Visco elastic · Visco elastic Computations of P olymeric Wire-coating Flo ws H. Matallah, P.T o...

Viscoelastic Computations of Polymeric

Wire-coating Flows

H. Matallah, P. Townsend and M.F. Webster

Institute of Non-Newtonian Fluid Mechanics

University of Wales, Swansea SA2 8PP

submitted to IJNME

June 2, 2000

Abstract

This study considers both a single and multi-mode viscoelastic analysis for wire-

coating ows. The numerical simulations utilise a �nite element time-stepping technique,

a Taylor-Petrov-Galerkin/pressure-correction scheme employing both coupled and decou-

pled procedures between stress and kinematic �elds. An exponential Phan-Thien/Tanner

model is used to predict pressure-drop and residual stress for this process. Rheomet-

rical data �tting is performed for steady shear and pure extensional ows, considering

both high and low density polyethylene melts. Simulations are conducted to match exper-

imental pressure-drop/ owrate data for a contraction ow. Then, for a complex industrial

wire-coating ow, stress and pressure drop are predicted numerically and quanti�ed. The

bene�ts are extolled of the use of a multi-mode model that can incorporate a wide-range

discrete relaxation spectrum to represent ow response in complex settings. Contrast

is made between LDPE and HDPE polymers, and dependency on individual relaxation

modes is identi�ed in its contribution to overall ow behaviour.

Keywords: Wire-coating, tube-tooling, multi-mode, �nite element, Taylor-Galerkin, Re-

covery

1 INTRODUCTION

In the �eld of polymer coating of wires, experimental studies are extremely di�cult to per-form due to the small size of the dies and minute volume of polymer melt within the region ofinterest. This leads to the need for numerical simulation as a predictive aid to optimise theprocess via die design and ow modelling. This allows for the analysis of variation in certainkey parameters, without having to resort to trial-and error expensive and di�cult experi-ments. There are two basic types of cable or wire-coating die designs commonly employed.These are represented schematically in Figure 1, and are termed pressure and tube-toolingdesigns. In pressure-tooling, the melt is driven under pressure making contact with the wireinside the die. In contrast, for tube-tooling, the melt is drawn down by the motion of thewire and the melt is extruded beyond the die. In both instances, the geometry is annular incross section. The setting of the ow geometry is found to be crucial in obtaining optimalcoatings.

There have been a number of studies that have addressed the modelling of wire-coating ows; see for example Fenner and Williams [1], Caswell and Tanner [2], Mitsoulis et al. [3, 4],Huang et al. [5], Binding et al. [6] for pressure-tooling, and Gunter et al. [7], Mutlu et

1

al. [8, 9, 10] for tube-tooling. These have provided some progress within the inelastic, non-isothermal and viscoelastic regimes. In a wider ow and processing context, some recentattempts have been made to embrace di�erential multi-mode simulations, see for examplethe works of Baaijens [11], Baaijens et al. [12], Azaiez et al. [13] and Gupta et al. [14]. Sucha multi-mode approach for di�erential constitutive models is more costly, yet can providea more accurate representation of the material rheology. The present study addresses thesigni�cance of employing a multi-mode, as opposed to a single-mode, di�erential constitutivemodelling approach to predict numerically the behaviour of tube-tooling wire-coating owsfor two polymer melts, low and high density polyethylene. We concentrate on the quality andsigni�cance of numerical predictions attainable from multi-mode approximations in contrastto single-mode alternatives. The modelling assumes incompressible melt ow, isothermalconditions, no-slip within the die, and an estimated location for the free-surfaces, see Refs.[8, 9].

We are concerned with two parts of the tube-tooling wire-coating process, namely owwithin the die itself and a draw-down ow beyond the die onto the wire cable. The owis generated by a pressure head and the dragging action of the moving wire. This type oftooling design is prevalent for wide bore cables, or outer coatings of multi-cable combinations.Cable speed is taken as 0:334m=s. The main goal of modelling such a complex industrialproblem is to achieve process optimisation. This is governed typically by minimisation ofthe ow-induced residual stress locked into the coating, and in addition, acceptable levels ofpressure-drop across the die. The ow response of the coating material and the die design [15]both have a role to play in this optimisation procedure. Pressure-drop predictions providesome means of correspondence between simulation and practice.

An exponential Phan-Thien/Tanner (PTT) di�erential constitutive model is selected torepresent the rheometrical behaviour of low and high density polyethylene melts (LDPE andHDPE) in steady shear and uniaxial ow. Typically, such material samples display bothshear thinning and strain softening properties and an exponential PTT model is capable ofreproducing such behaviour in qualitative form [16, 17]. Simple shear and uniaxial owsare used to evaluate the PTT parameter set (�, �), for which a close �t to the experimentaldata must be established. To calibrate pressure drops as a function of owrate and compareexperimental with predicted observations, a nineteen to one axi-symmetric contraction ow isadopted. For this ow, both single and multi-mode models are implemented. Once optimalityin parameter set is established, we turn attention to tube-tooling ow and the multi-modesimulations for LDPE and HDPE polymer coatings.

The numerical implementation is a Taylor-Petrov-Galerkin/pressure-correction schemethat involves a �nite element time-stepping technique [18], in conjunction with a recoveryscheme to capture continuous velocity gradients. Both decoupled and coupled numericalapproaches have been employed previously for a single mode analysis [8], where the resultswere found to be comparable. Here, only coupled solutions are considered for single-modecomputations. For e�ciency and pragmatism, multi-mode calculations are performed via adecoupled approach. This strategy is supported by the fact that in many ows the kinematicsthemselves do not vary signi�cantly with variation of material [19], and hence a reasonableapproximation is a linearisation of the system, adopting frozen coe�cients and Picard itera-tion. This implies segmenting the equations of the complete system into those for stress andkinematics separately, computing each to a steady state with frozen coe�cients. We notehere that the choice of initial frozen kinematic �elds are taken as associated with viscoelasticand shear thinning behaviour. Such a pragmatic approach is adopted similarly by others (seeSchoonen et al. [20], but with �xed inelastic kinematics).

Single mode modelling is pragmatic and yet e�ective in terms of relatively low computa-

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tional cost. A owrate-maximum shear rate relationship may be established for a particularmaterial and ow, such as within a contraction ow. Once this behaviour has been deter-mined, such knowledge may be employed to seek an optimal rheometrical parameter �t withina neighbourhood of the maximum shear rate, referencing standard viscometric ows. Thismodel speci�cation may then be utilised e�ectively to predict behaviour in a more generalcomplex ow, such as in a wire-coating setting. In contrast, multi-mode modelling with singlechoice of rheological parameters provides a closer match to the shear viscosity data over arange of shear rates. This facilitates consistent and wide ranging application for complex ows without parameter adjustment. In addition it is possible in the multi-mode context,to identify the contribution of each mode to the total stress and hence determine the mostdominant component.

We commence with a single-mode analysis for a Phan-Thien/Tanner (PTT) model andconsider the �tting of this model to the rheometrical data of Walters et al. [21], supplied fortwo independent case studies. The �rst is essentially a parameter �tting study for viscometric ows, steady simple shear and pure extension. Fitting of the various material parameters (�,� and �1) is investigated for shear viscosity in shear ow. Likewise, elongational viscositybehaviour is charted under steady uniaxial extension, according to a Binding analysis forcontraction ow [22]. The second case study, involves simulations for a more complex ow,the contraction ow [21]. Here, selected combinations of material parameters, that representdi�erent uids recommended from the �rst case study, is employed at di�erent owrate set-tings. Of course, ideally one requires a model that for one set of parameters approximates owover a wide range of conditions. The single mode case proves inadequate for this task. Quan-titative agreement is sought on predicted pressure-drop against the available experimentalpressure-drop/ owrate data.

