A numerical study of the measurement of elongational viscosity of polymeric fluids in asemihyperbolically converging die∗

K. Feigl1,†, F.X. Tanner1, B.J. Edwards2 and J.R. Collier21 Michigan Technological University, Department of Mathematical Sciences, 1400 Townsend Drive, Houghton, MI 49931-1295, USA

2 The University of Tennessee, Department of Chemical Engineering, Knoxville, TN 37996-2200, USA(July 29, 2003)

A method for measuring the elongational viscosity of polymer melts and solutions has been generalizedand evaluated by means of numerical simulations. The method involves passing a material through a cylindri-cal, converging die whose semihyperbolic shape mandates a shear-free, or nearly shear-free, flow within thedie, assuming wall slip. From the analysis of the relevant flow equations in the die, an expression for elonga-tional viscosity is derived under less restrictive conditions than in previous studies. This expression consistsof two terms, one of which is a measurable effective elongational viscosity defined in terms of the change inpressure over the die, the volumetric flow rate and the Hencky strain determined by the geometry. To evaluatethis method, finite element techniques are used to calculate the flow of a low-density polyethylene melt in twosemihyperbolically converging dies. After confirming that purely elongational flow is produced within the die,assuming wall slip, the effective elongational viscosity is computed from the calculated flow field and thesevalues are compared with the values of elongational viscosity found by integrating the constitutive equation forthe material in elongational flow. Over the wide range of elongation rates considered, very good agreement isfound between these two sets of values when the time associated with the effective viscosity is appropriatelyspecified. Further, a similar analysis for a Newtonian fluid showed that the effective elongational viscosity sat-isfies the Trouton ratio over the range of elongation rates considered. These results indicate that the measuredeffective elongational viscosity is an excellent approximation to the material’s true elongational viscosity. Con-sequently, semihyperbolically converging dies can be used effectively to obtain transient elongational viscositymeasurements at constant strain or constant strain rate.

Keywords: Elongational viscosity, semihyperbolically converging die, polymer melts, finite element method,integral constitutive equation

∗To appear in Journal of Non-Newtonian Fluid Mechanics†Corresponding author. Fax: 906-487-3133. Email: feigl@mtu.edu

1

I. INTRODUCTION

A major challenge in polymer rheology is the characterization of polymer melts and solutions in elongational flow fields. Theimportance of such a characterization stems from the fact that shear rheometry alone is inadequate to describe polymer behaviorin most types of flow fields. This, combined with the fact that most flows encountered in the processing of polymers are eithermixtures of shear and elongational flow or, as in the case of fiber spinning, injection molding and film blowing, consist primarilyof elongational flow, makes elongational rheometry a necessity for the polymer processing industry. Likewise, elongationalrheometry is important to those working in the theory, modeling and simulation of polymeric liquids. Knowledge of materialproperties, such as elongational viscosity, are needed by these researchers to build and test models and theories and to confirmcomputer simulations of various flow processes.

Despite the great need for elongational measurements, elongational rheometry has lagged behind shear rheometry. This isdue to a variety of difficulties encountered in the former, such as the difficulty to generate a steady and controlled elongationalflow field, the difficulty to compensate for any shear effects which may simultaneously occur, and difficulties in sample produc-tion. Nevertheless important contributions to elongational rheometry have been made by several research groups which havedeveloped and/or improved apparatus and flow analyses for the measurement of elongational properties. (See [1,2] for a reviewof experimental methods). Notable is the work of Meissner and coworkers [3] – [5] who developed rheometers to measure theelongational viscosity of polymer melts in uniaxial, biaxial and planar elongational flow.

Perhaps more widely used—but more controversial—techniques for measuring elongational viscosity involve the use of dieentrance flows in which an elongational viscosity, or effective elongational viscosity, is defined in terms of one or more pressuredrops, the volumetric flow rate and other material and/or geometrical parameters. These techniques are attractive because theyonly require a capillary-type rheometer, but they are associated with some of the difficulties mentioned above.

Techniques based on abrupt die entrance flow were first developed by Cogswell [6] and Binding [7,8] and, over the years,these techniques and their underlying assumptions and simplifications have been extensively investigated, modified and debatedin the literature. In particular, the work of Gotsis and Odriozola [9] made significant advancements toward understanding theusefulness of abrupt and tapered die entrance flow in extracting elongational viscosity data. Their approach relies on a modifiedBinding analysis with simple power-law models for the shear viscosity and first normal stress coefficient (as a function of shearrate) and a power-law model for the extensional viscosity (as a function of strain rate). Empirical evidence that the latter modelis valid for a limited range of strain rates, and provided that the total strain is kept constant, was given for several materials in theform of data taken in a Meissner-type device. The parameters in the power-law model for elongational viscosity are determinedgiven the parameters in shear models, the geometrical parameters of the die, the measured entrance pressure drop and the flowrate. By integrating the elongational viscosity data taken from the Meissner-type device over strain, the authors then interpretedthis die entrance data as an average, over strain, of the material’s true transient elongational viscosity. For two of the threematerials considered and over certain ranges of strain rate, these two sets of data compared relatively well as a function of strainrate.

Very recently, Zatloukal et al. [10] proposed and verified improvements to abrupt die entrance techniques to eliminate some ofthe deficiencies reported in the previous investigations. Specifically, these authors proposed a model for their newly introduced“entrance viscosity” parameter to account for observed entrance effects, along with a so-called “effective entry length correc-tion” to improve the predictions of the abrupt die entrance techniques at low strain rates. Their simulations showed that thesemodifications lead to improved determination of elongational viscosity.

Converging die entrance geometries have also been used to extract elongational viscosity data. Both James et al. [11,12] andCollier and coworkers [14] – [17] proposed the utilization of a converging die whose shape is defined byR2z= constant, whereR is the radius of the die at axial positionz. This semihyperbolic shape allows an elongational flow field to be produced withinthe die, under conditions to be elucidated in this study. An essentially shear-free flow field results if the significant shear effectsare restricted to a thin boundary layer along the wall due to, for example, a sufficiently high Reynolds number or sufficientwall slip. Collier et al. [17] have shown that boundary slip is likely occurring in these dies since they have observed sharkskinextrudates from the dies as well as the total irrelevance of a lubrication skin layer between the bulk polymer and the walls of thedie.

Although both research groups used the semihyperbolically converging die as an elongational rheometer, their analysis andderived expression for the elongational viscosity differed. The validity of the assumptions and analysis of James et al. hasbeen investigated by Park et al. [13] using numerical simulations of the fluid M1 in the converging die entrance geometry. Thislatter group of researchers reported a lack of agreement between their simulations and the measured elongational viscositiesobtained by James et al. [12]. In fact, they noted that the two sets of curves exhibited different trends. Park et al. attributed thesediscrepancies to the overestimation by James et al. of one term in their elongational viscosity expression.

Under clearly defined assumptions, Collier et al. [17] derived from the flow equations an expression for the elongationalviscosity involving the sum of a measurable “effective” viscosity and a term involving the change in enthalpy. The effectiveelongational viscosity is a function of the pressure drop over the die, the volumetric flow rate, and the Hencky strain value of thedie geometry. Collier et al. assumed that the constitutive equation for the fluid’s extra-stress tensor was a generalized Newtonian

2

one, and then supplemented the Cauchy momentum equation with a so-called “orientation body force” representing the elasticcharacter of the fluid. Herein, this assumption is removed and the analysis is shown to be valid for a very general and thoroughlyviscoelastic constitutive equation for the extra-stress tensor.

The purpose of this paper is twofold. First, the expression used by Collier et al. [17] in the determination of the elongationalviscosity has been re-derived under less restrictive conditions which are valid for a general viscoelastic fluid. Second, numericalsimulations have been performed in order to test the assumptions and analysis, and to illustrate that the enthalpy term in theexpression for the elongational viscosity can be neglected and that the measured effective elongational viscosity represents anexcellent approximation to the true elongational viscosity of a material. Specifically, finite element techniques are used to solvefor the flow of a low-density polyethylene (LDPE) melt in two semihyperbolically converging dies, corresponding to Henckystrains of 6 and 7. Assuming slip along the walls of the die, it is first confirmed that the desired elongational flow field is producedwithin the die. The effective elongational viscosity is then computed from the calculated solution and these values are comparedwith values of elongational viscosity found by integrating the constitutive equation used for the LDPE melt in elongational flow.

