Visco-Elastic Analysis of Polymer Melts in Complex Flows

8/4/2019 Visco-Elastic Analysis of Polymer Melts in Complex Flows

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Visco-elastic analysis of polymer melts in complex flows

W.M.H. Verbeeten, A.C.B. Bogaerds, G.W.M. Peters, and F.P.T. Baaijens

Eindhoven University of Technology, Materials Technology,

PO Box 513, NL-5600 MB Eindhoven,

The Netherlands.

March 1, 1999

Abstract

A mixed low-order finite element technique based on the DEVSS/DG method has been developed for the

analysis of two and three dimensional visco-elastic flows in the presence of multiple relaxation times. In order to

evaluate the predictive capabilities of some nonlinear constitutive relations, results of calculations are compared

with experiments for two complex flows. The well-known differential Giesekus and Phan-Thien Tanner model

and the recently introduced Feta-VD model are investigated. The latter model provides enhanced independent

control of the shear and elongational properties. Experiments and calculations are performed for the steady

shear flow around a symmetric confined cylinder and in a cross-slot flow for an LDPE polymer melt.

In particular in elongational dominated regions, the numerical / experimental evaluation shows that the

multi-mode Giesekus and the PTT models are unable to describe the stress related experimental observations.

The Feta-VD model proves to perform significantly better in these regions. However, a price is paid for this

model by an overprediction of the stresses in shear dominated regions.

1 Introduction

Visco-elasticity frequently has a significant effect on polymer melt flows in industrial applications.

Most present research on calculations of steady visco-elastic flows has been performed on 2D bench-

mark problems like the falling sphere in a tube problem (e.g. Lunsmann et al. [1993], Baaijens [1994],

Sun and Tanner [1994], Yurun and Crochet [1995], Baaijens et al. [1997]) or the planar contraction

problem (e.g. Yurun and Crochet [1995], Guenette and Fortin [1995], Azaiez et al. [1996], Baaijens

et al. [1997]). Also periodic flows such as the corrugated tube flow (e.g. Pilitsis and Beris [1991], Van

Kemenade and Deville [1994], Talwar and Khomami [1995b], Szady et al. [1995]) and the flow past

an array of cylinders (e.g. Talwar and Khomami [1995a], Souvaliotis and Beris [1996]) have been

extensively investigated. These flows are generally characterised by steep stress gradients near curvedboundaries and geometrical singularities, which require the use of highly refined meshes. Accurate

and realistic flow analysis of both polymer solutions and melts compels the use of multiple relaxation

times. When using mixed finite element methods this results in a very large number of degrees of free-

dom. Thus, solution efficiency, both in terms of CPU time and memory requirement, is an important

issue.

Over the last decade, a lot of research has been performed on solving the governing equations in

an accurate and stable manner and yet still being able to efficiently handle multiple relaxation times.

Two basic problems need to be resolved. First, the presence of convective terms in the constitutive

equation, which relative importance growths with increasing elasticity, causes problems. Second,

the choice of discretisation spaces of the independent variables (velocity, pressure, extra stresses and

auxiliary variables) are not independent. Several techniques have been proposed to overcome these

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problems. Some of the most effective mixed finite element methods presently available to handle the

convective terms, employ the Streamline Upwind Petrov Galerkin (SUPG) method of Marchal and

Crochet [1987], or the Discontinuous Galerkin (DG) method of Fortin and Fortin [1989], based on theideas of Lesaint and Raviart [1974]. The lack of ellipticity of the momentum equation is frequently

resolved by using the Explicitly Elliptic Momentum Equation (EEME) formulation introduced by

King et al. [1988], the Elastic Viscous Stress Split (EVSS) formulation [Rajagopalan et al., 1990], or,

more recently, the Discrete Elastic Viscous Stress Splitting (DEVSS) method of Guenette and Fortin

[1995].

In a mixed velocity-pressure-stress formulation, interpolation of the different variables has to

satisfy a compatibility condition. Marchal and Crochet [1987] introduced a four-by-four bi-linear

subdivision for the stresses on each bi-quadratic velocity element to obtain a stable discretisation

scheme. The EEME method has been shown to give accurate and stable results by introducing a

second-order elliptic operator to the momentum equation. This method however, is restricted to UCM-

like nonlinear constitutive equations and excludes the use of a solvent viscosity. The EVSS methodis obtained by splitting the deviatoric stress into a viscous and an elastic contribution. An adaptive

strategy in combination with a modified SUPG method as proposed by Sun et al. [1996] has given

stable results for the falling sphere in a tube benchmark problem. A disadvantage of all of these

