Das unterschiedliche Schwellverhalten eines extrudierten Schmelzestranges bei Variation des Relaxationsspektrums

The die swell of viscoelastic fluids after leaving a die depends not only on the geometry, temperature effects and the mass flow rate, but also on the specific properties of the media. Among these specific properties is the relaxation spectrum which determines the relaxation of stresses in the fluid. It is this relaxation that produces a die swell. This paper investigates theoretically the manner in which modifications of the relaxation spectrum affect the viscosity curve and the die swell behaviour. In particular it is shown which modifications of the spectrum have a large influence on the die swell and which are relatively unimportant. It is also shown how the time intervals in which the main die swell occurs dependend on the shear rates in the die and which parts of the relaxation spectrum are responsible for the size of the die swell.     

用PSM模型模拟聚合物熔体的长口模挤出胀大

 以作者提出的模拟黏弹流体挤出胀大流动的方法为基础，计算了 IUPAC-LDPE熔体经过长圆形口模的挤出胀大. 计算结果同文献的报道 基本一致，表明作者提出的方法适于黏弹聚合物熔体的挤出胀大模拟.     

Numerical simulation of delayed die swell

In a recent paper, Joseph et al. showed that, for a number of viscoelastic fluids, one can observe the phenomenon of delayed die swell beyond a critical extrusion velocity, or beyond a critical value of the viscoelastic Mach number. Giesekus had also observed that delayed die swell is a critical phenomenon.In the present paper, we find a set of material and flow parameters under which it is possible to simulate delayed die swell. For the viscoelastic flow calculation, we use the finite element algorithm with sub-elements for the stresses and streamline upwinding in the discretized constitutive equations. For the free surface, we use an implicit technique which allows us to implement Newton's method for solving the non-linear system of equations. The fluid is Oldroyd-B which, in the present problem, is a singular perturbation of the Maxwell fluid. The results show very little sensitivity to the size of the retardation time. We also show delayed die swell for a Giesekus fluid.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Viscoelastic simulations of stick‐slip and die‐swell flows

Numerical solutions of viscoelastic flows are demonstrated for a time marching, semi‐implicit Taylor–Galerkin/pressure‐correction algorithm. Steady solutions are sought for free boundary problems involving combinations of die‐swell and stick‐slip conditions. Flows with and without drag flow are investigated comparatively, so that the influence of the additional component of the drag flow may be analysed effectively. The influence of die‐swell is considered that has application to various industrial processes, such as wire coating. Solutions for two‐dimensional axisymmetric flows with an Oldroyd‐B model are presented that compare favourably with the literature. The study advances our prior fixed domain formulation with this algorithm, into the realm of free‐surface viscoelastic flows. The work involves streamline‐upwind/Petrov–Galerkin weighting and velocity gradient recovery techniques that are applied upon the constitutive equation. Free surface solution reprojection and a new pressure‐drop/mass balance scheme are proposed. Copyright © 2001 John Wiley & Sons, Ltd.     

用PSM模型模拟聚合物熔体轴对称收缩流动

采用模拟黏弹流体挤出胀大的方法，计算了IUPAC-LDPE熔体经过4：1轴对称收缩流道的流动．计算的相对涡强度、入口校正和献中的结果基本一致，给出的流场也显示出计算结果是合理的．表明该方法能够适用于用积分型PSM模型表征的黏弹流体在收缩流道内的流动模拟．     

A finite element method for computing the flow of multi-mode viscoelastic fluids: comparison with experiments

The numerical computation of viscoelastic fluid flows with differential constitutive equations presents various difficulties. The first one lies in the numerical convergence of the complex numerical scheme solving the non-linear set of equations. Due to the hybrid type of these equations (elliptic and hyperbolic), geometrical singularities such as reentrant corner or die induce stress singularities and hence numerical problems. Another difficulty is the choice of an appropriate constitutive equation and the determination of rheological constants. In this paper, a quasi-Newton method is developed for a fluid obeying a multi-mode Phan-Thien and Tanner constitutive equation. A confined convergent geometry followed by the extrudate swell has been considered. Numerical results obtained for two-dimensional or axisymmetric flows are compared to experimental results (birefringence patterns or extrudate swell) for a linear low density polyethylene (LLDPE) and a low density polyethylene (LDPE).     

