3D Viscoelastic analysis of a polymer solution in a complex flow

3D Viscoelastic analysis of a polymer solution in a complex¯ow

Arjen C.B. Bogaerds, Wilco M.H. Verbeeten, Gerrit W.M. Peters,Frank P.T. Baaijens *

Materials Technology, Faculty of Mechanical Engineering, Centre for Polymers and Composites, Dutch Polymer Institute, Eindhoven

University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received 30 October 1998

Abstract

A mixed low-order ®nite element technique based on the Discrete Elastic Viscous Stress Splitting (DEVSS)/Discontinuous Galerkin

(DG) method has been developed for the analysis of three-dimensional viscoelastic ¯ows in the presence of multiple relaxation times. In

order to evaluate the predictive capabilities of some established nonlinear constitutive relations like the Giesekus and the Phan-Thien±

Tanner (PTT) model results of 3D calculations are compared with experimental results in a cross-slot ¯ow geometry. Moreover, the

performance of a new viscoelastic constitutive equation that provides enhanced independent control of the shear and elongational

properties is investigated. Steady shear ¯ows and a combined shear/elongational ¯ow are analyzed for a polyisobutylene solution. A

general method is introduced to compare calculated stresses along the depth of the ¯ow with birefringence measurements using the

stress optical rule. In particular at and downstream of the stagnation point in the cross-slot ¯ow geometry, the numerical/experimental

evaluation shows that the multi-mode Giesekus and the PTT model are unable to describe the stress-related experimental observations.

The new viscoelastic constitutive relation proves to perform signi®cantly better for this stagnation ¯ow. Ó 1999 Elsevier Science S.A.

All rights reserved.

Keywords: 3D-Viscoelastic; Discontinuous Galerkin method; Constitutive equations; Complex ¯ow

1. Introduction

Most present research on calculations of steady viscoelastic ¯ows has been performed on 2D benchmarkproblems like the falling sphere in a tube problem (e.g. [16,3,26,33,5]) or the four-to-one contractionproblem (e.g. [33,9,1,5]). Also periodic ¯ows such as the corrugated tube ¯ow (e.g. [19,20,32,29,27]) and the¯ow past an array of cylinders (e.g. [11,28,24]) have been extensively investigated. These ¯ows are generallycharacterized by steep stress gradients near curved boundaries and geometrical singularities, which requirethe use of highly re®ned meshes. Accurate ¯ow analysis of both polymer solutions and melts compels theuse of multiple relaxation times. When using mixed ®nite element methods this results in a very largenumber of degrees of freedom and solution e�ciency, both in terms of CPU time and memory requirementis an important issue.

Over the last decade, much research has been performed on solving the governing equations in anaccurate and stable manner and still being able to e�ciently handle the multitude of stress unknowns.For a recent review on mixed ®nite element methods for viscoelastic ¯ow analysis, see Ref. [4]. Two

www.elsevier.com/locate/cmaComput. Methods Appl. Mech. Engrg. 180 (1999) 413±430

* Corresponding author. Tel.: +31-40247-4888; fax: +31-40244-7355.

E-mail address: [email protected] (F.P.T. Baaijens)

0045-7825/99/$ - see front matter Ó 1999 Elsevier Science S.A. All rights reserved.

PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 1 7 6 - 0

basic problems needed resolution: (i) the presence of convective terms in the constitutive equation whoserelative importance grows with increasing Weissenberg number and (ii) the choice of discretizationspaces of the independent variables (velocity, pressure, extra stresses and auxiliary variables). Severaltechniques have been proposed to overcome these problems and some of the most e�ective mixed ®niteelement methods presently available either employ the Streamline Upwind Petrov Galerkin (SUPG)method of Marchal and Crochet [17] or the Discontinuous Galerkin (DG) method of Fortin and Fortin[8] based on the ideas of Lesaint and Raviart [14] to handle the convective terms, while the lack ofellipticity of the momentum equation is resolved by using the Explicitly Elliptic Momentum Equation(EEME) formulation introduced by King et al. [12], the Elastic Viscous Stress Split (EVSS) formulation[21], or, more recently, the Discrete Elastic Viscous Stress Splitting (DEVSS) method of Gu�enette andFortin [9].