methods is the continuous interpolation of the extra stress and the subsequent large sets of global

unknowns upon discretisation. On the other hand, the Discontinuous Galerkin method employs a

discontinuous interpolation of the stress variables which leads to a more easily satisfied compatibility

condition for stress and velocity and a substantial reduction of global degrees of freedom when an

implicit/explicit scheme is used. Here, a combination of the DG and DEVSS methods is applied

to the visco-elastic flow simulations. A change of variable leads to an extra stabilising equation. By

using an implicit/explicit handling of the advective part, the extra stresses can be eliminated at element

level. This results in the DEVSS/DG method introduced to the calculation of visco-elastic flows byBaaijens et al. [1997] and has successfully been used by Beraudo et al. [1998].

In this work, an efficient numerical scheme, based on the DEVSS/DG method, is presented for the

analysis of 2D and 3D multi-mode visco-elastic flows. Furthermore, a numerical/experimental evalua-

tion is presented for two steady shear complex flows: a flow around a symmetric confined cylinder and

in a cross-slot flow. The behaviour of several constitutive equations is evaluated in these flows. Two

well-known constitutive relations of differential type (the Giesekus and the Phan-Thien Tanner model)

are applied together with a recently proposed model by Peters et al. [1999] that provides enhanced

independent control of shear and elongational properties. Although the computational procedure al-

lows for transient calculations, only steady flows will be investigated using a time-marching scheme

to reach the steady state solution.

2 Problem definition

Isothermal and incompressible fluid flows, neglecting inertia, are described by the equations for con-

servation of momentum (1) and mass (2):

= 0 , (1) u = 0 , (2)

where is the gradient operator, denotes the Cauchy stress tensor and u the velocity field. TheCauchy stress tensor is defined by equation (3):

= pI+ , (3)

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with a pressure term p = 13

tr() and the extra stress tensor, which has to be defined by a consti-tutive model.

2.1 Constitutive models

The extra stress tensor can be further divided into a Newtonian solvent and a visco-elastic part. For

a realistic description of the visco-elastic contribution, a multi-mode approximation of the relaxation

spectrum is often necessary for most polymeric fluids. This can be expressed as the sum of the separate

visco-elastic modes:

= 2sDu +

Mi=1

i , (4)

where s denotes the viscosity of the purely viscous or solvent mode, Du =1

2(L + Lc) the rate of

deformation tensor, in which L denotes the velocity gradient tensorL = (u)c. i is the visco-elasticcontribution of the ith relaxation mode and M the total number of different modes.

Within the scope of this work, a sufficiently general way to describe the constitutive behaviour

of a single mode is obtained by using a differential equation based on the generalised Maxwell-type

equation:

i +

i

(i)+ f(i,Du) = 2G(i)Du , (5)

with (i) and G(i) the stress dependent relaxation time and modulus, respectively. The upper

convected time derivative of the stressi is defined as:

i= it

+ u i L i i Lc , (6)

where t denotes time. The functions f(i,Du), (i) and G(i) depend upon the chosen constitutiveequation. Notice, that by choice of f(i,Du) = 0, (i) = i and G(i) = Gi, with i and Gi the(constant) relaxation time and modulus of the ith mode, respectively, the Upper Convected Maxwellmodel is retrieved.

In this work, two conventional non-linear constitutive models are investigated for rheological

behaviour. For the Giesekus model, the functions are defined as:

f(i,Du) =

Giii i , (7)

(i) = i , (8)

G(i) = Gi , (9)

where is a material parameter.For the exponential PTT model (PTTa), the functions are defined as:

f(i,Du) = Du i + i Du , (10)

(i) = ie Gi

I, (11)

G(i) = Gi , (12)

with and material parameters, and I = tr(i) the first invariant ofi. More detailed informationon these constitutive equations can be found in e.g. Tanner [1985], Bird et al. [1987] and Larson

[1988].

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Also, a new visco-elastic constitutive relation is investigated, which has recently been proposed

by Peters et al. [1999] and Schoonen et al. [1998]. It is based on the upper convected Maxwell model,

so f(i,D

u) = 0 is chosen. To incorporate more flexibility, both the relaxation time function (i)and the modulus function G(i) are chosen to be non-linear function of the visco-elastic stress i.

For steady simple shear flows, with shear rate , the shear stress can be written as:

12 = G = . (13)

Hence, the viscosity of the fluid is represented by the product of the modulus and the relaxation time

= G. Now, the simple shear viscosity () = 12 can be chosen such that it describes accuratelythe measured shear viscosity. A suitable choice, for instance, is the extended Ellis model:

(i) = G(i)(i) =Gii

1 + A |II|G2i ab

. (14)

Here, II is the second invariant ofi defined as II =1

2(I2

tr(2i )). Note, that for infinitesimal

strains (|II|/G2

i 1) the linear Maxwell model is recovered, as desired.