The ultimate objective of this work is to relate the signi�cance and sensitivity of thematch for the chosen constitutive model to the ow response of the polymer melts in ques-tion, under tube-tooling processing conditions. Proceeding in a structured manner from thepreliminary case studies above, a single-mode analysis is conducted for this complex indus-trial ow. Particular attention is paid to pressure-drop and stress build-up across the owfrom entry to exit. A second aspect to this work carries over to a multi-mode analysis, underwhich the three problem settings outlined above are revisited, see [13, 23] for motivation.From a single-mode analysis it is observed that pressure-drop prediction is sensitive to thedetermination of the material parameters of the PTT model. These are governed by themaximum shear rate sustained in the process at any speci�c owrate. Nevertheless, for thecomplex industrial ows of interest, that involves ow through a tube-tooling die and draw-down section, experimental data is unavailable. Hence for guidance, we turn to a comparisonbetween single and multi-mode model simulations to provide greater insight on the underly-ing ow behaviour of principle signi�cance.

2 GOVERNING EQUATIONS AND CONSTITUTIVE MODEL

The ow of shear and elongational thinning LDPE and HDPE uids is modelled using amulti-mode Phan-Thien/Tanner (PTT) model, invoking both single and multi-mode approx-imations. The momentum and continuity equations are given as

�@uuu

@t= r � �� uuu � ruuu�rp; (1)

r � uuu = 0; (2)

3

with velocity vector uuu, uid density �, pressure p, stress �� , time t and rate of deformationtensorDDD = (ruuu+ruuuy)=2. Within the single mode approximation, the stress �� is decomposedinto two parts, viscous and polymeric parts viz

�� = ��1 + 2�2DDD; (3)

for which �2 is a solvent viscosity. The extra stress tensor ��1 is then de�ned by

f��1 + �12

��1 = 2�1DDD; (4)

2

��1 =

�1�

�

2

�r

�� 1 +�

2

�� 1 =

r

�� 1 + �(DDD � ��1 + ��1 �DDD); (5)

where f is given by

f = exp

��1

�1trace(��1)

�; (6)

see [16, 24]. Material parameters �1 and �1, represent relaxation time and polymeric viscosity,respectively. The zero shear viscosity is then

�0 = �1 + �2: (7)

Model parameters (� � 0), (0 � � � 2) are non-dimensional parameters that can be evaluatedby �tting to experimental data. r and � are upper and lower-convected derivatives, whosecombination introduces second normal stress e�ects.

The PTT model has been chosen because of its shear thinning and strain softening prop-erties, as displayed by the melts in question. Within the complex ows of interest both shearand extension are present. It is instructive, therefore, to analyse the PTT model responsein pure shear and extension in isolation. With this in mind, we �rst present the theoreticalresponse of this model in ideal ows, for which we are able to conduct a multi-variate sensi-tivity analysis in (�, �, �1) against experimental values of shear viscosity �s and extensionalviscosity �e given by Walters et al. [21]. It is found appropriate to �rst seek an optimal �tto the shear viscosity, prior to qualifying goodness of �t to extensional viscosity, see below.

In the multi-mode context, the equations of state are given for each mode (i) as

fi�� i + �i12

�� i = 2�i1DDD; (8)

where the total stress is�� =

Xi

�� i; (9)

and fi is de�ned as above, but with respect to each mode (i). Accordingly, �i1 represents therelaxation time of each mode. A vanishing solvent viscosity is considered in the multi-modecase i.e., �2 = 0 appropriate for polymer melts, so that here the partial zero-shear viscosityfor each mode (i) is �i0 = �i1. In this case, � is the only parameter that is varied, as � ischosen to be zero, see on. Hence Eqs. 8 and 9 are solved with Eq. 1 and 2 for momentumand continuity.

In completely general form, retaining a solvent contribution we consider a steady simpleshear ow with shear rate _ . Following [25], the representation of the total shear viscosity �sover all modes is the sum of the partial shear viscosities, viz

�s( _ ) = �2 +Xi

�i1fi

f2i + (�i1 _ )2�(2� �)

; (10)

4

where each fi satis�es

ff2i + (�i1 _ )2�(2� �)gln(fi)� 2�(�i1 _ )

2(1� �) = 0: (11)

Accordingly, �rst and second normal stress-di�erences, N1 and N2 respectively, are de�nedas

N1( _ ) =Xi

2�i1�i1 _

2

f2i + (�i1_ )2�(2� �)

; (12)

N2( _ ) = ��

2N1( _ ): (13)

In contrast, for a single mode approximation, the summation collapses to a single term.In steady uniaxial extension, the elongational viscosity �e is a function of extension rate

_�, of the form

�e( _�) = 3�2 +Xi

2�i1f2i � 2�i

1_�(1� �)

+�i1

fi + �i1_�(1� �)

; (14)

where each fi is now determined from

ff2i � �i1 _�(1� �)� 2(�i1 _�)2(1� �)2gln(fi)� 6(�i1 _�)

2�(1� �) = 0: (15)

Equations (11) and (15) are solved in each case for the relevant fi through a Newton-Raphsoniteration, from which �s, �e, and N1 may be determined. We note that via Eq. 13 the secondnormal stress-di�erence, N2, may be expressed via dependency on � and N1. In this form, itmay be observed that as � tends to zero then so does N2. Hence, in the multi-mode context,where � is taken as zero for simplicity, this implies vanishing N2, which in turn justi�es thesetting of �2 = 0.

3 NUMERICAL SCHEME

Single-mode solutions are generated employing a coupled procedure previously described indetail in [18]. The coupled scheme involves solving for kinematics and stress simultaneously.The implementation is a fractional-stage time stepping scheme, woven around a pressure-correction method, that involves some three stages within each time step. The �rst stagesolves for velocity and stress, in a predictor-corrector doublet. The second stage, solves for apressure temporal increment. The third stage computes a correction to the velocity �eld, en-forcing incompressibility at each time step to the order of the scheme. The scheme embodiesimplicit and explicit treatment simultaneously, and hence is of classical semi-implicit type.Di�usion terms are approximated with a Crank Nicolson discretisation over a time step, whichintroduces implicitness and stability for these viscous ows. A direct method of solution isemployed to solve for the pressure equation step, whilst indirect Jacobi iteration is invokedfor the remaining stages. Also, the bene�ts are realised of stability enhancing recovery-basedmethods and consistent streamline upwinding procedures, that are incorporated within theimplementation [26, 18]. For these single-mode coupled calculations a continuation procedurein relaxation time parameter is employed to reach a speci�c value of �spec

1, chosen appropri-

ately, (see on for discussion).With a multi-mode decoupled procedure, a frozen kinematic �eld corresponding to a

single-mode solution (�spec1 ) is used in Eq. (8) to solve for stress components and each mode(i) of the PTT model in parallel, corresponding to (�i1, �

i1) parameters. For simpli�cation, in

the multi-mode context, the solvent part of the viscosity �2 = 0, and hence for the polymericpart of each mode (i), �i1 may be replaced by the partial zero-shear rate viscosity �i, and the

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relaxation time �i1 by �i. Non-dimensional numbers are de�ned as follows. For each mode(i), a Weissenberg number is

Wei = �iUch

Lch

; (16)

where, Uch, Lch are velocity and length scales, respectively. An average single mode �spec1

isestimated from the experimental data following the procedures outlined in Ref. [27]. Thisvalue is gathered from the base material function �ts to satisfy �s, �e, N1. For the LDPEpolymer, �spec

1is taken as 5s, giving a Weissenberg number of Wespec of 28 according to Eq.

