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J. Non-Newtonian Fluid Mech. 144 (2007) 170–177

xploring the utility of an axisymmetric semi-hyperbolic die for determiningthe transient uniaxial elongation viscosity of polymer melts

Anurag Pandey, Ashish Lele ∗Complex Fluids and Polymer Engineering Group, Polymer Science and Engineering Division, National Chemical Laboratory, Pune 411008, India

Received 16 November 2005; received in revised form 4 September 2006; accepted 23 December 2006

bstract

The estimation of elongation viscosity of polymeric fluids from pressure drop measurements across converging dies remains an attractiveethod due to its cost effectiveness. However, the elongation viscosity so measured can be contaminated by large shear effects at the walls unless

pecific stress boundary conditions can be imposed. Such experiments are in general difficult to perform. In this work we examine an alternative

impler method that relies on a combination of experimental pressure drop measurements in semi-hyperbolic dies and matching viscoelastic CFDimulations to obtain the transient elongation viscosity data. The calculations of the transient uniaxial elongation viscosity from this methodologyor an LDPE melt are compared with data obtained using extensional rheometers in order to explore the utility of the strategy.

2007 Elsevier B.V. All rights reserved.

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eywords: Elongation viscosity; Hyperbolic die; Simulations

. Introduction

Significant advances have been made in measuring the elon-ation viscosity of viscoelastic polymeric fluids in recent years.everal different types of elongation rheometers such as theheometrics Meissner Elongation rheometer (RME), Capillaryreakup Elongation Rheometer (CaBER), filament stretching

heometer and the Sentmanat Elongation Rheometer (SER), toame a few, have been developed and some of these are com-ercially available. In general however, the measurement of

longation viscosity using elongation rheometers is difficult dueo the presence of free surface [1], de-cohesion [2], inertia [3],nd effects [4] and cohesive failure [5].

Converging dies offer an attractive means for estimating thelongation viscosity of polymeric fluids primarily because ofheir cost effectiveness compared to the elongation rheometers.owever, the main difficulty in using converging dies concerns

he decontamination of shear contributions (which are generallyot known independently) from the measured pressure drop so

s to yield the true elongation viscosity. Cogswell [6] suggestedhat the total pressure drop across a sudden contraction coulde decomposed into a shear component and an elongation com-

∗ Corresponding author. Tel.: +91 20 2590 2199; fax: +91 2590 2618.E-mail address: ak.lele@ncl.res.in (A. Lele).

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377-0257/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.jnnfm.2006.12.007

onent, thereby providing a simplified way of estimating thelongation viscosity. However, yet another problem with con-erging dies is that fluid elements in different parts of a die areubjected to different deformation histories as they flow throughhe die. This is because: (a) the velocity field in most experi-

ental set ups is two dimensional and each velocity componentepends on more than one spatial variable, and (b) even if shearffects are countered (let us say by imposing specific bound-ry conditions), the fluid elements do not experience constantlongation rates along the flow direction in most convergingeometries. The only geometry that can impose constant elon-ation rate on fluid elements under certain boundary conditionss a hyperbolic geometry given by

2 = C

z+ B(1)

ere, R(z) is the radius of the die at an axial location z, and C andare constants that depend on the entry radius, R0, exit radius,

e, and length, L, by the following equations [7]:

= LR2

0R2e (2a)

R20 − R2

e

= LR2

e

R20 − R2

e(2b)

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A. Pandey, A. Lele / J. Non-Newto

Kim et al. [8] and Pendse and Collier [9] proposed a methodor estimating an effective elongation viscosity of polymeric flu-ds from the measured pressure drop across a semi-hyperbolicie. This technique was also used to measure the effective elon-ation viscosity of cellulose solutions [10–12]. Hyperbolic diesre today commercially available as accessories for capillaryheometers (from TA Instruments and Celsium Technologies)nd may be used to estimate the effective elongation viscosityf polymeric fluids.

The effective elongation viscosity is defined as [7]

eff = −�Pεhε

(3)

Here, the overall Hencky strain of the die, εh, and the elon-ation rate, ε, are defined as [7]

h = lnR2

0

R2e

(4a)

˙ = Q

πC= Q

πR20L

[exp(εh) − 1] (4b)

here Q is the volumetric flow rate in the die. Eqs. (3)–(4b) haveeen derived assuming a full-slip boundary condition at the dieall [7], which is equivalent to assuming a potential hyperbolicow.