Since only the viscosity function is determined, a choice remains to be made for the relaxation time

function (or modulus function). Here, the variable drag (VD) model [Peters, 1994] is chosen as

relaxation time function:

(i) = ie

q

IGi I

Gi , (15)

with a material parameter. This model is called the Feta-VD model, for equation (14) fixes theshear viscosity (hence the prefix Fixed eta). In contrast to the PTTa model, the shear viscosity of the

Feta-VD model is not sensitive to the material parameter , while the first normal stress differenceN1 = 11 22 is less sensitive to . This material parameter can now be used to fit the elongationalbehaviour of the material. Thus, the shear and elongational properties can be controlled more inde-

pendently. The three non-linear functions for the Feta-VD model now read:

f(i,Du) = 0 , (16)

(i) = ie

q

IGi I

Gi , (17)

G(i) =Gie

IGi

q

IGi

1 + A

|II|G2i

a

b

. (18)

3 Computational method

The modelling of polymer flows gives rise to some considerable characteristic problems. Looking

more closely at the governing equations (1), (2) and (5), it is obvious that the use of multiple relax-

ation times inevitably leads to a very large system of equations, when the extra stress variables are

considered as global degrees of freedom. Another problem and a challenging field of investigation is

the loss of convergence of the numerical algorithm for increasing elasticity in the visco-elastic flow

(increasing Weissenberg number We).There are several computational methods available today that are more or less capable of effi-

ciently handling the above problems. The method used here is known as the Discrete Elastic Viscous

Stress Splitting / Discontinuous Galerkin method (DEVSS/DG). It is basically a combination of the

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Discontinuous Galerkin method, developed by Lesaint and Raviart [1974], and the Discrete Elastic

Viscous Stress Splitting technique of Guenette and Fortin [1995]. The combination of the two meth-

ods into the DEVSS/DG method was first applied by Baaijens et al. [1997].

3.1 DEVSS/DG method

This technique finds its basis in the classical Galerkin Finite Element Method. Consider a spatial

domain that is divided into K elements (e) such that =

e with boundary .Guenette and Fortin [1995] proposed a new mixed formulation by introducing an L2 projection

of the rate of deformation tensor to yield a discrete approximation D (equation (22)), in combina-

tion with a stabilisation term in the discrete momentum equation (2(Du D), equation (19)). Thestabilising auxiliary viscosity in equation (19) can be varied in order to give optimal results. Fol-lowing Guenette and Fortin [1995] and Baaijens et al. [1997], =

Gii is chosen and found to

give satisfactory results.

Based on the ideas of Lesaint and Raviart [1974], and first applied for the analysis of visco-

elastic flows by Fortin and Fortin [1989], a discontinuous interpolation is applied to the extra stress

variables. They are now considered as local degrees of freedom and can be eliminated at element

level. Upwinding is performed on the element boundaries by adding integrals on the inflow boundary

of each element and thereby forcing a step of the stress at the element interfaces (equation (21)).

Time discretisation of the constitutive equation is attained using an implicit Euler scheme, with the

exception that exti is taken explicitly (i.e. exti

= exti (tn)). Hence, the termS

d

i: u n(exti ) d

has no contribution to the Jacobian which allows for local elimination of the extra stress. The mixed

weak formulation now reads:

Problem DEVSS/DG: Given (u, p,i, D) at t = tn, find a solution at t = tn+1 such that for alladmissible test functions (v, q,Sdi ,G),

Dv , 2sDu + 2Du D

+

Mi=1

i

v , p

= 0 , (19)

q , u

= 0 , (20)

S

d

i,i +

i

(i)+ f(i,Du) 2G(i)Du

Ke=1

ein

Sd

i: un (i

ext

i) d = 0 i

1, 2, . . . , M

, (21)

G , D Du

= 0 , (22)

where ( , ) denotes the L2-inner product on the domain , D =1

2

+ ( )c

with = u , v,

an auxiliary viscosity, ein is the inflow boundary of element e, n the unit vector pointing outward

normal on the boundary of the element (e) and exti denotes the extra stress tensor of the neighbour-

ing, upwinding element.