16. Similarly, �spec1

is 9s for HDPE polymer, from which Wespec equals 50.The total zero-shear viscosity is given viz

�0 =Xi

�i: (17)

Also, the shear elastic modulus gi for each mode (i), is de�ned as

gi =�i

�i: (18)

Time steps of O(10�3 � 10�4) are used and convergence to a steady state is monitoredvia a relative temporal increment norm on the solution taken to a tolerance of O(10�6). Asummary of the steps in the numerical procedure is provided as follows:

Stage 1: calculate a Newtonian �eld (uuu, p), as in Refs. [28].Stage 2: Starting from the Newtonian solution and quiescent initial stress conditions a

single-mode viscoelastic solution is calculated for �spec1

. The Recovery coupled scheme is usedwith a solvent viscosity of �2 = 0:01�0. Thus, viscoelastic kinematics are derived.

Stage 3: Commencing from such a single-mode solution, each of the three stress modesis calculated in a decoupled sense, on resetting the solvent viscosity to zero and freezingthe viscoelastic kinematics. For each mode (i), the viscosity, pressure and stress are non-dimensionalised as follows

(�i)? =�i

�i0

= 1; p = �iUch

Lch

p? and �� i = �iUch

Lch

��?i (19)

Stage 4: The total stress is then calculated through Eq. 9. Updated velocity and pressure�elds are recomputed by freezing the total stress. Here nondimensionalisation is performedaccording to the total zero-shear viscosity �0 given by Eq. 17. After a single pass, thisprocedure is noted to correct the pressure �eld, meanwhile variation in the velocity �eld isfound to be insigni�cant, so that further computation is found subsequently to be unnecessary.

4 PARAMETER FITTING FOR STEADY SHEAR AND PURE EXTENSION

4.1 Single Mode analysis

Under a single mode analysis, we consider only the case of an LDPE polymer. The inves-tigation into parameter sensitivity commences with the polymeric viscosity coe�cient �1.The zero shear viscosity is estimated as 54; 700Pa.s by �tting to the shear data and a singlerelaxation mode is evaluated as �1 of 5s for LDPE, as cited above (see also [7, 8, 21]. Thedensity � for both polymers considered is 760Kg=m3.

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4.1.1 �1 variation

Theoretically in steady shear ow, by increasing the parameter �1 and keeping (�, �) �xed, theLDPE uid thins at high shear rates and accordingly, will give rise to diminishing pressure-drop due to decreasing ow resistance. When �1 = 0, there is no polymeric contribution.In Figure 2a, we indicate that the �1 parameter (recorded in non-dimensional form) playsan important role when the range of the shear rates is high. As the shear rates observedexperimentally [21] increase, it is necessary to adjust �1 to match the data for any isolatedshear rate extrema. The inadequacy of a �xed parameter is clearly apparent here. Exper-imental data were measured for shear viscosity at shear rates between 0:1s�1 to 1s�1 and10s�1 to 103s�1. On the other hand in steady uniaxial extension, the extensional viscosityfor the LDPE uid exhibits strain-softening, as illustrated in Figure 2b. The PTT modeldisplays a slight increase of the extensional viscosity at extension rates less than unity, beforedecreasing to a high strain rate limiting plateau. The experimental measurements for elon-gational viscosity were taken between strain rates of 10s�1 and 103s�1. We comment thatthe Binding Analysis, used here to derive the experimental �e data, is an approximate theory[22, 29, 30, 31]. The only di�erence observed in the �t for di�erent settings of �1 is in the tailof the curve at high strain rates. As in Figure 2, larger values of �1 that asymptote to unity,tend to inherit viscosity behaviour of lesser �1 values over lower shear rate ranges. However,it is necessary to gradually increment values of �1 towards unity for reasons of numericalconvergence in the continuation procedure.

4.1.2 � variation

We proceed to analyse the e�ect of increasing � at �xed (�, �1). In steady shear ow, theviscosity of the PTT model decreases correspondingly, but at a lower rate than with increaseof �1. This may be observed in Figure 3a for (�, �1)=(0.02, 0:997�0). Shear thinningbehaviour of this form, with increasing �, will generate a decrease in pressure-drop. In Figure3a with � = 0:15, the PTT model provides a near-optimal least squares �t across the completerange of shear rates displayed, for a match to the experimental data at a maximum valueof _ = 103s�1. In contrast for a value of � = 0:7, Figure 3a displays a best �t for shearrates less than 10s�1. Switching to uniaxial extension, Figure 3b demonstrates reductionin elongational viscosity with increase in �, yet without in uencing the asymptotic plateauat high strain rates. Clearly, here there is only limited experimental data available and themodel can only give a qualitative representation of elongational behaviour.

4.1.3 � variation

Finally, adjusting the parameter � for constant values of �1 and �, gives a decrease in theshear viscosity for increasing �, as displayed in Figure 4. At high values of � = 1:0, thereis barely any change in the shear viscosity for 0 � � � 0:1 as observed in Figure 4b. Somedeparture from the experimental values is noted at the lower value of � = 0:15 for � � 0:1.The elongational viscosity is una�ected by such minor adjustment in parameter �, as seen inFigure 5; that is with the exception of � = 1 (annotated), for which a constant elongationalviscosity is derived. The comment above regarding quality of representation also applies here.

4.2 Multi-mode analysis

Table 2 provides material data for LDPE and HDPE polymers at a temperature of 2000C[32], in the form of relaxation times and partial zero-shear rate viscosities for a three-mode

7

approximation. The �ts to the the shear data yield zero-shear viscosities of 105,390 and139,184 Pa.s for LDPE and HDPE, respectively. Note, the variation to the single-mode caseis due to the di�erence in parameters that alters the �ts. Any solvent viscosity contributionis taken as minuscule for the polymer melts of interest, simply a mathematical convenienceto both aid data �tting and numerical convergence. For a melt any solvent contribution isinsigni�cant in practice; hence in the multi-mode context, �2 is assumed to vanish. Here,this is made practically possible from a numerical standpoint via the decoupled approachemployed. Also, the second normal stress di�erence (N2) is negligible compared with the�rst normal stress di�erence (N1). Hence from Eq. 13, the value of the non-dimensionalparameter � is small, and to simplify the analysis can be taken as zero. To estimate theremaining parameter �, the shear viscosity is plotted against the shear rate in Figure 6a forLDPE and Figure 7a for HDPE with various values of �. For both materials, the best �tfor 102 � _ � 103s�1 corresponds to a value of � of unity. At such a value, we observefrom both Figures 6b and 7b, the elongational viscosity (�e) �ts provide appropriate trendsin general behaviour, but gives a lower estimated value than the limited set of experimentaldata. This is due to the inability of the PTT model to match both shear and elongationaldata simultaneously.

From the knowledge gained through the single-mode study, we observe that the key factorto estimating the pressure-drop accurately in complex ows (see below) is the quality of �t tothe shear viscosity 'in situ' at the associated maxima of shear rate. Bearing this point in mind,in Figure 8 for LDPE we compare shear and elongational viscosity �ts to the experimentaldata for both the single mode of �1 = 5s (from above) and a three-mode model. In general,and accordance with Figure 8a for shear viscosity, the multi-mode case provides the better�t to the experimental data, throughout a wide range of shear rate. The multi-mode trendsof response for elongational viscosity of Figure 8b is a much better re ection of the actual uid properties than those observed with a single-mode approximation. Nevertheless, thematch here is somewhat adrift from the experimental data values, derived from the Bindinganalysis. The same �tting procedure has also been carried out using a seven mode model. Itis of note that no substantial improvement of �t is obtained if seven modes are employed inplace of three.