A major concern in using Eqs. (3)–(4b) for the calculation ofhe elongation viscosity of fluids is that it may not be straight-orward to implement a potential hyperbolic flow in typicalxperimental conditions because of the following reasons. Oneay argue that in order to maintain a steady homogeneous uni-

xial extensional converging flow the boundary conditions thatave to be satisfied are:

(a) Tangential velocity at the wall should be equal to the velocitythat would have existed for a streamline given by ψ|wall =−εR2z/2 = −εC/2. This tangential velocity can be shownto be equal to

Vt|wall =[ε

2

√r2 + 4z2 + 2zV0√

r2 + 4z2

]wall

= ε

[R2

0(C + 4z3) + 4z2C

2√z(C + 4z3)

](5a)

b) The normal stress (Tnn) and tangential stress (Tnt) along thestreamlines should satisfy [13]:

Tnn = r2

4z2 + r2 (Tzz − Trr) + Trr,

Tnt = 2zr

4z2 + r2 (Tzz − Trr) (5b)

Eq. (5b) implies that the tangential stress boundary condition

n an axisymmetric hyperbolic die should be given by

nt|wall = 2z3/2C1/2

4z3 + C(Tzz − Trr)|wall = 2z3/2C1/2

4z3 + Cεηe (5c) f

T

luid Mech. 144 (2007) 170–177 171

In order to satisfy the velocity boundary condition, the fluidas to slip at the die wall with a velocity equal to that given byq. (5a). While it is true that certain entangled flexible polymers

ike polyethylene do tend to slip at die walls, however, their slipelocity is dictated by the coupled dynamics of bulk chains andhains adsorbed on the die wall [14,15]. Consequently, wall slipay only be partial and the slip velocity may not be necessarily

qual to that given by Eq. (5a). Thus in general, the velocityoundary condition will not be satisfied.

The stress boundary condition given by Eqs. (5a) and (5c)s even more difficult to implement because it demands that theriction law at the die wall (relating wall velocity to frictionalorce) should produce a tangential stress which is equal to thativen by Eq. (5c). The frictional stress at the wall may be giveny a friction law such as

nt|wall = βVt|wall = βε

[R2

0(C + 4z3) + 4z2C

2√z(C + 4z3)

](6)

Eqs. (5c) and (6) have different z-dependences implyinghereby that the stress boundary condition would be difficulto satisfy over the entire wall.

Thus, it would be difficult to implement a steady-state homo-eneous uniaxial extensional flow in a converging hyperbolicie in a typical extrusion experiment. Consequently, it woulde incorrect to calculate the true elongation viscosity of a fluidrom the experimental pressure drop–flow rate data obtained forhyperbolic die using Eqs. (3)–(4b). An alternative simple strat-gy could be to measure the pressure drop across the hyperbolicie for various flow rates and use matching CFD simulationsmploying more realistic no-slip or partial slip boundary con-itions at the die walls to obtain the elongation flow relatedarameters of a chosen constitutive equation. The transientniaxial elongation viscosity can then be predicted from theonstitutive equation itself. Additionally, the CFD simulationsan be used to understand the centerline elongational flow field.he symmetry condition for an axisymmetric hyperbolic dieemands that the flow along the centerline of the die is irrota-ional and is characterized by a constant elongation rate for mostart of the die except close to the entrance region. The simulatedenterline data can be used to calculate the elongation viscosityf the fluid flowing through the hyperbolic die. We have shownhat the elongation viscosity so obtained for a cellulose solutions at least an order of magnitude lower than the effective elon-ation viscosity predicted by Eq. (3) indicating the significantontribution of shear stresses to the total pressure drop [16]. Inhe present study we explore this strategy further by comparinghe predictions of the centerline elongation viscosity obtainedrom CFD simulations in an axisymmetric hyperbolic die withhe uniaxial elongation viscosity data obtained from elongationheometers for a strain hardening low density polyethylene melt.