A mixed finite element scheme for viscoelastic flows with XPP model

A mixed finite element formulation for viscoelastic flows is derived in this paper, in which the FIC （finite incremental calculus） pressure stabilization process and the DEVSS （discrete elastic viscous stress splitting） method using the Crank-Nicolson-based split are introduced within a general framework of the iterative version of the fractional step algorithm. The SU （streamline-upwind） method is particularly chosen to tackle the convective terms in constitutive equations of viscoelastic flows. Thanks to the proposed scheme the finite elements with equal low-order interpolation approximations for stress-velocity-pressure variables can be successfully used even for viscoelastic flows with high Weissenberg numbers. The XPP （extended Pom-Pom） constitutive model for describing viscoelastic behaviors is particularly integrated into the proposed scheme. The numerical results for the 4：1 sudden contraction flow problem demonstrate prominent stability, accuracy and convergence rate of the proposed scheme in both pressure and stress distributions over the flow domain within a wide range of the Weissenberg number, particularly the capability in reproducing the results, which can be used to explain the ＂die swell＂ phenomenon observed in the polymer injection molding process.     

Polypropylene-polyethylene blends

Summary Die swell behaviour and morphology of melt blends of isotactic polypropylene (PP) and high density polyethylene for pure polymers and blends with 25, 50 and 75 weight % PP are described in the present study. A light interference contrast microscopy technique was used for the morphological characterization of melt blends and extrudate samples of the blends obtained with an Instron capillary rheometer. The results indicate that the domains from blends where the dispersed phase has higher viscosity than the continuous phase remain as continuous domains in the extrudate whereas domain destruction takes place when blends where the continuous phase has the higher viscosity are extruded.The die swell behaviour as well as the fiber forming properties of extrudates of melts having unstable domains extruded at high shear stresses resemble the behaviour of homopolymers, whereas samples with stable domains are significantly different, die swell increases with temperature at constant shear stress and stable fibers cannot be obtained after necking.With 10 figures and 1 table     

Finite element simulation of die swell for a Maxwell fluid

The finite element formulation recently developed [1] for viscoelastic fluids of singly memory type has been extended to handle problems involving free surfaces. Previously [1], flows without free surfaces were demonstrated by solving the die entry and reverse entry flows. The single integral Maxwell fluid is retained here as the constitutive model for this further test of the technique. New work in this paper includes an algorithm to track particles on the free surface and a new free surface updating scheme. For the present die swell problem, the method successfully converged up to Deborah number of about 1. As in the previous problems, convergence appears to be limited by numerically induced incompatibility of large strains. The resulting errors in the computed stress field have a more pronounced impact on free surface flows.     

Delayed die swell

The experiments reported here establish that there is a general critical condition associated with die swell which we call delayed die swell. This condition is defined by a critical speed which is the area-averaged velocity, the extrusion velocity, at the exit of the pipe when the swell is first delayed. The delayed swell ratio and delay distance first increase for larger, post-critical values of the extrusion velocity; then the increases are terminated either by instabilities or by smoothing. The maximum post-critical velocity at the pipe exit was always greater than the shear wave speed measured on the shear-wave-speed meter. The post critical area averaged velocity at the position of maximum swell before termination was always less than the shear wave speed. There were always points in the region of swelling where the ratio of the local velocity to the shear wave speed, the viscoelastic Mach number, was unity. The swelling of the jet is a nonlinear phenomenon which we suggest is finally terminated either by instability or when the variations of the velocity, vorticity and stress field are reduced to zero by the inward propagation of shear waves from the free surface of the jet. This propagation is generated by discontinuous “initial” data along χ in which the prescribed values of velocity at the boundary change from no-slip in the pipe to no-shear in the jet. The measurements raise the possibility that the delay may be associated with a change of type from supercritical to subcritical flow.     

