Numerical simulation of tube-tooling cable-coating with ...

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© 2013 The Korean Society of Rheology and Springer 197 Korea-Australia Rheology Journal, Vol.25, No.4, pp.197-216 (November 2013) DOI: 10.1007/s13367-013-0021-x www.springer.com/13367 Numerical simulation of tube-tooling cable-coating with polymer melts A. Al-Muslimawi 1,3 , H.R. Tamaddon-Jahromi 2 and M.F. Webster 2 1 College of Science, Mathematics, Swansea University, Swansea, SA28PP, UK 2 Institute of Non-Newtonian Fluid Mechanics, College of Engineering, Swansea University, Swansea, SA28PP, UK 3 Mathematics Department, College of Science, Basra University, Iraq (Received September 17, 2013; final revision received October 2, 2013; accepted October 4, 2013) This study investigates the numerical solution of viscous and viscoelastic flows for tube-tooling die-extru- sion coating using a hybrid nite element/nite volume discretisation (fe/fv). Such a complex polymer melt extrusion-draw-coating flow displays a dynamic contact line, slip, die-swell and two separate free-surfaces, presenting an inner and outer conduit surface to the melt-coating. The practical interest lies in determining efficient windows for process control over variation in material properties, stressing levels generated and vacuum pressure levels imposed. The impact of shear-thinning is also considered. Extensive reference is made throughout to viscous inelastic counterpart solutions. Attention is paid to the influence and variation in relevant parameters of Weissenberg number (We), solvent-fraction (β) and second normal difference (N 2 ) (ξ parameter for EPTT). The impact of model choice and parameters upon field response is described in situ through, pressure-drops, rates of deformation and stress. Various numerical alternative strategies, their stability and convergence issues are also addressed. The numerical scheme solves the momentum-con- tinuity-surface equations by a semi-implicit time-stepping Taylor-Galerkin/pressure-correction (TGPC) finite element (parent-cell) method, whilst invoking a sub-cell cell-vertex fluctuation distribution finite vol- ume scheme for the constitutive stress equation. The hyperbolic aspects of the constitutive equation are addressed discretely through upwind Fluctuation Distribution techniques, whilst temporal and source terms are consistently accommodated through medium-dual-cell schemes. The dynamic solution of the moving boundary problem may be resolved by either separating the solution process for each free-surface section (decoupling), or coupling both sections and solving simultaneously. Each involves a surface height location method, with dependency on surface nodal velocities and surface element sections; two such schemes are investigated. Dedicated and localised shock-capturing techniques are introduced to handle solution sin- gularities as disclosed by die-swell, slip and moving contact lines. Keywords: Tube tooling, cable coating, die extrusion, free surface, finite element/volume viscoelastic mod- elling 1. Introduction Cable coating procedures play an important role in the modern industrial process environment. This process involves the extension of a molten polymer over a moving cable, and typically requires a cable preheater, an extruder with a cross-head shaped die, a cooling trough for the extruded cable, and a take-up and payoff device (see, for example, Nadiri and Fenner, 1980; Mitsoulis, 1986; Mit- soulis et al., 1988). The subject matter of cable-coating, with dual free-surface location, die-swell and slip, dynamic contact point singularity and influence of vac- uum effects, remains a significant challenge, yet despite its commercial importance, relatively little tube-tooling cable-coating research has been reported in the literature. A finite element analysis of wire-coating for power law fluid was presented by Caswell and Tanner (1978). In this study, finite element approximation was able to accom- modate awkward geometries and non-Newtonian fluid properties in a realistic manner, and produce streamline and stress patterns within the die. Kuyl (1996, 1997) reported predictions for both isothermal and non-isother- mal high-speed tube-coating melt flow using a finite ele- ment method. He concluded that various parameters influence the melt cone shape and stability; also, pre- dicting the distance from the die-exit to the contact-point. Under isothermal analysis, Kuyl (1996) demonstrated that applying a vacuum both shortens and stabilizes the contact length. Through his non-isothermal work, Kuyl (1997) showed that cooling the melt cone extends the contact length; but also cone cooling destabilizes the cone length. Theoretically, Slattery et al. (1999) derived an analytic solution for cable-coating draw-down of an extruded annular melt. The analysis is for Newtonian, power-law # This paper is based on an invited lecture presented by the correspond- ing author at the 13th International Symposium on Applied Rheology (ISAR), held May 23, 2013, Seoul. *Corresponding author: [email protected]

Transcript of Numerical simulation of tube-tooling cable-coating with ...

Page 1: Numerical simulation of tube-tooling cable-coating with ...

© 2013 The Korean Society of Rheology and Springer 197

Korea-Australia Rheology Journal, Vol.25, No.4, pp.197-216 (November 2013)DOI: 10.1007/s13367-013-0021-x

www.springer.com/13367

Numerical simulation of tube-tooling cable-coating with polymer melts

A. Al-Muslimawi1,3

, H.R. Tamaddon-Jahromi2

and M.F. Webster2

1College of Science, Mathematics, Swansea University, Swansea, SA28PP, UK2Institute of Non-Newtonian Fluid Mechanics, College of Engineering, Swansea University, Swansea, SA28PP, UK

3Mathematics Department, College of Science, Basra University, Iraq

(Received September 17, 2013; final revision received October 2, 2013; accepted October 4, 2013)

This study investigates the numerical solution of viscous and viscoelastic flows for tube-tooling die-extru-sion coating using a hybrid nite element/nite volume discretisation (fe/fv). Such a complex polymer meltextrusion-draw-coating flow displays a dynamic contact line, slip, die-swell and two separate free-surfaces,presenting an inner and outer conduit surface to the melt-coating. The practical interest lies in determiningefficient windows for process control over variation in material properties, stressing levels generated andvacuum pressure levels imposed. The impact of shear-thinning is also considered. Extensive reference ismade throughout to viscous inelastic counterpart solutions. Attention is paid to the influence and variationin relevant parameters of Weissenberg number (We), solvent-fraction (β) and second normal difference (N2)(ξ parameter for EPTT). The impact of model choice and parameters upon field response is described insitu through, pressure-drops, rates of deformation and stress. Various numerical alternative strategies, theirstability and convergence issues are also addressed. The numerical scheme solves the momentum-con-tinuity-surface equations by a semi-implicit time-stepping Taylor-Galerkin/pressure-correction (TGPC)finite element (parent-cell) method, whilst invoking a sub-cell cell-vertex fluctuation distribution finite vol-ume scheme for the constitutive stress equation. The hyperbolic aspects of the constitutive equation areaddressed discretely through upwind Fluctuation Distribution techniques, whilst temporal and source termsare consistently accommodated through medium-dual-cell schemes. The dynamic solution of the movingboundary problem may be resolved by either separating the solution process for each free-surface section(decoupling), or coupling both sections and solving simultaneously. Each involves a surface height locationmethod, with dependency on surface nodal velocities and surface element sections; two such schemes areinvestigated. Dedicated and localised shock-capturing techniques are introduced to handle solution sin-gularities as disclosed by die-swell, slip and moving contact lines.

Keywords: Tube tooling, cable coating, die extrusion, free surface, finite element/volume viscoelastic mod-

elling

1. Introduction

Cable coating procedures play an important role in the

modern industrial process environment. This process

involves the extension of a molten polymer over a moving

cable, and typically requires a cable preheater, an extruder

with a cross-head shaped die, a cooling trough for the

extruded cable, and a take-up and payoff device (see, for

example, Nadiri and Fenner, 1980; Mitsoulis, 1986; Mit-

soulis et al., 1988). The subject matter of cable-coating,

with dual free-surface location, die-swell and slip,

dynamic contact point singularity and influence of vac-

uum effects, remains a significant challenge, yet despite

its commercial importance, relatively little tube-tooling

cable-coating research has been reported in the literature.