Because of the time effects in viscoelastic fluids, the stress, and hence elongational viscosity, within the converging die isgenerally transient, not steady. For a typical residence time,tR, of a fluid particle passing through the die, the fluid will nothave achieved a steady state by the time it exits unless its largest relaxation time is significantly less thantR. The measuredelongational viscosity must therefore be interpreted as a value at one particular time. As described in this paper, the theorydictates that this time corresponds to the time at which the mean extra-stress or elongational viscosity is reached within theconverging die, assuming that constant-rate elongational flow is instantaneously reached within the die. In practice, this is nota useful way to determine the appropriate time, in part because it requires knowledge of the evolution of the extra-stress withinthe die and in part because constant-rate elongational flow is not instantaneously achieved within the die. From the simulations,two ways of specifying this time are indicated, one of which is attractive from a practical point of view. Both ways result inexcellent agreement between the effective elongational viscosity computed from the simulations and the viscosity computed bydirect integration of the constitutive equation over the range of strain rates considered, from 0.1s−1 to 25s−1.

Furthermore, simulations of a Newtonian fluid, where time effects are absent, showed that the effective elongational viscositysatisfies the Trouton ratio. That is, the effective elongational viscosity was three times the specified shear viscosity over therange of elongation rates considered.

The remainder of the paper is organized as follows. Section II provides the theoretical background of the semihyperbolicallyconverging die and the analysis of the flow equations from which the expression for elongational viscosity was derived. Thenumerical methodology used to evaluate this analysis is given in Section III, while the results and a discussion follow in Sec-tion IV. Although no comparison is made between simulations and experimental data in the present paper, Section V showsthat experimental data taken from a different LDPE melt follow the same trend seen in the simulations. In particular, previouslyreported experimental data for viscosity versus elongation rate from [18] in dies of varying Hencky strain are converted intotransient viscosity curves at constant strain and constant strain rate. A summary in Section VI concludes the paper.

II. THEORETICAL BACKGROUND

A. Flows of Constant Elongation Rates

The elongational viscosity of a fluid is defined for an elongational (shear-free) flow as

ηe =τ11− τ22

ε,

whereτ11 is the normal stress in the flow direction,τ22 is the stress in the transversal direction andε is the elongation rate. Themeasurement ofηe is a challenging task, which is considerably simplified if the elongation rateε is constant throughout theregion of investigation. A device which satisfies this requirement is the semihyperbolically converging die. In the following,the basic idea behind this device is reviewed. In particular, it is shown that an incompressible, irrotational tubular flow with aconstant gradient in the main flow direction results in a purely elongational flow of constant elongation rate and that the geometryof this tube must correspond to a semihyperbolically converging die. The analysis is presented in the axisymmetric cylindricalcoordinates(r,θ,z), where the azimuthal derivative∂∂θ and the azimuthal velocity,vθ, are zero.

Consider a flow with a constant gradient in the main flow direction such that

ε =∂vz

∂z, (1)

whereε is the elongation rate. Ifε is a constant over the entire domain of interest, then the velocity component in thez-direction,vz, is given by

3

vz = εz+ f (r), (2)

where f (r) is a function ofr only. The general form for the radial velocity component,vr , follows from the incompressibility

assumption,1r∂(rvr )

∂r + ∂(vz)∂z = 0, and is

vr = − εr2

+1r

g(z), (3)

whereg(z) is a function ofzonly.If, in addition, the flow is irrotational, i.e.,∇×v = 0, then the velocity components are given by

vr = − εr2

(4)

vz = εz+d, (5)

whered is a constant representing the initial axial velocity. Equations (4) and (5) are obtained by virtue of

(∇×v)θ =∂vr

∂z− ∂vz

∂r= 0,

which leads to the relation

r f ′(r) = −g′(z).

Since the left hand side of this equation is only a function ofr, it follows that r f ′(r) = c for some constantc, and henceg′(z) = −c. Integration of these two equations yields

vr = − εr2

+cln r +dr

vz = εz−cz+dz,

wheredr anddz are constants of integration. Now, sincevr is finite whenr = 0, it follows that the constantc = 0. Further, sinceby symmetryvr = 0 whenr = 0, the constantdr = 0, and Eqs. (4) and (5) follow.

Observe that a direct consequence of Eqs. (4) and (5) is that the flow is purely elongational. Indeed, the rate-of-strain tensor(in cylindrical coordinates) is

γ = ∇v+(∇v)T =

−ε 0 0

0 −ε 00 0 2ε

,

which proves the claim.Next, it is shown that the shape of such a tube must correspond to a semihyperbolically converging die. The volumetric flow

rate,Q, through a tube of varying cross-section with radiusR= R(z), is given by

Q = 2π∫ R

0vzrdr = πvzR

2

becausevz is assumed constant over the cross-section. Consequently, using Eq. (5), one obtains the general equation for thetubular radius,

R2(z) =C

z+B, (6)

where the constantsC andB are determined from the inlet radius,Ro, atz= 0, and the exit radius,Re, atz= L. They are givenby

C = LR2

oR2e

R2o−R2

e(7)

B = LR2

e

R2o−R2

e. (8)

4

In order to simplify the following analysis, a shift in thez-coordinate byz+B is introduced. This places the die entry atzo = Band the exit atze = zo +L. In these coordinates, the equation for the semihyperbolically converging die becomes

R2(z) =Cz

(9)

and the velocity field is the same as in Eqs (4) and (5) withd = 0,

vr = − εr2

(10)

vz = εz. (11)

The previous discussion shows that a steady, incompressible, irrotational tubular flow with a constant elongation rate through-out the flow requires the shape given by Eq. (9). Conversely, a steady, incompressible, irrotational flow through a semihyperbol-ically converging die defined by Eq. (9) exhibits a velocity field given by Eqs. (10) and (11), and therefore, according to Eq. (1),the flow has the constant elongation rateε. This statement follows from the fact that the stream function,Ψ(r,z) = −εr2z/2,of an axisymmetrical incompressible flow satisfies the Laplace equation and the boundary conditions of the semihyperbolicallyconverging die, and consequently gives the desired velocity field by means of the relationsvr = 1

r∂Ψ(r,z)

∂z andvz = − 1r

∂Ψ(r,z)∂r .

A direct consequence of this analysis is that the axial velocity,vz, must be constant over the entire cross-section (cf. Eq. (11)).This is of course only satisfied for full-slip boundary conditions. In practice, the above analysis will still be a good approximationfor flows with nearly constant velocity profiles, such as in flows with boundary slip or in flows with a very thin wall shear layer.

B. Determination of the Elongational Viscosity

The analysis presented in this section leads to an expression which suggests a simple experimental determination of theelongational viscosity,ηe, by means of a semihyperbolically converging die. This expression is formally identical to the onederived by Collier et al. [17], but it uses less restrictive assumptions. In particular, the analysis is conducted directly for a generalextra-stress tensor without resorting to the concept of a so-called “orientation body force,” or body force representing the forcenecessary to orient the material [17]. Furthermore, a careful discussion of the time-dependence of the derived elongationalviscosity expression is now included.

The basic equations which describe the flow under consideration are the conservation equations for mass, momentum andenergy given by

DρDt

= −ρ∇ ·v (12)

ρDvDt

= −∇P+ ∇ ·τ (13)

ρDHDt

= −∇ ·q +τ : ∇v +DPDt

, (14)

whereρ is the mass density,v the velocity,P the pressure,τ the extra-stress tensor,H the enthalpy per unit mass,q the heatconduction per unit volume, andDDt = ∂

∂t + v · ∇ is the substantial derivative. In these expressions, the pressure,P, is notthe thermodynamic pressure. Under the divergence-free condition, the pressure takes on the significance of a scalar field, thegradient of which guarantees satisfaction of the imposed constraint. Hence the scalar contribution to the extra-stress tensor,τ ,i.e., its trace, is effectively incorporated into the pressure. Thus, without loss of generality,τ can be taken as traceless.

The conservation equations are subjected to the following assumptions whose justifications are discussed in Collier et al. [17]:

1. The flow is steady (∂∂t = 0), incompressible (∇ ·v = 0), irrotational (∇×v = 0), isothermal (∇ ·q = 0) and shear-free.