On the behavior of viscoelastic free jets with elliptical cross section

In [1,2] we develop a comprehensive theory of one space dimensional closure models (closed systems of 1-D equations for the unknown, or modal, variables) for free viscoelastic jets. These closure models are derived via asymptotics from the full 3-D boundary value problem under the conditions of a Von Kármán-like flow geometry, a Maxwell-Jeffreys constitutive model, elliptical free surface cross section, and a slender jet scaling. The focus of the present paper is to determine the consequences and predictions of the lowest order system of equations in this asymptotic analysis. For the special cases of elliptical inviscid and Newtonian free jets, subject to the effects of surface tension and gravity, our model predicts oscillation of the major axis of the free surface elliptical cross section between perpendicular directions with distance down the jet, and draw-down of the cross section, in agreement with observed behavior. In the absence of surface tension the transformation from a cross section with major axis in one direction to a cross section with major axis in the perpendicular direction occurs only once, in agreement with the observation of Taylor [4]. In viscoelastic regimes, our model predicts swell of the elliptical extrudate and distortion of the elliptical extrudate cross section from the dimensions of the die aperture.     

The squeeze-film flow of a viscoelastic fluid

We report on the use of a numerical method to solve the (inertia-less) squeeze-film flow problem for a viscoelastic fluid. The method is based on a Boundary Element formulation and relies on a time-marching scheme. The viscoelastic fluid is modelled by a constitutive law that allows for a shear stress overshoot mechanism in a suddenly started shear flow. The results show convincingly that the load enhancement sometimes observed experimentally is due to stress overshoot. A simple explanation for the enhancement is suggested; the stress overshoot appears quickly and takes a long time to die away, so that steady-state viscous behaviour is not very relevant.     

An upper and lower bound for the die swell of the plane jet

Summary A lower and upper bound for the die swell of a plane jet of a viscoelastic liquid is obtained and applied to a Newtonian jet. Extension of the bounds for other fluids is suggested.

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Zusammenfassung Eine untere und obere Grenze für die Aufweitung eines ebenen viskoelastischen Flüssigkeitsstrahls werden angegeben und auf einen newtonschen Freistrahl angewandt. Die Anwendung der angegebenen Kriterien auf andere Flüssigkeiten wird angeregt.

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With 2 tables     

基于格子Boltzmann方法的挤出胀大的数值模拟

将单相格子Boltzmann方法(lattice Boltzmann method, LBM)引入到粘弹流体的瞬态挤出胀大的数值模拟中,建立了基于双分布函数的自由面粘弹性流动格子Boltzmann模型．分析得到的流道中流动速度分布和构型张量结果与理论解十分吻合．对粘弹流体瞬态挤出胀大过程进行了模拟,并分析了运动粘度比和剪切速率对挤出胀大率的影响,得到的胀大率结果与理论分析和其它模拟结果基本一致．表明给出的LBM可以捕捉挤出胀大的瞬态效应．     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.     

Quasistatic periodic contact problem for a viscoelastic layer,a cylinder,and a space with a cylindrical cavity

This paper considers the contact problem of interaction of a rigid die, a rigid band, and a rigid insert with a viscoelastic layer, a viscoelastic cylinder, and viscoelastic space with a cylindrical cavity, respectively. It is assumed that the die, band, and insert move at a constant velocity along the boundaries of the viscoelastic bodies. In the first stage, the displacement of the boundaries of the above-mentioned bodies is determined as a function of the applied normal loads ignoring friction in the contact area. In the second stage, integral equations are derived to determine contact pressure in the contact problems. In the third stage, approximate solutions of the integral equations are constructed using a modified Multhopp-Kalandia method.     