A finite element analysis of wire-coating for power law

fluid was presented by Caswell and Tanner (1978). In this

study, finite element approximation was able to accom-

modate awkward geometries and non-Newtonian fluid

properties in a realistic manner, and produce streamline

and stress patterns within the die. Kuyl (1996, 1997)

reported predictions for both isothermal and non-isother-

mal high-speed tube-coating melt flow using a finite ele-

ment method. He concluded that various parameters

influence the melt cone shape and stability; also, pre-

dicting the distance from the die-exit to the contact-point.

Under isothermal analysis, Kuyl (1996) demonstrated that

applying a vacuum both shortens and stabilizes the contact

length. Through his non-isothermal work, Kuyl (1997)

showed that cooling the melt cone extends the contact

length; but also cone cooling destabilizes the cone length.

Theoretically, Slattery et al. (1999) derived an analytic

solution for cable-coating draw-down of an extruded

annular melt. The analysis is for Newtonian, power-law

# This paper is based on an invited lecture presented by the correspond-ing author at the 13th International Symposium on Applied Rheology(ISAR), held May 23, 2013, Seoul.*Corresponding author: [email protected]

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A. Al-Muslimawi, H.R.Tamaddon-Jahromi and M.F.Webster

198 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

and Noll simple fluids, and excludes the effect of vacuum.

In this study, the solution is developed with the assump-

tion that pressures inside and outside the film (coating) are

equilibrated (balanced and equal); a process in which

these pressures differ, such as a vacuum inside the film, is

treated as a limiting case. Moreover, Webster and Ngam-

aramvaranggul (2000) used a semi-implicit Taylor Galer-

kin/pressure-correction finite element scheme (STGFEM)

to solve annular incompressible coating flows for New-

tonian fluids, associated with coatings of wire and cable,

fibre-optic cables or glass-rovings Both annular tube-tool-

ing and pressure-tooling type extrusion-drag flows were

investigated for viscous fluids. For free surface capture, a

modified technique was employed to derive the shape of

the die extrudate, involving remeshing, a three-stage iter-

ative cycle, and using appropriated convergence criterion.

Moreover, the effects of die-walls and Navier slip con-

ditions on the stress singularities at the die exit were also

accounted for. In addition, Matallah et al. (2000) used the

finite element method to simulate viscoelastic wire-coat-

ing employing coupled and decoupled schemes. There, a

multi-mode EPTT model was used to emphasise the influ-

ence of die-design on optimal process settings. The draw-

down residence time, which dictates the dominance of

certain modes within the relaxation time spectrum, was

found to be a major factor to influence the decay of resid-

ual stressing in the coating. Further work was conducted

by Matallah et al. (2002) to include a comparison of both

single and multi-mode viscoelastic analysis for such wire-

coating flows. There it was shown that the differences in

findings appeared minimal from single to multi-mode

modelling, specifically in the draw-down and coating

regions. Similarly, Hade and Giacomin (2000) presented

analysis for isothermal power-law tube-coating flow,

under vacuum or with an externally applied pressure. This

study was concerned with the effect of shear-thinning on

the drawing force and contact-length of the melt-cone.

These authors found that, shear-thinning properties tend to

increase the drawing force, and for power-law index

m>0.5, only slightly affects the contact-length. In addi-

tion, an analytic solution was also derived by Hade and

Giacomin (2001) for the reduced problem, of isothermal

Newtonian conduit coating-drawing flow, under vacuum

but devoid of extrudate swell, and under a set of pre-

scribed flow assumptions. This result established the melt

cone shape as the solution of a Boundary Value Problem

(BVP), composed of a system of two coupled ordinary

differential equations, from which by iteration various

draw-down ratios and balances may be established. This

result has been found useful in providing a starting point

for the conduit-cone free-surface initial positioning. More-

over, with a view to viscoplastic analysis, Mitsoulis and

Kotsos (2007) studied the wire-coating flow of non-New-

tonian pseudoplastic and viscoplastic fluids using a Her-

schel-Bulkley model and finite element approximation.

These authors observed that yielded/unyielded zones exist

only beyond the die-exit, where the coating fluid moves

on the wire as a rigid body.

In brief, the present finite element/volume study covers

a steady isothermal analysis for the full and combined

tube-tooling/draw-down annular flow – hence, a predom-

inately die shear-flow to a draw-coating flow. This

includes practical considerations of dual inner-outer free-

surfaces (dynamic location), extrudate swell and slip,

dynamic contact point representation (melt-to-cable) and

effects of vacuum imposition. Initially, a Newtonian anal-

ysis has been conducted to construct an essential kine-

matic basis and steady, zero-vacuum solution, under

prescribed flow-rate and cable-speed process conditions.

This may then be utilised as a base-flow state and internal

starting condition for the time-stepping algorithm to

explore other flow scenarios, encompassing vacuum

imposition, viscous inelastic and viscoelastic melt flow

representations.

2. Governing System of Equations

Isothermal flow of an incompressible viscoelastic fluid

can be modelled through a system of generalised differ-

ential equations comprising those for momentum trans-

port, conservation of mass and viscoelastic stress equation

of state (constitutive law). The flow problems considered

here are modelled as annular and two-dimensional (axi-

symmetric coordinate frame of reference). In the absence

of body forces, such a system can be represented as

(1)

, (2)

where t represents temporal differentiation, conventional

spatial differential operator notation is assumed, and field

variablesu, p and T represent the fluid velocity, hydro-

dynamic pressure and the extra-stress tensor. In addition,

the non-dimensional Group number of Reynolds number

may be defined as ,withcharacteristicscalesof

for the fluid density, U for velocity (cable-speed), l for

length (l is draw-down and coating-length, l/U for process

residence-time scale), and a zero shear-rate

Here, is a polymeric viscosity and is a solvent vis-

cosity, so that a meaningful quantity is the solvent fraction

. For a viscoelastic fluid, the extra stress tensor T

may be further spit into a solvent and polymericpart,

defined as:

(3)

where d=( u+ u†)/2 is the rate of deformation tensor

(superscript denotes a matrix tensor transpose). For inelas-

∇ u⋅ 0=

Re∂u∂t------ ∇= T Reu ∇ u p∇–⋅ ⋅–⋅

Re ρUl µo⁄( )=

ρ

µo µp µs+=

µp µs

β µs µo⁄=

T 2µsd τ+=

∇ ∇

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Numerical simulation of tube-tooling cable-coating with polymer melts

Korea-Australia Rheology J., Vol. 25, No. 4 (2013) 199

tic fluids the polymeric extra-stress component vanishes;

as such, the viscosity is considered as a nonlinear function

of the shear and strain rates of the rate-of-strain tensor d.

Then, the extra-stress tensor T maybedescribed function-

ally as: , where, and represent shear-rate

and strain-rate, respectively. Here, two different inelastic

shear-extensional viscosity model fits to experimental data

are proposed, namely Fit-I and Fit-II (see Binding et al.,

1996). The functionality in Fit-I and Fit-II shear-thinning

inelastic models provides the following material shear vis-

cosity and extensional viscosity dependence:

(a) Fit-I model

(4)

(b) Fit-II model

(5)

Here, Fit-I offers a single power-index response, iden-

tical in shear and extension; this is generalized under Fit-

II, enjoying independent choice of power-indices, m1,

m2; hence offering this contrast to the approximation.