2. The inertial terms are negligible which impliesρ DvDt = 0.

In contrast to the assumptions on the extra-stress tensor stated in [17] which lead to∇ ·τg = 0, whereτg denotes the generalizedNewtonian stress tensor, the only condition on the extra-stress tensor in this study is:

3. τrr = τθθ. Hence, because the trace ofτ is zero,τrr = −τzz/2.

5

In the semihyperbolically converging die, Assumption 1 implies that constant-rate elongational flow defined by Eqs. (10) and(11) is strictly imposed. Therefore, the analysis below assumes the flow is strictly elongational flow with constant strain rateε.

Since Assumption 3 is always satisfied in elongational flow, a direct consequence of this assumption is

τ : ∇v =32τzzε. (15)

Consequently, under Assumption 1 and Eq. (15), the energy equation Eq. (14) becomes

32

τzzε = (v ·∇)(H −P)

= − ε2

r∂∂r

(H −P)+ εz∂∂z

(H −P), (16)

whereH = ρH is the volume-specific enthalpy and, in general,τzz= τzz(r,z), H = H(r,z) andP = P(r,z).At this point, Collier et al. [17] made two additional assumptions to simplify Eq. (16), namely,

• ∂P∂r = 0, which follows from an argument involving the integral of the general pressure expression, and

• |vr∂H∂r | � |vz

∂H∂z |, which follows from a similar integration argument and other flow considerations.

Under these assumptions, Eq. (16) reduces to

∂∂z

(H −P) =32

τzz

z. (17)

A similar simplified expression can be derived without these assumptions if the energy equation is written in streamlinecoordinates,(r(z),z), wherer(z) =

√C/z is a streamline for a nonnegative constantC. Specifically, along a streamline, Eq. (16)

can be written as

ddz

(H −P) =32

τzz

z, (18)

whereH = H(r(z),z), P = P(r(z),z) andτzz(r(z),z). Integrating Eq. (18) along the length of the converging die yields

∆H −∆P=32

∫ ze

zo

1z

τzzdz=32

ε∫ te

toτzz(t)dt, (19)

where∆H = H(ze)−H(zo), ∆P = P(ze)−P(zo) < 0 and the subscriptso ande denote the die entrance and exit, respectively.The last integral in Eq. (19) is over the residence time,t = t(z), of a fluid element in the converging die, wheret(z) = to +(1/ε)ln(z/zo). This expression fort(z) is a consequence of Eq. (11).

If τzz is constant in the elongational flow within the converging die, as it is for a generalized Newtonian fluid, then eitherintegral in Eq. (19) can be evaluated analytically to get an expression forτzz and hence for the constant elongational viscosity.For a viscoelastic fluid, the stress and elongational viscosity are generally time-dependent. If the relaxation times of a viscoelasticfluid are sufficiently small, then the above approach represents a good approximation given that the stresses reach constant valuesquickly within the die. However, in general, the stress of a viscoelastic fluid must be considered to be transient within the die.Since it is reasonable to assume thatτzz is continuous in the intervalto ≤ t ≤ te, by virtue of the Mean Value Theorem, Eq. (19)can be written as

∆H −∆P=32

ε(te− to)τzz(t) (20)

wheret ∈ (to, te) is the time at whichτzz achieves its average value in this time interval.Regardless of whether or not extra-stress is constant within the die,τzz takes the form

τzz=23

∆H −∆Pεh

, (21)

whereεh = ε(te− to) is the Hencky strain, which is equivalent to

εh = ln(Ao/Ae) = ln(R2o/R2

e) = ln(ze/zo), (22)

6

whereA represents the cross-sectional area. Furthermore, from the velocity field given in Eqs. (10) and (11), it follows that theresidence timetR = te− to is the same along all streamlines and may be written as

tR =1ε

ln(R2o/R2

e) =1ε

ln(ze/zo) =εh

ε. (23)

To be consistent with the start-up of elongational flow, the entry time is set toto = 0, implying thatte = tR.By Assumption 3 and Eq. (21), the elongational viscosity in cylindrical coordinates is given by

ηe =τzz− τrr

ε=

3τzz

2ε=

∆H −∆Pεεh

. (24)

Observe that the pressure,P, and the extra-stress tensor,τ , are related by the momentum equation,∇ · τ = ∇P, whereτ isdescribed by an appropriate constitutive equation.

If the effective elongational viscosity,ηef, is defined as

ηef = −∆Pεεh

, (25)

thenηe can be written as

ηe = ηef +∆Hεεh

. (26)

For a viscoelastic fluid,τzz, τrr , ηe andηef in Eqs (21),(24)–(26) represent the values of these quantities at timet ∈ (0,tR) atwhichτzz achieves its average in the interval 0≤ t ≤ tR. They may also be interpreted as values of these quantities after applyinga strain ofε = εt to the fluid, since a constantε is assumed. In general, sincet < tR, it follows that this applied strain,ε, isless than the Hencky strain of the die,εh, that is, less than the total strain experienced by the fluid as it passes through the die.However, as previously indicated, if the relaxation times of the fluid are small enough so that the stress reaches its steady valuewithin the die sufficiently quickly, then one may taket ≈ tR, so thatε ≈ εh.

Finally, sinceve = εze = ε(zo +L) = vo + εL, and because of mass conservation,ve = vo(Ao/Ae) = voexp(εh), the elongationrate can be expressed in terms of geometric quantities and the initial velocityvo (or equivalently by the volumetric flow rateQ),as

ε =vo

L(exp(εh)−1) =

QπR2

oL(exp(εh)−1). (27)

If ∆H is negligible, i.e.,ηe≈ ηef, then Eq. (26) suggests a method for determining the elongational viscosity, by measuring thepressure difference in a flow within a semihyperbolically converging die, whereε is given by Eq. (27). In the actual rheometer,fluid is pushed from a tube of constant radius into the semihyperbolically converging die. Strictly speaking, the assumptionsand analysis described above are valid in this rheometer only if constant-rate elongational flow defined by Eqs (10) and (11) isinstantaneously reached upon entry of the fluid into the die. The validity of these assumptions and analysis, together with theapproximationηe ≈ ηef, is corroborated by means of quasi-three-dimensional simulations of Newtonian and viscoelastic fluidsin the following sections.

III. NUMERICAL METHODOLOGY

The assumptions and analysis given in the previous section are evaluated by solving numerically for the flow of a polymericfluid in the semihyperbolically converging die by means of finite element techniques. The material considered is a low-densitypolyethylene (LDPE) melt which has been well characterized in [19] – [23].

A. Fluid and constitutive equation

In the numerical approach, the governing system of equations consists of the conservation equations of mass and momentumfor an incompressible, isothermal flow (cf. Eqs. (12) and (13)), together with a constitutive equation which models the extra-stress tensor,τ . The extra-stress tensor is taken to have the form of the factorized Rivlin-Sawyers [25] constitutive equation,given by

7

τ (t) =∫ t

−∞

[h1(I1, I2)

{B(t, t ′)−δ

}+h2(I1, I2)

{C(t, t ′)−δ

}]m(t − t ′) dt′. (28)

The second-order tensorsB andC = B−1 are the Finger strain and Cauchy strain tensors, respectively, andδ is the second-orderunit tensor. The damping functions,h1 andh2, are strain-dependent functions of the invariants,I1 = tr(B) andI2 = tr(C), of theFinger tensor. The time effects are expressed in the Maxwell linear viscoelastic memory functionm(t − t ′) given by

m(t − t ′) =K

∑k=1

ηk

λ2k

e−(t−t′)/λk , (29)

whereλk andηk, k = 1, ...,K, are a set of relaxation times and partial viscosities for the material.Under the assumption that the ratio,θ, of second normal stress difference,N2, to first normal stress difference,N1, is inde-

pendent of strain, i.e. thatθ ≡ N2/N1 is constant, it follows that the damping functionsh1 andh2 are proportional, and can berelated to a damping functionh(I1, I2) by

h1(I1, I2) = (1+ θ)h(I1, I2)h2(I1, I2) = θh(I1, I2) .