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3.2 Solution strategy

In order to obtain an approximation of Problem DEVSS/DG, a 2D domain is divided into K quadri-

lateral and a 3D domain into K hexahedral elements. A choice remains to be made about the order ofthe interpolation polynomials of the different variables with respect to each other. As is known from

solving Stokes flow problems, velocity and pressure 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 compatibility condition in order to obtain stable results. Baai-

jens et al. [1997] have shown that for 2D problems, discontinuous bi-linear interpolation for extra

stress, bi-linear interpolation for discrete rate of deformation and pressure with respect to bi-quadratic

velocity interpolation (figure 1, left) gives stable results. This approach is extrapolated to a third

dimension and satisfying the LBB condition, spatial discretisation is performed using tri-quadratic

interpolation for velocity, tri-linear interpolation for pressure and discrete rate of deformation while

the extra stresses are approximated by discontinuous tri-linear polynomials (figure 1, right). Integra-

tion of equation (19) to (22) over an element is performed using a quadrature rule common in finiteelement analysis (3 3 or 3 3 3 Gauss rule, for 2D and 3D respectively).

u

i, p, D

ui, p, D

Figure 1: Mixed finite element. Left: u bi-quadratic, p, D bi-linear, i discontinuous bi-linear. Right: u tri-quadratic, p, D tri-linear, i discontinuous tri-linear.

To obtain the solution of the nonlinear equations, a one step Newton-Raphson iteration process is

carried out. Consider the iterative change of the nodal degrees of freedom (, u, D, p) as variablesof the algebraic set of linearised equations. This linearised set is given by:

Q Qu 0 0

Qu Quu QuD Qup

0 QDu QDD 0

0 Qpu 0 0

u

D

p

=

f

fu

fD

fp

, (23)

where f ( = , u, D, p) correspond to the residuals of equations (19) to (22), while Q followfrom linearisation of these equations. Due to the fact that ext has been taken explicitly in equation

(21), matrix Q has a block diagonal structure which allows for calculation of Q1

on the element

level. Consequently, this enables the reduction of the global DOFs by static condensation of the extra

stress. Despite this approach, still a rather large number of global degrees of freedom remains per

element, as it is depicted in figure 1 (43 or 137 DOF/element, in the 2D and 3D case respectively). A

further reduction of the size of the Jacobian is obtained by decoupling problem 23. First, the Stokes

problem is solved (u, p) after which the updated solution is used to find a new approximation for D.

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The following problems now emerge:

Problem DEVSS/DGa: Given (u, p,i, D) at t = tn, find a solution at t = tn+1 of the algebraic set:Quu QuQ

1

Qu Qup

Qpu 0

u

p

=

fu QuQ

1

f

fp

, (24)

and Problem DEVSS/DGb: Given D at t = tn and u at t = tn+1, find D from:

QDD D = fD . (25)

Notice that fD is now taken with respect to the new velocity approximation, i.e. fD(un+1, D

n) rather

than fD(un, D

n). The nodal increments of the extra stress are retrieved element by element following:

= Q1

f , (26)

with f also taken with respect to the new velocity approximation (f(n, un+1)).

Using the above procedure, for 3D calculations still a substantial number of unknowns remain

per element. Although direct solvers often prove to be more stable in comparison to iterative solvers,

they become impractical for the 3D visco-elastic calculations due to excessive memory requirements

which inevitably leads to the application of iterative solvers. To solve the non-symmetrical system

of problem DEVSS/DGa an iterative solver is used based on the Bi-CGSTAB method of Van der

Vorst [1992]. The symmetrical set of algebraic equations of problem DEVSS/DGb is solved using a

Conjugate-Gradient solver. Incomplete LU preconditioning is applied to both solvers. It was found

that solving the coupled problem (hence, solving for u, D, p at once) led to divergence of the solverwhile significantly better results were obtained for the decoupled system. In order to enhance the

computational efficiency of the Bi-CGSTAB solver, static condensation of the center-node velocityvariables results in filling of the zero block diagonal matrix in problem 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 con-

ditions must be imposed on the boundaries of the flow channels. At the entrance and the exit of the

flow channels the velocity profiles are prescribed. And at the entrance, the steady state stresses are

prescribed along the inflow boundary.

4 Rheological characterisation

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. It has been extensively characterised

by Schoonen [1998] and is also given in Peters et al. [1999]. The parameters for a four mode Maxwell

model are obtained from dynamic measurements. The non-linear parameter for the Giesekus model

is determined on the steady shear data. Two sets of parameters for the exponential PTT model are

determined. The first set is fitted only on the steady shear data, whereas the second set is also fitted on

elongational measurements. For the Feta-VD model, the Ellis parameters (A, a and b) are determinedon the steady shear data. The fit for this model is based on the steady shear data and the steady planar

elongational data, taken from measurements in a cross-slot device.

