Viscoelastic analysis of complex polymer melt ﬂows using ...peters/Papers/2002/VerbeetenEA_ComplFlowXPP_2002.pdf · Viscoelastic analysis of complex polymer melt ﬂows using

J. Non-Newtonian Fluid Mech. 108 (2002) 301–326

Viscoelastic analysis of complex polymer melt flows usingthe eXtended Pom–Pom model

Wilco M.H. Verbeeten, Gerrit W.M. Peters∗, Frank P.T. BaaijensMaterials Technology, Faculty of Mechanical Engineering, Eindhoven University of Technology,

P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract

The ability of the multi-mode eXtended Pom–Pom model to predict inhomogeneous flows of a polyethylene meltis investigated. Two benchmark problems are examined: the confined flow around a cylinder, and the flow througha cross-slot device. Numerical results for the eXtended Pom–Pom model are compared to experimental data, andpredictions of the Giesekus and exponential Phan-Thien Tanner (PTT-a) models. Characteristic features observedexperimentally in the benchmark flows are described well by all three models. The eXtended Pom–Pom modelperforms most satisfactorily, both with respect to the rheological data and the inhomogeneous flow data.© 2002 Elsevier Science B.V. All rights reserved.

Keywords:Constitutive models; Differential models; eXtended Pom–Pom model; PTT-a model; Giesekus model; Polymermelt; Flow around a cylinder; Cross-slot flow; Experimental results; Quantitative comparison

1. Introduction

A quantitative description of the rheological behaviour of polymer melts is crucial in understandingthe relation between processing and product properties. As an intermediate step between the well-definedrheometrical flows and complicated industrial processing flows, simplified, experimentally accessible,inhomogeneous flows that exhibit a combination of transient shear and elongational deformation areinvestigated. The viscoelastic analysis of these flows allows the assessment of constitutive models andnumerical predictions for prototype industrial flows.

One of the main problems in constitutive modeling is to obtain a correct description of the transientnonlinear behaviour in elongational and shear flows simultaneously. Well-known and widely used models,such as the multi-mode exponential Phan-Thien Tanner (PTT-a) and the Giesekus model yield unsatisfac-tory results in the prediction of inhomogeneous flows[2]. Recently, a new class of constitutive equationshas been introduced based on the tube model of Doi and Edwards[9], which is able to simultaneously

∗ Corresponding author. Fax:+31-40-244-7355.E-mail address:[email protected] (G.W.M. Peters).URL: http://www.mate.tue.nl/

0377-0257/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S0377-0257(02)00136-2

302 W.M.H. Verbeeten et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 301–326

describe the transient shear and elongational behaviour of polyethylene melts. Some of them start offin an integral version, e.g. the Pom–Pom model[22,14,6], but an approximation of the differential typeis also given for computational convenience. Others are only available as an integral model, e.g. theMolecular Stress Function (MSF) model[29], and no differential approximation exists. Although thenewly developed deformation fields method by Peters et al.[24] is an attractive way to calculate modelsof the integral type, numerical methods using differential constitutive equations are still more efficient.Verbeeten et al.[33] introduced the eXtended Pom–Pom model, a modification of the original differentialPom–Pom model of McLeish and Larson[22], which is able to describe the behaviour of polyethylenemelts over a wide range of rheometric data, including reversed flow. This last rheological flow is stillrather troublesome for most integral models. In this work, we will restrict ourselves to the investigationof differential constitutive equations.

Bishko et al.[5] already showed results for the original differential Pom–Pom model in a complex4:1 contraction flow. Although only a single-mode version of the Pom–Pom model was used, goodqualitative predictions were obtained for highly branched polymers. Their computational results arepromising and predicted all the specific features observed in experiments for LDPE melts. However,a full comparison with experimental complex flow data was not possible, since a multi-mode versionis necessary for an accurate description of the rheology of polymer melts. Furthermore, due to the ex-cessive shear-thinning behaviour of the original differential Pom–Pom model in fast shear flows, theshear stress versus shear rate curve shows a maximum. This induces a non physical constitutive in-stability. A multi-mode version will avoid this problem on a sufficiently wide range of shear rates,because modes with higher relaxation times will dominate after the shear-thinning behaviour sets infor the faster modes, resulting in a less shear-thinning behaviour. Moreover, as the shear-thinning be-haviour of the eXtended Pom–Pom model is less extreme, this particular model effectively avoids thisinstability.

Like in Bishko et al.[5], most studies of polymer melts are conducted on the benchmark contraction orcontraction/expansion problem, e.g. Xue et al.[35], Wapperom and Keunings[34], and Alves et al.[1]for purely numerical research, Martyn et al.[18–20] for a mostly experimental investigation in a threedimensional geometry, and Béraudo et al.[4] for a combined numerical/experimental study. Unfortunately,material elements over the centerline of contraction problems are extended only moderately, because theelongational rate is witnessed during a too short time interval. Much less research has been performed onthe flow around a confined cylinder, e.g. Peters et al.[25] and Renardy[28] for numerical studies, andSchoonen[30], Baaijens et al.[2], Hartt and Baird[13] and Baaijens[3] for a combined experimentaland numerical investigation. A viscoelastic analysis of a polymer melt in a cross-slot device has onlybeen published once to our knowledge[27]. These latter two geometries impose large strain rates on thematerial over an extended period of time inducing finite strains and are therefore discriminating towardsconstitutive models with different steady elongational properties.

The objective of this study is to assess the performance of the multi-mode eXtended Pom–Pom model[33] for two benchmark flows, i.e. the confined flow around a cylinder and the cross-slot flow, andcompare it with experimental results taken from Schoonen[30]. Both geometries were designed to havenearly two-dimensional flow kinematics and birefringence, i.e. a depth-to-height aspect ratio of at leasteight in the whole flow domain, such that comparison with two-dimensional calculations is allowed[30,31]. Furthermore, the results will be compared against the performance of the exponential PTT andthe Giesekus model. The differences and similarities between the two prototype industrial flows will beexamined with respect to strain rates, stresses and their history in a Lagrangian sense. We will discuss

W.M.H. Verbeeten et al. / J. Non-Newtonian Fluid Mech. 108 (2002) 301–326 303

the advantages and drawbacks of these flows when used for testing constitutive models by comparingexperimental and numerical results.