Modeling and simulation of three‐dimensional extrusion swelling of viscoelastic fluids with PTT,Giesekus and FENE‐P constitutive models

The investigation of the extrusion swelling mechanism of viscoelastic fluids has both scientific and industrial interest. However, it has been traditionally difficult to afford theoretical and experimental researches to this problem. The numerical methodology based on the penalty finite element method with a decoupled algorithm is presented in the study to simulate three‐dimensional extrusion swelling of viscoelastic fluids flowing through out of a circular die. The rheological responses of viscoelastic fluids are described by using three kinds of differential constitutive models including the Phan‐Thien Tanner model, the Giesekus model, and the finite extensible nonlinear elastic dumbbell with a Peterlin closure approximation model. A streamface‐streamline method is introduced to adjust the swelling free surface. The calculation stability is improved by using the discrete elastic‐viscous split stress algorithm with the inconsistent streamline‐upwind scheme. The essential flow characteristics of viscoelastic fluids are predicted by using the proposed numerical method, and the mechanism of swelling phenomenon is further discussed.Copyright © 2013 John Wiley & Sons, Ltd.     

Stress analysis for linear viscoelastic materials

Summary The relations between stress and strain prescribing linear viscoelastic behavior are discussed from the standpoint of application to problems of stress analysis. This consideration involves some important differences from the assessment of linear viscoelastic laws for the representation of material properties only.The use of the differential operator relation between stress and strain is usually most convenient, although the integral operators comprising the creep function or relaxation function can also sometimes be conveniently utilized. Examples of these are given, including the problem of indentation, in which the region of contact of a rigid spherical indentor under constant load can be expressed directly in terms of the creep function. When differential operator relations, which correspond to the usually considered viscoelastic models, are used, it is shown that a consistent application of the theory of delta functions and the associated symbolic differentiation permits initial values to be determined for the evaluation of the resulting differential equations. No mathematical difficulties arise if the loading functions are not smooth, as might be anticipated by the high order time derivatives which appear in the viscoelastic relations. An application of this theory is given which falls outside the scope of theLaplace transform type of analysis. The latter is often used to deal with this kind of problem, but has a restricted field of application.

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Zusammenfassung Die vorliegende Arbeit behandelt die Beziehungen zwischen Spannung und Dehnung in linear viskoelastischen Körpern vom Standpunkt der Spannungsberechnung. Diese Problemstellung geht wesentlich über die Erwägungen, die beim Studium der materiellen Eigenschaften auftreten, hinaus.Die Benützung der Differentialoperatorbeziehung zwischen Spannung und Dehnung ist meist günstiger, obgleich die Integraloperatoren, die aus Kriechfunktionen oder Relaxationsfunktionen bestehen, manchmal auch zweckdienlich benützt werden können. Als Beispiel der Anwendung dieser Integraloperatoren wird das Stempelproblem gelöst, wobei die Ausdehnung des Kontaktbereichs des festen, kugelförmigen Stempels unter konstanter Last direkt durch die Kriechfunktion ausgedrückt wird.Bei der Benützung von Differentialoperator-beziehungen (die den bekannten viskoelastischen Modellen entsprechen) wird gezeigt, daß eine Anwendung der Theorie der Deltafunktionen die Bestimmung der Anfangsbedingungen für die Differentialgleichungen zuläßt. Auch wenn die Belastungsfunktionen nur stückweise glatt sind, treten keine Schwierigkeiten auf, trotz der höheren Ableitungen, die in den viskoelastischen Stoffbeziehungen auftreten. Ein zweites Beispiel zeigt eine Anwendung der Deltafunktionen auf ein Problem, das der Behandlung mit Hilfe derLaplace-Transformation nicht zugänglich ist.

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The results presented in this paper were obtained in the course of research sponsored by the Office of Ordnance Research, U. S. Army under Contract No. DA-19-020-ORD-4550.     

Extrudate swell through an orifice die

The extrudate swell of a viscoelastic fluid through an orifice die is investigated by using a mixed finite element and a streamline integration method (FESIM), using a version of the K-BKZ model. The free surface calculation is based on a local mass conservation scheme and an approximate numerical treatment for the contact point movement of the free surface. The numerical results show a vortex growth and an increasing swelling ratio with the Weissenberg number. Convergence with mesh refinement is demonstrated, even at a high Weissenberg number of O(587), where the swelling ratio reaches a value of about 360%. In addition, it is found that the effective flow channel at the entrance region next to the orifice die is reduced due to the enhanced vortex growth, which may be a source of flow instability.