The parameter fits for the different numerical models at T = 190 [C] are given in table 1. Figure 2shows the rheological behaviour of the four numerical models and measured data in simple shear.

Predictions in extension for the different models are depicted in figure 3. All models show similar

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Maxwell parameters Feta-VD Giesekus PTTa-1 PTTa-2

i Gi [Pa] i [s] A, a, b , , ,

1 7.598 104

2.097 103

2 1.664 104 2.767 102

3 3.518 103 2.711 1012, 2, 1 0.12, 0.44 0.29 0.15, 0.08 0.004, 0.08

4 3.174 102 2.474 100

Table 1: Material parameters of different models for LDPE melt (T = 190 [C], = 0.9377 [s]).

102

100

102

101

102

103

104

Steady shear viscosity for LDPE melt at T=190oC

Shear rate [s1]

Viscosity[Pas]

FETA (=0.12,=0.44)

Giesekus (=0.29)

exp. PTT (=0.150,=0.08)

exp. PTT (=0.004,=0.08)

Maxwell model

Steady shear

Literature (cone/plate)

Literature (capillary)

102

100

102

100

102

104

106

First Normal Stress Difference for LDPE melt at T=190oC

Shear rate [s1]

FirstNormalStressDifferenceN1[Pa]

FETA (=0.12,=0.44)

Giesekus (=0.29)

exp. PTT (=0.150,=0.08)

exp. PTT (=0.004,=0.08)

Maxwell model

Steady shear

Literature (cone/plate)

Figure 2: Model predictions and measurements of steady shear data (left) and first normal stress

difference (right) of LDPE melt at T = 190 [C].

101

100

101

102

103

103

104

105

106

107

Transient uniaxial elongational viscosity for LDPE melt at T=120

o

C

Time t [s]

Viscosityu+[

Pas]

= 0.03 [s1]

= 0.10 [s1]

= 0.30 [s1]

= 1.00 [s1]

FETA (=0.12,=0.44)

Giesekus (=0.29)

exp. PTT (=0.150,=0.08)

exp. PTT (=0.004,=0.08)

Maxwell model

102

101

100

101

102

103

104

105

106

Planar elongational viscosity for LDPE melt at T=150

o

C

Strain rate [s1]

Viscosityp

[Pas]

Crossslot

FETA (=0.12,=0.44)

Giesekus (=0.29)

exp. PTT (=0.150,=0.08)

exp. PTT (=0.004,=0.08)

Figure 3: Model predictions and measurements of transient (left) and planar elongational data (right)

of LDPE melt at T = 120 [C] and T = 150 [C], respectively.

results for the shear data. The agreement with measurements is rather well in shear for all these

fits. However, in extension, the models show quite different behaviour. For the transient elongational

behaviour, only the PTTa-2 fit ( = 0.004, = 0.08) can predict the upswing reasonably well. Secondbest is the Feta-VD model. However, for the planar extension, the elongational thickening behaviour

for this PTTa-2 model is overpredicted. Here, the Feta-VD model gives the best agreement. The sharp

increase clearly visible for different modes, does not seem to be very natural.

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5 Complex flows of LDPE melt

A comparison is made between numerical and experimental results for two complex flows. First, the

flow around a symmetric confined cylinder is reported in detail, both for 2D and 3D calculations.

Then, the visco-elastic flow through a cross-slot device is presented. Both flow geometries are known

to have a simple shear region, a solely elongational region, and a combined shear / elongational region,

which makes them to be complex flow geometries.

The experimental data of the flows described in this work consists of two parts [Schoonen, 1998].

First, fieldwise velocity measurements have been carried out using Particle Tracking Velocimetry

(PTV). Second, Flow Induced Birefringence (FIB) has been used to measure the stresses over the

depth of the flow. For these stress measurements, a 10mW HeNe laser (wave length 0 = 633 [nm])was used. The FIB measurements can be compared with the calculations by means of the empirical

stress optical rule. For a projection of the birefringence tensor in a plane perpendicular to the optical

path, the empirical stress optical rule reads:

sin2 = 2k0dCxy , (27)

cos 2 = k0dCN1 , (28)

with the rotation angle, the phase retardation, k0 the initial propagation number (k0 = 2/0), dthe thickness, C the stress optical coefficient, xy the mean plane shear stress and N1 = xxyy themean first normal stress difference of the element. A stress optical coefficient of 1.53 109 [Pa1]

was determined for this material. Isochromatic lines are observed for retardation levels that equal

multiples of2. Then, equations (27) and (28) reduce to a single equation for the isochromatic lines