These parameters may be evaluated by fitting to the exper-

imental data, as shown in Figs. 1a and 1b. Fit-I introduces

the extension-rate dependence via cosh-functionality, so

that large and early rise in extensional viscosity data may

be accommodated. Under inelastic Fit-I, the model param-

eters taken are: m=0.5, λ=1, and K=1; whilst under inelas-

tic Fit-II, the parameter choice is: m1=0.3, m2=0.4, K1=20

and K1=0.45. This model selection permits a three-way

comparison between: Newtonian, Fit-I and Fit-II models;

also exposing the attractive independent shear-extensional

T 2µ γ· ε·,( )d= γ· ε·

µs γ·( ) µ0 1 Kγ·( )2

+( )m 1–( ) 2⁄

=

µE ε·( ) 3µ0 mλε·( ) 1 3 Kε·( )2+( )

m 1–( ) 2⁄cosh=

µs γ·( ) µ0 1 K1γ

·( )2+( )

m1

1–( ) 2⁄

=

µE ε·( ) 3µ0 1 3 K1ε·( )2

+( )m1

1–( ) 2⁄

1 K2ε·( )2

+( )m2

1–( ) 2⁄

=

Fig. 1. (Color online) Experimental viscosity data: Inelastic; a) Fit-I, b) Fit-II; c) viscoelasic; EPTT model fits.

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A. Al-Muslimawi, H.R.Tamaddon-Jahromi and M.F.Webster

200 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

fitting under Fit-II, not realized within Fit-I. Note, the

order of magnitude difference in extrapolated levels of

zero shear-rate viscosity for the two model fits. Here, the

common basis of zero shear-rate viscosity is taken as that

of the constant Newtonian value and level based on Fit-I,

so that the comparison of Fit-I and Fit-II is meaningful.

Note also, the drop of three decades from first to second

viscosity plateau levels – which poses extreme imposition

on the computational modelling for viscoelastic repre-

sentation of these melt flows through reduction in the dif-

fusion contribution to the momentum equations (see

below for solvent fractions β=10-1, 10-2, 10-3).

The particular choice of viscoelastic (memory-stress)

constitutive model for polymer melts is that of Phan-Thien

and Tanner (1997) in exponential form (EPPT), derived on

the basis of a system network (rubber). In contrast to mod-

els, such as constant shear viscosity Oldroyd-B, this EPPT

version supports shear-thinning and strain hardening/soft-

ening behaviour, held applicable therefore for typical melt

response over a suitable processing range of deformation

rates. The differential constitutive equation for the extra-

stress of the EPTT model expressed in single-relaxation

time mode form is:

, (6)

where ,

represents a linear (Gordon-Schowalter) combination of

the lower-convected (co-variant) and upper-convected

(contra-variant) derivatives, defined as:

and

. (7)

Here, an additional dimensionless Group parameter is

introduced in the form of a Weissenberg number

, which is a function of single-approximation

material relaxation time, , and a characteristic average

deformation rate, this given by the designated velocity and

length scales. Note, here the correspondence to a Deborah

number, since a characteristic process time (based on

decay of residual stressing in the coating) may also be

taken as the particle residence time over the draw-down/

coating length ( ). The nonlinear function f is defined

in terms of the trace of stress, trace (τ) as:

(8)

At any particular setting of We, µ0, the model param-

eters that control the material characteristic response, via

shear and elongational properties of the fluid are: , ξ,

and β parameters. Covering both shear and extensional

viscosity experimental data, an optimal match in a least-

squares sense, is taken with a parameter set of ( =

0.5, ξ=0.15, β=10-3),see Fig. 1c.

3. Problem Specification and Numerical Schemes

A schematic flow diagram for the tube-tooling problem

with its finite element mesh discretisation (draw-down

section) is displayed in Fig. 2. The flow enters the annular

die-tube AB, then a converging cone-section BC and a

land-region CD; being subsequently drawn-down by the

moving cable-carcass into a converging cone DE, and

before being taken up within the coating section on the

cable EF. Boundary conditions applied at the die walls

prior to die-exit are taken generally as slip and no-slip.

Within the draw-down section and cable-coating region

the boundary condition settings are as follows: at the

outer-conduit (top) free-surface section, tractions vanish

and natural conditions apply (zero gauge-pressure, atmo-

spheric); this is also the case on the inner-conduit (bottom)

free-surface section under zero-vacuum conditions (oth-

erwise, a differential inner-outer pressure-difference across

the conduit is applied). At the outflow, a plug flow is

imposed travelling with the moving cable-carcass. The

moving cable also corresponds to the lower domain melt-

boundary, first making contact with the cable at the

dynamic contact point (line), where wetting of the cable

may occur. Furthermore, all numerical solutions pertain to

Newtonian, inelastic, and viscoelstic creeping flows at

minimal levels of inertia, equating to Re=10-4.In addition,

to construct the essential basis for steady-state solutions

and domain-internal starting conditions (kinematics, at

fixed flow-rate and cable-speed), the flow problem is first

solved for Newtonian fluid properties. Then, from this

Newtonian position, inelastic and viscoelastic flow solu-

tions are instigated. A regular block-structured mesh is

employed in the finite element discretisation, 2004 ele-

ments and 4355 nodes, with 27306 total number of degree

of freedom for u, p, and τ. This is a medium meshing

option, that has been assessed for spatial accuracy in prob-

f

τ Weτ+ 2 1 β–( )d =

τ

2---–⎝ ⎠

⎛ ⎞τ ξ

2---τ+=

τ∂τ∂t----- u τ u τ τ u∇⋅–⋅∇–∇⋅+=

τ∂τ∂t----- u τ u τ τ u∇⋅+⋅∇+∇⋅+=

We λ1U l⁄=( )λ1

l U⁄

fεEPTT

1 β–( )---------------Wetrace τ( )exp=

εEPTT

εEPTT

Fig. 2. (Color online) Schematic flow diagram.

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Korea-Australia Rheology J., Vol. 25, No. 4 (2013) 201

lem representation against both a finer and coarser

meshes; the medium mesh provides accuracy to within

O(0.1%) on solution variables and is the preferred choice

on efficiency grounds.

The general framework of the time-marching hybrid fe/

fv scheme employed here involves two distinct aspects.

First, velocity and pressure are computed via a semi-

implicit incremental pressure-correction procedure with

finite element spatial discretization. Secondly, a finite vol-

ume based fluctuation distribution scheme is adopted for

computation of the hyperbolic extra-stress equations. The

algorithm consists of a two-step Lax-Wendroff time-step-

ping procedure, extracted via a Taylor series expansion in

time. Here, first velocity and stress components are pre-

dicted to a half time-step (Stage 1a), and then, updated

over the full time-step (Stage 1b). To ensure satisfaction of

the incompressibility constraint, pressure at the forward

time-step is derived from a Poisson equation (Stage 2),

with velocity corrected at a final stage (Stage 3) to uphold

continuity. In either fe or fv form, nodal updates are

enforced through assembling the system of equations for

each element. On velocity components, this is performed

via inversion of the diffusion-augmented mass-matrix,

through an efficient element-by-element Jacobi iteration.