Marchal and Crochet used the inconsistent SU method and emphasized that in a mixed velocity-pressure-stress formulation interpolation of the deviatoric stress cannot be chosen independent and has tosatisfy a compatibility condition. In order to ful®ll this condition they used a four-by-four bi-linearsubdivision for the stresses on each bi-quadratic velocity element. This however leads to a very highnumber of degrees of freedom of the stresses, especially when multiple relaxation times are considered.The EEME method has been shown to give accurate and stable results by introducing a second-orderelliptic operator to the momentum equation. This method however, is restricted to UCM-like nonlinearconstitutive equations and excludes the use of a solvent viscosity. The EVSS method is obtained bysplitting the deviatoric stress into a viscous and an elastic contribution. An adaptive strategy in com-bination with a modi®ed SUPG method as proposed by Sun et al. [25] has given stable results for thefalling sphere in a tube benchmark problem. A disadvantage of all of these methods is the continuousinterpolation of the extra stress and the subsequent large sets of global unknowns upon discretization.The Discontinuous Galerkin method on the other hand employs a discontinuous interpolation of thestress variables which leads to a more easily satis®ed inf-sup condition for stress and velocity and asubstantial reduction of global degrees of freedom when an implicit/explicit scheme is used. In this work,the numerical method applied to the viscoelastic ¯ow simulations is basically a modi®ed formulation ofthe DG method. The DG method has been extended with the DEVSS formulation for which a change ofvariable leads to an extra stabilizing equation. Furthermore, using an implicit/explicit handling of theadvective part of the constitutive equation, this leaves the DEVSS/DG method which has been introducedto the calculation of viscoelastic ¯ows by Baaijens et al. [5] and has recently been successfully used byB�eraudo et al. [6].

In this work, an e�cient numerical scheme, based on the DEVSS/DG method, is developed for theanalysis of 3D multi-mode viscoelastic ¯ows. Also, a numerical/experimental evaluation is presented forboth steady shear ¯ows as well as a steady inhomogeneous ¯ow in a cross-slot device to study the behaviorof di�erent constitutive models under such circumstances. To achieve this, a method based on Muellercalculus is implemented to retrieve measurable optical quantities from computed stress ®elds by assumingthe stress optical rule to hold. Two of the more popular constitutive relations of di�erential type (theGiesekus and the Phan-Thien±Tanner (PTT) model) are applied together with a recently proposed modelby Peters et al. [18] that provides enhanced control of shear and elongational properties. Although thecomputational procedure allows for transient calculations, only steady ¯ows will be investigated using atime-marching scheme to reach the steady-state solution.

2. Problem de®nition

Here, only steady incompressible, isothermal and inertia-less ¯ows are considered. In the absence ofbody forces, these ¯ows can be described by a reduced equation for conservation of momentum (1) andconservation of mass (2):

~r � r �~0; �1�

~r �~u � 0; �2�

414 A.C.B. Bogaerds et al. / Comput. Methods Appl. Mech. Engrg. 180 (1999) 413±430

with ~r the gradient operator, and ~u the velocity ®eld. The Cauchy stress tensor r is de®ned as

r � ÿpI � s; �3�with pressure p and the extra stress tensor s. For most realistic viscoelastic ¯uids it is often necessary toapproximate the relaxation spectrum with a discrete set of viscoelastic modes. The total extra stress forthese so-called multi-mode applications can be expressed as the sum of the stresses belonging to the sep-arate viscoelastic modes, hence

s �XM

i�1

si; �4�

where M denotes the total number of different modes. Within the scope of this work, a suf®ciently generalway to describe the constitutive behavior of one individual mode is obtained by using a constitutiveequation of the differential form

sr � fc�s;D� � fd�s�

k� 2GD; �5�

with k and G the relaxation time and the modulus of this mode and D the rate of deformation tensor de®nedas D � 1

2�~r~u� �~r~u�c� with � �c denoting the conjugate of a second order tensor. The upper convected time

derivative of the extra stress tensor is de®ned as

sr � os

ot�~u � ~rsÿ L � sÿ s � Lc; �6�

with L the velocity gradient tensor L � �~r~u�c. Both functions fc�s;D� and fd�s� depend upon the chosenconstitutive model. Notice that for fc�s;D� � 0 and fd�s� � s the Upper Convected Maxwell (UCM) modelis obtained. Some conventional nonlinear models that are applied in this work for the rheological char-acterization of the viscoelastic ¯uid, are the Giesekus model which is de®ned as

fc � 0; fd � s� �

Gs � s; �7�

and the linear PTT model

fc � n�D � s� s �D�; fd � 1��

GIs

�s; �8�

with Is the ®rst invariant of s (i.e. Is � tr�s�) and e and n adjustable parameters. More detailed informationon the rheological behavior of the constitutive equations can be found in Tanner [30], Bird et al. [7] andLarson [13].