In the caseN2 ≡ 0, thenθ = 0 and the second term in Eq. (28) vanishes. The following two-parameter expression forh(I1, I2)proposed by Papanastasiou et al. [22] is used

h(I1, I2) =α

α−3+ βI1+(1−β)I2, (30)

whereα is a parameter governing shear flow andβ is a parameter governing elongational flow.The fluid used in this study is a low-density polyethylene (LDPE) melt at 160◦C. This fluid was characterized rheologically

by Meissner [19,20] and Laun [21] who collected experimental data. All parameters in the constitutive equation, Eq. (28), werefound by fitting them to rheological data taken from the LDPE melt. The relaxation spectrum{λk,ηk} in Eq. (29) was foundby fitting these parameters to linear viscoelastic data, in particular the storage and loss moduli measured in small amplitudeoscillatory shear flow. Table I shows the relaxation spectrum derived for the LDPE melt at 160◦C [21,23]. The parameters inthe damping function, Eq. (30), were found by fitting these parameters to shear and elongational data. They were found to beα = 14.38 andβ = 0.018 [22,24]. The stress difference ratio was taken to beθ = −0.10.

Figure 1 shows some predictions of the above constitutive equation to the rheological behavior of the LDPE melt in steadyshear and elongational flow. In particular, the model predicts that a steady elongational viscosity is attained for all elongationrates.

B. Finite element method

The conservation equations for mass and momentum, together with the constitutive equation, Eq. (28) and appropriate bound-ary conditions, are solved using a finite element method (FEM) described by Bernstein and coworkers [26] – [28]. The basicalgorithm decouples the calculation of the extra-stress tensor from the calculation of the velocity and pressure fields in the fol-lowing two-step iterative manner: in one step, the extra-stress tensorτ in Eq. (28) is computed from a known discrete velocityfield. In the second step, velocity and pressure fields are updated by solving the conservation equations where the extra-stresscomputed in the first step enters the momentum equation as a known pseudo-body force. Therefore, the extra-stress must becomputed at all Gaussian quadrature points inside each element.

The finite element used in this study is the crossed triangle macroelement which consists of a quadrilateral divided intofour triangles formed by the diagonals of the quadrilateral. The velocity approximation is piecewise linear on the trianglesand continuous over the domain (P1−C0), and the pressure approximation is piecewise constant on the triangles (P0−C−1).In axisymmetric geometry, the velocity approximation is valid for the so-called “reduced” velocity field as described in [27]and [28]. After calculation this reduced velocity field is transformed back to the true velocity fieldv. An essential featureof this crossed triangle element is that, given a discrete velocity field, the stress can be computed as accurately as desiredfrom Eq. (28). The reason is that the piecewise linear velocity approximation and resulting (almost everywhere) point-wiseincompressibility allow the particle paths, residence times and strain tensors needed in the stress integral to be determinedanalytically in each triangle. These quantities can then be extended triangle by triangle while a numerical integration scheme isapplied to evaluate the stress integral. As a postprocessing step, the triangle-wise constant pressure and stress fields, along withthe velocity gradient field, are projected onto the space of continuous bilinear polynomials whose nodes are the vertices of the

8

quadrilateral macroelements. Rigorous convergence results for the crossed triangle element have been established in both planarand axisymmetric geometry [28].

Once the FEM solution has been calculated, numerical particle tracking techniques (NPT) can be used to compute the shearand elongation rates along particle paths from the discrete velocity solution (see [29,30]). This involves computing the paththat a fluid element follows from a given initial position and calculating the velocity gradient at each computed path point. Byrotating a local coordinate system moving with the particle so that the 1-direction is the direction of flow and the 2-direction isthe direction in which velocity varies, the velocity gradient can be transformed into the form where the diagonal entries representthe elongation rates in the directions of the rotated axes, and the only nonzero off-diagonal entry is the shear rate. This gives

ε1 =12e1 · γ ·e1 , ε2 =

12e2 · γ ·e2 , γ = e1 · γ ·e2 (31)

whereγ = ∇v +(∇v)T is the rate of strain tensor and the basis vectors,e1 ande2, of the particle’s local coordinate system are

e1 =1

||v||(

v1v2

)=

(cosθsinθ

)and e2 =

1||v||

( −v2v1

)=

( −sinθcosθ

)(32)

whereθ is the angle of rotation in the two-dimensional computational domain.In the shear-free flow given in Eqs. (10) and (11), the elongation rateε corresponds to the elongation rate in the primary flow

direction,ε1, along thez-axis. Furthermore, except close toz= 0 in this flow, ε1 ≈ ε ≡ ∂vz/∂z since the streamlines becomenearly parallel to thez-axis.

As indicated above, one advantage of the crossed triangle element is that the equations for the particle path (or streamline)and the transit time of a particle in a given triangle can be solved analytically in each triangle given a discrete velocity field.The path can then be pieced together element-by-element, ensuring at least two particle points per element. The accuracy of ourFEM and NPT procedures has been illustrated by Feigl and coworkers [27,29,30], where good comparisons have been foundwith experimental data for various LDPE melts.

IV. RESULTS AND DISCUSSION

Two die-entry geometries were considered in this study. In the first, the radius,Ro, of the entry tube (or reservoir), the finalradius,Re, of the converging die and the axial length,L, of the converging die areRo = 10mm,Re = 0.5mm, andL = 25mm,respectively. In the second geometry,Ro andL remain the same, while the final radius of the converging die is changed toRe = 0.3mm. GivenRo, Re andL, the shape of the die follows Eqs (6)–(8). According to Eq. (22), the first geometry correspondsto a Hencky strain ofεh = 6 and the second to a Hencky strain ofεh = 7. In both cases, the axial length of the entry tube wastaken to be 30mm. It is assumed that the flow field in the die geometry is axisymmetric (noθ-dependence) and torsion-free(vθ = 0). The computational domain is therefore taken to be a two-dimensional slice through the centerline of the die geometry,where the centerline acts as a computational boundary. For each geometry, calculations were performed on two meshes. Themesh and geometrical parameters are summarized in Table II, and Figure 2 shows the computational domains. There was littleor no significant difference in the results on the two meshes.

The boundary conditions imposed were fully-developed flow along the inflow boundary and axisymmetric boundary condi-tions along the centerline. To simplify the imposition of a boundary condition along the outflow boundary, a tube of constantradius equal toRe was appended to the converging die. This tube was taken to have a length of 35mm, which was assumed longenough for a fully-developed profile to be specified along the outlet. Along the walls of the tubes, two types of boundary con-ditions were considered: in one case, no-slip was imposed along the entire wall, and in the other case slip was allowed to occurwithin the dies. The slip conditions used here were zero shear rate and zero normal velocity along the walls of the dies. Thezero shear rate condition was implemented in the layer of elements abutting the die walls, or more specifically along the edgesof these elements which were normal to the boundary. This was accomplished by forcing the value of velocity at a boundarynode to be the same as that of the adjacent interior node lying on the same normal to the boundary. In the present study, no otherspecific wall slip law was considered. The effects, if any, of the specific slip law used, the amount of slip (full or partial) andperiodic stick-slip conditions are the subjects of ongoing investigations.

A key assumption in the theoretical analysis presented in Section II is that the flow within the converging die is shear-free andhas velocity components given by Eqs. (10) and (11). The analysis should still provide a good approximation if the shear effectsare small or are restricted to a thin boundary layer along the wall. However, if the shear stresses are dominant throughout theflow, then the previous analysis will not be valid, as is shown next. First consider the situation where the fluid does not slip alongthe wall. Figure 3(a) shows the axial velocity profiles of the LDPE melt within the die of Hencky strainεh = 6 at various crosssections for a volumetric flow rate ofQ = 10mm3/s. The figure shows that there is significant shear clearly present throughoutthe die, so that the assumption of a flow field given by Eqs. (10) and (11) within the die is not even approximately satisfied.

9

Therefore, the subsequent analysis presented in Section II is not valid and Eq. (26) does not represent a means to determineelongational viscosity. For comparison with the case of wall slip discussed below, Figure 3(b) shows the elongation ratesε1in the direction of flow along several streamlines whenεh = 6 andQ = 10mm3/s. These values were computed according toEq. (31) using the particle tracking technique discussed in Section III-B. The dashed horizontal line indicates the value of theelongation rateε in elongational flow, computed by Eq. (27). It is noteworthy that along each streamline,ε1, given by Eq. (31),becomes nearly constant within the die. However, unlike in elongational flow,ε1 does not reach the same value (specificallyε)along all streamlines. Rather, the values ofε1 are distributed aroundε, with ε1 decreasing as the radial distance of the streamlineincreases.