5 CONTRACTION FLOW

Armed with the parameter �ts to steady shear and pure extension, we proceed to analysea contraction ow, testing our simulated solution pressure-drops across the domain againstthose observed experimentally at a series of di�erent owrates [21].

5.1 Problem speci�cation

A schematic ow diagram for the axi-symmetric 19:1 contraction ow with its associated�nite element mesh is represented in Figure 9. The �nite element discretisation is representedby 1830 elements and 3829 nodes. This problem is representative of ow in a contractionCapillary rheometer. The ow may be assumed to be symmetric and hence only half thedomain is considered. Characteristic length and velocity scales are taken as the radius Rc ofthe die tube and mean exit velocity Uc, whilst the zero-shear rate viscosity �0 is consideredas the characteristic viscosity. A non-dimensional Reynolds number is de�ned as

Re =�UcRc

�0; (20)

8

and, for a single relaxation time, a Weissenberg number is taken as

We = �spec1

Uc

Rc

: (21)

For the multi-mode case, see above for comparable de�nitions. With respect to boundaryconditions, no-slip conditions are taken on the downstream tube walls, and symmetry con-ditions apply on the ow centreline. The upstream length is chosen to be su�ciently large68Rc, to establish fully developed Poiseuille ow in the inlet region. The downstream dielength is 40Rc. At the domain exit, a �xed pressure datum is adopted for consistency anda fully developed Poiseuille ow prevails. Each individual owrate is treated as a separateproblem to which a steady state is sought. For a single-mode, initial conditions are takenof a Newtonian kinematic �eld, coupled to a fully relaxed stress �eld with appropriate in-let boundary modi�cation. In the multi-mode case, initial kinematics are supplied from aprior single-mode computation. Then for each stress component mode, it is found suitable toimpose relaxed inlet boundary conditions that rapidly adopt their appropriate levels in theentry ow region.

5.2 Results for a single mode model

To establish a pressure drop versus owrate relationship numerically and to compare thisto experimental observations, three owrates have been employed for di�erent PTT uidparameter sets (�, �), in combination with �1. A given owrate, Q, covers a particular shear-rate range. The shear viscosity �s( _ ) may be adjusted through �1, � and � parameters to �tthe experimental data. Indeed, rising �1 tending towards �0, governs the second Newtonianplateau level for �s( _ ). Hence, elevating �1, e�ectively extends the range of �t for the �s( _ )function over wider _ ranges. Thus the larger the value of Q, the more �1 must be elevatedto enhance the �s( _ ) �t to the data. The classi�cation of various test uids, with �1-valuesrising up to �0, highlights Fluids B-H as indicated in Table 1. There, �1 = 0:875�0 forFluid B, �1 = 0:99�0 for Fluid C, �1 = 0:995�0 for Fluid D, �1 = 0:999�0 for Fluid E,�1 = 0:998�0 for Fluid F and �1 = �0 for Fluids G and H. We note that, lower �1-values aidsnumerical convergence as the solvent contribution incorporates damping into the system. Forthe single mode approximation, a single relaxation time of �spec

1= 5s is adopted to represent

the relaxation time spectrum for an LDPE polymer (see Ref. [32]). Reynolds numbers are ofO(10�6), whilst Weissenberg number values at various owrates are summarised in Table 3,and pressure-drop results in Table 4.

Overall, a summary of our �ndings reads as follows. By comparing the numerical andexperimental data for these three owrates, and investigating the parameter sensitivity ofshear viscosity on �1, � and � at di�erent shear rates, we conclude that the most in uentialparameter governing �p is �1; the remaining parameters (�, �) provide �ne tuning of the�t. The choice of parameter �1 depends on the value of maximum shear rate _ max attainedfor any given owrate. Hence with knowledge of _ max, �1 may be determined by choosing avalue to �t the experimental data of [21]. In Figure 10, maximum shear rates observed for thecontraction ow are plotted against owrate, from which a quadratic relationship emerges.From this, one may predict _ max for a given owrate and hence extract a particular �1-�t.

It is clear that with a single-mode approximation, it is necessary to continually adjust the uid parameters at each owrate setting, to adequately reproduce quantitative pressure-droppredictions. This keeps pace with the shear rate maxima observed, though does not providea single uid model representation suitable across a range of owrates. This is a drawback tothe use of a single mode PTT model. Essentially, what is being achieved here is to provide

9

a localised �t to data around the range of shear rates that dominates the process at eachparticular owrate. As regards incrementation of �1 towards unity, this leads to more severenumerical di�culties in convergence to steady state. Thus care in continuation with thisparameter is advisable purely on numerical grounds. At this juncture, it is natural to widenthe study to embrace a multi-mode approximation and follow a similar line of investigation.

5.3 Results for a three-mode model

With a three-mode PTT approximation, results for both LDPE and HDPE grade polymersare presented and compared to the experimental data for the contraction ow. Comparisonagainst the single-mode approximation is made only for the LDPE polymer, and � is takenas zero (see above, N2 << N1). Our observations are that computation times double fromsingle to three-mode model calculations.

5.3.1 LDPE uid

With the LDPE polymer a �rst owrate of Q2 = 39:9mm3=s is considered, for which thevalues of Weissenberg numbers for the corresponding three modes are given in Table 3. TheReynolds number is Re = 1:8 � 10�7 and the experimental pressure-drop is 10.9MPa. With uid G1 o�ering a parameter combination (�, �, �1)=(1.0, 0.0, �0), a simulated pressure-dropof 10.3MPa is predicted. This contrasts to the best single mode computation of 11.7MPa for uid D2 with (�, �, �1)=(0.3, 0.02, 0.995�0).

In contrast, at a second lesser owrate of Q1 = 2:37mm3=s, and the same parametercombination (�, �, �1)=(1.0, 0.0, �0), i.e. uid G1, a simulated pressure-drop of 5.90MPais generated; an over-estimation compared to the experimental data of 3.27MPa. This is inaccordance with the shear viscosity �t, where for the shear rate range 10 � _ � 102s�1, thebest �t renders a value of � = 2:5, as in Figure 6a. For uid G2 of � = 2:5, the pressure-drop is 4.00MPa compared to 4.09MPa for the single mode calculation of uid C1. Figure12a summarises graphically the comparison of numerical and experimental data at both owrates, including the single mode results. The values are tabulated in a uni�ed manner inTable 4. Conventional use of the PTT class of models, is restrictive with a constant valuefor the parameter �. If this parameter is taken in the range 1:0 � � � 2:5, pressure-drop willbe better estimated by the multi-mode than the single mode model for the owrate range2:37 � Q � 40mm3=s. We observe in Figure 12a, that the discrepancy from the experimentalvalue is huge for the single mode uid C1 { parameter set (�, �, �1)=(0.15, 0.02, 0.99�0) atQ2 = 39:9mm3=s.