A new class of viscoelastic constitutive relations that incorporate more ¯exibility was recently proposedby Peters et al. [18] and Schoonen et al. [22]. Their method involves a generalization of the UCM model byallowing both the relaxation time and the modulus to be a function of the extra stress, i.e. k�s� and G�s�.Both k�s� and G�s� are chosen such that in the limit of infenitesimal strains the linear Maxwell model isrecovered. For steady planar shear ¯ows, with shear rate _c, it holds that

s � Gk _c; �9�with s the shear stress. To obtain the simple shear viscosity function �g � Gk�, the empirical Cox±Merz ruleis approximated by a modi®cation of the Ellis model, hence

Gk � G0k0

1� A IIs=G20� �a� b �10�

with IIs � 12�I2

s ÿ tr�s � s�� the second invariant of the extra stress tensor and G0; k0 initial linear materialparameters. Since only the viscosity function is determined, a choice remains to be made for the relaxationtime (or the modulus). A suitable choice can be

A.C.B. Bogaerds et al. / Comput. Methods Appl. Mech. Engrg. 180 (1999) 413±430 415

k � k0

1� �e=G0�Is; �11�

which is the same function as is used in the linear PTT model. This model has been introduced as the Feta-PTT model for its shear viscosity is ®xed by Eq. (10) (hence the pre®x Fixed eta) and thus not sensitive tovariations of the nonlinear parameter e. While the ®rst normal stress coef®cient only slightly depends on e,the elongational viscosity proves to be signi®cantly more sensitive to variations of this parameter. As aresult, shear and elongational properties can be controlled more independently.

3. Computational method

The modeling of polymer ¯ows gives rise to some considerable characteristic problems. Looking moreclosely at the governing equations (Eqs. (1), (2) and (5)) it is obvious that the use of multiple relaxationtimes inevitably leads to a very large system of equations when the extra stress variables are considered asglobal degrees of freedom. Another problem and a challenging ®eld of investigation is the loss of con-vergence of the numerical algorithm for increasing elasticity in the viscoelastic ¯ow.

There are several computational methods available today that are more or less capable of e�cientlyhandling the above problems. The method used in this report is known as the method DEVSS/DG. It isbasically a combination of the DG method that was developed by Lesaint and Raviart [14] and the DEVSStechnique of Gu�enette and Fortin [9]. The DEVSS/DG was ®rst applied to 2D viscoelastic ¯ows in Baaijenset al. [5] and can be stated as:

Problem DEVSS=DG: Find si, ~u, �D and p such that for all admissible test functions Si,~v, G and q,

Si; sir

�� fc�si;Du� � fd�si�

kiÿ 2GiDu

�ÿXK

e�1

ZCe

inflow

Si :~u �~n si

ÿ ÿ sexti

�dC � 0 8 i 2 1; 2; . . . ;Mf g; �12�

Dv; 2�g �Du

:ÿ �D� �

XM

i�1

si

!ÿ �~r �~v; p� � 0; �13�

�G; �D:ÿDu� � 0; �14�

�q; ~r �~u� � 0; �15�where �: ; :� denotes the L2-inner product on the domain X, sext

i the extra stress tensor of the neighboringelement, ~n the unit vector pointing outward normal on the boundary of the element (Xe) andDu � 1

2�~r~u� �~r~u�c� with ~u �~u;~v.

As proposed by Gu�enette and Fortin [9] a stabilization term has been added to the momentum equation�2�g�Du ÿ �D�, Eq. (13)) in combination with an L2 projection of the rate of deformation tensor to yield adiscrete approximation of �D (Eq. (14)). The stabilizing parameter �g in Eq. (13) can be varied in order to giveoptimal results. Following Gu�enette and Fortin [9] and Baaijens et al. [5], �g �PG0k0 is chosen and foundto give satisfactory results.

Based on the ideas of Lesaint and Raviart [14], a discontinuous interpolation is applied to the extra stressvariables which are now considered as local degrees of freedom and can be eliminated at the element level.Upwinding is performed on the element boundaries by adding integrals on the in¯ow boundary of eachelement and thereby forcing a step of the stress at the element interfaces (Eq. (12)). Time discretization ofthe constitutive equation is attained using an implicit Euler scheme, with the exception that sext

i is takenexplicitly (i.e. sext

i � sexti �tn�). Hence, the term

RSi :~u �~n�ÿsext

i �dC has no contribution to the Jacobian whichallows for local elimination of the extra stress.


In order to obtain an approximation of problem DEVSS/DG, the 3D domain is divided into K hex-ahedral elements. A choice remains to be made about the order of the interpolation polynomials of thedifferent variables with respect to each other. As is known from solving Stokes ¯ow problems, velocity andpressure interpolation cannot be chosen independently and has to satisfy the Ladyzenskaya±Babuska±Brezzi condition. Likewise, interpolation of velocity and extra stress has to satisfy a similar compatibilitycondition in order to obtain stable results. Baaijens et al. [5] have shown that for 2D problems, discon-tinuous bi-linear interpolation for extra stress, bi-linear interpolation for discrete rate of deformation andpressure with respect to bi-quadratic velocity interpolation gives stable results. Hence, extrapolating thisapproach to a third dimension and satisfying the LBB condition, spatial discretization is performed usingtri-quadratic interpolation for velocity, tri-linear interpolation for pressure and discrete rate of deformationwhile the extra stresses are approximated by discontinuous tri-linear polynomials (Fig. 1). Integration ofEqs. (12)±(15) over an element is performed using a quadrature rule common in ®nite element analysis(3� 3� 3-Gauss rule).