As noted in the introduction, the material is believed to experience at least some form of slip along the boundary [17]. Ergo,the fluid is allowed to slip along the tube walls after entering the die, as indicated above. A typical example of the velocity fieldproduced with the converging die is shown in Figure 4. This figure shows the velocity profiles within the die of Hencky strainεh = 6 for a volumetric flow rate ofQ = 10mm3/s. Figure 4(a) shows the axial velocity profiles as a function ofr for severalaxial cross sections, and Figure 4(b) shows the radial velocity profiles as a function ofz for several radial cross sections. Thegraphs indicate that, after an initial transition phase,vz andvr are constant along each axial and radial cross section, respectively.That is,vz = vz(z) andvr = vr(r), as is predicted by the analysis in Section II. Moreover, the graphs indicate that∂vz/∂z and∂vr/∂r also become constant throughout the converging die.

The observation that∂vz/∂z is constant within the die is confirmed in Figure 5, where∂vz/∂z is graphed as a function ofzfor several radial cross sections forQ = 10mm3/s andQ = 100mm3/s in the die of Hencky strainεh = 6. Also shown are thegraphs of the elongation rate,ε1, in the flow direction along several streamlines. Except close to the die entry, the curves in eachplot are graphically identical, as expected. The dashed horizontal line in the graphs represents the value of the elongation rateε computed by Eq. (27). At a Hencky strain ofεh = 6, these values areε = 0.51s−1 and ε = 5.1s−1 for Q = 10mm3/s andQ = 100mm3/s, respectively. Likewise, the graphs of∂vr/∂r (not shown), as a function ofr, reach the constant value of−ε/2at a givenQ.

These results indicate that, after an initial transition phase, elongational flow defined by Eqs. (10) and (11) has been achievedwithin the semihyperbolically converging die forεh = 6. The results were nearly identical on both the standard and the refinedmesh. Furthermore, analogous results were found for the geometry corresponding to the Hencky strain ofεh = 7.

The initial transition phase between when the fluid enters the converging die and when elongational flow is reached looksrelatively short when the elongation rates are plotted as a function of axial position, as shown in Figure 5. However, when theelongation rates are plotted as a function of transit time within the converging die, one sees that the time required for the constantelongation rate to be reached can represent a significant portion of the total transit time of a fluid element within the die. This isillustrated in Figure 6 where the elongation rates along the centerline are shown forQ = 10mm3/s andQ = 100mm3/s in thedie ofεh = 6. Timet = 0s corresponds to the entry of the fluid element into the die.

As a final indication of the flow field within the die, the normal stress difference,τzz− τrr , and the shear stress,τrz, are givenin Figure 7 along several streamlines forQ= 10mm3/s andQ= 100mm3/s in the die ofεh = 6. Timet = 0s again correspondsto the entry of the fluid element into the die. The streamlines are labeled by their radial distance from the centerline in thefully developed upstream flow. One sees that the shear stresses remain small relative to the normal stress differences. The timeat which the normal stress difference begins to decrease along a particular streamline is the time at which the fluid elementhas reached the end of the converging die. One sees from these graphs that the transit time varies along different streamlines.Theoretically, if constant-rate elongational flow is instantaneously reached, then the transit times along all streamlines wouldbe identical, namely,tR = εh/ε, which for the cases shown in Figure 7 are approximately 11.8s forQ = 10mm3/s and 1.18sfor Q = 100mm3/s. The fact that the simulations do not reproduce this theoretical value is the result of the differences in thetheoretical velocity field and the simulated velocity field at the beginning of the die entry. In particular, the transit times alongmost streamlines in the simulations are smaller than this theoretical value. This is due to either a larger die entrance velocity orshorter streamlines than given by the theoretical flow. For example, along the centerline, where the theoretical and simulatedstreamlines coincide, the velocity with which the fluid enters the die is larger than the theoretical velocity, which is close to zero,resulting in a shorter transit time along the centerline in the simulations (see Figure 6). Also plotted in Figure 7 is the normalstress difference,τzz− τrr , computed by direct integration of the constitutive equation in elongational flow.

Having established that, after an initial transition phase, elongational flow exists within the converging die with the predictedelongation rateε given in Eq. (27), the formula derived in Section II for the effective elongational viscosity,ηef, is now underconsideration. This equation is given in Eq. (25) to beηef = −∆P/(εεh), where∆P= Pe−Po is the pressure difference betweenthe outlet pressure,Pe, and the inlet pressure,Po, (∆P is negative). The Hencky strainεh is known from the geometry (cf.Eq. (22)), and the (constant) elongation rateε is known from the geometry and the volumetric flow rateQ as shown in Eq. (27).

This equation is tested forηef by computing values of∆P from the numerical pressure fields given by the finite elementcalculations, and substituting these values into the expression−∆P/(εεh). Then, the resulting values ofηef are comparedwith values of elongational viscosity,ηe = (τzz− τrr )/ε, computed by integrating the material’s constitutive equation in thehomogeneous elongational flow field given in Eqs. (10) and (11). In this elongational flow, the general constitutive equationEq. (28) yields

10

τzz(t)− τrr (t) =∫ t

−∞

[h1(I1, I2)

{e2ε(t−t′)−e−ε(t−t′)

}+h2(I1, I2)

{e−2ε(t−t′) −eε(t−t′)

}]m(t − t ′) dt′, (33)

where the invariants areI1 = e2ε(t−t′) +2e−ε(t−t′) andI2 = e−2ε(t−t′) +2eε(t−t′).For each Hencky strain, several elongation rates, ranging fromε = 0.1s−1 to ε = 25s−1, were considered. A desired elongation

rate in the calculation is produced by using Eq. (27) to determine the necessary volumetric flow rate,Q. A list of the elongationrates and corresponding volumetric flow rates used in the calculations are given in Table III for each value of the Hencky strain.

As discussed in Section II, the elongational viscosity,ηe given in Eq. (24), or the effective elongational viscosity,ηef, definedin Eq. (25), represent the respective value at a particular time,t. Under the conditions that constant-rate elongational flow isstrictly and instantaneously reached within the converging die, then due to the integration of the energy equation,t = t, wheret ∈ (0, tR) is the time at which the average value ofτzz, or η, is reached within the converging die. In practice, it is reasonableto expect that constant-rate elongational flow is not strictly and instantaneously reached within the converging die. This isconfirmed in Figures 6 and 7. Moreover, even if such a flow were so produced, it would be a challenge to determinet.

Here, two ways of associating a time to the computed or measured effective viscosity are considered. In the first way, thistime is taken to be the residence time of a fluid element in the region of constant strain rate along a streamline. The time in thetransition region before the constant strain rate is reached is neglected. Comparison of the values ofηef = −∆P/(εεh) computedfrom the finite element calculations with values of transient elongational viscosity computed from the material’s constitutiveequation is shown in Figure 8. Each curve in this graph represents the transient elongational viscosity computed by integratingthe constitutive equation, Eq. (33), in elongational flow for a givenε. The symbols represent values ofηef = −∆P/(εεh) plottedas a function of particle residence time in the region of constant elongation rate, as described above. Due to variations inthis residence time along different streamlines, this time is indicated as a range of values by a short horizontal line. The circlescorrespond to values ofηef in a die ofεh = 6 and the squares correspond to values in a die ofεh = 7. The figure shows remarkablygood agreement betweenηef and the “exact” values ofηe computed by integrating Eq. (33). This result strongly supports thethesis that the effective elongational viscosityηef represents the true elongational viscosity,ηe, or, at least, that it provides anexcellent approximation toηe. Hence∆H ≈ 0, and can be neglected in practice.

Although it is relatively easy to extract the above residence times from the simulation data, it is perhaps not practical to doso in experiments. A more practical and desirable approach for representing the transientηef would be to plot these values as afunction ofatR, wheretR = εh/ε is the theoretical residence time, known from the die geometry and volumetric flow rate, anda is a positive horizontal shift factor which would need to be determined from other flow and fluid considerations and possiblyfrom the die geometry, i.e.a = a(εh). Support for this approach of representingηef is given in Figure 9. This figure shows acomparison ofηef from the simulations and the values of elongational viscosity from integration of the constitutive equation,where the former are plotted as a function ofatR in which the shift factor is taken to bea = 1/3 for both die geometries. Thevalue of the shift factor was chosen based on the good agreement found in Figure 8 and the observation thattR/3 roughly fellwithin the range of streamline residence times used to representηef in this figure. Although it remains to determine a practicalway for determining the shift factor, Figure 9 indicates that calibratingηef at one time might be sufficient to shift all the measureddata, at least for a given die geometry.