For the numerical part of this study, the Discrete Elastic Viscous Stress Splitting technique in combina-tion with the Discontinuous Galerkin method (DEVSS/DG) is used. InSection 2, the problem definitionand the constitutive equations are outlined. The computational method is briefly described inSection 3and can be found in more detail in Bogaerds et al.[7]. The material characterization and the performanceof the three constitutive models in simple rheometric flows is given inSection 4. Results for the twoprototype industrial flow geometries are presented inSection 5, followed by a discussion and conclusionson the different flows. It is shown that all three models are able to qualitatively, and to a large extent alsoquantitatively, predict the velocities and stresses in the different complex flow geometries. In general, theeXtended Pom–Pom model shows the best quantitative comparison.

2. Problem definition

Isothermal and incompressible fluid flows, neglecting inertia, are described by the equations for con-servation of momentum(1) and mass(2):

∇ · σ = 0, (1)

∇ · u = 0, (2)

where∇ is the gradient operator,σ denotes the Cauchy stress tensor andu the velocity field. The Cauchystress tensorσ is defined for polymer melts by:

σ = −pI + 2ηsD +M∑i=1

τ i . (3)

Here,p is the pressure,I the unit tensor,ηs denotes the viscosity of the purely viscous (or solvent) contri-bution,D = (1/2)(L + LT) the rate of deformation tensor, in whichL = ( ∇u)T is the velocity gradienttensor and(·)T denotes the transpose of a tensor. The visco-elastic contribution of theith relaxation modeis denoted byτ i andM is the total number of different modes. The multi-mode approach is necessary formost polymeric fluids to give a realistic description of stresses over a broad range of deformation rates.The visco-elastic contributionτ i still has to be defined by a constitutive model.

2.1. Constitutive models

Within the scope of this work, a sufficiently general way to describe the constitutive behaviour of asingle-mode is obtained by using a differential equation based on the generalized Maxwell-type equation:

∇τ i + f GS(τ i ,D)+ λ(τ i)

−1 · τ i = 2GiD, (4)

withGi the plateau modulus of theith mode obtained from the linear relaxation spectrum determined by

dynamic measurements. The upper convected time derivative of the stress∇τ i is defined as:

∇τ i = τ i − L · τ i − τ i · LT = ∂τ i

∂t+ u · ∇τ i − L · τ i − τ i · LT, (5)


wheret denotes time. Both tensorial functionsf GS(τ i ,D) andλ(τ i) depend upon the chosen constitutiveequation. Notice, that by choice off GS(τ i ,D) = 0 andλ(τ i) = λi , with λi the ith linear relaxationtime, the Upper Convected Maxwell (UCM) model is retrieved.

For the Giesekus model we have:

f GS(τ i ,D) = 0, λ(τ i)−1 = 1

λi

[αi

Giτ i + I

], (6)

with αi a material parameter to be fitted on the rheological data. For the exponential PTT-a model it holdsthat:

f GS(τ i ,D) = ξi(D · τ i + τ i · D), λ(τ i)−1 = 1

λi[e(εi/Gi)Tr(τ i )I ], (7)

with ξi the slip parameter from the Gordon-Schowalter derivative andεi a material parameter, both to befitted on the rheological data.

A new class of constitutive relations based on the tube model and a simplified topology of branchedmolecules was recently proposed by McLeish and Larson. The basic idea of the model is that the rheo-logical properties of entangled polymer melts mainly depend on the topological structure of the polymermolecules. The simplified topology consists of a backbone with a number of dangling arms at bothends and is called a Pom–Pom molecule. The backbone is confined by a tube formed by other back-bones. A key feature is the separation of relaxation times for the stretch and orientation. For moredetails we refer to[22,33]. Verbeeten et al.[33] incorporated local branch-point displacement as intro-duced by Blackwell et al.[6] and modified the orientation equation of the original differential versionto the eXtended Pom–Pom model to overcome three problems: solutions in steady state elongationshow discontinuities; the equation for orientation is unbounded for high strain rates; the model doesnot have a second normal stress difference in shear. A branched molecule can be represented by sev-eral equivalent Pom–Pom modes. The extra stress equation for the eXtended Pom–Pom (XPP) model isdefined as:

τ i = σ −GiI = 3GiΛ2i Si −GiI , (8)

with Λi the backbone tube stretch, defined as the length of the backbone tube divided by the length atequilibrium, andSi denoting the orientation tensor. The evolution equations for the orientation, based onthe Giesekus model as given inEq. (6), and backbone tube stretch are given by:

∇Si + 2[D : Si ]Si + 1

λb,iΛ2i

[3αiΛ

4i Si · Si + (1−αi − 3αiΛ

4iTr(Si · Si))Si − (1−αi)

3I

]=0, (9)

Λi = Λi [D : Si ] − evi(Λi−1)

λs,i(Λi − 1), vi = 2

qiλb,i = λi. (10)

Here,λb,i is the relaxation time of the backbone tube orientation equal to the linear relaxation timeλi , λs,i

is the relaxation time for the stretch,vi a parameter denoting the influence of the surrounding polymerchains on the backbone tube stretch, andqi is the amount of arms at the end of a backbone.

The XPP model can also be written in a fully equivalent single-equation fashion[33], which has thesame structure as the Giesekus and PTT-a models. This facilitates implementation in existing software


packages. In that case the functions fromEq. (4)are defined as:

f GS(τ i ,D) = 0, λ(τ i)−1 = 1

λb,i

[αi

Giτ i + F(τ i)I +Gi(F (τ i)− 1)τ−1

i

], (11)

with

F(τ i) = 2rievi(Λi−1)

(1 − 1

Λi

)+ 1

Λ2i

[1 − αiTr(τ i · τ i)

3G2i

](12)

and

Λi =√

1 + Tr(τ i)

3Gi, ri = λb,i

λs,i, vi = 2

qiλb,i = λi. (13)

Notice, that for this definition of the backbone stretch, we may run into numerical problems if1 + (Tr(τ i)/3Gi) < 0. Physically, Tr(τ i) can not become smaller than 0. However, numerically wehave encountered these unrealistic values at the front and back stagnation points in the flow around acylinder at sufficiently high Weissenberg numbers. Unphysical negative backbone stretch values in thedouble-equation XPP model were also encountered at the exact same locations for higher Weissenbergnumbers than presented in this work. This is a numerical artifact, similar to the negative values for the traceof the extra stress, Tr(τ i), encountered in the Giesekus and PTT-a models for the Weissenberg numbershown in this paper, independent of the meshes used. In this respect, the cross-slot flow is a smooth flowsince no a-physical stretches are predicted, either for the single-equation and the double-equation XPPmodel. We would like to remark, that these unrealistic negative backbone stretch values have not beenencountered while computing homogeneous rheometrical flows.