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 �ds; du; d�D; dp� as variables of thealgebraic set of linearized equations. This linearized set is given by

Qss Qsu 0 0Qus Quu Qu �D Qup

0 Q�Du Q�D �D 00 Qpu 0 0

0BB@1CCA

ds

du

d�D

dp

0BB@1CCA � ÿ

fs

fu

f �D

fp

0BB@1CCA; �16�

where fa �a � s;~u; �D; p� correspond to the residuals of Eqs. (12)±(15), while Qab follow from linearization ofthese equations. Due to the fact that sext has been taken explicitly in Eq. (12), matrix Qss has a block di-agonal structure which allows for calculation of Qÿ1

ss on the element level. Consequently, this enables thereduction of the global DOFs by static condensation of the extra stress. Despite this approach, still a ratherlarge number of global degrees of freedom remains per element as it is depicted in Fig. 1 (137 DOF/ele-ment). A further reduction of the size of the Jacobian is obtained by decoupling problem (16). First, the`Stokes' problem is solved (~u; p) after which the updated solution is used to ®nd a new approximation for �D.The following problems now emerge:

Problem DEVSS=DGa: Given s;~u; �D and p at t � tn, ®nd a solution at t � tn�1 of the algebraic set

Quu ÿ QusQÿ1ss Qsu Qup

Qpu 0

� �du

dp

� �� ÿ fu ÿ QusQ

ÿ1ss fs

fp

� �: �17�

Fig. 1. Mixed ®nite element, ~u ! tri-quadratic, p; �D ! tri-linear, si ! discontinuous tri-linear.


Problem DEVSS=DGb: Given �D at t � tn and ~u at t � tn�1, ®nd d�D from

Q�D �D d�D � ÿf �D: �18�Notice that f �D is now taken with respect to the new velocity approximation, i.e. f �D�~un�1; �Dn� rather thanf �D�~un; �Dn�. The nodal increments of the extra stress are retrieved element by element following:

ds � ÿQÿ1ss fs; �19�

with fs also taken with respect to the new velocity approximation (fs�sn;~un�1�).Using the above procedure, still a substantial number of unknowns remain per element. Although direct

solvers often prove to be more stable in comparison to iterative solvers, they soon become impractical for3D viscoelastic calculations due to excessive memory requirements which inevitably lead to the applicationof iterative solvers. To solve the nonsymmetrical system of problem DEVSS/DGa an iterative solver hasbeen used based on the Bi-CGSTAB method of van der Vorst [31]. The symmetrical set of algebraicequations of problem DEVSS/DGb has been solved using a Conjugate-Gradient solver. Incomplete LUpreconditioning has been applied to both solvers. It was found that solving the coupled problem (hence,solving for du; d�D; dp at once) led to divergence of the solver while signi®cantly better results were obtainedfor the decoupled system. In order to enhance the computational e�ciency of the Bi-CGSTAB solver, staticcondensation of the center-node velocity variables results in ®lling of the zero block diagonal matrix inproblem DEVSS/DGa and, in addition, achieves a further reduction of global degrees of freedom.

Finally, to solve the above sets of algebraic equations, both essential and natural boundary conditionsmust be imposed on the entrance and exit of the ¯ow channels. At the entrance and exit of the ¯ow channelsthe velocity unknowns are prescribed. Also, at the entrance, the known shear rates enable the calculation ofthe `steady state' stresses by solving the constitutive equations which are then prescribed along the in¯owboundary.

4. Rheological characterization

The polymer solution consists of 2.5% Polyisobutylene (Oppanol B200, BASF) dissolved in tetradecane(2.5% PIB/C14). This solution has been extensively characterized and documented by Schoonen et al. [22].Results of the material characterization are listed in Table 1. The ¯uid has been characterized by theGiesekus model and the linear PTT model (once with a nonzero parameter n and once with n set to zero).The moduli and the relaxation times are obtained from dynamic measurements and the nonlinear pa-rameters (e; n) have been ®tted on steady shear data only. Fig. 2 shows the rheological behavior of both theGiesekus and the linear PTT model in simple shear together with the measured shear data. It can be seenthat the PTT model (set 1) agrees very well with experimental observations of steady shear viscosity. Be-cause of the limited range covered by the relaxation times, deviations from measured data start at shearrates of approximately _c > 2 � 102 �sÿ1�. Predictions for the di�erent models in extension are depicted inFig. 3. As can be seen from this ®gure the Giesekus model and, to a lesser degree, the PTT model with thenonzero n parameter, show an elongational thickening behavior for increasing extension rates whereas thePTT model with the zero n parameter shows an elongational thinning behavior.