Similar theory is applicable for stress components, but

with extracted diagonalised mass-matrix (collapsed) form

to overcome singularity under fv-discretisation. This algo-

rithm has been discussed extensively in the literature to

which the reader is referred (see, for example, Wapperom

et al., 1998; Aboubacar et al., 2001; Belblidia et al., 2008;

Webster et al., 2004, 2005). In the numerical procedures,

typical time steps of O(10-4) are taken for inelastic and vis-

coelastic analysis, and O(10-3) under the Newtonian study.

The time-stepping termination criteria are governed

through a relative solution temporal increment norm in a

least squares sense.

3.1. Time dependent free-surface prediction In addition, this article documents the application of

free-surface methodology, to determine the location of the

dynamic free surface position for the melt-cone and coat-

ing. Two phases have been considered to resolve the prob-

lem and determine a suitable shape for the annular die-

extrudate (melt conduit). The first phase, seeks a solution

Fig. 3. (Color online) a) Time dependent free-surface predictions, b) dynamic contact point.

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202 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

when free-surface movement is suppressed, and for which

a domain-interior solution is generated. As such, the inter-

nal-domain solution variables are determined first without

boundary movement. Then, at a second phase, the solution

to the free surface problem may be tackled commencing

from this interior solution, using coupled and decoupled

schemes.

The free-surface location is performed via a finite ele-

ment analysis (Chandio et al., 2002, on related reverse-

roller coating) and by employing a surface height func-

tion, h(x,t), which is determined via solution of the fol-

lowing equation (see Sizaire, 1997):

(9)

where is the velocity vector and is the

radial height, see Fig. 3a. Internal (jet) domain remeshing

is performed after each time-step to avoid excessive dis-

tortion of elements in the boundary zones.

3.2. Dynamic contact point adjustmentIn this study an important factor to take into consid-

eration throughout the coating process is the approxima-

tion of the dynamic contact point solution, where the melt

interfaces meet the cable with the moving substrate sur-

face. Here, the dynamic contact point (line) is allowed

freedom of movement under slippage, through wetting or

peeling of adjacent surface sections, onto or off the mov-

ing cable-carcass. In this manner, the contact point is tra-

versed relatively smoothly, so that, when the contact angle

(θ) is adjusted, as subtended by the adjacent free-surface

line-segment (linear straight-sided surface element

approximation), any node on that bottom free-surface line-

segment (see for example, nodes 3, 2, and 1 in Fig. 3)

which makes contact with the cable is assumed to wet the

cable boundary. Thereupon, cable conditions then apply to

such segments and nodes. Under such a procedure and due

to the sucking-drawing down of the polymer melt onto the

cable, the horizontal distance between the die-exit of the

tube and the contact-point on the cable slightly reduces

throughout the coating process.

3.3. Slip conditions on the die-wallA further factor to consider relates to the influence of

slip boundary conditions, as pertaining to and applied at

the die-wall location. Based on a momentum balance at

the wall, it can be shown that the slip velocity may be

taken as a function of the velocity gradient at the wall:

(10)

where is the slip coefficient and is the shear stress.

So, for example with Newtonian fluids, uslip is given by:

(11)

where, δ is referred to as the slip-length and / is the

die-wall shear velocity-gradient. This is the basis employed

and reported upon below under a selection of sampled

results.

4. Stability and Convergence of Numerical Com-putations

4.1. Strain-rate stabilization (SRS)Considering equation (2) above, once spatial discreti-

sation has been established, localised singularity capturing

for die-swell and contact-point solution, may be invoked

through strain-rate stabilization. This may be introduced

through the weak-form integral (D-Dc) term, which takes

the form:

(12)

over the domain Ω with weighting functions . Here,

under the spatial approximations offered and at any time tn,

D represents the discontinuous rate of deformation under

fe-approximation, and Dc its continuous recovered equiv-

alent, based on localized velocity-gradient recovery (Bel-

blidia et al. 2008). Theoretically, this weak-form

dissipative term in the Strain-Rate Stabilisation (SRS) for-

mulation has the dual effect of: (i) controlling cross-stream

solution propagation (absent in most currently favoured

schemes) and (ii) easing compatibility relationships

between function spaces on stress and velocity gradients

(extended LBB-condition satisfaction; see Brezzi and For-

tin, 1991; Guénette and Fortin, 1995; also Belblidia et al.,

2008). In this manner and consistent with the weak-form

problem, discontinuities in the velocity gradients are nat-

urally incorporated within the formulation to locally rep-

resent the singularities present; as performed under shock-

capturing, but without necessitating explicit knowledge of

the singularity form and its representation.

4.2. Numerical stability and treatment of themomentum diffusive matrix (S-Matrix)

Under the present temporal schema, the generalised dif-

fusion terms in the momentum equation are treated in a

semi-implicit manner in order to enhance temporal sta-

bility. Hence, by adopting a Crank-Nicolson representa-

tion for such inelastic diffusion terms at stage 1 of TGPC

fractional-stages, the following fully-discretised equations

may be derived (see Hawken et al., 1991):

Stage 1a:

, (13)

Stage 1b:

∂h∂t------ vz

∂h∂z------ ur–+ 0=

u ur vz,( )= h h x t,( )=

uslip βslipτrz=

βslip τrz

uslip δ∂u∂r------=

∂u ∂r

ϕi2αµs∇ D Dc–( )n⋅ ΩdΩ

ϕi x( )

2Re

t∆----------M

1

2---S+ U

n1

2---+

Un

–⎝ ⎠⎛ ⎞ SU ReN u( )U+[ ]– L

TP+

n=

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Korea-Australia Rheology J., Vol. 25, No. 4 (2013) 203

, (14)

where S represents the symmetric positive semi-definite

momentum diffusive matrix (positive definite augmented

mass-matrix), defined under axisymmetric frame of ref-

erence, as:

,

,

, (15)

,

.

To enhance numerical stability under iteration for low

solvent-viscosity levels and inelastic settings (as with

melts), and hence to determine optimal scheme config-

uration for time-stepping and iterative convergence, dif-

ferent system-matrix filters have been investigated. This

has lead to five alternative scheme approaches being

considered (see below), when applied to the diffusive

matrix (S) component of the augmented mass-matrix on

the left-hand side (lhs) of Stage-1, equations (13) and

(14). Under free-surface problem numerical solution, this

is particularly important for low solvent fractions (β<10-

2),and indeed for more severe problem settings of low

power-law index (m<0.5) or stronger vacuum level set-

tings (∆P<-80). These schematic approaches may be

summarised as:

Approach 1 (Full-S-matrix): the diffusive matrix (lhs-S)

remains unaltered, as in equation (15).

Approach 2 (Full-S-matrix_r(lhs)=1): Here, the radial-

factor (r) is replaced by unity in the integrand components

of lhs-S alone (as in Cartesian reference).

Approach 3 (Diag-S-matrix_r(lhs)=1): As Approach 2,

by discarding off-diagonal Srz-matrixcomponenttermsb-

utalsodiagonalisedlhs-Swithr=1.

Approach 4 (Newtonian S-matrix_r(lhs)=1): Here, a

Newtonian approximation is adopted on the lhs-S-matrix

with r=1. This diagonalises the lhs-S-matrix, by applica-

tion of continuity and the chain-rule to off-diagonal Srz-

matrix component terms.

Approach 5: (Full-S-matrix_Gauss_Samp): As Approach

1, but with varying quadrature points (GQP) for numerical

integration per element.