. Experimental

The low density polyethylene (LDPE) sample was obtainedrom BASF and is comparable to the Melt I sample [17].he linear viscoelastic properties of LDPE were obtained from

1 nian Fluid Mech. 144 (2007) 170–177

saUfi1ctmwewqotsoFt

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Table 1Relaxation spectrum at 150 ◦C

Relaxation time, λi (s) Modulus, gi (Pa)

0.01 61390.30.05202 18912.70.27059 11308.81.40753 4703.937.32169 1569.74

38.0859 349.593

1

fi2wtfi

72 A. Pandey, A. Lele / J. Non-Newto

mall amplitude oscillatory shear measurements performed onn ARES strain-controlled rheometer (Rheometrics Scientific,SA) using 25 mm diameter disposable aluminum parallel platextures. Disk samples were molded in a compression press at50 ◦C and used as such for rheological measurements. A forcedonvection oven supplied with nitrogen gas was used to con-rol the temperature in the ARES rheometer and simultaneously

aintain an inert atmosphere around the samples. Measurementsere carried out over a temperature range of 150–230 ◦C. At

very temperature in this range, the linear viscoelastic regionas delineated by a strain sweep experiment after which a fre-uency sweep experiment was carried out over a frequency rangef 0.01–100 rad/s with appropriately chosen strain amplitude inhe linear regime. A mastercurve was obtained by horizontallyhifting the frequency sweep data to a reference temperaturef 150 ◦C. The shifted data and the shift factors are shown inig. 1. An eight mode relaxation spectrum was obtained from

he mastercurve and is shown in Table 1.A semi-hyperbolic die (Rheometrics Scientific, USA) having

n entry radius of 10 mm, length of 25 mm and a Hencky strainf 7 was used for pressure drop measurements. The die was

ig. 1. (a) Master curve of dynamic data time-temperature superposed to aeference temperature of 150 ◦C. The line through the viscosity data showsredictions of the K-BKZ model for a damping function of k = 0.24 (b) shiftactors for LDPE at 150, 170, 190, and 210 ◦C.

cilrsatflut

PUmgpctofcomiazan

Fo

198.115 19.4486030.55 0.99232

xed to the end of the barrel of a capillary rheometer (Rheovis100, CEAST, Italy) using an adapter. Extrusion experimentsere conducted at 150, 170, 190 and 210 ◦C for prescribed pis-

on speeds and the axial force was measured using a load cellxed atop the piston. The pressure upstream of the die entry wasalculated from the cross-sectional area of the barrel assum-ng insignificant pressure loss in the barrel itself because of thearge barrel diameter compared to the average diameter of theapidly converging hyperbolic die. For some flow rates a pres-ure transducer was also used to measure directly the pressuret approximately 25 mm upstream of the die entry. It was foundhat the two pressure readings differed by less than 10%. Theow curves at various temperatures were horizontally shiftedsing the same shift factors as obtained from the dynamic datao obtain a master curve.

CFD simulations were performed using the commercialolyflow finite element code (Version 3.10.4 of Fluent Inc.,SA). The semi-hyperbolic die geometry was modeled andeshed using GAMBIT (Fluent Inc., USA). The computational

eometry mirrored the hyperbolic die used for experimentalressure drop measurements in that it had a straight cylindri-al entry section of 10 mm radius and 10 mm length followed byhe hyperbolic profile. The straight section represented the barrelf the capillary rheometer. A few simulations were performedor higher flow rates using a barrel length of 30 mm in order toheck the effects of entrance flow. It was found that the resultsbtained for barrel lengths of 10 and 30 mm were identical. Theesh, the axial coordinates and the boundary conditions used

n simulations are shown in Fig. 2. The origin of the coordinate

xes was located on the axis of symmetry at the die exit. Thus,= 0, 25, 35 mm indicates the exit, entry into the hyperbolic diend entry into the straight barrel section, respectively. It may beoted that the choice of the origin in the simulations is different

ig. 2. Mesh and boundary conditions used in Polyflow simulations for the dief εh = 7.

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dttsdbetween the experimental and simulated values were obtainedusing β = 0.002–0.0002 for flow rates less than 100 mm3/s. Forflow rates higher than this value the experimental pressure dropwas found to be lower than the simulated values; this may be