Table 1

Material parameters of 2.5% polyisobutylene dissolved in tetradecane (2.5% PIB/C14) at T � 20°C, obtained from Schoonen et al. [22]

Mode Maxwell Giesekus PTT (set 1) PTT (set 2)

G (Pa) k (s) e �ÿ� e �ÿ� n�ÿ� e �ÿ� n�ÿ�

1 3:3 � 101 3:6 � 10ÿ3

2 3:6 � 100 3:1 � 10ÿ2 0.30 0.80 0.00 0.42 0.07

3 1:8 � 100 1:6 � 10ÿ1

4 5:1 � 10ÿ2 5:6 � 10ÿ1


Fig. 2. Steady shear viscosity (left) and ®rst normal stress coe�cient (right) of 2.5% PIB/C14 together with experimental data, ±±±

Giesekus, � � � PTT (n� 0), ± ± PTT (n 6�0).

Fig. 3. Predictions for 2.5% PIB/C14 in uniaxial (left) and planar elongation (right), ±±± Giesekus, � � � PTT (n� 0), ± ± PTT (n6��0).

Fig. 4. Material functions for increasing nonlinear parameter e of the Feta-PTT model (e � 0:04; 0:06; 0:08; 0:10; 0:12). Steady shear

viscosity (left) and ®rst normal stress coef®cient (right).


Determination of the parameters of the Feta-PTT model is more troublesome. Parameters of themodi®ed Ellis model can easily be determined from measured viscosity data (Fig. 4 (left), A; a; b �16; 2; 0:45). However, since the nonlinear parameter e only slightly in¯uences the shear properties of themodel, additional elongational data is required to obtain a ®t for this parameter. Fig. 4 (right) shows thein¯uence of variations of e on the ®rst normal stress coef®cient whereas Fig. 5 shows the sensitivity of theelongational viscosities towards e. Additional information on the behavior of the ¯uid in elongation isobtained from Schoonen et al. [22] who were able to estimate the planar elongational viscosity at a numberof elongational rates from the stagnation ¯ow described in this work. It can be seen that e � 0:06 yields thebest ®t of the elongational data whereas a reasonable ®t of the ®rst normal coef®cient is obtained.

5. Flow-induced birefringence

The experimental data of the ¯ows described in this work consists of two parts. First, pointwise velocitymeasurements have been carried out using Laser Doppler Anemometry (LDA) and second, Flow InducedBirefringence (FIB) has been used to measure the stresses over the depth of the ¯ow. Generally, the stressstate will not be constant over the depth of the ¯ow cell due to the in¯uence of both con®ning walls.Therefore, in order to present a valid comparison between measured and computed stress related quantities,the ¯ow-induced birefringence measurements need some further attention.

5.1. General Mueller/Stokes approach

Physically, the change of polarization of light traveling through the ¯ow cell is caused by refractivegradients induced by the alignment of the polymeric molecules. Hence, upon emerging from the ¯ow cell, therelative phase di�erence (retardation angle) of the extraordinary and ordinary component of the light di�ersfrom its initial value. Fig. 6 shows the change of polarization within a small detail of the ¯ow cell. The twoprincipal electro-magnetic components will move di�erently along the optical path which causes a relativephase di�erence. The optical axis of this detail of the ¯ow is rotated (rotation angle v) relative to a ®xed axis.

Mathematically, the state of polarization and the change of polarization caused by a retarder can bedescribed by the Stokes vector and the Mueller matrix (4� 4) characteristic for this retarder [10]. Hence,the change of polarization of light traveling through the total ¯ow cell can be expressed as

Sout � MfcSin; �20�with Mfc the Mueller matrix of the ¯ow cell and Sin, Sout the Stokes vectors of respectively the incoming andoutgoing light beam. A general di�erential method to calculate Mfc was developed by Azzam [2]. For the

Fig. 5. Uniaxial (left) and planar (right) elongational viscosity for increasing nonlinear parameter e of the Feta-PTT model

(e � 0:04; 0:06; 0:08; 0:10; 0:12).


viscoelastic ¯ows described in this work, the change of polarization of light traveling through the opticalanisotropic medium with continuously varying properties along the optical path is approximated by adiscrete set of 2D optical elements (Fig. 7). Thus, the Mueller matrix of the ¯ow cell is now given bymultiplication of the subsequent Mueller matrices of these 2D optical elements,

Mfc � MN MNÿ1 . . . M2 M1; �21�where N is taken equal to the number of numerical elements along the optical path. Each optical element ischaracterized by a phase retardation (d) and an orientation angle (v) which leads to the following Muellermatrix,

Mi�d; v� �1 0 0 00 c2

2v � cds22v �1ÿ cd�s2vc2v ÿ sds2v

0 �1ÿ cd�s2vc2v s22v � cdc2

2v sdc2v

0 sds2v ÿ sdc2v cd

0BB@1CCA

i

; �22�

where cu � cos u and su � sin u (u � d; 2v). Mechanical and optical properties can be coupled by means ofthe empirical stress optical rule, which relates the deviatoric part of the refractive index tensor (n) to theextra stress tensor by a characteristic stress optical coe�cient (C),

Fig. 6. Change of polarization of light within a detail of a retarder. Extraordinary (e) and ordinary (o) wave move differently through

the optical element causing a relative phase difference. The optic axis is rotated at a rotation angle v relative to a ®xed frame.