5.3.2 HDPE uid

For the HDPE polymer, computations are conducted for owrates of Q1 = 2:37mm3=s andQ2 = 34:0mm3=s. At the former owrate, the �tting of the PTT model to the experimentalshear viscosity, as in Figure 7, provides an equally good dual set of parameters with either (�,�, �1)=(2.5, 0.0, �0) or (�, �, �1)=(3.0, 0.0, �0), i.e. uidH2 or H3. Simulated pressure-dropsare 7.40 MPa for uid H2 � = 2:5, and 6:90MPa for uid H3 � = 3:0, that compare against anexperimental value of �p = 5:83MPa. It is implied that additional increase in the parameter�, will lead to further decrease in pressure-drop.

At a second owrate of Q2 = 34:0mm3=s, and according to Figure 7, the best �t tothe shear viscosity data, results in the set of parameters (�, �, �1)=(1.0, 0.0, �0) for therange 102 � _ � 103s�1, i.e. uid H1. The simulated pressure-drop is 19.2MPa, whilst the

10

experimental value is 19.9MPa. This evidence is taken as endorsement for the choice of multi-mode representation with uids G1 and H1, that is wholy acceptable across the owratesselected. A summary of simulated and experimental pressure-drop values is provided forHDPE in Table 5 and Figure 12b.

6 WIRE-COATING FLOW

Having established the goodness of �t to experimental pressure-drop data for a complex con-traction ow, we now proceed to the industrial ow of interest that provides the motivationfor this study. The ow quantities of speci�c relevance are residual ow-induced stress andpressure-drop. The former is fresh information to be gathered for this problem that is impor-tant to control coating properties. These quantities have impact on the optimisation of theprocess design. Only uid G1 (i.e. one set of parameters (�, �)=(1,0)) is adopted through-out the study of tube-tooling wire-coating for LDPE and H1 for HDPE polymer, since areasonable �t is achieved in the shear-rate range under consideration ( _ � 103s�1).

6.1 Problem speci�cation

A schematic ow diagram for the tube-tooling problem with its �nite element mesh is dis-played in Figure 13. The ow enters the annular tube AB, then a converging cone sectionBC and a land region CD, and is draw-down by the wire in a converging cone form DE, andcoating part of the wire EF, as shown in Figure 13a. Due to symmetry, it is necessary toonly model one half of the domain. Characteristic length and velocity scales are taken asthe horizontal distance between the die exit of the tube and the contact point on the wire,i.e., the draw-down length Ldraw and the velocity of the wire Uwire, respectively. The zero-shear viscosity �0 is considered as the characteristic viscosity. Here, the same de�nition forReynolds and Weissenberg numbers is taken as for the contraction ow problem. Boundaryconditions are given as follows. No-slip conditions are taken on tube walls for the die tube,ABCD and D'C'B'A'. At the entry AA' a fully developed Newtonian annular velocity pro�lecorresponding to a speci�c ow rate Q is imposed, (Ur = 0 and Vz = Vz(r)). From thisvelocity pro�le, a PTT stress pro�le is generated analytically. Along the free surface, DE andF'D', a plug-like velocity pro�le is assumed, and at the outlet DD', the uid is considered tobe moving on the wire with the same velocity (plug ow). A uniform structured mesh is usedin the �nite element discretisation, 2680 elements and 5649 nodes, (thoroughly investigatedin our prior studies, see Mutlu et al. [8, 9]). To capture sharp velocity gradients near singularregions, di�erent meshes were employed elsewhere (see [8, 33]). In the present study only the�nest mesh of Ref. [8] is employed, for which the number of elements per region is given inTable 7.

6.2 LDPE uid

Table 6 provides values of Weissenberg numbers for an LDPE polymer, with material param-eters given in Table 2, for a standard owrate (1Q) and double owrate (2Q). The Reynoldsnumbers are Re = 1:5 � 10�4 and Re = 3:0 � 10�4, respectively. Figure 14 plots the pressure-drop line for both 1Q and 2Q, along a sample line tangential to the ow. Simulated pressure-drop values are 6.90 and 8.40MPa, respectively, and are compared to 7.49 and 11.2 MPausing uid C1 for the single mode approximation (as [8] using �1 = 0:99�0, � = 0:15 and� = 0:02). Pressure-drop results are tabulated in Table 8 accordingly. For the single mode,pressure-drop at 2Q is one and a half times that at 1Q, whilst for a multi-mode model, thisfactor reduces to 1.22. Overall, the trend in variation of pressure-drop is similar for single

11

and multi-mode models in the tube-tooling die sections, notwithstanding the elevation of thesingle-mode 2Q result. In contrast in the draw-down section for the single mode, pressure-drop is almost constant for 1Q, but for 2Q, it increases around the location where the uidmeets the wire. Traveling with the wire, pressure diminishes to zero. However for a multi-mode model, the pressure-drop decreases continuously with only a slight change in the slopeon encountering the wire. These di�erences in response for these models, are due to thevariation in radial shear stress gradients, particularly at die exit, that are about one third ofthe level observed in the inlet tube-tooling section. In Figure 15, the shear stress is plottedalong a sample line tangential to the ow. A constant value is observed in the inlet tube,followed by a change of sign over the tube-tooling cone, with an increase in the absolute valueas the shearing increases. In the land section, the degree of shearing increases again, sharplyat the start, rising to reach a maximum at the end. There is then a dramatic sharp dropand oscillation in shear stress at the inlet of the draw-down cone. Thereafter, it decreasesgradually but smoothly along the draw-down cone, to increase slightly when the wire is metand subsequently remains steady. This behaviour is similar for both owrates, only di�eringin the absolute level of the stress.

For the normal stress component in the axial direction, �zz, a sharp oscillation is experi-enced at the inlet of the tube-tooling cone, followed by a smooth increase, a sudden decreaseat the inlet of the land region, an increase towards its exit and a sharp drop at the inlet ofthe draw-down cone. Over this section the stress component decreases, but with a smallervariation rate than that observed upstream in the tube-tooling cone. There is a marginalincrease when the uid meets the wire. A relaxation of stress occurs within the coating owon the wire, with larger slope than in the draw-down cone. A build-up of shear and normalstresses is observed in the tube-tooling cone and the land region, followed by relaxation inthe draw-down and wire coating ow sections.

For stress, the trends are identical for single to multi-mode cases, the di�erences lie in thestress levels, that are dictated by the inlet ow. At inlet and for the shear rates that applythere, see Figure 8, a closer match to �s is observed with a single mode model, yielding alower value of shear stress (via viscosity) than with the multi-mode model. We note also thatin the converging tube the rate of increase of �zz in the multi-mode case is greater than thatcorresponding to the single mode instance. This is due to the in uence of the shortest mode�1 = 0:017s; see below for further comment on the contributions due to the separate modes.

6.3 HDPE uid

Results for the HDPE polymer are also presented at the same owrates as above for LDPE.In this case, the parameter � = 1:0 represents the best �t to the shear viscosity, as in Figure7, hence uid H1. Predicted pressure-drops are 12.4 and 15.4MPa for the two associated owrates, respectively. Figure 16 shows a sample line plot for pressure-drop at both owrates,and a similar trend to that for the LDPE polymer is observed. We note that pressure-dropat the double owrate is 1.24 times as great as that at the standard owrate; the factorrelating pressure-drops is similar to that found for LDPE, of 1.22. It is conspicuous thatcorresponding pressure-drops for HDPE polymer are almost double those for LDPE polymer.According to Figure 17, line plots for shear and normal stress components along a sampleline tangential to the ow, the general behaviour of stress is similar to that for LDPE, thoughscales are doubled in shear stress and increase by about 20% in normal stress. These �ndingsare in keeping with general expectations for these materials and ows.