A disadvantage for a two-dimensional numerical implementation, concerning both versions, is thenon-zero third stress componentτzz, contrary to the Giesekus and PTT-a models. Compared to thesingle-equation form, the double-equation formulation has an extra equation for the stretch, resultingin a larger system matrix in FE codes. However, the equation for the third orientation componentSzz canbe eliminated, since it is known that Tr(Si) = 1. As we eliminate the extra stresses, or, equivalently, theorientation tensor and the stretch, at the element level (seeSection 3), the computational cost of thesetwo extra equations is relatively low. Therefore, elimination ofSzz is not performed.

From the computational point of view the double-equation XPP model is preferred because computa-tions do not fail if during the iterative process negative values of the backbone stretch are computed. Thesingle-equation XPP model causes the computations to stop if(Tr(τ i)/3Gi) < −1, which is a-physicalsince Tr(τ i) should be positive, but may occur during the iterative process.

3. Computational method

The DEVSS/DG method, Baaijens et al.[2], which is a combination of the Discrete Elastic ViscousStress Splitting (DEVSS) technique of Guénette and Fortin[10] and the Discontinuous Galerkin (DG)method, developed by Lesaint and Raviart[16], is used. Application to the double-equation XPP model isdiscussed in this section. The weak formulation and solution strategy for the Giesekus and PTT-a modelsis similar and can be found in more detail in Bogaerds et al.[7] and Baaijens et al.[2].


3.1. DEVSS/DG method

Consider the problem described by theEqs. (1), (2), (9) and(10). Following common FE techniques,these equations are converted into a mixed weak formulation:

Problem DEVSS/DG. Find (u, p, D,Si , Λi) for any timet such that for all admissible test functions(v, q,E, si , li),(

Dv,2ηsD + 2η(D − D)+M∑i=1

Gi(3Λ2i Si − I )

)− (∇ · v, p) = 0, (14)

(q,∇ · u) = 0, (15)

(E, D − D) = 0, (16)

(si ,

∇Si + 2[D : Si ]Si + 1

λb,iΛ2i

[3αiΛ

4i Si · Si + (1 − αi − 3αiΛ

4iTr(Si · Si))Si − (1 − αi)

3I

])

−K∑e=1

∫Γ ein

si : u · n(Si − Sexti )dΓ = 0 ∀i ∈ 1,2, . . . ,M, (17)

(li , Λi −Λi [D : Si ] + evi(Λi−1)

λs,i(Λi − 1)

)

−K∑e=1

∫Γ ein

li u · n(Λi −Λexti )dΓ = 0 ∀i ∈ 1,2, . . . ,M. (18)

Here,(·, ·) denotes theL2-inner product on the domainΩ, Dv = (1/2)( ∇v + ( ∇v)T), η an auxiliaryviscosity,Γ ein is the inflow boundary of elementΩe, n the unit vector pointing outward normal on theboundary of the element (Ωe) andSext

i andΛexti denote the orientation tensor and backbone stretch of the

neighboring, upstream element.The stabilization term 2η(D − D) has been added to the momentumEq. (14), where the discrete

approximationD is obtained from anL2 projection of the rate of deformation tensor (Eq. (16)). Theauxiliary viscosityη = ∑M

i=1Giλi is found to give satisfactory results.Time discretisation of the constitutive equation is obtained using an semi-implicit Euler scheme. In

this scheme most of the variables are updated implicitly with the exception of the external componentsSexti andΛext

i which are taken explicitly (i.e.Sexti = Sext

i (tn) andΛexti = Λext

i (tn)). Hence, the terms∫si : u · n(−Sext

i )dΓ and∫li u · n(−Λext

i )dΓ have no contributions to the Jacobian which allows forelimination of the orientation tensor and backbone stretch at element level. Furthermore, the equation forthe discrete rate of deformationD is decoupled from the ’Stokes’ problem and thus updated explicitly.

3.2. Solution strategy

In order to obtain an approximation ofProblem DEVSS/DG, a 2D domain is divided into quadrilateralelements. A bi-quadratic interpolation for the velocityu, bi-linear for the pressurep and discrete rate


of deformationD, and a discontinuous bi-linear interpolations for the orientation tensorSi and stretchΛi are known to give stable results (see[7] and[2]). Integration ofEqs. (14)–(18)over an element isperformed using a 3× 3-Gauss quadrature rule common in FE analysis.

To obtain the solution of the nonlinear equations, a one step Newton-Raphson iteration process iscarried out. Consider the iterative change of the nodal degrees of freedom(δu, δp, δD, δS, δΛ) as variablesof the algebraic set of linearized equations. This linearized set is given by:

Quu Qup QuD QuS QuΛ

Qpu 0 0 0 0

QDu 0 QDD 0 0

QSu 0 0 QSS QSΛ

QΛu 0 0 QΛS QΛΛ

δu

δp

δD

δS

δΛ

= −

fu

fp

fD

fS

fΛ

, (19)

with fα(α = u, p, D,Si , Λi) correspond to the residuals of equationsEqs. (14)–(18), whileQαβ followfrom linearisation of these equations. Due to the fact thatSext

i andΛexti have been taken explicitly in

Eqs. (17) and (18), the matricesQSS,QSΛ,QΛS , andQΛΛ form a block diagonal structure which allows

for calculation of

(QSS QSΛ

QΛS QΛΛ

)−1

on element level. This matrix is recomputed at each Newton iteration.

Consequently, this enables the reduction of the global DOF’s by static condensation of the orientationtensor and backbone stretch block.

Application of a iterative solvers, such as Bi-CGSTAB, to the resulting set of equations proved unsuc-cessful. Therefore, a decoupling procedure is adopted. First, the ‘Stokes’ problem (u, p) is solved usingD of the previous solution, after which the updated solution for velocities and pressure is used to find anew approximation forD. The following two separate problems now emerge:

Problem DEVSS/DGa. Given (u, p, D,Si , Λi) at t = tn, find a solution att = tn+1 of the algebraic set:Quu − (QuSQuΛ)

(QSS QSΛ

QΛS QΛΛ

)−1 (QSu

QΛu

)Qup

Qpu 0

(δu

δp

)

= −

fu − (QuSQuΛ)

(QSS QSΛ

QΛS QΛΛ

)−1(fS

fΛ

)

fp

, (20)

and

Problem DEVSS/DGb. GivenD at t = tn andu at t = tn+1, find δD from:

(QDD)(δD) = −(fD +QDuδu). (21)

Notice that the right hand side is now taken with respect to the new velocity approximation. The no-dal increments of the orientation tensor and backbone stretch are retrieved element by element


following:(δS

δΛ

)= −

(QSS QSΛ

QΛS QΛΛ

)−1(fS

fΛ

), (22)

with fS andfΛ also taken with respect to the new velocity approximation (fS(Sni , un+1

), fΛ(Λni , un+1)).