A. Pandey, A. Lele / J. Non-Newto

han that assumed in Eq. (1), where z = 0 implied the entrance tohe hyperbolic geometry. A quadrilateral mapping scheme wassed for meshing. Grid skewness was restricted to 0.629 with lesshan 0.39% mesh elements having equi-angle skewness of morehan 0.5. The simulation method used a quadratic representationf the velocity field and a linear-continuous representation of theressure field. The boundary conditions used were: symmetryondition along the centerline, no-slip condition at the die wall,inflow’ condition at the inlet to the straight barrel section and anoutflow’ condition at the exit of the hyperbolic die. The inflownd outflow conditions imply the imposition of a fully developedow at the entry and exit cross-sections, which implies that theradients of all flow variables except pressure are forced to beero at these boundaries. This is equivalent to attaching suffi-iently long pipes of constant diameters at the entry and exit inrder to ensure fully developed flow conditions. All simulationsere run in two steps. The first step is an evolution step in which

he viscoelastic contribution in the constitutive equation is builttep wise to ensure convergence. The second step uses the con-erged solution of the first step as an initial guess and runs ateady-state simulation until final convergence is achieved. Inolyflow, convergence is based on the calculation of a globalelative error for each field (pressure, velocity, etc). The relativerror is calculated as the ratio of the difference in the value ofgiven field at every node between two successive iterations

nd the maximum value of the field. The convergence crite-ia used in our simulations were 0.01 for evolution and 0.005or steady-state simulation. Preliminary simulations were per-ormed to check the sensitivity of mesh density to calculatedata. The total CPU simulation times (evolution + steady state)n a 32 bit, 2.66 GHz, P4 processor machine for meshes of 243,76, 900 and 1296 elements were 125, 402, 682 and1157 s. Aesh of 1296 elements was chosen to ensure grid insensitivity

o simulated values of pressure drop, centerline elongation ratend normal stresses as well as to ensure convergence.

Transient uniaxial elongation viscosity data was kindly sup-lied to us by Dr. Martin Sentmanat. The measurements werearried out using an SER accessory [18,19] attached to a Rheo-etrics RDA-II rheometer. Transient elongation viscosities were

alculated from torque values measured at 150 ◦C for elongationates of 0.5, 1, 3 and 5 s−1. The data at 0.5 s−1 elongation ratebtained from the SER was compared with data obtained inde-endently on an RME rheometer at the same elongation rate andt the same temperature. The RME data was kindly supplied tos by Dr. D.J. Groves and Prof. T.C.B. McLeish of the Universityf Leeds, UK. The two data sets agreed very well. Extensionaliscosity data at lower elongation rates of 0.01, 0.03, 0.1 and.3 s−1 were also supplied to us by the University of Leeds.

. Results and discussions

As mentioned earlier, our proposed methodology for extract-ng elongation viscosity of LDPE from pressure drop data across

he semi-hyperbolic die involves the use of CFD flow sim-lations in the die. We need a constitutive equation for theimulations and to this extent the strategy is limited by the modelsed. In this work we have adopted the K-BKZ equation with the

Fd

luid Mech. 144 (2007) 170–177 173

agner irreversible damping function [20]. The model is giveny

-- =∫ t

−∞

∑i

gi

λiexp

[−(t − t′)λi

]exp(−k√I − 3)B-- (t, t′) dt′

(7a)

nd

= βIB + (1 − β)IIB (7b)

Here, τ-- is the stress tensor, B-- the Finger tensor, {λi, gi} theiscreet set of relaxation spectrum (see Table 1), and IB, IIBre the first and second invariants of the Finger tensor, respec-ively. The model is characterized by two parameters, k and β.he parameter k governs the shear-thinning tendency of the fluid,hile the parameterβ influences the elongation behaviour. Elon-ation hardening is attributed to values of β→ 0, while β→ 1ives elongation thinning behaviour.

We determine k by fitting the predictions of the constitutiveodel under steady shear flow conditions to the complex viscos-

ty versus frequency data obtained from the mastercurve of thescillatory shear experiments (in doing so, we assume implicitlyhe validity of the empirical Cox–Merz rule) and to the pressurerop data in the hyperbolic die. Fig. 1 shows the model fit to theomplex viscosity data. The value of k chosen was 0.15. Slightlyigher values of k = 0.18–0.24 gave a better prediction at higherhear rates. However, the pressure drop data in the hyperbolicie is better predicted by k = 0.15.

Next, we obtain the value ofβ by fitting the simulated pressurerop values to the experimentally determined flow curve data forhe semi-hyperbolic die. Fig. 3 shows a comparison betweenhe experimental pressure drop data and the simulated pres-ure drop values for various values of β. The simulated pressurerop increases with decreasing values of β. Good comparisons

ig. 3. Comparison between experimental flow curve and simulated pressurerop data for various values of the parameter β.