( = k2, k = 1, 2, 3, . . . ):N2

1+ 42

xy=

k0dC

, k = 1, 2, 3, . . . . (29)

5.1 Flow around a cylinder

An investigation of the planar flow around a symmetric confined cylinder is carried out. This flow

is characterised by a compression of the polymer towards the cylinder, shearing along the cylinders

surface and the material is stretched in the wake of the cylinder. Figure 4 shows the 3D geometry, the

mesh and characteristics used to analyse numerically this flow. For symmetry reasons, only a quarter

of the whole geometry has been modelled. At the entrance and exit, a fully developed velocity profile

is prescribed, taken from a flow through a rectangular duct. Using the velocity profile at the entrance,the steady state stresses are computed and given as natural boundary conditions at the inlet. On rigid

walls, the no-slip condition is used. Along the center plane, symmetry conditions are prescribed.

To characterise the flow, the dimensionless Weissenberg number is chosen, which denotes the

amount of elasticity in the flow:

We =u2D

R, (30)

where is the mean relaxation time ( = (

2iGi)/(

iGi)), u2D is the 2D mean velocity and R

is the radius of the cylinder. The experiments are performed at a temperature of170 [C] at four flowrates. The characteristics are given in table 2. For the 3D calculations, only the lowest flow rate for

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z

H

x: flow direction

RR = 1.1875 mmD = 40 mmH = 4.95 mm

Dy

#Elements = 2856

#Nodes = 26295

#DOF(u, p) = 74069

#DOF(D) = 22512

#DOF() = 548352

Figure 4: FE mesh, the geometry and characteristics for a planar flow around a symmetric confined

cylinder.

u2D [mm/s] || = u2D/R [s1] || =

u2D/R [kPa] We []

0.96 0.81 3.74 1.4

1.98 1.67 7.71 2.9

5.23 4.40 20.36 7.7

7.55 6.36 28.11 11.1

Table 2: Characteristics for flow around a cylinder (T = 170 [C], = 1.7438 [s])

the Giesekus model is shown, as it is only to point out, that the third dimension for this flow geometry

is negligible.

In figure 5, the normalised velocities in x-, y- and z-direction at cross-section x/R = 1.5 aredepicted, together with the dimensionless normal stress differences and the main shear stress xy. As

can be seen, the influence of the front and back wall on the profiles is rather limited. Only over a

small depth near those confining walls, the profiles change from zero to their maximum. Even at

this cross-section, where the influence of the cylinder is rather large, the z-velocity is at most only10% of the velocity in x-direction. Therefore, a nominally 2D geometry can be assumed. For further

comparison, 2D calculations suffice.

In figure 6, the mesh and characteristics are shown for the 2D calculations. Again for symmetry

reasons, only half of the geometry is meshed. Similar boundary conditions as for the 3D flow are

prescribed. Along the centreline of the flow around a cylinder, the velocities were measured. In fig-

ure 7, the measured velocities for the four flow rates together with the Giesekus and PTTa-1 numerical

results are shown. Unfortunately, the PTTa-2 model failed to give converged solutions and therefore

is not presented here. The velocity profiles predicted by the models are similar and in good agreement

with the experimental velocities. For We = 7.7 and We = 11.1, the PTTa-1 model predicts a higherovershoot downstream near the cylinder than the Giesekus model. The latter seems to be more in

agreement with the experimental data.

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W e = 1 : 4

x/R

y/Rexperiment

2

0

0

12 3

-4 -2 0 2 4 6 8 10 12

1

0122

34

43

F e t a - V D 1

G i e s e k u s

P T T a - 1

Figure 8: Measured (top) and calculated isochromatic fringe patterns at We = 1.4 (T = 170 [C], = 1.7438 [s], u2D = 0.96 [mm/s], Feta-VD: , = 0.12, 0.44, Giesekus: = 0.29, PTTa-1:, = 0.15, 0.08). The numbers indicate the fringe order.

The measured and calculated isochromatic fringe patterns for all four Weissenberg numbers are

depicted in figures 8 to 11. At first sight, the agreement between the measured and calculated isochro-

matic fringe patterns seems rather good. However, if we have a closer look, some differences can

be detected. In the simple shear region (x/R = 4), both position and number of predicted fringesare very close to the measured pattern for the Giesekus and PTTa-1 models. This holds for all four

flow rates. However, the Feta-VD model overpredicts the amount of fringes, especially for the higher

Weissenberg numbers. This can be explained by looking at figure 2. In the shear rate region of the