To solve the non-symmetrical system ofProblem DEVSS/DGa an iterative solver is used based onthe Bi-CGSTAB method of Van der Vorst[32]. The symmetrical set of algebraic equations ofProblemDEVSS/DGb is solved using a Conjugate-Gradient solver. Incomplete LU preconditioning is applied toboth solvers. In order to enhance the computational efficiency of the Bi-CGSTAB solver, the velocityvariables of the center node are eliminated at element level by static condensation. Besides a reductionof the number of degrees of freedom, the key result of this is that the zero lower block diagonal matrixrelated to the pressure variables becomes non-zero, substantially improving the rate of convergence ofthe iterative solver.

Finally, to solve the above sets of algebraic equations, both essential and natural boundary conditionsmust be imposed on the boundaries of the flow channels. At the entrance and the exit of the flow channelsthe fully developed velocity profiles are prescribed. At the entrance, the values of the orientation tensorand backbone stretch of a few elements downstream the inflow channel are prescribed along the inflowboundary, imposing a periodic boundary condition for these stress variables. In this way, the orientationtensor and backbone stretch at the inflow boundary develop similarly as the stress variables in the rest ofthe flow domain.

4. Material characterization

The polymer melt that is investigated in this work is a commercial grade low density polyethylene(DSM, Stamylan LD 2008 XC43), further referred to as LDPE. This long-chain branched material hasbeen extensively characterized by Schoonen[30] and was also used in Peters et al.[27].

In the linear viscoelastic regime, all three constitutive models investigated in this work reduce to thelinear Maxwell model. In all cases, four modes have used to describe the linear viscoelastic data, obtainedfrom dynamic measurements at a temperature ofT = 170C. The non-linearity parametersq and theratioλb/λs for the XPP model andα for the Giesekus model are determined using the transient uniaxialelongational data only. For the PTT-a modelε andξ are identified by using transient uniaxial elongationaland shear data. Since second planar elongational or second normal stress difference data is not available forthis material, the anisotropy parameterα in the XPP model is chosen as 0.3/q. Anisotropy is decreasingfrom the free ends inwards, and by choosingα as an inverse function of the number of armsq, this isindeed accomplished (see[33]). All parameters are given inTable 1.

Fig. 1shows the rheological behaviour of the three constitutive models and measured data in uniaxialand simple shear behaviour. The ‘steady state’ data points in the uniaxial elongational viscosity plot arethe end points of the transient curves. Since the samples break at the end of the transient measurements,it is hard to say if steady state has truly been reached. However, sometimes it is claimed that steady statecoincides with break-up of the samples[21].

Within the experimental range, all three models give a reasonable to good prediction of major portions ofthe transient uniaxial elongational viscosity curves. Significant differences are observed in the steady-state


Fig. 1. Transient (a) and quasi-steady state (b) uniaxial viscosityηu, transient (c) and steady state (d) shear viscosityηs, andtransient (e) and steady state (f) first normal stress coefficientΨ1 at T = 170C of the XPP, Giesekus and PTT-a models forDSM Stamylan LD2008 XC43 LDPE melt.


Table 1Linear and non-linear parameters for fitting of the DSM Stamylan LD 2008 XC43 LDPE melt

i Maxwell parameters XPP Giesekus Exponential PTT

Gi (Pa) λi (s) qi λb,i/λs,i αi αi εi ξi

1 7.2006× 104 3.8946× 10−3 1 7.0 0.30 0.30 0.30 0.082 1.5770× 104 5.1390× 10−2 1 5.0 0.30 0.30 0.20 0.083 3.3340× 103 5.0349× 10−1 2 3.0 0.15 0.25 0.02 0.084 3.0080× 102 4.5911× 100 10 1.1 0.03 0.04 0.02 0.08

Tr = 170C; vi = 2/qi ; activation energy:E0 = 48.2 kJ/mol; WLF-shift parameters:C1 = 14.3 K, C2 = 480.8 K.

curves. The XPP model captures the available data points best, while the PTT-a model demonstrates anon-smooth behaviour, reflecting the individual relaxation times. This could be improved by choosing alarger number of relaxation times. Based on measurements of other low density polyethylene melts (see[23,11]), the uniaxial elongational viscosity at high strain rates is expected to be elongational thinning.Therefore, the plateau value predicted by the Giesekus model is not realistic.

For simple shear, all models show a rather good agreement with the measured data (Fig. 1c and d).The Giesekus model somewhat over predicts the measurements. At shear rates exceeding 102 (s−1), thecurves for all models drop rapidly (seeFig. 1d), corresponding to the longest relaxation time. Sincethese high shear rates do not occur in the experiments, this does not cause difficulties in the numericalsimulation of these flows. Although difficult to observe inFig. 1c, the PTT-a model predicts oscillationsin transient start-up of simple shear. These oscillations occur because the non-linearity parameterξ hadto be chosen larger thanε to obtain a good fit of the shear data. These oscillations are a drawback ofthe PTT-a model.Appendix A elaborates more on this issue for a different low density polyethylenemelt.

All three models slightly over predict the first normal stress coefficient. In the start-up region of thefirst normal stress coefficient curve, the different relaxation times of the XPP model can be detected (seeFig. 1e). This could be avoided by choosing more Maxwell modes. As mentioned before, this would alsoresult in a smoother curve for the steady state uniaxial viscosity predicted by the PTT-a model. However,as computational time increases significantly with every mode added, four Maxwell modes is consideredas a satisfactory optimum between a good description of the rheological behaviour and computationalcost.