174 A. Pandey, A. Lele / J. Non-Newtonian Fluid Mech. 144 (2007) 170–177

Fv

psnCltiw

wvavhmodeolctdcbdbpd

btsshTrt

Fs

tsIntldflggfdhtt

nqo

ig. 4. Comparison between RME, SER data on transient uniaxial elongationiscosity and K-BKZ model predictions for β = 0.002.

ossibly due to partial slip at the die wall at high flow rates. Fig. 3hows that for β = 0.0002 the simulated pressure drop values areearly identical to those predicted forβ = 0.002. Given the higherPU time required and increased difficulty in convergence for

ow values of β, we have chosen to use β = 0.002 in all of our fur-her simulations. We note that this value is close to zero, whichmplies that the melt should show strain hardening behaviourhile flowing in the semi-hyperbolic die as will be seen later.To check the accuracy of the values of k and β so obtained

e compare the predictions of the transient uniaxial extensioniscosity using Eqs. (7a) and (7b) with the SER and RME datas shown in Fig. 4. The extensional data clearly shows a lineariscoelastic regime at small times followed by a strong strainardening regime for all elongation rates used in the experi-ents. The comparison shown in Fig. 4 suggests that the value

f β = 0.002 does a reasonable job of predicting the extensionalata. We have further checked the sensitivity of the predictedlongation viscosity to the parameter β and find that as the valuef β decreases the elongation viscosity becomes increasinglyess sensitive to β. For values of β < 0.002 the elongation vis-osity does not change as much as it does for values of β higherhan this. This sensitivity is reflected in the way the pressurerop varies with β. However, we find that the elongation vis-osity is more sensitive to β than is the pressure drop, which isecause of the dominance of shear effects on the overall pressurerop in the die. We propose that an appropriate value of β cane obtained with reasonable confidence by fitting the simulatedressure drop values to the experimental pressure drop–flow rateata for the hyperbolic die.

We now focus our attention along the centerline of the hyper-olic die. Fig. 5(a) shows the simulated elongation rate alonghe centerline for a flow rate of 0.002 cm3/s, while Fig. 5(b)hows the corresponding first normal stress difference for theame flow rate. Also shown in Fig. 5(a) and (b) are results for a

4

ypothetical Newtonian fluid having a viscosity of 3 × 10 Pa s.he simulated centerline elongation rate showed three distinct

egimes (see Fig. 5(a)): Regime-I represents the entry region intohe die during which the elongation rate increases from zero at

c

ε

ig. 5. Evolution of (a) centerline elongation rate and (b) centerline first normaltress difference for a Newtonian fluid and LDPE melt.

he entry into the straight section of the barrel (z = 35 mm, nothown) to the expected value inside the hyperbolic die. Regime-I represents the region in the die where the elongation rate isearly constant, while Regime-III represents the region close tohe die exit (i.e., near z = 0 mm). In the third regime the simu-ations predict a sharp decrease in the elongation rate, which isue to the particular outflow boundary condition imposed on theow at the die exit. Clearly, the flow variables cannot have zeroradients along axial direction at the die exit in a convergingeometry. A more appropriate flow simulation should includeree surface flow outside the die instead of assuming a fullyeveloped flow at the exit. However, we neglect the exit effectsere since our primary interest was to investigate the flow insidehe die. Hence the only simulation results of interest to us arehose in Regime-II.

The centerline elongation rate in Regime-II for the Newto-ian fluid reached a constant value of 0.558 s−1 which agreeduantitatively with that obtained from the lubrication analysisf Newtonian flow in a hyperbolic die with no-slip boundary

onditions [21] given by

˙|centerline = 2Q

πR20L

[exp(εh) − 1] (8)

nian Fluid Mech. 144 (2007) 170–177 175

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gFe1dubdrecvec

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rTtfl

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η

ia

t

loeTotsaemt

Fhs

ftmhbS

η

vETtSggaiiieftTamp

afl

t

A. Pandey, A. Lele / J. Non-Newto

Note that the centerline elongation rate for lubrication flows twice that for the inviscid case given by Eq. (4b). Further,q. (8) is useful in providing a good guess to estimate the flow

ate required to achieve a desired centerline elongational rate foriscoelastic flow. The normal stress difference for the Newtonianuid also attained a constant value of 5.03 × 104 Pa, which whenivided by the elongation rate gave exactly the Trouton viscosityf the fluid (ηe = 3η) thus calibrating our simulations.