Fig. 7. Flow through a rectangular duct (¯ow direction perpendicular to the paper), optical properties are represented by N optical

elements with N equal to the number of elements along the depth of the duct in the numerical simulation.


n � Cs: �23�Now, consider only the projection of the birefringence tensor in the plane perpendicular to the optical path,say for example the xy-plane, the empirical stress optical rule for one optical element leads to:

sin 2vd � 2k0 dCsxy ; �24�

cos 2vd � k0 dCN1; �25�with k0 the initial propagation number �k0 � 2p=k0�, d the thickness, sxy the mean plane shear stress andN1 � sxx ÿ syy the mean ®rst normal stress difference of the ith element. Application of Eqs. (24) and (25) tothe Mueller matrix of a single element layer (Mi), enables the calculation of a discrete approximation of theMueller matrix of the total ¯ow cell (Mfc) and hence, allows for a numerical/experimental evaluation of theoptical properties.

5.2. FIB measurements for polymer solutions

The setup, used for the 2.5% PIB/C14 solution, gathers pointwise optical data and is shown in Fig. 8.Here, only a short outline of the experimental setup is presented, for a detailed description on the FIBexperiments see Schoonen et al. [22]. Unpolarized monochromatic light from a source with intensity Iin

(k0� 632.8 (nm)) travels through the total setup. Using standard expressions for the Mueller matrices of theoptical elements [10], the intensity of the transmitted light can be described by

Iout � Iin

41� �M42 cos�4xt� �M43 sin�4xt��; �26�

with x the rotating frequency of the half wave plate and M42, M43 the (4,2) and the (4,3)-component of theMueller matrix of the ¯ow cell. For low-viscosity viscoelastic ¯ows that induce little retardation upon thetransmitted light, it can be shown that M42 and M43 reduce to integrals along the optical path:

M42 � 2k0CZ

sxy dz; �27�

M43 � ÿk0CZ

N1 dz: �28�

This approximation has, for instance, been used in Li et al. [15] and Schoonen et al. [22].

6. Flows of a polyisobutylene solution

A comparison is presented between numerical and experimental results for a steady 3D shear ¯ow in arectangular tube (slit ¯ow), Fig. 9 (left), and a steady combined complex ¯ow, Fig. 9 (right), through across-slot device. For the cross-slot ¯ow, due to its nonhomogeneous nature (material near the center will

Fig. 8. Experimental (FIB) setup for 2.5% PIB/C14 solution, light travels through a linear polarizer (p) at 0°, a rotating half-wave plate

(r), a collimating lens (l), the ¯ow cell (fc), a quarter-wave plate (q) at 45° and again through a linear polarizer at 0°.


experience a much higher strain rate than near the in- or outlet), the behavior of the constitutive models canbe evaluated for complex ¯ows. The aspect-ratio of both main axes of the rectangular cross section of thechannel has been chosen close to unity and thus, a full 3D ¯ow ®eld is obtained.

6.1. Slit ¯ow

Numerical investigations of a steady shear ¯ow have been performed on a rectangular channel with adepth to height ratio of 2. Fig. 10 shows the geometry and mesh used to analyze this ¯ow. For reasons ofsymmetry only one-quarter of the total channel has been modeled. At the entrance and the exit, initially, afully developed Newtonian velocity pro®le [23] is prescribed with a 2D mean velocity ��u2D� at symmetryplane z � 0. This characteristic mean 2D ¯ow follows from experimental observations and was determinedat �u2D � 26 � 10ÿ2 (m/s). As a next step, the converged solution at the middle of the ¯ow cell is taken asboundary conditions for a recurring computation. In this way the slit ¯ow is iterated until the non-Newtonian steady state solution is obtained, usually two or three iteration steps su�ce. An alternative tothis procedure (but one that has not been implemented in our code yet) is the application of periodicalboundary conditions for the in- and out¯ow unknowns. In this way fully developed ¯ows may be obtainedin a more natural way by prescribing a ¯ow rate rather than velocities. A dimensionless ¯ow strength forthis shear ¯ow can be obtained by means of the Weissenberg number

We ��k�u2D

H� 2:54; �29�

with �k a viscosity averaged relaxation time (�k �P�k2G�=P�kG�).Results of calculated fully developed velocity pro®les along the y-axis and the z-axis for the different

established constitutive models are shown in Fig. 11 together with the measured values. As can be seen fromthis ®gure, both the Giesekus and the PTT model with the nonzero n parameter ®t the measured velocitydata reasonably well.