5. Complex flows

A comparison is made between numerical and experimental results for two complex flows: flow arounda cylinder between two parallel plates and flow through a cross-slot device. These benchmark flow geome-tries are known to have regions with pure simple shear, pure planar elongation and combinations thereof.Velocities have been measured using particle tracking velocimetry, while flow induced birefringence incombination with the stress-optical rule is used for comparison with numerical stress computations. Foreach geometry, we will present results for one flow rate only. These flow rates differ for the cylinderand cross-slot flow geometry, and have been chosen such that the the maximum planar elongational rateachieved in both geometries is in the same order of magnitude, about 2 s−1 for the cylinder problem and


about 1.5 s−1 for the cross-slot geometry. For more details on the experimental aspects and extendedexperimental results, we refer to Schoonen[30].

Both geometries are designed to have near two-dimensionality, i.e. a depth-to-height aspect ratio ofat least 8:1 throughout the whole flow domain. Schoonen[30] demonstrated that with an aspect ratio of8:1 the influence of the confining front and back walls is about 6% maximum on the isochromatic stresspatterns within the flow rate range used in this investigation. This was confirmed in numerical studiesby Schoonen et al.[31] and Bogaerds et al.[7]. This is a sufficiently small deviation to be nominallytwo-dimensional and compare experimental results to two-dimensional calculations. The general rule fordeviations due to three-dimensional effects is that the integrated effect of experimental stresses (whichrepresents the birefringence patterns) is lower near walls and higher over centerlines in comparison totwo-dimensional calculations.

To characterize the strength of the different flows, the Weissenberg number is defined as:

Wi = λu2D

h. (23)

Here,λdenotes the viscosity averaged relaxation time for the material(λ=

(∑Mi=1 λ

2i Gi

)/(∑M

i=1 λiGi

)),

u2D is the two-dimensional mean velocity andh a characteristic length of the flow geometry.To link the calculated stresses to the measured isochromatic fringe patterns, the semi-empirical stress-

optical rule is used. It states that the deviatoric part of the refractive index tensor is proportional to thedeviatoric part of the stress tensor:

nd = Cσ d, (24)

in whichC is the stress-optical coefficient, identified to equal (see[27]):

C = 1.47× 10−9 Pa−1. (25)

The stress-optical coefficient is slightly dependent on temperature[17].For two-dimensional flows, the stress-optical rule can be simplified to:

τFRG =√

4τ 2xy +N2

1 = kλ

dC, (26)

with τFRG defined as the isochromatic fringe stress,τxy the shear stress in thexy-plane,N1 = τxx− τyy thefirst normal stress difference,λ the wave length of the light used in the measurements,d the light pathlength in the birefringent medium, which in these cases are the depths of the flow cells, andk the fringeorder of the observed dark fringe bands where extinction of the light occurs.

5.1. Flow around a cylinder

A schematic representation of the planar cylinder flow geometry is shown inFig. 2(a). Experimentswere performed at a temperature of 170C, resulting in a viscosity averaged relaxation time for thematerial of λ = 1.7415 s. The radius of the cylinder isR = 1.1875 mm, the channel height equalsH = 4.95 mm, and the depth of the flow celld = 40 mm. The depth-to-height ratio results in 8.08,creating a nominally two-dimensional flow. The height-to-diameter ratio equals 2.08. The characteristic


Fig. 2. Schematic (a) and detail of FE meshes (b, c, d) of the planar flow around a cylinder. The origin of the coordinatesystem is at the center of the cylinder.T = 170C, λ = 1.7415 s,λmax = 4.5910 s,u2D = 1.975 mm/s,R = 1.1875 mm,H = 4.95 mm≈ 4R, d = 40 mm.

mean two-dimensional velocity at the inflow isu = 1.975 mm/s. As a characteristic length, the radius ofthe cylinderR is chosen, giving the flow averaged Weissenberg number:

Wi = 2.9. (27)

Notice, that the averaged Weissenberg number of the longest relaxation time used in the calculations isWi = 7.6, and may even be much higher locally.

Only half of the geometry needs to be analyzed due to symmetry. The center of the cylinder is placedatx/R = 0,y/R = 0, while the inlet is atx/R = −8 and the oulet atx/R = 15. The periodic boundaryfor the inlet stresses is situated atx/R = −5. At the entrance and exit, a fully developed velocity profileis prescribed. No slip boundry conditions are imposed on the top confirning wall the cylinder, and thevelocity iny-direction is suppressed for the symmetry liney/R = 0. The time step equals2t = 0.001 sfor all three constitutive models.

Mesh refinement has been investigated for the XPP model using three different meshes, as depicted inFig. 3. The mesh characteristics are given inTable 2. Due to the thin stress layer along the cylinder wall,

Fig. 3. Calculated isochromatic fringe stresses in the planar flow around a cylinder along the cylinder wall for the meshes C1,C2 and C3 using the XPP model at Wi= 2.9. T = 170C.


Table 2Mesh characteristics of the flow around a cylinder

Mesh C1 Mesh C2 Mesh C3

#Elements 920 1830 2,600#Nodes 3,847 7,547 10,661#DOF(u, p) 6,858 13,378 18,853#DOF(D) 3,012 5,832 8,193#DOF(S,3) = 4 × #elements× 20 73,600 146,400 208,000Smallest element size at front/back stagnation point (×10−4R2) 9.0 4.0 1.1Drag force on cylinder 293,547.5 292,694.8 292,693.8

this is a numerically difficult region. Therefore, to be confident that mesh convergence has been achieved,the isochromatic fringe stresses along the cylinder wall for the three meshes are plotted inFig. 3. Takingthe finest mesh C3 as a reference, the largest relative difference of the fringe stress for mesh C1 is 3.31%,while for mesh C2 this equals 0.38%. Furthermore, the drag force on the cylinder is calculated given inTable 2. The relative difference between the drag forces is 0.29% in case of mesh C1 and 0.00034% formesh C2. We feel that results for mesh C2 are sufficiently converged with respect to mesh refinement.For convenience, mesh C2 is used for further investigations.

With the current parameter setting, the PTT-a model encountered convergence problems and steady-statecould not be reached. Computations for this model diverged aftert = 9.1 s, which is less than twice themaximum relaxation time. Calculations using finer meshes did extend the calculations to larger times,but could not avoid divergence. By using a less strain hardening parameter set convergence problemswere not encountered within the range of Weissenberg numbers investigated. We therefore suspect thatthe high strain hardening behaviour of the material and the associated parameter set, is one of the keyreasons for the convergence problems.