For the viscoelastic LDPE melt the simulated centerline elon-ation rate in Regime-II never truly attained a constant value.or the various flow rates used in this work the variation in thelongation rates in Regime-II were found to be of the order of0–12% around that given by Eq. (8). The first normal stressifference for the LDPE melt also did not reach constant val-es for any of the flow rates explored in this work, increasingy 60–80% in Regime-II thus indicating a non-linear depen-ence on the elongation rates. That the normal stresses did noteach constant values is not surprising since the LDPE melt isxpected to show a non-linear strain hardening behaviour, whichould prevent the elongational stresses from reaching a steadyalue during the residence time of the fluid in the die. In ourxperiments the mean residence time of the fluid in the die wasalculated from the volume (V) and the flow rate (Q) as

res = V

Q= πC

QlnL+ B

B(9)

The volume of the die used was V = 0.05 cm3 and the flowates used in our experiments were Q = 4.2 × 10−5 to 0.02 cm3/s.his gives a mean residence time of 2.5–1200 s, which is smaller

han the longest relaxation time of the fluid (see Table 1) for mostow rates except the smallest used in our experiments.

The local value of the first normal stress difference and theorresponding elongation rate at every axial location along theenterline can be used to calculate the local elongation viscositys per

e|z = Tzz − Trr

ε

∣∣∣∣z

(10)

The local axial coordinate at which the elongation viscositys calculated can be converted into the life time of a fluid elementlong the centerline by using

z =∫ L−z

L

dz′

Vz|centerline(11)

In Eq. (11) Vz|centerline is the axial velocity along the center-ine. The limits of the integral in Eq. (11) imply the assumptionf a zero reference time as being the time at which the fluidnters the hyperbolic geometry (i.e., t = 0 when z = L = 25 mm).he choice of this reference point is somewhat arbitrary and webserved that a horizontal shift factor (a) was necessary to obtainhe correct ‘transient’ viscosity data using Eqs. (10) and (11) ashown later. The shift factor in a way accounts for the finite,

lbeit small, deformation that the melt experiences before itsntry into the die. Indeed our simulations show a distinct defor-ation of the streamlines in the region of the barrel just above

he entrance of the die. Feigl et al. [7] also used a shift factor

cttf

ig. 6. Comparison between simulated centerline elongation viscosity in a semi-yperbolic die and RME, SER transient uniaxial elongation viscosity data. Alsohown is the effective elongation viscosity calculated using Eq. (3).

or matching the effective elongation viscosity obtained fromheir simulations with viscosity predictions of the constitutive

odel. Better ways of estimating the shift factors are needed,owever, for the present we have chosen to identify shift factorsy comparing the centerline elongation viscosity data with theER and RME data as described below.

Fig. 6 shows predictions of the local elongation viscositye|z in the semi-hyperbolic die obtained from Eq. (10) plottedersus atz where tz is the local residence time obtained fromq. (11). These are compared with the SER and RME data.he flow rates in the simulations were chosen so as to match

he average centerline elongation rates with those used in theER and RME experiments. Eq. (8) was used as a guide touess the flow rates, which were then refined slightly so as toive a maximum of 6% root mean square error between theverage simulated elongation rates (in Regime-II) and those usedn the SER and RME experiments. The simulated values shownn Fig. 6 correspond to only the Regime-II data; the data pointsn Regime-I and Regime-III are not plotted for reasons explainedarlier. The horizontal shift factors (a) were found to increaserom 1.1 to 2.0 as the centerline stretch rate increased from 0.01o 5 s−1 thus showing a weak dependence given by a ε0.09

centre.he appropriately shifted simulated centerline viscosity captureslmost quantitatively the strain hardening nature of the LDPEelt, which is a direct result of the fact that the value of the

arameter β is close to zero as described earlier.Fig. 6 also shows the plot of the effective elongation viscosity

s calculated from Eq. (3) versus the mean residence time of theuid calculated as

res|inviscid = V

Q= πC

QlnL+ B

B= εh

ε(12)

The numbers shown adjacent to each data point denote the

orresponding stretch rates calculated from Eq. (4b) for each ofhe flow rates used in our experiments (see Fig. 3). It is interestingo note that if the transient elongation viscosity data obtainedrom the SER experiments were to be extrapolated to longer

1 nian F

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lphwEl0dctttdddTtd

76 A. Pandey, A. Lele / J. Non-Newto

imes then the effective elongation viscosity data, when slightlyhifted horizontally, would appear to lie on the SER data curves.imilar trends were also observed by Edwards et al. [22]. We find

his surprising because the calculation of effective elongationiscosity intrinsically assumes a full slip boundary conditionhereas our experimental pressure drop–flow rate data does not

ndicate wall slip, except perhaps a partial slip at the higher flowates. Therefore, the observed apparent matching between theffective elongation viscosity and the SER or RME data may beortuitous. However, this deserves a more detailed study.