FIB measurements (|| = 0.81 6.36 [s1]), the shear viscosity of all models are almost equal.However, for the first normal stress difference, the Feta-VD model predicts higher values and thus

will show more fringes in shear dominated regions. This overprediction of fringes is also observed at

cross-section x/R = 0.In the wake of the cylinder, the stresses predicted by the Giesekus and PTTa-1 models relax rather

fast. The experimentally observed relaxation is significantly slower. In this elongation dominated

region, the Feta-VD model follows the experiments much better. If the planar elongational data in

figure 3 is observed, the Giesekus model predicts a slight elongational thickening behaviour, which is

not sufficient to capture the upswing. The PTTa-1 model even has a elongational thinning behaviour

at the measured elongational rate. Only the elongational thickening behaviour of the Feta-VD model

seems to be sufficient to capture the stress upswing in the wake of the cylinder. If the parameters of

the models are changed, such that the upswing is captured, still the relaxation of the stresses can not

be predicted satisfactory. In the Feta-VD model, the Variable Drag part is responsible for a realistic

stress relaxation.

Overall, the Giesekus model shows the best agreement in the shear dominated regions. Whereas

the Feta-VD model gives the best predictions for the elongation dominated region.

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W e = 2 : 9

x/R

y/R

0

2

-4

experiment

-2 0

0

12 3

4

23

5 6

2 4 6

1

0

1234567

8 10 12

F e t a - V D 1

G i e s e k u s

P T T a - 1


W e = 7 : 7

0 2 4 6 8 1210-2-4

0

experiment

2

x/R

y/R2

0

1

3 4 5

1 2 34

9876 3

0

12

1

128 7 6 5 4 3

F e t a - V D 1

G i e s e k u s

P T T a - 1


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W e = 1 1 : 1

0

2

x/R

y/R

0 2 4

7

6 8 1210-2-4

0

1

2

34 5

6 7 910

811

2 345

1 10

1234561011 9 8

e x p e r i m e n t

F e t a - V D 1

G i e s e k u s

P T T a - 1


5.2 Cross-slot flow

As a second complex flow, the cross-slot flow is investigated. Here, two liquid flows approach each

other in opposing directions, meet in a stagnation point, and leave in perpendicular opposing direc-

tions. In the stagnation point, the material experiences an infinite extensional strain. Over the centre-

line towards and away from the stagnation point, the material is only compressed and stretched. In

the in- and outflow rectangular ducts, a pure shear flow is present. In the region around the stagnation

point, the flow is a mixture of shear and elongation. Figure 12 shows the geometry, mesh and charac-

teristics used to analyse numerically this flow. For symmetry reasons, only one quarter of the whole

geometry is modelled. On rigid walls, no-slip conditions are used. Along centrelines, symmetry con-

ditions are prescribed and a fully developed velocity profile is taken as boundary conditions at the in-

and outflow. The steady shear stresses are derived and prescribed as natural boundary condition at the

inflow, based on the fully developed velocity profile.

For characterisation of the flow, the radius R in equation 30 is replaced by half the height of theinflow channel h = 1

2H:

We =u2D

h. (31)

The experiments are performed at a temperature of150 [C] at two flow rates. The characteristics aregiven in table 3.

Over the centreline, the velocities are measured and shown in figure 13, along with the calculated

velocities for the different models. Again, the PTTa-2 model gave convergence problems. Near the

stagnation point, the residence time of the material approaches infinity. Therefore, a relatively small

14


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D

H

R

x

z

y

R = 1.25 mm

: inflow plane

D = 40 mm

H = 5 mm

: outflow plane

: flow direction

#Elements = 1904

#Nodes = 7875

#DOF(u, p) = 13976

#DOF(D) = 6102

#DOF() = 91392

Figure 12: Detail of FE mesh, the geometry and characteristics for a cross-slot flow.

u2D [mm/s] || = u2D/h [s1] || =

u2D/h [kPa] We []

3.0 1.20 11.6 4.1

4.4 1.76 17.0 6.0

Table 3: Characteristics for cross-slot flow. (T = 150 [C], = 3.4338 [s])

4 3 2 1 0 1 2 3 48

6

4

2

0

2

4

6

8

y / H 0 x / H

Vy

0

Ux

[mm/s]

Velocity along centerline u2D = 3.0 [mm/s]

Experiment

FetaVD, , = 0.12,0.44

Giesekus, = 0.29

exp. PTT1, , = 0.15,0.08

4 3 2 1 0 1 2 3 48

6

4

2

0

2

4

6

8

y / H 0 x / H

Vy

0

Ux

[mm/s]

Velocity along centerline u2D = 4.4 [mm/s]

Experiment

FetaVD, , = 0.12,0.44

Giesekus, = 0.29

exp. PTT1, , = 0.15,0.08

Figure 13: Measured and calculated velocities along the centreline. (T = 150 [C], = 3.4338 [s],u2D = 3.0, 4.4 [mm/s])

amount of polymer melt flows through this area, and only a limited number of data points could be

measured around this stagnation point. The velocity profile in the upstream part is for all models in

good agreement with the experimental data. The Feta-VD model predicts an undershoot, which is not

observed in the other models nor in the measured data. In the downstream part, the area of strong

elongation, the models overestimate the velocity. Also, an overshoot is calculated, which is largest for

the PTTa-1 model. This overshoot is experimentally not seen.