Fig. 10. FE mesh of a rectangular duct (D/H� 2). The ¯ow direction is de®ned by the positive x-axis, #elements� 640, #nodes� 6069,

#DOF(~u; p)� 17178, #DOF(�D)� 5346, #DOF(s� � 4� #elements� 48 � 122880.

Fig. 9. Steady shear ¯ow through a rectangular duct (left) and steady extensional ¯ow generated by the impingment of two rectangular

¯ows (right).


Fig. 12 shows the calculated fully developed stresses at a cross section of the slit for the Giesekus model.The, in principle, discontinuous stresses are averaged over the nodes and scaling is performed followingjsj �P�kG�i �u2D=H . It can be seen from this ®gure that both con®ning walls have a large in¯uence on thecalculated stress ®eld. Especially the stress components relevant for birefringence measurements, i.e. N1 and

Fig. 12. Calculated fully developed velocity pro®le and stress components for the Giesekus model (�u2D � 26 � 10ÿ2 (m/s),

jsj � 18:9 (Pa)).

Fig. 11. Calculated and measured fully developed velocity pro®le along y-axis (left) and z-axis (right) (�u2D � 26 � 10ÿ2 (m/s)), ±±±

Giesekus, � � � PTT (n � 0), ± ± PTT (n 6� 0).


to a lesser degree sxy , are in¯uenced by these walls. Rather than simple extinction near the upper and lowerwalls, as is observed for sxy , additional nonzero values of N1 are observed due to the shear rate perpen-dicular to the xy-plane. Thus, large deviations can be expected for these integrated stresses along the depthof the ¯ow compared to 2D calculations. This is also con®rmed in Fig. 13 where integrated stresses arecompared to stresses at the mid plane (z � 0). Hence, this ¯ow has to be treated as truly 3D and cannot becompared using 2D viscoelastic calculations.

Comparison of the pointwise optical data with the calculated stresses is performed by means of theempirical stress optical coe�cient. This material constant has been determined by Schoonen et al. [22]which is adopted here (C � 2:505 � 10ÿ9 (m2/N)). The calculated optical properties for the di�erent modelsare presented in Fig. 14 together with measured values of the optical signal. It can be seen that all modelspredict shear stresses that are in good agreement with experimental observations. Predictions of the ®rstnormal stress di�erence, on the other hand, are best for the Giesekus model while the PTT model under-predicts and the Feta-PTT over-predicts the shear induced normal stresses.

6.2. Cross-slot ¯ow

Flow of the PIB/C14 solution through a cross-slot device has been analyzed. The geometry and the meshused for this ¯ow are depicted in Fig. 15. Again, due to symmetry, only a fraction of the total ¯ow is

Fig. 13. Integrated principal stress di�erence (N1) (left) and shear stress (sxy) (right) over the depth of the slit for the Giesekus model

(±±±) compared to 2D stresses (± ±) (jsj � 18:9 (Pa)).

Fig. 14. Calculated and experimental optical data for di�erent constitutive models, M42 (� R sxydz) (left) and M43 (� R N1dz) (right),

±±± Giesekus, � � � PTT (n� 0), ± ± PTT (n6�0), ± � ± Feta-PTT.


modelled (1/8). Nonzero boundary conditions for the velocity at the in- and outlet are obtained from thepreviously described slit ¯ow. Using R as a typical length scale rather than H yields We � 1:27.

As, for the slit ¯ow, the best overall agreement with experiments was observed for the Giesekus model,this model, together with the new Feta-PTT model, is used for calculations of the cross-slot ¯ow. Fig. 16shows the velocity pro®les at out-¯ow cross section x=R � 1:5 calculated with the Giesekus model. Obvi-ously the predicted maximum velocity is in good agreement with experimentally observed values. However,along the y-axis a more ¯attened velocity pro®le has been measured. For comparison, added to this ®gureare the steady-state velocity pro®les, as obtained from slit ¯ow calculations. It can be seen that the velocityat this cross section still exhibits extensional effects. This is also con®rmed in Fig. 17 which shows thecalculated and measured velocity as well as the calculated strain rate ( _exx) along the in¯ow axis towards thestagnation point and from there along the out¯ow axis. Velocity, strain rate and principal stress differencealong the same axes and over the depth of the ¯ow are shown in Fig. 18. Along the in¯ow planes, elon-gational rates increase from zero towards a local maximum at approximately y=R � 0:8. A maximum isreached at the stagnation point ( _exx� 22.9 (s-1)). At the out¯ow planes, rather than a local maximum, asmall plateau is observed after which the elongational rates rapidly decrease towards a negative value.Thus, the ¯uids elasticity causes the velocity to reach a maximum value where the channel straightens again(x � R� H=2). The in¯uence of both con®ning walls on the ®rst normal stress difference can clearly be

Fig. 15. FE mesh of cross-slot device (D=H � 2), in¯ow along y-axis, out¯ow along x-axis, #elements� 2835, #nodes� 25631,

#DOF(~u; p)� 71988, #DOF(�D)� 21600, #DOF(s� � 4� #elements� 48 � 544320.