12

6.4 Analysis of separate modes

To understand the contribution of each mode (i) to the total stress of the multi-mode model,both �rz and �zz are plotted for the standard owrate, LDPE polymer in Figure 18. Of theindividual modal contributions, the two shorter relaxation time modes dominate. The shearstress is dominated by the shortest relaxation time. This is true for all cases studied, coveringboth materials and owrates, as con�rmed in Figure 18 for LDPE and Figure 19 for HDPE.

Alternatively, for the �zz component, according to Figure 18 and LDPE, the shortestrelaxation time dominates in the tube-tooling cone; this mode dictates the shape of thetotal stress in all the sections of the ow. In contrast, the second shortest relaxation timedominates in the inlet tube and draw-down sections, including the ow on the wire. Forthe HDPE polymer, as in Figure 19, the shortest relaxation time dominates throughoutthe tube-tooling die sections, whilst in the draw-down, the second shortest relaxation timedominates.

7 CONCLUSIONS

A contraction ow has been used e�ectively to demonstrate how single and multi-mode PTTmodels perform in quantitatively replicating pressure-drop/ owrate data. Essentially this isgoverned by the goodness of �t to the shear viscosity. In this regard, the multi-mode instanceperforms better across the range of shear rates of interest. In the single-mode case reasonablecorrespondence has been derived, presupposing a judicious choice of averaged relaxation time,which has been possible here. Also, the choice of material parameter combination mustvary with each owrate (and its associated maximum shear rate) to adequately re ect theexperimental data. This generates di�erent uid model representations and demonstratesthe shortcomings of an averaged mode approximation. The multi-mode alternative, performsbetter in this respect with a �xed uid parameter set.

In the more generalised context of the industrial tube-tooling wire-coating ow, the di�er-ences in results on stress distributions and pressure-drop between these modelling approachesis more stark. Here, the multi-mode approximation reveals the �ne detail of stress responsethroughout the ow, peak values attained and dominance of individual components. Thekey point emerges that the shorter relaxation time modes are observed to dominate withinthe process. These modes are of the order of less than one second, being synonymous witha representative particle residence time as it travels through the ow domain. Stress andpressure-drop levels are practically doubled for the HDPE above the LDPE polymer, whichis consistent with practical industrial experience for these materials. Also pressure-drop vari-ation with owrate doubling is held to be more realistic with the multi-mode option. Thedi�erences from single to multi-mode total stress appear minimal in the draw-down andcoating regions, taking into account the elevations in stress levels within the steady shear dieentry ow as discussed above. Hence, at a �xed owrate, reasonable estimates of residualstress levels in the coating may be made from a single mode analysis, with the proviso thatan appropriate selection of relaxation time is taken.

8 ACKNOWLEDGEMENTS

The authors wish to gratefully acknowledge the �nancial support and collaboration of BICCCables Ltd. Thanks is also due to Professor Ken Walters, FRS, Professor Russell Davies andDr David Binding for their support work on rheological characterisation.

13

References

[1] R.T. Fenner and J.G. Williams. Trans. J. Plastics Inst., pages 701{706, 1967.

[2] B. Caswell and R.I. Tanner. `Wirecoating die design using �nite element methods'.Polymer Eng. Sci., 18 (5):416{421, (1978).

[3] E. Mitsoulis. `Fluid ow and heat transfer in wire coating: A review'. Advances in

Polymer Technology, 6 (4):467{487, (1986).

[4] E. Mitsoulis, R. Wagner, and F.L. Heng. `Numerical simulation of wire-coating low-density polyethylene: Theory and experiments'. Polymer Eng. Sci., 28 (5):291{311,(1988).

[5] H.C. Huang, P. Townsend, and M.F. Webster. Numerical Grid Generation in Compu-

tational Fluid Dynamics and Related Fields. N.P. Weatherill and P.R. Eiseman and J.Hauser and J.F. Thompson, Pineridge Press, Swansea, 1994.

[6] D.M. Binding, A.R. Blythe, S. Gunter, A.A. Mosquera, P. Townsend, and M.F. Webster.`Modelling polymer melt ows in wirecoating processes related �elds'. J. Non-NewtonianFluid Mech., 64:191{206, (1996).

[7] S. Gunter, P. Townsend, and M.F. Webster. `Simulation of some model viscoelasticextensional ows'. Int. J. Num. Meth. Fluids, 23:691{710, (1996).

[8] I. Mutlu, P. Townsend, and M.F. Webster. `Simulation of cable-coating viscoelastic owswith coupled and decoupled schemes'. J. Non-Newtonian Fluid Mech., 74:1{23, (1998).

[9] I. Mutlu, P. Townsend, and M.F. Webster. `Computation of viscoelastic cable coating ows'. Int. J. Numer. Meth. Fluids, 26:697{712, (1998).

[10] I. Mutlu, P. Townsend, and M.F. Webster. `Mathematics of �nite elements andapplications-highlights 1996'. ed. J.R. Whiteman, Wiley, Chichester, 18:299{312,(1997).

[11] H.P.W. Baaijens. Evaluation of constitutive equations for polymer melts and solutionsin complex ows. PhD. Thesis, Eindhoven University of Technology, Netherlands, 1994.

[12] F.P.T. Baaijens, S.H.A. Selen, H.P.W. Baaijens, G.W.M. Peters, and H.E.H. Meijer.`Viscoelastic ow past a con�ned cylinder of a LDPE melt'. J. Non-Newtonian Fluid

Mech., 68:173{203, (1997).

[13] J. Azaiez, R. Gu�enette, and A. Ait-Kadi. `Entry ow calculations using multi-modemodels'. J. Non-Newtonian Fluid Mech., 66:271{281, (1996).

[14] M. Gupta, C.A. Hieber, and K.K. Wang. `Viscoelastic modelling of entrance ow usingmultimode Leonov model'. Int. J. Numer. Meth. Fluids, 24:493{517, (1997).

[15] H. Matallah. Ph.D. Thesis. University of Wales, Swansea, 1998.

[16] N. Phan-Thien and R.I. Tanner. `A new constitutive equation derived from networktheory'. J. Non-Newtonian Fluid Mech., 2:353{365, (1977).

[17] P. Saramito and J.M. Piau. `Flow characteristics of viscoelastic uids in an abruptcontraction by using numerical modelling'. J. Non-Newtonian Fluid Mech., 52:263{288,(1994).

14

[18] H. Matallah, P. Townsend, and M.F. Webster. `Recovery and stress-splitting schemesfor viscoelastic ows'. J. Non-Newtonian Fluid Mech., 75:139{166, (1998).

[19] J.R.A. Pearson and S.M. Richardson. Computational Analysis of Polymer Processing.Applied Science Publishers Ltd, London and New York, 1983.

[20] J.F.M. Schoonen, F.H.M. Swartjes, G.W.M. Peters, F.P.T. Baaijens, and H.E.H. Meijer.À 3D numerical/experimental study on a stagnation ow of a polyisobutylene solution'.J. Non-Newtonian Fluid Mech., 79:529{561, (1998).

[21] K. Walters, D.M. Binding, and R.E. Evans. `Modelling the rheometric behaviour of3 polyethylene melts'. Technical report, Institute of non-Newtonian Fluid Mechanics,University of Wales, Aberystwyth, May 10, 1994.

[22] D.M. Binding. Àn approximate analysis for contraction and converging ows'. J. Non-Newtonian Fluid Mech., 27:173{189, (1988).

[23] L.M. Quinzani, G.H. Mckinley, R.A. Brown, and R.C. Armstrong. `Modeling the rheol-ogy of polyisobutylene solutions'. J. Rheol., 34:705{748, (1990).