Fig. 4shows the normalized velocity profiles at five different cross-sections (x/R = −4, −2.5,−1.5,1.5,2.5) and the velocities over the centerline (y/R = 0). The measured data points show a slightasymmetry at cross-sections closest to the cylinder, indicating a minor misplacement of the cylinder. Ingeneral, the predicted velocities are in good agreement with the measured data for the XPP and Giesekusmodels. Over the centerline, the models underpredict the measured velocity profile downstream of thecylinder, although the centerline exit velocity is predicted correctly.

The measured and calculated isochromatic stress patterns for the XPP and Giesekus models are shownin Fig. 5. Both models predict very similar stress patterns, and especially for the XPP model all detailsare predicted correctly in this qualitative picture. The Giesekus model predicts one fringe too many inthe region along the cylinder and in the downstream section due to the overprediction of the first normalstress coefficient as shown inFig. 1f.

Fig. 6 shows a more quantitative comparison of the stresses over the centerliney/R = 0 and alongthe linex/R = 0. Fig. 5aindicates that both models accurately predict the fringes over the upstreamcenterline. Along the downstream centerline, the models overpredict the experimental data.

The curved wall of the cylinder induces a stress boundary layer that is difficult to resolve numer-ically. It is also difficult to access experimentally. By using a laser in the optical set-up, fringe tran-sitions are observable within 0.02 mm of the cylinder, which was not possible using a conventionalmercury lamp as a light source. Nevertheless, it is quite difficult to count the fringes very close to the


Fig. 4. Calculated and measured velocity profiles in the planar flow around a cylinder over five cross-sections (a)(x/R = −4,−2.5,−1.5,1.5,2.5), and over the centerline (b) (y/R = 0) at Wi = 2.9. T = 170C.

Fig. 5. Calculated and measured isochromatic fringe patterns of the planar flow around a cylinder for the XPP model (a) and theGiesekus model (b) at Wi= 2.9. T = 170C.


Fig. 6. Calculated and measured stresses in the planar flow around a cylinder over the centerline (y/R = 0) (a) and over thecross-section (x/R = 0) (b) for the XPP and Giesekus molels at Wi= 2.9. T = 170C.

cylinder due to a small misalignment of the optical set-up (as was already indicated by the velocityprofiles).

Notice, that it was impossible to count fringes up to the downstream stagnation point. Here, the fringesshow vague transitions, probably accounted for by viscous heating or beam deflections due to temperatureeffects[26,30]. Peters and Baaijens[26] numerically investigated the influence of viscous heating in aflow around a cylinder for a one-mode PTT model with typical polystyrene melt parameters. They foundan increase in temperature at the back of the cylinder and in its wake. Due to this rise in temperaturethe viscosity decreased and thus higher velocities and lower stresses were predicted. Already at Wi= 4a reduction of the stress up to 40% and differences in downstream peak velocities of∼ 8% was foundin their calculations. The temperature dependency of the viscosity of a polystyrene melt, however, ishigher than for a long-chain branched melt. Therefore, the influence of viscous heating on the viscosity,velocity profile and stresses decreases for an LDPE melt. However, the presence of temperature effectsis experimentally clearly shown inFig. 7, which is taken from Schoonen[30]. To come to strong conclu-sions that the deviations between the calculations and the experimental data is truly due to temperatureeffects, more investigations are needed. A full analysis of the light path through a polymer melt withrefractive index gradients should be performed, as is done for an axisymmetric flow cell by Harrison et al.[12].

Along the cross-section atx/R = 0 (seeFig. 5b) the prediction of the XPP model is in good agreementwith the experimental data fory/R > 1.15. Close to the cylinder wall the XPP model predicts a stress peak,that is not observed experimentally. The Giesekus model behaves in a similar fashion, but overpredictsthe experimental data. This is expected because in this part of the domain the flow is shear dominatedand for simple shear flows the current parameter set used for the Giesekus model is known to overpredictnormal stress in shear. Like the XPP model, the Giesekus model predicts a stress peak near the cylinderwall. Because of experimental difficulties, like the misalignment of the cylinder, it is impossible to detectif such a distorted pattern is also present in the experiments. Since the shear rate at the cylinder wallγ ≈ 10–15 s−1, the peaks are not caused by a constitutive instability. Nor can it be accounted for by an


Fig. 7. Recorded deflection of a laser beam iny-direction (∆ys) along the centerline of the planar flow around a cylinder of anLDPE melt. The recording screen is placed at 1.86 m from the flow cell. The dimensions for the laser beam spot is scaled downin x-direction by a factor of 20.

insufficient refinement of the mesh, but rather as an artifact of the high strain hardening materials, sinceit is predicted by both models.

To see if fluid elements in this particular complex flow can reach their elongational planar steady state,a particle is followed along the downstream centerline starting just downstream the rear stagnation pointatx/R = 1.01. The first normal stress difference as a function of the Lagrangian time for the XPP modelis plotted as the solid curve inFig. 8. This curve, showing a typical peak value, is the same as plotted inFig. 5a, however now as a function of time instead of position. The dashed line is the build-up in stress if

Fig. 8. Calculated first normal stress difference in the planar flow around a cylinder over the downstream centerline (y/R = 0,x/R ≥ 1.01) as a function of Lagrangian time (—) and first normal stress difference at constant maximum strain rate asexprienced over the downstream centerline (- - -) for the XPP model at Wi= 2.9. T = 170C.


a constant strain rate equal to the maximum strain rate as experienced over the downstream centerline isimposed on the material, so that it is able to reach its steady state value.Fig. 8shows that for this complexflow at this particular flow rate the planar elongational steady state stress is not reached.

5.2. Cross-slot flow

The planar cross-slot flow is a complex flow geometry with a stagnation point that is easily accessibleexperimentally. Material flows in from two opposite sides, and flows out perpendicular to the inflowagain at two opposite sides (Fig. 8a) . Experiments were performed at a temperature ofT = 150C.Using the given activation energy fromTable 1and the temperature dependent Arrhenius function, linearparameters can be transformed to this temperature. This results in a viscosity averaged relaxation time forthe material ofλ = 3.4294 s. Half the height of the channels ish = 2.5 mm and the depth isd = 40 mm,giving a depth-to-height ratio of 8. The corners are rounded with a radius ofh/2 = 1.25 mm. The meantwo-dimensional in- and outflow velocities areu = 3.10 mm/s, which results from taking the same pumpflow rate as for the flow around a cylinder. If half the channel height is chosen as a characteristic length,the flow averaged Weissenberg number results:

