 1828 observed in the scattered radiation. It is shown that the scattered-light techniques can become a powerful and indispensable tool of flow-birefringence techniques, if the actual patterns of light scattering are taken into account. Proof is given that the light scattering and the related birefringence cannot be described by Rayleigh's mathematical model of scattering (Part 1, Ref. 51). The optical effects of the modulation of the primary beam and the scattered beam, called scattered primary isochromatics and scattered secondary isochromatics, are described in terms of the parameters of the system. A practical example is presented. Samples of typical recordings of light-intensity modulation by typical flow patterns are given in the form of scattered primary and secondary isochromatics and integrated isochromatics. One type of intensity modulation enables the determination of shear-strain rate at an interior point in the flow field. Using the determined shear-strain rates at the interior points, a velocity profile along the axis of symmetry of a rectangular-conduit flow was obtained. The results so obtained were satisfactory when compared with the results of direct volumetric measurement. The other type of intensity modulation reveals the nonuniform distribution of birefringence along the path of the transmitted light in the rectangular-conduit flow. The results presented in Part 1 of this paper, are complementary to the results presented below.     

Considerations of the “freezing-in” of flow-induced orientation in polymer melts by vitrification with application to processing

We develop the implications of the experimentally tested hypothesis that (i) birefringence developed during flow is quantitatively frozen-in during vitrification of glass-forming polymer melts and (ii) that the rheo-optical law may be combined with a knowledge of the stress field existing immediately prior to vitrification to yield birefringence distributions. This hypothesis is applied to various problems including multiaxial stretching of sheets, melt spinning, tubular film extrusion and injection molding. Special problems concerned with internal temperature distributions are discussed. We examine difficulties which may arise in application of the hypothesis due to residual thermal stresses. Comparisons are made to other methods of representing orientation development during flow.     

Slip at the interface between immiscible polymer melts I: method to measure slip

Slip at the interface between immiscible polymer melts remains poorly understood. A method that relies solely on rheological measurements to obtain the interfacial slip velocity uses the slip-induced deviation in the flow variables. To use the method, accurate estimates of the flow variables under the assumption of no-slip are necessary. Although such estimates can be easily derived under some cases, in general, this is not straightforward. Therefore, methods to determine the interfacial slip velocity without using estimates for the flow variables under no-slip conditions are desirable. In this work, we focus on investigations of slip at the interface between two immiscible polymer melts undergoing two-phase coaxial flow. To enable such investigations, we have adapted the Mooney method, usually used to investigate wall slip, to investigate polymer/polymer interfacial slip. Using this method, we have measured the slip velocity at the interface between polypropylene and polystyrene as a function of the interfacial stress. To determine the validity of the modified Mooney method, we also determine the slip velocity using the slip-induced deviation in the flow variables. To enable this determination, we use polypropylene and polystyrene with almost identical shear rate-dependent viscosities over a range of shear rates. The slip velocity obtained from the modified Mooney method displayed excellent agreement with that determined using the deviation from no-slip. In agreement with prior work, the dependence of the slip velocity on the interfacial stress is a power-law. Our investigation spans a sufficiently wide range of interfacial stress to enable the direct observation of two power-law regimes and also the transition between the two regimes. We also find that the power-law exponent of approximately 3 at low stresses decreases to approximately 2 at high stresses.     

Theoretical analysis of non-uniform extensional flows in relation to processing rheology and predictions of some constitutive equations

In this paper, a characteristic equation involving the stream function, already given by one of the authors in a previous work for classifying axisymmetric incompressible flows, is re-considered. Non-uniform nearly extensional flows are derived as particular solutions from this equation. Using experimental data in the literature for polymer solutions and melts, it is proved that particular solutions of the characteristic equation lead to kinematics very close to those encountered in the fiber-spinning process. The kinematic equations satisfactorily correlating the fiber-spinning data are used in order to determine the ability of constitutive equations to predict realistic stresses in the flow domain. The rheological parameters of the fluids, obtained from experiments, are used for computation of differential and integral constitutive equations in the spinning conditions. Comparisons with the stress response of adequate constitutive equations are given and discussed.Also affiliated to: Université Joseph Fourier Grenoble I and Institut National Polytechnique de Grenoble, Associé au CNRS (URA 1510)     