It is desirable to confirm the above result in the linear viscoelastic region, in particular the thesis thatηef ≈ ηe. For the twogeometries considered in this paper, this would require very small volumetric flow ratesQ. At such low flow rates, large relativeerrors arose in the calculations, making a comparison meaningless. Therefore, it is verified instead thatηef ≈ ηe in the caseof a Newtonian fluid. The analysis and results for the Newtonian fluid is similar to that for the LDPE melt. Specifically, onecan produce similar velocity fields as seen in Figure 4, and constant elongation rates are achieved within the die. As expected,the constant elongation rate computed from the finite element calculation was independent of fluid. That is, for a givenQ andεh, the graphs ofε as a function ofz for a Newtonian fluid are graphically identical to those for the LDPE melt. Unlike in thecase of the LDPE melt, the values ofηef computed from the pressure fields in the Newtonian calculations represent steady-stateelongational viscosities. Moreover, these values should be independent of elongation rate and should equal three times the shearviscosity. Figure 10 confirms that these conditions are satisfied. This figure shows values of theeffectiveTrouton ratioηef/η asa function ofε, whereη is the Newtonian shear viscosity.

In summary, in this section it has been shown that the effective viscosity given by Eq. (25) is an excellent approximation ofthe elongational viscosity. That is, the relationshipηef ≈ ηe holds for both Newtonian and viscoelastic fluids.

V. TRANSIENT VISCOSITY CURVES

The simulation results of the previous section provide strong evidence that the semihyperbolically converging die allowsmeasurement of elongational viscosity. As there is no pressure data from the IUPAC LDPE melt used in this study in thesemihyperbolically converging die, comparison between simulation and experiment is not possible. Nevertheless, experimental

11

support for the measurement method is offered in this section by showing that data measured from another LDPE melt in thesemihyperbolically converging die exhibits the expected trends.

Patil [18] and Collier and coworkers [16,17] used the semihyperbolically converging die to measure the effective elongationalviscosity of various polymer melts. In all cases, the measured values were plotted as a function of elongation rate at constantεh. Figure 11 shows the results for one of the LDPE melts at 175◦C considered by Patil for dies of Hencky strainεh = 4,5,6,7.The power-law trend exhibited by the data in this figure is typical of the trend observed in the other materials considered in thesemihyperbolically converging die [16–18]. This same trend has also been seen in the elongational viscosity data taken withMeissner-type devices [9]; this agreement supports the validity of the semihyperbolically converging die approach. Based on thesimulation results of the previous section, it is now apparent that the effective elongational viscosity in these graphs, taken in thesemihyperbolically converging die, is actually the true elongational viscosity, or at least an excellent approximation. However,representation of the data, as shown in Figure 11, can be somewhat misleading in that these plots give the impression that theviscosity values are at steady state. Given the short lengths of the dies (25 mm), it is not likely that a viscoelastic fluid will attainits steady state within the die. Indeed, for this to be true, the largest relaxation time of the polymer would have to be significantlyless than the die residence time, defined in Eq. (23) of Section II to betR = εh/ε.

A more explicit way to represent the data is to shift the abscissa according totR = εh/ε to plot viscosity versus time. This isrepresented in Figure 12 for the data shown in Figure 11. These curves give the viscosity of the material after applying a certainstrain,ε, to it. As discussed in Section II-B, it is generally true for a viscoelastic fluid thatε = εt < εh, wheret < tR is the timeat which the mean stress, or viscosity, is reached within the die. Each data point on a given curve corresponds to a differentvolumetric flow rate or strain rate. The curves show a linear trend for eachεh with the position of the curves increasing withincreasedεh. This behavior is consistent with the simulation results, as seen in Figures 8 and 9.

Perhaps the most useful way to interpret the data is to convert the results of Figure 12 to the form used in Figures 8 and 9, thatis, to plot viscosity versus time at constant strain rate. This can be accomplished using several different Hencky dies and fittingspline curves to interpolate the data of Figure 12. The results are shown in Figure 13. For each strain rate, there are four datapoints corresponding to the four dies. The time in this figure is taken to be the theoretical residence timetR = εh/ε. Accordingto the simulation results, these curves should further be shifted horizontally to the left. The appropriate shift factor remains tobe determined. Nevertheless, it is clear that the data in Figure 13 exhibits the characteristic trend of transient elongational data.Thus, it demonstrates that the semihyperbolically converging dies yield transient elongational viscosity data at constant strainrate over a very wide range of extension rates.

VI. SUMMARY AND CONCLUSIONS

An attractive technique for measuring elongational viscosity, introduced by Collier and coworkers [14] – [17], has beengeneralized and evaluated using numerical simulations. The technique involves extruding a material through a cylindrical,semihyperbolically converging die. The shape of the converging die allows a shear-free, or nearly shear-free, flow to be producedwithin the die. The analysis of the flow equations has been generalized to use less restrictive assumptions than reported inprevious studies. As before, the resulting expression for elongational viscosity involves the sum of a measurable effectiveviscosity and a term involving a change in enthalpy. The effective elongational viscosity, defined asηef = −∆P/(εεh), involvesonly the change in pressure over the die, the volumetric flow rate, and the Hencky strain determined by the geometry.

Numerical simulations showed that the term involving the enthalpy could be neglected and thatηef provides a very goodapproximation to the true elongational viscosity of the material. This was done by using finite element techniques to calculatethe flow of a LDPE melt in the semihyperbolically converging die (assuming wall slip within the die) and computingηef asdefined above. Two ways of associating a time toηef were considered: in one case, time was taken to be the residence time offluid particles in the region of constant elongation rate within the die, and in the second case, time was taken to beatR wheretR is the theoretical residence time defined by the geometry and flow rate anda > 0 is a constant shift factor applied to all data.In both cases, excellent agreement was found between the results of the simulations and the elongational viscosity computed byintegrating the constitutive equation for the LDPE melt in elongational flow. Moreover, experimental data taken from a differentLDPE in the semihyperbolically converging die showed the same trend as that seen in the simulation data. Finally, simulationsof a Newtonian fluid also showed that theηef satisfied the Trouton ratio.

Strictly speaking, the shape of the die produces elongational flow only when the fluid fully slips along the wall, thus elimi-nating all shear effects. The analysis of the flow equations will still provide a good approximation if this assumption is slightlyrelaxed and the flow is nearly shear-free. An essentially shear-free flow field results, for example, if the shear effects are re-stricted to a thin layer along the wall. In practice, this has been achieved through the use of a lubricant (cf. [14]– [16]). In theabsence of lubricants, a nearly shear-free flow can also occur if the fluid exhibits slip along the walls of the die, as reportedin [17]. That study shows experimental evidence that materials such as LDPE melts do slip along the walls of the die even atmoderate flow rates.

12

The semihyperbolically converging die method has several advantages over other methods for measuring elongational viscos-ity. First, compared to many other elongational rheometers, experiments in the semihyperbolically converging die are relativelyeasy and inexpensive to perform. The apparatus itself can be constructed simply by replacing the die in a capillary rheometerwith a semihyperbolically converging die and placing a pressure transducer on the wall at the die entry. Taking the pressureat the outlet to be atmospheric provides the pressure difference∆P, while the volumetric flow rate and the geometry of thedie provide the elongation rateε and Hencky strainεh = 2ln(Ro/Re). The effective viscosity,ηef = −∆P/(εεh), is thereforeimmediately computed. Furthermore, there is no sample preparation required. This is often a time-consuming and delicate taskassociated with other methods. Also, some other devices require temperatures not to exceed the melting point, which can be asevere limitation in industrial applications. Converging dies are not restricted to these relatively low temperatures. In addition,experiments have shown that high elongation rates can be reached in the semihyperbolically converging die. Collier et al. [17]reported elongation rates up to 533 s−1, which is well within the range of elongation rates encountered in industrial processes.This is in sharp contrast to other commercial elongational rheometers, which can only reach elongation rates up to one orderof magnitude lower than those encountered in industry. Other specific advantages of the semihyperbolically converging dieapproach over abrupt die entrance methods is that no models for shear or elongational viscosity and no additional viscometricdata are needed in the former. (In abrupt die entrance methods, shear data is needed to fix the parameters in the shear modelsused in the approach.) It was also shown that the semihyperbolically converging dies can be used to obtain transient elongationalviscosity data at constant strain or constant strain rate. Thus, they provide an attractive alternative for measuring elongationalviscosity of polymeric fluids.