Wi = 4.3, (28)

while the averaged Weissenberg number for the longest relaxation time equals Wi= 11.2.A detail of the FE mesh used for the calculations of all three models is given inFig. 8b. Only a quarter

of the geometry needs to be analyzed due to symmetry. The stagnation point is located at the origin of thecoordinate system. The inlet is placed aty/h = 10 and the outlet atx/h = 20. The periodic boundaryfor the inlet stresses is situated aty/h = 6. At the entrance and exit, a fully developed velocity profileis prescribed. No slip boundary conditions are imposed on the top confining wall, and the velocity inx-

Fig. 9. Schematic (a) and detail FE mesh (b) of a quarter of the planar cross-slot flow. The origin of the coordinate systemis in the stagnation point.T = 150C, λ = 3.4294 s,u2D = 3.10 mm/s,h = 2.5 mm,R = 1.25 mm = h/2, d = 40 mm,#elements= 1904, #nodes= 7875, #DOF(u, p) = 13976, #DOF(D) = 6102, #DOF(S,Λ) = 4× #elements× 20 = 152320,smallest element size located at stagnation point: 7.7 × 10−4h2.


Fig. 10. Calculated and measured velocity profiles in the planar cross-slot flow over five cross-sections iny-direction (a)(y/h = 3.2, 2.5, 2.0, 1.5, 0.5), over six cross-sections inx-direction (b) (x/h = 0.6, 1.0, 1.4, 1.8, 2.4, 3.2), and over thesymmetry lines (c) (y/h = 0, x/h = 0) at Wi = 4.3. T = 150C.

andy-directions is suppressed for the symmetry linesx/h = 0 andy/h = 0, respectively. The timestep equals2t = 0.002 s for all three models. No convergence problems were detected for either of theconstitutive models.

Fig. 9ashows the measured and calculated velocity profiles at several cross-sections of the inflow chan-nel (y/h = 3.2, 2.5, 2.0, 1.5, 0.5). All three models show a perfect match with the measurements.In the outflow channel, velocities at different cross-sections (x/h = 0.6, 1.0, 1.4, 1.8, 2.4, 3.2) aremeasured and compared to the calculations, as given inFig. 9b. Again, all three models show an excellentagreement with measured data. The centerline velocities, inFig. 9c, show the same tendency. The largest


Fig. 11. Calculated and measured isochromatic fringe patterns of the planar cross-slot flow for the XPP model (a), the Giesekusmodel (b) and the PTT-a model (c) at Wi= 4.3. T = 150C.

difference can be detected over the downstream centerline, showing the highest overshoot for the PTT-amodel, although the difference is rather small.

The isochromatic light intensities for the experiments and three models are shown inFig. 11. All threemodels show a good agreement with the experiments, except for the fully developed in- and outflowregions. Here, all models overpredict the stresses by almost one fringe, being largest for the Giesekus


Fig. 12. Calculated and measured stresses in the planar cross-slot flow over the centerline (y/h = 0, x/h = 0) for the XPP,Giesekus, and PTT-a models at Wi= 4.3. T = 150C.

model and lowest for the XPP model. This is consistent with results for the first normal stress coefficientas given inFig. 1f.

The centerline stress build-up and relaxation is quantitatively shown inFig. 12. From this figure, itis clear that all three models correctly describe the first part of the stress build-up. The maximum isoverpredicted by all three models, and is largest for the PTT-a model, followed by the Giesekus model.The stress relaxation for the PTT-a model is a little too fast and therefore under predicts the experimentaldata downstream of the stagnation point. The XPP and Giesekus models follow the experimental dataexcellently a little away from the stagnation point. It was impossible to distinguish all individual fringesexperimentally very close to the stagnation point due to the high stress gradient. Furthermore, due to thehigh extension at the stagnation point, it might be that the stress-optical rule fails in this region. This mayaccount for the deviation between experiments and calculations.

To see if planar extensional steady state is reached, a particle is tracked along the centerline, startingat x/h = 1 × 10−6, y/h = 5 and passing very closely the stagnation point atx/h ≈ 1 × 10−3, y/h ≈1×10−3. The stress that is build up during flow over the centerline as a function of Lagrangian time for theXPP model is plotted as a solid curve inFig. 13. This curve is equal to the curve inFig. 12. The ‘plateau’corresponds to the stagnation region, where residence time is long. The dashed curve corresponds tothe stress that is build up if the maximum strain rate as experienced over the centerline is constantlyimposed on the material. It is obvious from the figure, that in the stagnation region, stresses do reachtheir planar elongational steady state values. The model even predicts a higher peak stress, meaning thatthe location of the highest strain rate does not necessarily have to be the location of the highest stress.From the last graph, we can conclude, that the cross-slot device can be used as a rheological tool andthe planar elongational steady state values can be extracted from this complex flow. It is, however, morecumbersome to determine experimentally the strain rate that the particle experiences in the stagnationregion. The strain rate could be determined by performing calculations, but logically it depends on themodel that is used. FromFig. 9c, an experimental strain rate can be estimated:εexp = 1.4 ∼ 1.5 s−1. TheXPP, Giesekus and PTT-a models predictεXPP = 1.53 s−1, εGsk = 1.49 s−1, andεPTT−a = 1.60 s−1.


Fig. 13. Calculated first normal stress difference in the planar cross-slot flow over the centerline (y/h = 0x/h = 0) as a functionof Lagrangian time (—) and first normal stress difference at constant maximum strain rate as experienced over the centerline(- - -) for the XPP model at Wi= 4.3. T = 150C.

6. Conclusions

The performance of the eXtended Pom–Pom (XPP) model is investigated in two complex flow geome-tries, the planar flow around a confined cylinder and through a cross-slot device. The numerical predictionsusing the XPP model are compared with experimental results and, where possible, calculations of theGiesekus and exponential PTT-a models. Key features observed experimentally in these complex flowsare qualitatively, and to a large extent quantitatively, predicted by all three models. This includes thestress build up due to planar extension and compression at or near the stagnation points followed by arelatively slow stress relaxation.

The XPP model shows the best overall description of the available rheometrical data and, therefore, themeasured velocity and birefringence data in the two complex flows. This is not trivial, since for this modelparameter identification is based on linear viscoelastic data and transient uniaxial elongational data only,while material particles witness complicated deformation histories. In particular at and near the stagnationpoints particles are subject to planar in stead of uniaxial elongation as in the rheometrical characterization.From the material particle point of view the inhomogeneous flows are transient, and therefore it is crucialto capture the transient shear and elongational rheology of the polymer melt accurately.