Fig. 6 shows that for a given hyperbolic die, a realistic com-arison between the RME/SER data and the simulated centerlinelongation viscosity can be made only over a limited time range.ealizing that this time span is a fraction of the total resi-ence time of the fluid in the die, we can extend the span bysing hyperbolic dies of different Hencky strains. Comparisonsetween effective elongation viscosity obtained from hyperbolicies of various Hencky strains and transient elongation viscosity

btained from a Meissner elongation rheometer were reportedy Edwards et al. [22]. Fig. 7 shows the results of simulationserformed on different hyperbolic dies of Hencky strains 1, 2,, 5, 7, and 9. All dies had the same entry radius of 10 mm and

ig. 7. (a) Simulated centerline elongation rates (approx ε = 0.3 s−1) in variousyperbolic dies, and (b) corresponding centerline elongation viscosity.

tioe

4

eativoMdarttostdasetw

A

wuTeti

luid Mech. 144 (2007) 170–177

ength of 25 mm, while their exit radii vary and consequently, thearameters B and C (see Eqs. (2a) and (2b)), which describe theyperbolic profile, were different. Flow simulations in the diesere performed at flow rates that were suitably chosen (usingq. (8)) so as to give similar elongation rates along the center-

ines of each die. These flow rates were 0.8, 0.215, 0.075, 0.009,.00123, and 0.000142 cm3/s, respectively, for the hyperbolicies of Hencky strains mentioned earlier. The comparison of theenterline elongation rates is shown in Fig. 7(a), while the cen-erline elongation viscosity is shown in Fig. 7(b). In Regime-IIhe centerline elongation rates in all dies are approximately equalo 0.3 s−1. Since the centerline elongation rate in a hyperbolicie at any given flow rate is not exactly constant along the flowirection, variations in the elongation rates between the dies ofifferent Hencky strains were maintained to be less than 20%.he elongation viscosity had to be shifted horizontally in order

o obtain nearly quantitative comparison with the RME/SERata over a wider time span as shown in Fig. 7(b). The shift fac-ors were found to increase from 1.0 to 1.6 as the Hencky strainncreased from 1.0 to 9.0, thus showing a stronger dependencen the Hencky strain (a ε0.3

h ) as compared to the dependence onlongation rate given earlier.

. Conclusions

In this work we have explored the utility of a simple strat-gy that employs a combination of pressure drop measurementscross a hyperbolic die and matched CFD simulations to estimatehe transient uniaxial elongation viscosity of a strain harden-ng LDPE melt. We have shown that the transient elongationiscosity so obtained compares almost quantitatively with databtained from the more sophisticated RME/SER rheometers.ore importantly, the calculation of the elongation viscosity

oes not need the assumption of the unrealistic full slip bound-ry conditions at the die wall. Although the proposed strategy iselatively simple to implement it has several drawbacks. First,he methodology is dependent on the constitutive model used forhe simulations. Thus, the quality of the data so obtained will benly as good as the model itself. However, an advantage of thetrategy is that all model parameters can be obtained self consis-ently from viscometric shear flow measurements and pressurerop data across the hyperbolic die without the need to performny additional measurements on an elongation rheometer. Theecond drawback is that a CFD code is required to obtain thelongation viscosity data from this strategy. The third disadvan-age concerns the need for a horizontal shift factor, for whiche do not have as of now a practical estimation method.

cknowledgements

We are grateful to Dr. Martin Sentmanat for providing usith the transient uniaxial elongation viscosity data obtainedsing the SER. We are also grateful to Dr. D.J. Groves and Prof.

.C.B. McLeish for supplying us the RME data. We acknowl-dge the generous support from Fluent Inc. for providing licenseo POLYFLOW. We also thank the anonymous referees for theirnsightful comments on our manuscript.

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