Equibiaxial extensional flow of polymer melts via lubricated squeezing flow. II. Flow modeling

A model for lubricated squeezing flow of a viscoelastic fluid is developed in order to study the viability of this flow as a rheological technique for generating equibiaxial extensional deformations in polymer melts. In this simple flow model, the melt, described by an upper-convected Maxwell fluid, is squeezed between thin films of a Newtonian fluid. Comparisons of the model predictions for constant strain rate and constant stress flows are made with experimental results presented in the first paper. Predictions from the model are able to describe the effects of lubricant viscosity and experimental configuration and indicate the technique fails for these flows at Hencky strains of approximately one. The cause for this failure is lubricant thinning, which leads to significant errors in both the measured stress difference and the strain. Received: 31 January 2000 Accepted: 31 May 2000     

The effect of slip in the flow of a branched PP melt: experiments and simulations

In this paper visualisation and direct velocity profile measurement experiments for a branched polypropylene melt in a 10:1 axisymmetric contraction demonstrate the onset of wall slip. Video processing of the flow shows the formation of vortices and their diminution with increasing flow rate. Numerical simulations using a multimode K-BKZ viscoelastic and a purely viscous (Cross) model—both of them incorporating a nonlinear slip law—were used to predict the flow kinematics and dynamics as well as to deduce the slip velocity function by performing fitting to the velocity profiles. It was found that the numerical predictions agree well with the experimental results for the velocity profiles, and vortex formation, growth and reduction. It is suggested that such experiments (visualisation of entrance flow and direct velocity profile measurement) can be useful in evaluating the validity of constitutive equations and slip laws in the flow of polymer melts through processing equipment.     

Transient 3D flow of polymer solutions: A Lagrangian computational method

We describe a computational method for the numerical simulation of three-dimensional transient flows of polymer solutions that extends the work of Harlen et al. [O.G. Harlen, J.M. Rallison, P. Szabó, A split Lagrangian–Eulerian method for simulating transient viscoelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995) 81–104]. The method uses a Lagrangian computation of the stress together with an Eulerian computation of the velocity field. Adaptive mesh reconnection based on Delaunay tetrahedra is used to ensure well-shaped elements. Additional shape-quality improvement procedures are developed to improve the algorithm. We validate the method for the benchmark problem of a rigid sphere falling in a cylindrical pipe. Inertia is neglected. We compare results for the axisymmetric case with previous work (using a FENE model), and then consider the off-axis non-axisymmetric case. In the latter case, we find that as the sphere falls, it drifts across the pipe, a phenomenon previously observed in experiments but not fully explained. The physical mechanisms that cause the time-dependent drift are identified, and a simple model based on the normal stresses in the fluid is shown to predict the magnitude of the drift velocity.We also consider a second benchmark problem involving a constriction in an axisymmetric pipe. Numerical difficulties associated with ill-shaped elements near the concave boundary arise for higher Weissenberg numbers. The merits and drawbacks of the new numerical method, and its applicability to various flow geometries are discussed.     

Assessment of nonlinear strain measures for extensional and shearing flows of polymer melts

From stress-strain experiments in extensional and shearing flows, nonlinear strain measures and effective damping functions are derived for a polyisobutylene melt. The strain measures determined in planar extensional flow and in simple shear flow coincide. Experimental results are compared with predictions of two molecular theories, the Doi-Edwards model and the molecular stress function approach of Wagner and Schaeffer. Discrepancies between theories and experiment lead to a reconsideration of the classification of extensional flows. The symmetry of the flow field is identified and quantified as an important parameter influencing the strain measure, and a unifying strain measure for general extensional and shearing flows of polymer melts is presented.