[24] N. Phan-Thien. À non-linear network viscoelastic model'. J. Rheol., 22:259{283, (1978).

[25] A. Arsac, C. Carrot, J. Guillet, and P. Revenu. `Problems originating from the use ofthe Gordon-Schowalter derivative in the Johnson-Segalman and related models in variousshear- ow situations'. J. Non-Newtonian Fluid Mech., 55:21{36, (1994).

[26] O.C. Zienkiewicz and J.Z. Zhu. `Superconvergence and the superconvergent patch re-covery'. Finite Element in Analysis and Design, 19:11{23, (1995).

[27] S. Gunter, M.F. Webster, and P. Townsend. BICC internal report. Computer ScienceDepartment, University of Wales, Swansea, 1995.

[28] D.M. Hawken, H.R. Tamaddon-Jahromi, P. Townsend, and M.F. Webster. À Taylor-Galerkin based algorithm for viscous incompressible ow'. Int. J. Num. Meth. Fluids,10:327{351, (1990).

[29] F.N. Cogswell. `Converging ow of polymer melts in extrusion dies'. Polymer Eng. Sci.,12:64{74, (1972).

[30] D.M. Binding. In: A.A. Collyer (Ed.), Techniques in Rheological Measurement. Chap-man and Hall, London, UK, Chap 1, 1993.

[31] D.M. Binding, M.A. Couch, and K. Walters. `The pressure dependence of the shear andelongational properties of polymer melts'. J. of Non-Newtonian Fluid Mech., 79:137{155, (1998).

[32] A.R. Davies, K. Walters, and D.M. Binding. Àn analysis of the discrete relaxationspectra of 3 polyethylene samples for BICC group', private communications. Technicalreport, Institute of non-Newtonian Fluid Mechanics, University of Wales, Aberystwyth,June 20, 1996.

[33] I. Mutlu, P. Townsend, and M.F. Webster. Àdaptive solutions for viscoelastic ows'.Commun. Numer. Meth. Eng., 12:643{655, (1996).

15

List of Tables

1 Classi�cation of uids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 We for contraction ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Pressure drop for contraction ow, LDPE polymer . . . . . . . . . . . . . . . 175 Pressure drop for contraction ow, HDPE polymer, multi-mode . . . . . . . . 176 We for tube-tooling ow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Number of elements per region . . . . . . . . . . . . . . . . . . . . . . . . . . 188 Pressure drop for tube-tooling ow . . . . . . . . . . . . . . . . . . . . . . . . 18

List of Figures

1 Schematic diagram of wire coating dies: (a) tube-tooling die; (b) pressure-tooling die. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Shear and elongational viscosities �ts; �1 variation, � = 0:15, � = 0:02. . . . . 203 Shear and elongational viscosities �ts; � variation, �1 = 0:997, � = 0:02. . . . 214 Shear viscosity �ts; � variation, �1 = 0:997. . . . . . . . . . . . . . . . . . . . 225 Elongational viscosity �ts; � variation, �1 = 0:997. . . . . . . . . . . . . . . . 236 Rheometrical data �ts with � variation for 3-mode LDPE. . . . . . . . . . . . 247 Rheometrical data �ts with � variation for 3-mode HDPE. . . . . . . . . . . . 258 Single and multi-mode �ts for LDPE. . . . . . . . . . . . . . . . . . . . . . . 269 Schematic ow diagram and �nite element mesh for 19:1 contraction geometry. 2710 Maximum shear-rate v owrate correlation for contraction problem, LDPE . 2811 Pressure drop v owrate for contraction problem, single mode model . . . . . 2912 Simulated and experimental pressure drops for contraction ow; a) LDPE, b)

HDPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3013 Schematic diagram and �nite element mesh for wire coating tube-tooling die. 3114 Tube-tooling ow, streamwise pressure pro�les, single and multi-mode models,

LDPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3215 Tube-tooling ow, streamwise stress pro�les, single and multi-mode models,

LDPE, a) �rz, b) �zz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316 Tube-tooling ow, streamwise pressure pro�les, standard and double owrate,

HDPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3417 Tube-tooling ow, streamwise stress pro�les, multi-mode, HDPE, a) �rz, b) �zz 3518 Tube-tooling ow, streamwise stress pro�les for individual stress mode, LDPE,

a) �rz, b) �zz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3619 Tube-tooling ow, streamwise stress pro�les for individual stress mode, HDPE,

a) �rz, b) �zz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

16

Fluid B �1 = 0:875�0 Fluid C �1 = 0:99�0 Fluid D �1 = 0:995�0B1: (0.1,0.01) C1 (0.15,0.02) D1 (0.15,0.02)B2: (0.3,0.01) C2 (2.0,0.02) D2 (0.3,0.02)B3: (2.0,0.01)

Fluid E �1 = 0:999�0 Fluid F �1 = 0:998�0 Fluid G �1 = �0, � = 0(0.3, 0.02) (0.15, 0.02) G1: � = 1:0

G2: � = 2:5G3: � = 3:0

Fluid H �1 = �0, � = 0H1: � = 1:0H2: � = 2:5H3: � = 3:0

Table 1: Classi�cation of uids

LDPE HDPE

�i(s) �i(Pa:s) gi =�i�i

�i(s) �i(Pa:s) gi =�i�i

Mode 1 0.017 1234 72588 0.017 2792 164222Mode 2 0.87 15982 18370 0.87 19595 22523Mode 3 33.9 88174 2601 59.57 116797 1961

Table 2: Material parameters

LDPE HDPE

Q mm3=s 2.37 39.9 2.37 34.0

Mode 1 0.10 1.73 0.10 1.47We Mode 2 5.30 88.4 5.30 75.3

Mode 3 204.6 3444 360.3 5156.7

Table 3: We for contraction ow

Qmm3=s �pExp (MPa) �pSim (MPa) �pSim (MPa)Single mode, �spec

1= 5s Multi-mode

2.37 3.27 4.09, uid C1 5.90, G1, � = 1:0We = 30:18, � = 0:15, � = 0:02 4.00, G2, � = 2:5

39.9 10.9 11.7, uid D2 10.3, G1, � = 1:0We = 508:0, � = 0:3, � = 0:02

Table 4: Pressure drop for contraction ow, LDPE polymer

Qmm3=s �pExp (MPa) �pSim (MPa)

2.37 5.83 7.40, uid H2, � = 2:56.90, uid H3, � = 3:0

34.0 19.9 19.2, uid H1, � = 1:0

Table 5: Pressure drop for contraction ow, HDPE polymer, multi-mode

17

LDPE HDPE

1Q 2Q 1Q 2Q

Mode 1 0.095 0.19 0.095 0.19We Mode 2 4.843 5.686 4.843 5.686

Mode 3 188.7 377.4 332.3 664.6

Table 6: We for tube-tooling ow

ABB'A' BCC'B' CDD'C' DEE'D' EFF'E'

680 680 120 800 400

Table 7: Number of elements per region

LDPE HDPE

1Q 2Q 1Q 2Q

�p (MPa) Single mode 7.49 11.2, uid C1 ��

Multi-mode 6.90 8.40, uid G1 12.4 15.4, uid H1

Table 8: Pressure drop for tube-tooling ow

18

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

a) Tube-tooling die

Melt ow

Wire

Melt ow

b) Pressure-tooling die

Melt ow

Wire

Melt ow

Figure 1: Schematic diagram of wire coating dies: (a) tube-tooling die; (b) pressure-toolingdie.