In addition, a single Jacobi-iteration version under each

of the above approaches, yields a resultant diagonalised

system-matrix (via the corresponding iteration matrix, see

Ding et al. 1992). Furthermore, a single Gauss-Quadrature

point approximation for the integrands per element, ren-

ders piecewise-constant viscosity approximation, as in a

constant Newtonian level viscosity per fe-triangle (parent)

control volume. Findings on the application of each of

these individual scheme variants are discussed below.

5. Discussion of Numerical Solutions

5.1. Viscous newtonian and inelastic fit approxima-tions and results

Surface friction effects: With specific attention paid to

surface slip for viscous flows, the influence of slip onset

( =40%),as opposed to no-slip conditions within the die,

is examined. Pure slip is defined as 100% slippage on the

die-walls (at full slip velocity, defined on the steady mean

velocity in the die, consistent with a plug-flow approx-

imation and vanishing deformation rates), total friction

equates to 0% slippage (no-slip) and the 40% slip con-

dition implies that 40% of the slip velocity is imposed.

The impact of imposing such slip conditions on pressure

profiles is to generate almost 12% reduction in pressure-

maxima (at die-entry, hence, in pressure-drop), see Fig.

4a. The maximum velocity attained at the land-region

reduces correspondingly by around 20%. In Fig. 4b, a

plug flow is seen to be reached by the middle of the land-

region and sustained beyond. Over the land die-region,

and on both top and bottom surfaces, as observed in Fig.

4c, there is around 70% decrease in exposure to shear-

deformation ( ) with the selected slip-BC imposed. This

translates to bottom surface shear rates of 15.5 units with-

out slip, and to 4.7 units with slip. Slip imposition also

impacts on deformation about the complex zone of land-

entry. In addition, central streamwise profile results reveal

slight reduction in the generalised measure of extensional

deformation (not shown). The shear stress levels and dis-

tribution in Fig. 4d, are held to be consistent with these

deformation rate profiles.

Pressure and shear-rate profiles: Resuming no-slip

approximation henceforth, Fig. 5a provides comparison in

pressure-drop between purely-viscous approximations in

Newtonian, inelastic Fit-I and Fit-II solutions. At the die-

inlet and for Newtonian solutions, an overall maximum of

2260 units is observed, while for inelastic Fit-I model this

peak is reduced by about a factor of four, and with Fit-II

model by about one-half. Fig. 5b provides associated

three-way sampled and localised profiles in shear-rate

( ). Comparing shear-rate over top and bottom surfaces,

similar but sign-reflected features are observed. At the end

of the die-inlet bottom surface, Fit-II generates the largest

maximum -level of 60 units, which explains the larger

Re

t∆------M

1

2---S+ U

*U

n–( ) SU– L

Tp+[ ]

nRe N u( )U[ ]

n1

2---+

–=

SSrr Srz

Szr Szz=

Srr µ 2∂φi

∂r-------

∂φj

∂r-------

∂φi

∂z-------

∂φj

∂z-------

φiφj

r2

--------+ + r ΩdΩ∫=

Srz µ∂φi

∂r-------

∂φj

∂z------- r Ωd

Ω∫=

Szr µ∂φi

∂z-------

∂φj

∂r------- r Ωd

Ω∫=

Szz µ∂φi

∂r-------

∂φj

∂r------- 2

∂φi

∂z-------

∂φj

∂z-------+ r Ωd

Ω∫=

β

Γ

Γ

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204 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

Fig. 4. (Color online) Surface friction effects, a) pressure, b) velocity, c) shear-rate, d) shear-stress profiles, inelastic (m=0.5).

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Korea-Australia Rheology J., Vol. 25, No. 4 (2013) 205

stressing levels amongst the inelastic solutions. Through-

out the converging section, there is a sharp drop at its

beginning, reaching a minimum over its middle section,

with rise over its latter section. Through the land region,

the degree of shearing rises to a constant plateau once

more. First, this is manifest with a sharp localized peak at

its start on the top surface, rising to reach a maximum. In

contrast and on the bottom surface, there is a correspond-

ing initial sharp decline followed by an obvious increase.

Subsequently, at the inlet of the draw-down cone, there is

a dramatic sharp drop and oscillation in shear-rate. On the

top-surface and for the Newtonian solution, the shear-rate

peak around the melt-cable contact-point observes a local

maximum of 28 units; whereas, on the bottom-surface the

largest apparent peak lies with Fit-II at 34 units. In all

cases and as anticipated with viscous solutions, stress fully

dissipates over the cable region.

Strain-Rate Stabilization (SRS): Here, the problem is

solved by appealing to singularity capturing and stabili-

zation techniques, referred to earlier as ‘Strain-Rate Sta-

bilization (SRS) (D−Dc). Two choices of stabilization

factor α=0.25, α=0.5 have been trialled. In this regard,

the focused interest lies in the effects of such strategies

upon the die-exit stress singularity and resulting swell.

According to these findings, a smaller level of swelling is

observed (not shown) when the singularity is additionally

accounted for through the (D-Dc) realisation. Moreover,

one can extract the effects of (D-Dc) treatment on the

dynamic contact point solution, which follows slower

movement under singularity capturing (i.e. the distance

from the die-exit to contact point is greater under (D-Dc)

treatment). In Fig. 6 with and without strain-rate stabi-

lization (D-Dc), shear-rate ( ) and shear stress (τrz) pro-

files along the top and bottom surfaces and strain-rate ( )

and normal stress (τzz) along draw-down centre are pre-

sented for the Newtonian fluid. From these profiles, nota-

ΓΣ

Fig. 5. (Color online) a) Pressure drop along centreline, b) shear-rate, Newtonian, Fit-I, Fit-II.

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206 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

ble reduction is displayed in shear-rate ( ), shear stress

(trz) at the top surface, and strain-rate ( ) along the draw-

down centre (almost 60% reduction in shear-rate peak

with D-Dc). Therefore, this technique is observed to have

a significant impact on capture of peak stress and strain

rates exiting the die on the top surface (though note, there

is little influence from the variation in level of constant

factor α). Alternatively, on the bottom surface, there are

insignificant variations observed in and τrz with α=0.5;

while for α=0.25, there is notable change near the contact

point, with reduction in its magnitude. In the normal stress

τzz profile, one can note change only near the contact-point

region, where a larger normal stress is noted under (D-Dc)

treatment: lying around 4 units with D-Dc and 2.5 units

without D-Dc. These findings are anticipated to remain

localised within this viscous approximation.

Momentum diffusive matrix (S-Matrix) treatments:

History plots of the relative error increment norms in

velocity and pressure are provided in Fig. 7 for the four

different temporal-iterative approaches described above.

ΓΣ

Γ

Fig. 6. (Color online) , , τrz and τzz profiles: (D-Dc) scheme, α=0.25 and 0.5; Newtonian.Γ Σ

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These results reflect a higher rate of convergence for the

two Approaches (2 and 4) under lhs-S-matrix with r=1, as

compared to that extracted under Approach 1 with the full

lhs-S-matrix. As a consequence, larger time-steps are

required under Approach 1 (with full-S-matrix – rendering

more constraint on its time-stepping capability), as

opposed to with the alternative approaches. Note, almost

the same rate of time-stepping convergence is observed for

Approaches 2 and 4 (Full-S-matrix_r(lhs)=1 and Newto-

nian S-matrix_r(lhs)=1). Hence, Approach-2 is the pre-

ferred choice here. Table 1 presents the findings in direct

mode of comparison on the rates of convergence observed,

when employing different numbers of Jacobi mass-itera-

tions of 1, 3, and 5. Likewise and under Approach-5, the

rates of convergence are also displayed in Table 1 for dif-

ferent Gauss-Quadrature point approximations of GQP=1,

4, and 7 (equivalent to exact piecewise-constant, linear,

quadratic function approximations). Reduced integration is

implied for GQP<7. Here, findings reveal that numerical

solutions are stable for GQP=4, and GQP=7; whilst numer-

ical instability is encountered under the piecewise-constant

function approximations of GQP=1. This result would

indicate that reduced integration is not a favourable option

here, and that Newtonian constant-viscosity approximation

at the element level would demand much further mesh

refinement to offer any advantage.