Fig. 16. Calculated and measured velocity pro®le along y-axis (left) and z-axis (right) at out¯ow cross-section x=R � 1:5(�u2D � 26 � 10ÿ2 (m/s), Giesekus model). For comparison the fully developed velocity pro®les, obtained from slit ¯ow calculatations,

are added (± ±).


observed in this ®gure. Just like slit ¯ow calculations, a typical parabolic, shear induced, shape is observednear the in- and outlet of the ¯ow. The stress ®eld near the stagnation line is mainly dominated by planarelongation with a maximum at the stagnation point.

Comparison of the calculated stress with experiments is performed using the approach described inSection 5. Again, the stress optical coe�cient is taken at C � 2:505 � 10ÿ9 (m2/N). Fig. 19 (left) shows the

Fig. 18. Calculated velocity, strain rate (exx) and principal stress di�erence along symmetry planes x=R � 0 _ 1:5 P y=R P 0 (in¯ow)

and y=R � 0 _ 2 P x=R P 0 (out¯ow) (j _cj � �u2D=R � 13:0 �sÿ1�).

Fig. 17. Calculated and measured velocity along positive y-axis (in¯ow) towards the stagnation point and along positive x-axis

(out¯ow) (left) and strain rate ( _exx) along the same axes (right) (�u2D � 26 � 10ÿ2 (m/s), j _cj � �u2D=R � 13:0 �s�ÿ1).


measured optical data compared to calculated optical data along the in- and out-¯ow symmetry planes forboth the Giesekus and the Feta-PTT model. For the Giesekus model it is seen that along the in¯ow planesthe calculated principal stress difference is accurately described. However along the out¯ow axes the pre-dicted normal stresses deviate from experimental data. Although the overall shape is consistent with theexperiments, integrated ®rst normal stress difference along the stagnation line is about 30% of experi-mentally observed values. The Feta-PTT model, on the other hand yields a less exact description of themainly shear induced normal stresses along the in¯ow boundaries though a far more accurate ®t of themeasured data along the relaxation region is obtained. This is also shown in Fig. 19 (right) where at theout¯ow cross section x=R � 1:5 measured and calculated data are shown along the y-axis. For comparisonwith a steady-state shear ¯ow, see also Fig. 14. The performance of the new Feta-PTT model is furtherdemonstrated in Fig. 20 where experimental and numerical results are shown for different values ofthe dimensionless Weissenberg number. Included are measurements which were performed at�u2D � 13 � 10ÿ2 (m/s) yielding a Weissenberg number which is half the Weissenberg number of the previ-ously described ¯ow.

7. Conclusions and discussion

A mixed low-order ®nite element based on the DEVSS/DG method has been implemented for thecalculation of 3D viscoelastic ¯ows. Calculations have been performed on a steady shear ¯ow and a

Fig. 20. Calculated and measured optical signal (M43) along positive y-axis towards the stagnation point and along positive x-axis (left)

and at out¯ow cross section x=R � 1:5 (right) for the Feta-PTT model at different ¯ow rates, ±±±� �u2D � 13 � 10ÿ2 (m/s), ± � ±��u2D � 26 � 10ÿ2 (m/s).

Fig. 19. Calculated and measured optical signal (M43) along positive y-axis towards the stagnation point and along positive x-axis (left)

and at out¯ow cross section x=R � 1:5 (right), ±±± Giesekus, ± � ± Feta-PTT.


combined shear/elongational ¯ow of a polymer solution. For the evaluation of these ¯ows di�erent con-stitutive relations have been applied. The ¯ow of the solution through a rectangular duct with a height todepth ratio of 2, has been numerically evaluated using some established nonlinear constitutive models(Giesekus, PTT) and the, only recently introduced, Feta-PTT model. From pointwise birefringence mea-surements it follows that shear stresses are ®tted reasonably well by all models whereas considerable dif-ferences are observed between the different models for the shear-induced normal stresses. For the cross-slot¯ow, it is observed that the normal stresses, predicted with the Giesekus model are far too low, near thestagnation line, compared with experimental data. It is expected from predictions of the models in planarelongation that none of the ®ts performed with the PTT model is capable to predict the normal stressesinduced by the elongational component of the ¯ow due to the elongational thinning (or less elongationalthickening) of this model. The Feta-PTT model performed signi®cantly better in the complex stagnation¯ow than the Giesekus model. Although some accuracy is lost on the shear-induced normal stresses, a lot isgained with the prediction of the principal stress difference induced by the elongational component of the¯ow.

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3D Viscoelastic analysis of a polymer solution in a complex flow

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