There are several open issues which remain to be resolved. First, how sensitive is the effective elongational viscosity to thetype of slip which occurs along the die wall. Experimental data of Collier et al. [17] taken in the converging dies both with andwithout wall lubricant show little or no difference. Although there is evidence that the fluids did slip along the wall even in theabsence of lubricant, it is unlikely that the slip characteristics were the same as when a wall lubricant, or skin, was present. Thiswould, therefore, indicate some degree of insensitivity. Nevertheless, it is still of interest to determine the effect of different sliplaws, the degree of slip and stick-slip on the simulation results. This is the subject of ongoing investigations.

Further, a practical way of determining the appropriate time shifting of the experimental data needs to be determined. Thesimulations showed good agreement with the constitutive equation whenηef was plotted as a function ofatR. Since the theoreticalresidence timetR = εh/ε is known from the die geometry and the flow rate, what remains to be determined is the shift factor,a. In the simulations, one shift factor,a = 1/3, was applied to data from bothεh = 6,7, although it is possible thata = a(εh).Possible ways to calibrate the data are currently being investigated.

Finally, it should be determined how well the effective elongational viscosity measured in the semihyperbolically convergingdie compares with the elongational viscosity measured in other rheometers. This is also the subject of future work.

ACKNOWLEDGMENTS

Financial support of this project was provided by the Department of Chemical Engineering at The University of Tennessee.The experimental data was supplied by Dr. P. Patil, The University of Tennessee.

13

[1] C.W. Macosko, “Rheology: Principles, Measurements, and Applications,” VCH, New York, 1994.[2] F. Morrison, “Understanding Rheology,” Oxford University Press, New York, 2001.[3] J. Meissner, Development of a universal extensional rheometer for the uniaxial extension of polymer melts, Trans. Soc. Rheol., 16 (1972)

405–420.[4] J. Meissner, Experimental aspects in polymer melt elongational rheometry, Chem. Eng. Commun., 33 (1985) 159–180.[5] J. Meissner and J. Hostettler, A new elongational rheometer for polymer melts and other highly viscoelastic liquids, Rheol. Acta, 33

(1994) 1–21.[6] F.N. Cogswell, Converging flow of polymer melts in extrusion dies, Polym. Eng. Sci. 12 (1972) 64–73.[7] D.M. Binding, An approximate analysis for contraction and converging flows, J. Non-Newtonian Fluid Mech. 27 (1988) 173–189.[8] D.M. Binding, Further considerations of axisymmetric contraction flows, J. Non-Newtonian Fluid Mech. 41 (1991) 27–42.[9] A.D. Gotsis and A. Odriozola, The relevance of entry flow measurements for the estimation of extensional viscosity of polymer melts,

Rheol. Acta 37 (1998) 430–437.[10] M. Zatloukal, J. Vlcek, C. Tzoganakis and P. S´aha, Improvement in techniques for the determination of extensional rheological data from

entrance flows: computational and experimental analysis, J. Non-Newtonian Fluid Mech. 107 (2002) 13–37.[11] D.F. James, G.M. Chandler and S.J. Armour, A converging channel rheometer for the measurement of extensional viscosity, J. Non-

Newtonian Fluid Mech. 35 (1990) 421–443.[12] D.F. James, G.M. Chandler and S.J. Armour, Measurement of the extensional viscosity of M1 in a converging channel rheometer, J. Non-

Newtonian Fluid Mech. 35 (1990) 445–458.[13] H.J. Park, D. Kim, K.-J. Lee and E. Mitsoulis, Numerical simulation in converging channel flow of the fluid M1 using an integral

constitutive equation, J. Non-Newtonian Fluid Mech. 52 (1994) 69–89.[14] J.R. Collier, U.S. Pat. 5,357,784, 1994.[15] H.W. Kim, A. Pendse and J.R. Collier, Polymer melt lubricated elongational flow, J. Rheol. 38(4) (1994) 831–845.[16] A.V. Pendse and J.R. Collier, Elongational viscosity of polymer melts: A lubricated skin-core flow approach, J. Appl. Polym. Sci., 59

(1996) 1305–1314.[17] J.R. Collier, O. Romanoschi and S. Petrovan, Elongational rheology of polymer melts and solutions, J. Appl. Polym. Sci., 69 (1998)

2357–2367.[18] P. Patil, Measurement of elongational rheology of polymer melts using semi-hyperbolically converging dies.Ph.D. Dissertation, The

University of Tennessee (2002).[19] J. Meissner, Dehnungsverhalten von Poly¨athylen-Schmelzen, Rheol. Acta, 10(2) (1971) 230–242.[20] J. Meissner, Neue Messm¨oglichkeiten mit einem zur Untersuchung von Kunststoff-Schmelzen geeigneten modifizierten Weissenberg-

Rheogoniometer, Rheol. Acta, 14(3) (1975) 201–218.[21] H.M. Laun, Description of the non-linear shear behaviour of a low density polyethylene melt by means of an experimentally determined

strain dependent memory function, Rheol. Acta, 17(1) (1978) 1–15.[22] A.C. Papanastasiou, L.E. Scriven, and C.W. Macosko, An integral constitutive equation for mixed flows: viscoelastic characterization,

J. Rheol., 27(4) (1983) 387–410.[23] S. Dupont and M.J. Crochet, The vortex growth of a K.B.K.Z fluid in an abrupt contraction, J. Non-Newtonian Fluid Mech., 29 (1988)

81–91.[24] X.-L. Luo and R.I. Tanner, A streamline element scheme for solving viscoelastic flow problems. Part II: Integral constitutive models,

J. Non-Newtonian Fluid Mech., 22 (1986) 61–89.[25] R.S. Rivlin and K.N. Sawyers, Nonlinear continuum mechanics of viscoelastic fluids, Ann. Rev. Fluid Mech. 3, (1971) 117–146.[26] B. Bernstein, D.S. Malkus and E.T. Olsen, A finite element for incompressible plane flows of fluids with memory, Inter. J. Num. Meth.

Fluids, 5 (1985) 43–70.[27] B. Bernstein, K. Feigl and E.T. Olsen, Steady flows of viscoelastic fluids in an axisymmetric abrupt contraction geometry: A comparison

of numerical results, J. Rheol., 38(1) (1994) 53–71.[28] B. Bernstein, K. Feigl and E.T. Olsen, A first order exactly incompressible finite element for axisymmetric fluid flow, SIAM J. Num.

Anal., 33(5) (1996) 1736–1758.[29] K. Feigl and H.C.Ottinger, The flow of a LDPE melt through an axisymmetric contraction: A numerical study and comparisons to

experimental results, J. Rheol., 38(4) (1994) 847–874.[30] K. Feigl and H.C.Ottinger, A numerical study of the flow of a LDPE melt in a planar contraction and comparison to experiments, J.

Rheol., 40(1) (1996) 21–35.

14

TABLE I. Relaxation spectrum for IUPAC LDPE melt at 160◦C, (zero-shear-rate viscosityη0 = 35.8kPas) [21].

k λk [s] ηk/λk [Pa]1 7.01×10−5 1.29×105

2 7.01×10−4 9.48×104

3 7.01×10−3 5.86×104

4 7.01×10−2 2.67×104

5 7.01×10−1 9.80×103

6 7.01×100 1.89×103

7 7.01×101 1.80×102

8 7.01×102 1.00×100

TABLE II. Geometry and mesh parameters for the semihyperbolically converging die.

Hencky strain Ro Re L Number of macroelements[mm] [mm] [mm] Mesh 1 Mesh 2

εh = 6 10 0.5 25 5400 9750εh = 7 10 0.3 25 5450 10200

TABLE III. Flow cases considered in the semihyperbolically converging die.