Pressure vacuum imposition: To explain the influence

of different factors on the cable coating process, the New-

tonian problem is first resolved with negative pressure

(vacuum) on the bottom surface of the draw-down section

and by appealing to a coupled free-surface solution strat-

egy. This has the effect of imposing an inward sucking

force on the conduit, drawing it down upon the cable car-

cass. Then, in practice, the top free-surface boundary con-

ditions may still be retained for the draw-down conduit

perimeter, with various selected values of negative pres-

sure-differential being imposed on its inner (bottom) sur-

face. The effect of negative pressure-differential on the

contact-point position may be observed in Fig. 8a for

alternative settings of imposed negative pressure vacuums

∆P=0,-10,-30,-80,-200. There is insignificant adjust-

ment in contact-point position for ∆P=0 up to ∆P=-30; but

this state adjusts with further amplification of vacuum

pressure level. The results reveal that relative to a zero

vacuum conduit state (∆P=0), there is between 2% and

5% change in contact-point position for pressure vacuum

changes from ∆P=-80 to ∆P=-200, respectively. The rela-

Fig. 7. (Color online) History profiles of the relative increment error norms in velocity and pressure, mint=3.

Table 1. Rate of convergence (number of time-steps) with dif-

ferent numbers of mass-iterations, Approaches 1-5, GQP=7

Approach No. of mass iterationConvergence Rate

(Time Units)

1) Full-S-Mat

1 20

3 20

5 Not stable

2) Full-S-

Mat_r(1hs)=1

1 0.002

3 1.3

5 1.5

3) Diag-S-

Mat_r(1hs)=1

1 3.5

3 3.4

5 2.9

4) Newtonian

S-Mat_r(1hs)=1

1 0.002

3 1.6

5 1.6

5) Full-S-

Mat_Gauss_samp

1(GQP=1) Not stable

1(GQP=4) 20

1(GQP=7) 20

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208 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

tionship between imposed negative pressuredifferential

and uptake of contact-point position is clearly apparent in

the profile plot of Fig. 8a. From this profile, one can antic-

ipate a reduction in draw-down length as the pressure vac-

uum level rises. In addition, the pressure along the

centreline of the draw-down section is displayed in Fig. 8b

for various settings of inner-surface negative pressure.

Findings reflect that, as vacuum levels rise (inner pressure

falls), undershoot occurs near the dieexit and overshoot

around middle of the die.

Fig. 8. (Color online) a) Contact point position, b) pressure drop along centreline, c) coupled free-surface solutions, vacuum effects,

Newtonian.

Fig. 9. (Color online) a) Coupled b) decoupled free-surface solutions, vacuum effects, Fit-II, m=0.9.

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Inelastic m=0.9 solutions: Inelastic Fit-II solution pres-

sure fields for m=0.9 solutions are displayed Fig. 9a, for

the various alternative settings of imposed negative pres-

sure-differential ∆P=0, -60, -80. Here, it is also relevant

to show direct contrast between coupled and decoupled

free-surface strategy solutions (see on), to expose the pos-

itive benefits of the decoupled approach whilst more

severity is introduced into the problem through vacuum

pressure elevation (see arrow added in Fig. 9). From these

fields, one can observe a distinct impact on conduit shape

with increase of vacuum pressure level (inward sucking

pressure rises in magnitude). For stable steady inelastic

solutions, a critical vacuum pressure level of ∆P -80 is

achieved for Fit-II (see ∆P=-60 stable solution), which

occurs considerably sooner in vacuum pressure elevation

than under Newtonian rheology (∆P -200). Findings

reflect that the outer conduit top-surface shape is majorly

affected for ∆P≤-20, whilst the inner surface shape

remains relatively unaltered. There is also significant

downstream shift in contact point location, between

inelastic solutions with ∆P=-60 and ∆P=-80, approaching

critical conditions. Describing extrema in top-surface

shape change for the largest vacuum level imposed (∆P=-

80), there is greater concavity experienced in the swell

region post-die-exit, the conduit surface is more wavy

throughout its length, and there is obvious pinching in the

vicinity of the melt-cable contact-point. This structure rep-

resents a dramatic change when compared to the New-

tonian solution position, which showed a prominent

corresponding decrease in draw-down length. Taking into

account the variability in critical vacuum pressure levels

between Newtonian and inelastic representations, it is

quite apparent that inelastic solutions are more sensitive to

contact-point conditions and Newtonian solutions to die-

swell conditions. This contrast is drawn out in Fig. 8c and

Fig. 9a.

Decoupled free-surface solutions: As discussed above,

a decoupled free-surface approach has also been intro-

duced when required to improve the numerical capture of

the conduit surface shape (see for example, the degra-

dation in solution quality as shown in Fig. 9a with ∆P=-

80 at m=0.9 and gathered under the coupled freesurface

approach). Here, improved solution quality is accessible

with the decoupled free-surface approach, as in Fig. 9b,

∆P=-80, where significant improvement in free-surface

shape is thus observed. This free-surface location tech-

nique utilizes a decoupled (independent) approach in cal-

culation between the two independent surface sections,

top and bottom conduit surfaces. This technique has been

successfully trialled for Newtonian, inelastic and vacuum

pressure scenarios. With respect to bottom (inner) conduit

free-surface zone movement, findings reveal that conver-

gence occurs relatively rapidly to meet the specified con-

vergence tolerances. The solution pointlocation near the

contact-point takes nearly twice as long to settle down to

an invariant state, as does the pointlocation near the die-

exit. On convergence to steady state of the top (outer)

free-surface of the conduit, one concludes that much

longer computation times are required here, when com-

pared to those for the bottom surface. In Fig. 9b, and

under this decoupled approach, a solution for ∆P=-100 is

also obtained, whilst for the coupled approach, this was

not computationally possible (divergence ensued). Over-

all, the results reflect that rapid convergence is noted for

both top and bottom surfaces with the decoupled free-sur-

face approach in comparison to that for the coupled

approach. In addition, one observes a consistent trend of

an increase in time to achieve a common convergence

threshold upon decreasing m further away from unity

(notably, with increased relative thinning effects).