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3

0

4

1

5

1

3

32

2

21

1

0

1

2

3

10 2 3 4 5 6 7 8 9 10x/h

y/h

e x p e r i m e n t

F e t a - V D 1

G i e s e k u s

P T T a - 1


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2

01

4

1

34

23

1

23

4

0

1

3

0 1 2 3 4 5 6 7 8 9 10x/h

2y/

5

e x p e r i m e n t

F e t a - V D 1

G i e s e k u s

P T T a - 1


17


18/20

The measured and calculated isochromatic fringe patterns for the two flow rates are depicted in

figure 14 and 15. In the fully developed inflow region, the Feta-VD model overpredicts the amount

of fringes by almost a factor two. Similar as the overprediction in the flow around a cylinder, this iscaused by a higher prediction of the first normal stress difference by this model. Also, it can partly be

ascribed to the slightly higher mean velocity in the calculations (see figure 13). In this inflow region,

the Giesekus and PTTa-1 model slightly overpredict the amount of fringes, partly due to the higher

mean velocity.

Near the stagnation area, the Feta-VD model predicts the most fringes, which seems to be accurate.

The PTTa-1 model has the least fringes in this region. Over the downstream centreline, away from

the stagnation point, the PTTa-1 and Giesekus results relax too fast in regard to the experimental data.

The Feta-VD model does a better job, and shows a good agreement with the experiments in this case.

A better view of the performance over the centreline is given in figure 16. This figure clearly shows

the incapability of the Giesekus and PTTa-1 model to predict the upswing and relaxation.

10 5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

y/H x/H

N12+

4xy

2

[N/m2]

Experiment

FetaVD, , = 0.12,0.44

Giesekus, = 0.29

exp. PTT1, , = 0.15,0.08

10 5 0 5 10 15 200

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x 105

y/H x/H

N12+

4xy

2

[N/m2]

Experiment

FetaVD, , = 0.12,0.44

Giesekus, = 0.29

exp. PTT1, , = 0.15,0.08

Figure 16: Measured and calculated stresses over the centreline. (T = 150 [C], = 3.4338 [s],u2D = 3.0, 4.4 [mm/s])

6 Conclusions and discussion

A mixed low-order finite element based on the DEVSS/DG method is developed and implemented

for the calculation of 2- and 3-dimensional visco-elastic flows. Steady state fluid flows through two

complex flow geometries for several flow rates have been calculated for an LDPE polymer melt.Several different constitutive relations have been applied for the evaluation of these complex flows.

Two of them are established and well-known differential models, the Giesekus and exponential PTT

model, and a third one is the recently introduced Feta-VD model, also in a differential form.

By means of Particle Tracking Velocimetry (PTV) and Flow Induced Birefringence (FIB), exper-

imental values are obtained and compared to the calculations. For both complex flows, the velocity

and stress calculations with the Giesekus and PTT models show a good agreement between numerical

and experimental data in shear dominated regions. The Giesekus model gives a slightly better com-

parison than the PTT model. However, in the elongational dominated regions, the stress relaxation

of the Giesekus and PTT models is too fast compared to the experiments. Here, the upswing of the

stresses can not be captured due to the low planar elongational thickening and thinning behaviour

(at these rates) of the Giesekus and PTT model, respectively. Consequently, the relaxation is not in

18


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agreement with the experimental data either. In these elongational dominated regions, the Feta-VD

model performs significantly better. Its higher planar elongational thickening behaviour is sufficient

to capture accurately the stress upswing observed in the experiments. The Variable Drag part of themodel accounts for a satisfying stress relaxation. However, the Feta-VD model pays a price by loss

of accuracy in predictions of the shear induced first normal stress difference. The overprediction of

the first normal stress difference is the main cause of the discrepancy for the stresses in shear domi-

nated regions. This does not negatively influence the calculated velocity profiles, which are in good

agreement with the experimental values.

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Visco-Elastic Analysis of Polymer Melts in Complex Flows