A number of reasons can be pointed out that are responsible for the discrepancies between the ex-perimental and numerical data. First of all, the non-linearity parameters of the constitutive models areessentially fitted on the uniaxial rheological data only, in case of the XPP and Giesekus model, or incombination with steady shear first normal stress coefficient data for the PTT-a model. Notice that alldata is represented on a logarithmic scale. Experiments to measure the uniaxial elongational properties arecumbersome and relatively inaccurate. It is difficult to estimate the experimental error, but an inaccuracyof about 10% would not be surprising. Furthermore, only four modes are used to fit the rheological curveswith limited accuracy. It should, however, be realized that the number of relaxation times cannot be raisedarbitrarily, because their identification soon becomes ill-posed. This in particular imposes limits on theachievable smoothness of the steady uniaxial viscosity curve of the PTT-a model. Within the current


PTT-a model the relaxation time is an exponential function of the trace of the extra stress tensor. Perhapsother expressions would improve the behaviour of the model. Third, measurements in the complex flowgeometries contain errors. Clearly, although the geometries are designed to have near two-dimensionality,three-dimensional end effects are present in the experiments. In case of the depth-to-height aspect ratioused in the current experiment, about 8, Schoonen[30] and Bogaerds et al.[7] have shown that this mayinduce errors of about∼ 6% in the birefringence data. These effects cause the experimental results tobe lower near the walls and higher over the centerline in comparison to two-dimensional calculations.Other errors are due to a misalignment of the cylinder while evidence exists that temperature effects,causing viscous heating and beam deflections[30], may not be neglected. Finally, of course, the validityof the stress-optical rule may be questioned. In the present experiments, however, there is no immediateindication that this rule fails.

Although, both the planar flow around a cylinder confined by two parallel plates and the cross-slotdevice impose high strain rates on the material, important differences exist between these two complexflows. The thin stress boundary layer around the cylinder causes a number of difficulties. Experimentally,it is problematic to detect all stress fringes close to the cylinder, since a small optical misalignment causesreflection of the light and distortion of the fringe pattern. In the wake of the cylinder, heat transport isrelatively low and therefore this region is susceptible for temperature effects which may cause the fluidto be less viscous and resulting in higher velocities, lower stresses and optical distortions, which is notaccounted for in the calculations. The flow around the cylinder is difficult to resolve numerically[8], andindeed, convergence problems are encountered for the PTT-a model. For higher Weissenberg numbersthan shown in this work, divergence occurred for all three constitutive models. It is, at present, unclearhow to resolve this problem.

The cross-slot device has proven to be a rather friendly flow geometry. No severe experimental ornumerical problems were encountered. Compared to the cylinder flow it is easier to extract birefringenceand velocity data from the cross-slot device. Moreover, the cross-slot flow is more discriminating, because,unlike in other flow geometries such as the flow past a cylinder and contraction flows, steady state(planar) elongation is reached. This is interesting, because it would allow this flow device to be usedas a rheometrical instrument, provided that the stress-optical rule applies. Values extracted from thestagnation region could be used together with other rheometrical data to determine the non-linearityparameters more accurately. A significant advantage of the cross-slot device, compared to many otherelongational devices, is that it generates a confined flow. This is interesting, because less strain hardening,or even strain softening, materials are difficult or impossible to handle in traditional uniaxial extensionaldevices. Based on these consideration, the cross-slot flow geometry is the preferred geometry to testconstitutive models at non-viscometric flow conditions.

Appendix A. The rhological behaviour of the PTT-a model

To illustrate that the PTT-a model is only capable of satisfactory describing the rheological behaviourof low density polyethylene melts up to a certain point, this model is presented for the well-characterizedLupolen 1810H LDPE melt[11,15].

Two sets of parameters for the PTT-a model are given inTable 3. As can be clearly seen inFig. 14,set 1 correctly predicts the uniaxial elongation. Notice, that for 10−4 ≤ ε ≤ 10−1 the different modescan be detected in the steady state uniaxial viscosity plot. This is due to the excessive strain thinning


Fig. 14. Transient (a) and quasi-steady state (b) uniaxial viscosityηu, transient (c) and steady state (d) shear viscosityηs andtransient (e) and steady state (f) first normal stress coefficientΨ1 of the PTT-a model with set 1 for Lupolen 1810H melt atT = 150C. ε = 0.0030, 0.0102, 0.0305, 0.103, 0.312, 1.04 s−1. γ = 0.001, 0.01, 0.03, 0.1, 0.3, 1, 10 s−1.


Fig. 15. Transient (a) and quasi-steady state (b) uniaxial viscosityηu, transient (c) and steady state (d) shear viscosityηs andtransient (e) and steady state (f) first normal stress coefficientΨ1 of the PTT-a model with set 2 for Lupolen 1810H melt atT = 150C. ε = 0.0030, 0.0102, 0.0305, 0.103, 0.312, 1.04 s−1. γ = 0.001, 0.01, 0.03, 0.1, 0.3, 1, 10 s−1.


Table 3PTT-a parameter sets for fitting of the Lupolen 1810H melt atT = 150C

i Maxwell parameters PTT-a 1 PTT-a 2

G0,i (Pa) λ0,i (s) εi ξi εi ξi

1 2.1662× 104 1.0000× 10−1 0.150 0.10 0.150 0.0602 9.9545× 103 6.3096× 10−1 0.080 0.10 0.080 0.0303 3.7775× 103 3.9811× 100 0.040 0.10 0.040 0.0204 9.6955× 102 2.5119× 101 0.025 0.10 0.025 0.0155 1.1834× 102 1.5849× 102 0.007 0.10 0.007 0.0056 4.1614× 100 1.0000× 103 0.010 0.10 0.015 0.010

behaviour of the PTT-a model at high strain rates. The steady state shear viscosity and first normal stresscoefficients are also predicted correctly. Since the slip parameterξ is larger than the elongation parameterε, oscillations occur in the start-up shear responses. These oscillations are not seen in the experiments.For set 2, theξ parameter is reduced, such that oscillations in start-up shear responses disappear. Theuniaxial elongation is predicted correctly (seeFig. 15). Unfortunately, shear responses are overpredictedagain.

For such a strain hardening melt, the PTT-a model is unable to predict all features correctly. If elongationis predicted correctly, shear is overpredicted, or oscillations occur in the start-up shear behaviour. If shearis predicted correctly, elongation is underpredicted.

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