Free-surface line-segment mid-side node calculation-

coupled schemes: Furthermore, the coupled method has

been tested with two different free-surface location line-

segment oriented schemes, in order to more accurately

capture the temporal-update positions of the nodes that lie

on the dynamic free-surface. Firstly, from the prevailing

velocity field, nodal temporal-updates (hin+1) are com-

puted via equation (9). In the vertex-node oriented

scheme this provides the raw data on nodal position for

vertex nodes, so that mid-side node positions are deter-

mined by linear interpolation along each straight-line sur-

face-segment (Vertex_Node_FS_Cal). Secondly and by

way of alternative, a similar strategy is adopted by

appealing to equation (9) computation, but with solution

retained primarily on mid-side nodes (Mid-Side_Nodes-

FS-Cal), instead of vertex-nodes. Then in this form, the

average of each pair of surrounding mid-side node

heights is employed to determine the height of the com-

mon-connected vertex nodes. The principle behind such

an alternative is that surface dynamics is better controlled

by surface segments (and not nodal-points), and in addi-

tion mid-side node data (shared by maximum of two inte-

rior fe-elements) is often of superior quality to vertex

node data (potentially unlimited sharing between fe-ele-

ments on irregular meshing; see Hawken et al., 1991, for

superconvergence properties, fe-interpolation on tessel-

lations and solution sampling). The corresponding tem-

poral-history profiles of the relative error increment

norms in nodal surface height, velocity and pressure are

illustrated in Fig. 10 for both vertex and mid-side node-

based procedures. From these profiles, one can detect

similar trends and levels of convergence in all variables

and under both strategies up to time (tn=10 units) beyond

which faster convergence is clearly apparent under the

mid-side node strategy (equivalently, O(50%) less com-

pared to that enjoyed under the vertex-node strategy).

Hence, the mid-side node strategy is selected as the supe-

rior free-surface location scheme of choice.

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210 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

5.2. Predictions with viscoelastic approximations5.2.1. =0 (zero N2), Variation in and We

Pressure-drop profiles: In contrast to the foregoing vis-

cous modelling and under single relaxation-time mode

viscoelastic approximation, pressure profiles along the

centreline for a EPTT model are displayed in Fig. 11, with

different settings of β=10-1, 10-2, 10-3 for We=1, 5, 10,

20 at fixed εEPTT-value (εEPTT=0.5). These pressure pro-

files reveal a linear decline in pressure throughout the die-

inlet section; after which the pressure declines non-lin-

early over the converging die-section. Subsequently, the

pressure declines linearly within the land region. The

Fig. 10. (Color online) Temporal convergence history profiles: a) free-surface heights, b) pressure, and c) velocity, vertex node free-sur-

face calculation vs mid-side node free-surface calculation.

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Korea-Australia Rheology J., Vol. 25, No. 4 (2013) 211

shear-thinning behaviour of the EPTT model, with

increasing elasticity (We) and decreasing solvent-fraction

(β), generates a decline in the pressure-drop. For solvent-

fraction of β=10-3, the total pressure-drop reduces nearly

by a factor of eight from We=1 to We=20. The corre-

sponding reduction in total pressure-drop is almost 3.7 for

β=10-2 and 1.7 for β=10-1. Theoretically in steady shear

flow, by decreasing the solvent-fraction (β) and keeping

(εEPTT, ξ) fixed, the EPTT model thins at high shear-rates

and accordingly, will provide a decline in pressure-drop.

This is clearly apparent in Fig. 11 at different levels of

β=10-1, 10-2, 10-3 for any given value of We. In addition,

Fig. 11. (Color online) Pressure drop along centreline: EPTT (ε=0.5, ξ=0.0), variation in β and We.

Fig. 12. (Color online) Shear-rate surface profiles: EPTT, variation in β, We=5.

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212 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

the effect of second normal stress difference (N2) through

ξ is shown (β=10-3, ξ=0.15). Almost 20% reduction in

pressure-drop is now observed at this low level of solvent-

fraction when compared against solutions for (β=10-3,

ξ=0.0); see also below on the effect of second normal-

stress difference (parameter ).

Shear-rate surface profiles: In addition, corresponding

shear-rate ( ) profiles along the top and bottom surfaces,

are shown in the axial direction for different solvent-frac-

tion (β) and elasticity of We=5, see Fig. 12. Throughout

the converging section, there is a sharp drop in shear-rate,

reaching a minimum over its middle section. Beyond the

die-exit, once entering the draw-down flow, a sharp drop

in shear-rate is noted. Similar behaviour is observed in

both top and bottom surface shear-rate profiles. There is

only a gradual decrease in shear-rate over the draw-down

section, followed by a sharp decline when the polymer

meets the cable. Travelling with the cable, the rate of

decrease in shear-rates is minimal. Throughout the land

region, the degree of shearing adopts a constant plateau.

At the end of the die-inlet, solvent-fractions β=10-2 and

β=10-3generate the largest maximum -level of about 62

and 53 units for bottom and top surfaces, respectively. For

β=10-1, a similar trend is observed as with β=10-2 and

β=10-3, in both top and bottom surface shear-rate profiles

throughout the die-section, but observing lower shear-rate

values.

First normal stress difference profiles: First normal

stress difference (N1) profiles along the centreline are shown

in Fig. 13 for different settings of We=5, 10, 20 and β=10-1,

10-2,10-3 atfixed εEPTT-value (εEPTT=0.5) and ξ=0.0. Here,

the general trends and patterns in N1 profiles exhibit a con-

stant value along the inlet-tube, followed by a sudden

change in the solution with each adjustment in geometry.

Subsequently, and as the polymer departs from the die to

enter the draw-down section, a sharp solution decrease

within the draw-down is generated (same behaviour is

observed in τzz profiles, not shown). A smaller level of

stressing is established with larger values of We, at any

given solvent-fraction (β), due to the dominant influence of

shear-thinning. Maxima in first normal difference (N1) is

located at the draw-down section around the contact point

for all levels of solvent-viscosity (β) at We=5. Note that, a

relative decrease in N1, is noted as We elevates from 5 to 10.

There is nearly O(43%) decrease in N1, when comparing

peak draw-down zone values between We=5 to W=10 at the

solvent-fraction level of β=10-3, and O(41%) and O(38%)

for β=10-2 and β=10-1, respectively. The reduction in peak

draw-down zone value in N1 is about O(39%) from We=10

to We=20 at β=10-3. Once more, this reduction in N1 is asso-

ciated with the shear-thinning behaviour of the EPTT model

choice and with respect to decreasing the solvent-fraction

(β). A further point for discussion, new to this memory fluid

approximation, is the stress relaxation phenomenon. Relax-

ation in tensile stress occurs in the coating flow on the cable

and is linked to the relaxation time, for a fixed process time-

scale. The rapid decay of normal first normal difference

(N1) for the smaller value of We (shorter relaxation-time) is

Γ

Fig. 13. (Color online) First normal stress difference profiles,

EPTT, We, β –variation, ξ=0.0, εEPTT=0.5.

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much more marked, indicating that the stress gradient is

much greater in its early spatial development. Here, an

increasing value of We from 5 to 10, and 20, has the effect

of inhibiting stress decay. The implication is that for larger

values of We, longer times (hence, distances) will be

required for the stress to decay to an acceptable minimal

level.

5.2.2. Effect of second normal-stress difference ( -

variation)

In this section, the effect of second normal stress dif-

ference (N2) upon field response is analysed, in addition to

the dominant effect of the first normal stress difference

(N1). Recall that in pure shear flow and with the PTT

model, an increase in proportionally reduces N1, and

mildly increases N2. The flow, however, is proportional to

the local gradients of the individual stresses, which are

more affected by ξ-variation. Values of ξ larger than 0.2

(N2/N1=20%) lead to numerical convergence difficulties,

being subject to onset of the well-known Gordon-Schow-

alter instability (demonstrated in shear through the non-

monotonic nature of the shear stress-shear rate relation-

Fig. 14. (Color online) Shear stress surface profiles EPTT, We,ξ–variation, β=10-3, εEPTT=0.5.