19

1

2

3

4

5

6

-2 -1 0 1 2 3 4

123456

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

123456

Exp

a) Shear viscosity

Log( _ s�1)

Log(�s Pa.s)

[1] �1 = 0:875

[2] �1 = 0:99

[3] �1 = 0:995

[4] �1 = 0:997

[5] �1 = 0:998

[6] �1 = 0:999

b) Elongational viscosity

Log( _� s�1)

Log(�e3Pa.s)

Figure 2: Shear and elongational viscosities �ts; �1 variation, � = 0:15, � = 0:02.

20

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

a) Shear viscosity

Log( _ s�1)

Log(�s Pa.s)

[1] � = 0:15

[2] � = 0:3

[3] � = 0:5

[4] � = 0:7


Log( _� s�1)

Log(�e3Pa.s)

Figure 3: Shear and elongational viscosities �ts; � variation, �1 = 0:997, � = 0:02.

21

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

a) � = 0:15

Log( _ s�1)

Log(�s Pa.s)

[1] � = 0:

[2] � = 0:02

[3] � = 0:1

[4] � = 1:0

[4] [2][3]

[1]

b) � = 1:0

Log( _ s�1)

Log(�s Pa.s)

[1], [2]

[4][3]

Figure 4: Shear viscosity �ts; � variation, �1 = 0:997.

22

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

a) � = 0:15

Log( _� s�1)

Log(�e3Pa.s)

[1] � = 0:

[2] � = 0:02

[3] � = 0:1

[4] � = 1:0[4]

[4]

b) � = 1:0

Log( _� s�1)

Log(�e3Pa.s)

Figure 5: Elongational viscosity �ts; � variation, �1 = 0:997.

23

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

1234

Exp

a) Shear viscosity

Log( _ s�1)

Log(�s Pa.s)

[1] � = 0:1

[2] � = 0:5

[3] � = 1:0

[4] � = 2:5


Log( _� s�1)

Log(�e3Pa.s)

Figure 6: Rheometrical data �ts with � variation for 3-mode LDPE.

24

1

2

3

4

5

6

-2 -1 0 1 2 3 4

123

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

123

Exp

a) Shear viscosity

Log( _ s�1)

Log(�s Pa.s)

[1] � = 1:0

[2] � = 2:5

[3] � = 3:0


Log( _� s�1)

Log(�e3Pa.s)

Figure 7: Rheometrical data �ts with � variation for 3-mode HDPE.

25

1

2

3

4

5

6

-2 -1 0 1 2 3 4

12

Exp

1

2

3

4

5

6

-2 -1 0 1 2 3 4

12

Exp

a) Shear viscosity

Log( _ s�1)

Log(�s Pa.s)

[1] 1-mode �1 = 0:999,

� = 0:15, � = 0:02

[2] 3-mode � = 1:0,

� = 0:


Log( _� s�1)

Log(�e3Pa.s)

Figure 8: Single and multi-mode �ts for LDPE.

26

a) Schematic ow diagram

z

r

Ur = 0

Vz = V (r)

�� = ��(r) Solid wall

Vr = 0

Vz = V (r)

Ur = 0, @vz@r

= 0 (�rz = 0)

b) Finite element mesh

Figure 9: Schematic ow diagram and �nite element mesh for 19:1 contraction geometry.

27

2001801601401201008 06 04 02 000

500

1000

1500

2000

2500

3000

3500

4000

_ max (s�1)

Q (mm3=s)

"

estimated value

Figure 10: Maximum shear-rate v owrate correlation for contraction problem, LDPE

28

�p (MPa)

Q (mm3=s)

Figure 11: Pressure drop v owrate for contraction problem, single mode model

29

8 07 06 05 04 03 02 01 000

5

10

15

20

25

30

4 03 02 01 000

5

10

15

20

25

30

�p (MPa)

�1 = 0:99, � = 0:15, � = 0:02

�1 = 0:995, � = 0:3, � = 0:02

3-modes, � = 1:0

3-modes, � = 2:5

Exp [21]

Q (mm3=s)

a) LDPE

3-modes, � = 1:0

3-modes, � = 2:5

3-modes, � = 3:0

Exp [21]

�p (MPa)

Q (mm3=s)

b) HDPE

Figure 12: Simulated and experimental pressure drops for contraction ow; a) LDPE,b) HDPE

30

��

��

��

��

��

a) Schematic ow diagram

Wire

AB

C D

E F

F'

D'C'

B'A'

b) Finite element mesh

A

E

F'

F

D

D'

A'

D

D'

Figure 13: Schematic diagram and �nite element mesh for wire coating tube-tooling die.

31

-1.833 -1.167 -0.500 0.167 0.833 1.500

0

2

4

6

8

10

12

1-mode, 1Q

3-modes, 1Q

3-modes, 2Q

1-mode, 2Q

�p (MPa)

Axial distance, z?

Figure 14: Tube-tooling ow, streamwise pressure pro�les, single and multi-mode models,LDPE

32

-1.700 -1.060 -0.420 0.220 0.860 1.500

-0.2

-0.1

0.0

0.1

0.2

1-mode, 1Q

1-mode, 2Q

3-modes, 1Q

3-modes, 2Q

-1.700 -1.060 -0.420 0.220 0.860 1.500

0.0

0.1

0.2

0.4

0.5

0.6

1-mode, 1Q

1-mode, 2Q

3-modes, 1Q

3-modes, 2Q

�zz (MPa)

Axial distance, z?

�rz (MPa)

Axial distance, z?

Figure 15: Tube-tooling ow, streamwise stress pro�les, single and multi-mode models,LDPE, a) �rz, b) �zz

33

-1.833 -1.167 -0.500 0.167 0.833 1.500

0.0

4.0

8.0

12.0

16.0

20.0

1Q

2Q

�p (MPa)

Axial distance, z?

Figure 16: Tube-tooling ow, streamwise pressure pro�les, standard and double owrate,HDPE

34

-1.833 -1.167 -0.500 0.167 0.833 1.500

-0.2

-0.1

0.0

0.1

0.2

0.3

1Q

2Q

-1.833 -1.167 -0.500 0.167 0.833 1.500

0.0

0.2

0.4

0.6

0.8

1.0

1Q

2Q

�zz (MPa)

Axial distance, z?

�rz (MPa)

Axial distance, z?

Figure 17: Tube-tooling ow, streamwise stress pro�les, multi-mode, HDPE,a) �rz, b) �zz

35

-1.70 -1.06 -0.42 0.22 0.86 1.50

-0.10

0.05

0.20

mode1

mode2

mode3

-1.70 -1.06 -0.42 0.22 0.86 1.50

0.0

0.1

0.2

mode1

mode2

mode3

�zz (MPa)

Axial distance, z?

�rz (MPa)

Axial distance, z?

Figure 18: Tube-tooling ow, streamwise stress pro�les for individual stress mode, LDPE,a) �rz, b) �zz

36

-1.70 -1.06 -0.42 0.22 0.86 1.50

-0.1

0.0

0.2

0.3

mode1

mode2

mode3

-1.70 -1.06 -0.42 0.22 0.86 1.50

0.0

0.2

0.3

0.5

mode1

mode2

mode3

�zz (MPa)

Axial distance, z?

�rz (MPa)

Axial distance, z?

Figure 19: Tube-tooling ow, streamwise stress pro�les for individual stress mode, HDPE,a) �rz, b) �zz

37

Visco elastic · Visco elastic Computations of P olymeric Wire-coating Flo ws H. Matallah, P.T o...

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