Elongation rate Volumetric flow rateQ [mm3/s]ε [s−1] εh = 6 εh = 7

0.1 2.0 0.720.51 10.0 3.592.54 50.0 17.975.1 100.0 35.9410.0 196.84 70.7525.0 492.1

15

(a)

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

Shear rate [s−1

]

100

101

102

103

104

105

106

107

η [P

a s]

; Ψ1

[Pa

s2 ]

ηΨ1

η0

Ψ1,0

(b)

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

Elongation rate [s−1

]

103

104

105

106

107

Elo

ngat

iona

l vis

cosi

ty [P

a s]

3η0

FIG. 1. Predictions of the constitutive equation for the rheological behavior of the LDPE melt. (a) Shear viscosity (circles) and the firstnormal stress coefficient (squares). The dashed horizontal lines indicates the zero-shear-rate viscosity,ηo, and the zero-shear-rate first normalstress coefficient,Ψ1,0. (b) Steady elongational viscosity. The dashed horizontal line indicates the Trouton viscosity, 3ηo.

16

Re=0.5 mmRo=10 mm

L = 25 mm

Ro=10 mm

L = 25 mmRe=0.3 mm

FIG. 2. Geometries for the semihyperbolically converging dies for Hencky strains ofεh = 6 (top) andεh = 7 (bottom).

17

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8r [mm]

0

5

10

15

20

25

30

Axi

al v

eloc

ity [m

m/s

]

z=10mm

z=15mm

z=20mm

z=25mm

(b)

−30 −20 −10 0 10 20 30 40 50z [mm]

−1

−0.5

0

0.5

1

1.5

2

Elo

ngat

ion

rate

in 1

−di

rect

ion

[s−1 ]

FIG. 3. (a) Axial velocity profiles at various cross sections and (b) elongation rates along various streamlines of the LDPE melt with no-slipboundary conditions for a Hencky strainεh = 6 and a volumetric flow rate ofQ = 10mm3/s. The elongation rate curves in (b) correspond tothe streamlines (from top to bottom)r = 1,3,5,6,7,8mm, wherer is the radial position of the streamline in the fully-developed upstream flow.The dased horizontal line in (b) indicates the value ofε in elongational flow computed from Eq. (27).

18

(a)

0 0.2 0.4 0.6 0.8 1 1.2r [mm]

0

5

10

15

Axi

al v

eloc

ity [m

m/s

]

z = 5mm

z = 10mm

z = 15mm

z = 20mm

z = 25mm

(b)

−30 −20 −10 0 10 20 30 40 50z [mm]

−0.3

−0.2

−0.1

0.0

0.1

Rad

ial v

eloc

ity [m

m/s

]

FIG. 4. (a) Axial velocity profiles at various cross sections and (b) radial velocity profiles at various constant radii of the LDPE melt withthe slip boundary conditions discussed in Section IV for a Hencky strainεh = 6 and a volumetric flow rate ofQ = 10mm3/s. The curvesin (b) correspond to the radial positions ranging fromr = 0.1mm (top curve) tor = 1.0mm (bottom curve) with increments of∆r = 0.1mmproceeding from the top to the bottom curve.

19

(a)

−30 −20 −10 0 10 20 30 40 50z [mm]

−0.5

0

0.5

1

Elo

ngat

ion

rate

[s−1 ]

(b)

−30 −20 −10 0 10 20 30 40 50z [mm]

−2−1

0123456789

Elo

ngat

ion

rate

[s−1 ]

FIG. 5. Elongation rates of the LDPE melt with the slip boundary conditions discussed in Section IV for a Hencky strainεh = 6 and avolumetric flow rate of (a)Q = 10mm3/s and (b)Q = 100mm3/s.

20

−4 −2 0 2 4 6 8 10Transit time [s]

−2

−1

0

1

2

3

4

5

6

Elo

ngat

ion

rate

[s−1 ]

Q = 10 mm3/s

Q = 100 mm3/s

FIG. 6. Centerline elongation rates of the LDPE melt with the slip boundary conditions discussed in Section IV for a Hencky strainεh = 6and a volumetric flow rate ofQ = 10mm3/s andQ = 100mm3/s as a function of transit time, wheret = 0s corresponds to entry of a fluidelement within the converging die.

21

(a)

−4 −2 0 2 4 6 8 10 12Transit time [s]

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

τ zz−

τ rr, τ

rz [

kPa]

Constitutive equation r = 1 mm r = 4 mm r = 8 mm

τzz−τrr

τrz

(b)

−1.0 −0.5 0.0 0.5 1.0 1.5Transit time [s]

−1

0

1

2

3

τ zz−

τ rr, τ

rz [

kPa]

Constitutive equation r = 1 mm r = 4 mm r = 8 mm

τzz−τrr

τrz

FIG. 7. Evolution of the normal stress difference,τzz− τrr , and shear stress,τrz, along several streamlines for the LDPE melt with the slipboundary conditions discussed in Section IV for a Hencky strainεh = 6 and a volumetric flow rate of (a)Q= 10mm3/s and (b)Q= 100mm3/s.Time t = 0s corresponds to entry of a fluid element within the converging die. The streamlines are labeled by their radial position,r, in thefully-developed upstream flow. The solid curve in each graph representsτzz− τrr computed by integration of the constitutive equation inelongational flow with elongation rate corresponding to the values ofQ above, namely, (a)ε = 0.51s−1 and (b)ε = 5.1s−1.

22

10−3

10−2

10−1

100

101

102

103

Time [s]

102

103

104

105

106

107

Eon

gatio

nal v

isco

sity

[Pa

s]

ηef (εh=6)

ηef (εh=7)

Constitutive equation

25.010.0

5.1 2.55 0.510.1

FIG. 8. Comparison between the elongational viscosities for various elongation rates obtained by integration of the constitutive equation(solid curves) and by computation of the effective elongational viscosity,ηef, for Hencky strainsεh = 6 (circles) andεh = 7 (squares). Thevalues ofηef are plotted as functions of the residence time of a fluid element in the region of constant strain rate. The short horizontal linesconnecting two data points indicate the range of particle residence time in the die along the various streamlines. The dashed line indicates theTrouton curve for low shear rates.

10−3

10−2

10−1

100

101

102

103

Time [s]

102

103

104

105

106

107

Eon

gatio

nal v

isco

sity

[Pa

s]

ηef (εh=6)

ηef (εh=7)

Constitutive equation

25.010.0

5.1 2.55 0.510.1

FIG. 9. Comparison between the elongational viscosities for various elongation rates obtained by integration of the constitutive equation(solid curves) and by computation of the effective elongational viscosity,ηef, for Hencky strainsεh = 6 (circles) andεh = 7 (squares). Thevalues ofηef are plotted as functions ofatR, wheretR = εh/ε is the theoretical residence time of a fluid element within the converging die anda = 1/3 is a horizontal shift factor. The dashed line indicates the Trouton curve for low shear rates.

23

10−2

10−1

100

101

102

Elongation rate [s−1

]

0

1

2

3

4

5

6

(Effe

ctiv

e) T

rout

on r

atio

[−] Standard mesh

Refined mesh

FIG. 10. Effective Trouton ratio,ηef/η, for a Newtonian fluid with shear viscosityη, computed for the standard mesh and the refined mesh.The dashed line gives the theoretical value.

10−1

100

101

102

Strain rate [s−1

]

104

105

106

107

η ef [P

a s]

εh = 4εh = 5εh = 6εh = 7

FIG. 11. Experimental values of the effective elongational viscosity versus strain rate of a low density polyethylene melt at 175◦ C for fourdifferent Hencky strain dies as taken from [18].

24

10−2

10−1

100

101

102

Time [s]

104

105

106

107

Elo

ngat

iona

l vis

cosi

ty [P

a s] εh = 4

εh = 5εh = 6εh = 7

FIG. 12. Experimental values of the effective elongational viscosity versus time,tR, at constant strain for the polymer of Fig. 11.

10−2

10−1

100

101

Time [s]

104

105

106

107

Elo

ngat

iona

l vis

cosi

ty [P

a s] 1 s

−1

5 s−1

10 s−1

15 s−1

20 s−1

25 s−1

30 s−1

FIG. 13. Experimental values of the effective elongational viscosity versus time,tR, at constant strain rate for the polymer of Fig. 11.

25