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214 Korea-Australia Rheology J., Vol. 25, No. 4 (2013)

ship). Fortunately, values of (N2/N1) rarely exceed 0.2

(20%) and more often physically do not exceed 0.1 (10%).

Shear-stress surface profiles: τrz-profiles are displayed

in Fig. 14 for EPTTβ=10-3, We, ξ, ε=0.5, covering both

top and bottom surfaces and 0≤ξ≤0.15, 1≤We≤20. Here

again, profiles are similar over inner and outer surfaces,

differing in this respect, only around singularities and die-

design adjustments. At each We-value, there is only minor

adjustment beyond the die in each ξ–solution; the major

effect is noted within the shear flow and the die, where

with elevation in ξ-value larger shear-stress is detected, as

also noted at the die exit.

First normal stress difference profiles: Under ξ=0.15

setting, the elastic effects and first normal stress difference

(N1) comparison between the various mode solutions,

We=5, 10, 20, are shown in Fig. 15, considering the full

geometry-centreline, and its draw-down section zoomed.

N1-peak draw-down zone value at We=5 is 1.03 units,

whilst this reduces to 0.6 and 0.25 units for We=10 and

We=20, respectively. At the outlet, the N1 residual stress

value with a relatively sharp decay reaches to 0.25 units

for We=5 (almost 76% drop from its peak). In contrast, the

level of residual stress of 0.15 units for We=20 indicates

the relatively slow decay for this higher level of elasticity

(almost 40% drop from its peak) when comparing to

We=5 result. In addition, predicted flow response between

settings of ξ=0.15 and ξ=0.0 may be compared and con-

trasted. As shown in Fig. 1c, adjusting the parameter ξ, for

constant values of εEPTT=0.5and β=10-3, yields a decrease

in the shear viscosity for increasing ξ. This provides inter-

cepts with the experimental shear viscosity data at grad-

ually lower deformation rates, as ξ-value is raised. The

broadest and better least-squares fit to the data is seen to

correspond to ξ=0.15, with intercept lying at shear-rates

O(102).The elongational viscosity is practically unaffected

by such adjustment in ξ-parameter. Notably on numerical

convergence, it is observed that the largest critical level of

elasticity (Wecrit) for which numerical solution could be

attained with the EPTT(0≤ξ≤0.15) model was that of

Wecrit~O(20); and this lies in direct contrast to Wecrit~O(1)

for EPTT(=0.2). Under β=10-3, there is little significant

difference to be observed in N1 found between ξ=0.15 and

ξ=0.0 for the larger levels of elasticity We=10 and We=20

at the centreline position (see Figs. 13 and 15). Never-

theless, at the more modest level of We=5, there is a minor

reduction in N1-peak draw-down value noticed for ξ=0.15

when compared against the solution for ξ=0.0. Further-

more for ξ=0.0, and in the coating-outlet flow, the N1

residual-stress values with different setting of We=5, 10,

20 are observed to be 0.23, 0.21, and 0.13 units, respec-

tively. Equivalently, in direct contrast and at ξ=0.15, N1

residual-stress values, at the same position and the same

level of elasticity, are recorded as 0.25, 0.21, and 0.14

units, respectively. This would imply that, if anything,

residual-stress is tending to rise with inclusion of ξ 0 and

N2-effects. Again, this is consistent with findings noted

above on shear-stress in the extrudate. Similar trends in N1

are found for the top free-surface section as on the cen-

treline, when comparing ξ=0.15 and ξ=0.0 solutions. Fur-

thermore, at higher levels of elasticity (We=20), no

significant differences in N1 are observed between top-sur-

face and centreline solutions at either levels of ξ=0.0 and

ξ=0.15.

6. Conclusions

This paper has investigated the numerical solution of

viscous inelastic and viscoelastic flows for the combined

tube-tooling/draw down flow, principally through the vari-

ation of model-material parameters. Many factors that

have a bearing upon the process as a whole have been

investigated – including those of surface friction; vacuum

Fig. 15. (Color online) First normal stress difference (N1), β=10-3, ξ=0.15.

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Numerical simulation of tube-tooling cable-coating with polymer melts

Korea-Australia Rheology J., Vol. 25, No. 4 (2013) 215

pressure applied across the conduit; aspects of dynamic

contact-point and die-exit singularity capturing, via loc-

alised shock-capturing; coupled and decoupled free-sur-

face solution approaches; and the relative importance of

shear and extensional rheology. An exponential Phan-

Thein/Tanner (EPTT) viscoelastic and two viscous inelas-

tic models have been employed to predict pressure-drops

and residual stresses. Notably, under viscoelastic model-

ling with the EPTT model, close attention has been paid to

the influence and variation in Weissenberg number (We),

solvent-fraction (β) and second normal stress-difference

(N2).

A number of conclusions may be drawn from this study.

Across the geometry, there is a variation in shear and ten-

sile stress levels; yet this is not great. Stressing is observed

to build up in the converging die, relax in the early part of

the draw-down only to build up again over the remainder

of the draw-down cone; before finally relaxing in the

region on the cable. Regarding the effects of surface fric-

tion, one can gather that slip is a mechanism to signif-

icantly reduce pressure-drops in the die-section of the

flow-process. The influence of solution singularity cap-

turing has been successfully explored, both on the

dynamic contact point and the die-exit solution, using the

strain-rate stabilization (D-Dc) technique. In this respect,

it is demonstrated that such treatment can have a signif-

icant impact on peak stress and strain rates exiting the die,

which themselves influence the levels of die-swell

encountered. In addition, taking into account the vari-

ability in critical vacuum pressure levels between purely-

viscous Newtonian and inelastic representations, it is quite

apparent that inelastic solutions are more sensitive to con-

tact-point conditions and Newtonian solutions to die-swell

conditions. Throughout the study, some additional numer-

ical convergence aspects have also been addressed, spe-

cifically pertaining to temporal solution stability for these

free-surface problems, and as solvent fractions diminish,

power-indices lower or vacuum pressures rise. Moreover,

a new free-surface location technique has been estab-

lished, utilizing a decoupled (independent) approach in

calculation between top and bottom (outer and inner) con-

duit surfaces. This technique has been successfully trialled

for Newtonian, inelastic and vacuum pressure scenarios.

Further improvements in free-surface procedures are also

documented through mid-side node/line-segment oriented

approximations.

In the viscoelastic modelling context, and with shear-

thinning response of the EPTT model, increasing elasticity

(We) and decreasing solvent-fraction (β), is found to gen-

erate a decline in the pressure-drop. For solvent-fractions

of β=10-3, the total pressure-drop reduces nearly by a fac-

tor of eight from We=1 to We=20. In respect to stress pro-

files, a smaller level of stressing is established with larger

values of We, at any given solvent-fraction (β), due to the

dominant influence of shear-thinning. In summary, a

build-up of shear and normal stress is observed in the

tube-tooling cone, followed by relaxation in the draw-

down and cable-coating flow sections. In addition, second

normal stress difference effects are vitally important in

this problem, which have major impact on shear stress,

normal stress and first normal stress difference levels. Fur-

thermore, it is shown that a decoupled approach may pro-

vide a pragmatic strategy to stabilise the nonlinear

solution procedure, and hence aid in extracting predictive

solutions at low levels of solvent-fraction (β=10-3) and

under the influence of non-zero second normal stress-dif-

ference (N2).

Acknowledgment

The first author acknowledges financial support from

the Ministry of Higher Education, and Mathematics

Department, College of Science, Basra University, Iraq

during the course of this research.

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