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UNIVERSITY OF WALES SWANSEA

REPORT SERIES

Multi-mode simulation of filament stretching flows by

H. Matallah, K.S. Sujatha, M.J. Banaai and M.F. Webster

Report # CSR 11-2006

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Multi-mode simulation of filament stretching flows

H. Matallah, K.S. Sujatha, M.J. Banaai and M.F. Webster* Institute of non-Newtonian Fluid Mechanics,

Department of Computer Science, Swansea University,

Singleton Park, SA2 8PP, Swansea, United Kingdom

Abstract This study considers the continuous filament-stretching problem through a hybrid finite element/volume scheme. A range of strain-hardening fluid models are employed of Oldroyd-B, Giesekus, and linear Phan-Thien/Tanner form (LPTT). Both exponential and linear plate-retraction rates are investigated, indicating their different dynamics. In addition, we compare and contrast single versus multi-mode modelling, pointing to the importance of each mode. The influence of multi-mode representation is made apparent through the stress components generated and consequences thereof. Results at large levels of Hencky-strain are presented, displaying symmetrical behaviour in axial stress. There is no sign of bead-like structure formation with either Giesekus or LPTT (ξ=0) solutions, in contrast to that apparent with the LPTT (ξ=0.13) counterpart. Keywords: Multi-mode; Filament-stretching; strain-hardening; Linear and exponential-stretching 1. Introduction A number of studies have been conducted to solve viscoelastic filament stretching flows. This has been performed through several numerical techniques to overcome numerical instabilities that arise from a combination of elliptic and hyperbolic equation types. Such problems are complicated through the presence of many parameters governing transient, free-surface, body and surface tension forces. To model such flows, one needs realistic constitutive equations that faithfully reflect the fluid behaviour. In this manner, several integral and differential constitutive laws have been considered [1-4]. The FENE-CR model was implemented by Sizaire and Legat [1] through a two-dimensional finite element solver to predict strain-hardening/constant shear viscosity material response in an Eulerian configuration (L2 extensibility coefficient of 4325.5). In contrast, Bach et al. [2] used an integral constitutive model as a slight modification of the K-BKZ version. Such constitutive models were implemented under single-mode approximations. In addition, Yao el al. [5] investigated the transient viscoelastic behaviour in filament stretching devices under uniaxial elongation, following the cessation of stretching. A commercial software package POLYFLOW was employed in this regard adopting a Galerkin discretisation. Both multi- and single-mode solutions were presented in order to differentiate between the predicted viscoelastic behaviour and the role of the fluid relaxation spectrum. Shear, dynamic and transient stretching characteristics were successfully fitted by employing a wide relaxation spectrum for a multi-mode PTT model (exponential version, 5 and 9-modes) in the work of Langouche and Debbaut [6]. There, under a semi-analytical framework, excellent agreement was found between predictions and experimental data at moderate Hencky-strains, both for instantaneous extension and step-strain recovery experiments. Yet, beyond a Hencky-strain of 2.0 units,

* corresponding Author. Tel: +44 1792 295656; fax: +44 1792295708 Email: M.F.Webster@swan.ac.uk

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predictions overestimated experimental data. Subsequently, a finite element scheme was implemented to extend the analysis into two-dimensions. This allowed for the additional considerations of boundary conditions, surface tension and inertia, and succeeded in identifying the occurrence of inertial oscillations during start-up.

In the present study, we employ a novel hybrid finite element/volume scheme (hy-fV) developed in [7,15], and applied here specifically within the transient viscoelastic free-surface context. The scheme is centred about a number of key features. First, it is a time-stepping procedure of fractional-stage form on each Hencky-strain step, uniting incremental pressure-correction stages with Lax-Wendroff/Taylor-Galerkin time-splitting. Second, an Arbitrary-Lagrangian-Eulerian (ALE) formulation is preferred to a pure Eulerian alternative, where we appeal to a Compressed-Mesh (CM) spatial implementation as opposed to that of a Volume-of-Fluid (VOF) scheme. Third, our previous studies have identified the superiority of particle-tracking (dx/dt) over height function )/( th ∂∂ schema to determine the motion of the filament free-surface. Fourth, domain spatial discretisation is based on a finite volume sub-cell approximation for stress, with finite element technology invoked for velocity and pressure. Regarding ALE-implementations, these are frequently employed to track complex free-surface shapes that retain freedom of mesh movement. In such procedures, the mesh is shifted with a suitable mesh velocity. The ALE-formulation was introduced in the finite difference domain by Noh [8] and Hirt et al. [9], and was further extended into the finite element domain by Hughes et al. [10] for incompressible viscous flows.

There are many nonlinear constitutive models available today that may describe the rheology of interest. Here, we choose a Giesekus model as our base-case [11], in both single and multi-mode form, and contrast this against Oldroyd and Phan-Thien-Tanner (PTT) models. The Giesekus model can fit both linear and nonlinear shear rheology of most concentrated polymeric solutions, and is commonly employed to represent weakly strain-hardening fluids. For example in this respect, the Giesekus model has already been employed by various authors, such as Li et al. [12] and Yao et al. [5]. In their work, Li et al. utilised a higher-order-discrete-elastic-viscous-stress (hp-DEVSS) finite element method to solve an axisymmetric stagnation flow. The linear-stretching process has also been studied by Foteinopoulou et al. [13]. There, the main focus was to predict the growth of bubbles within a filament stretched between two plates moving under constant imposed velocities. Koplick and Banavar [14] have also studied this constant plate-velocity filament-stretching problem but on an atomic scale, investigating the interfacial rupture of the liquid bridge. In their work, the applied force is variable and fluctuates throughout the course of stretching. Below, having first established filament-stretching results by employing a single-mode representation, through Oldroyd-B, PTT (linear version LPTT) and Giesekus [15] models, we shift our attention to study the effect of having a wide relaxation spectrum. This is performed through a three-mode approximation, specifically appealing to the LPTT and Giesekus models. The numerical scheme employed is similar to that implemented under the single-mode form with some modifications to tackle small solvent viscosity contributions. Both linear and exponential-stretching configurations have been analysed and contrasted under the single-mode representation. Previously, the hybrid finite element/volume method has been effectively employed to solve different steady and transient viscoelastic problems [16,7,15]. In this earlier work [15], we have given attention to a 2-mode representation, where results were gathered at low-moderate Hencky-strain levels. In the present paper, we switch the focus

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of attention to a three-mode approximation, showing how high Hencky-strain levels may be accommodated. 2. Governing equations and numerical scheme The governing equations for the current problem involving of an incompressible viscoelastic isothermal fluid of density ρ, may be expressed through the momentum and continuity equations, respectively, as

gduuu ρµρρ ++⋅∇+∇−∇⋅−=

∂∂

)2( ττττspt

, (1)

0=⋅∇ u , (2)

where, u, p, τ represent the velocity, the hydrodynamic pressure and the extra-stress tensor, respectively. The zero-shear viscosity is separated into a polymeric (µp) and a solvent (µs) contribution, so that ;0 ps µµµ += the rate-of-deformation tensor is defined as d=(L+L†)/2

with L†= u∇ , the velocity gradient. Gravity may be represented by g, hence the gravity vector in a (r, z)-coordinate system may be expressed as g=(0,g)†.

For the multi-mode instance with M-modes, and considering each mode (i) with its partial viscosity (µi) and relaxation time (λi), the extra-stress tensor of the Giesekus [11] and LPTT [17,18] models may be represented through the general form as

iii

iiiiiiiiiii

ii f

tττττττττττττττττττττττττττττττττττττ ⋅−⋅+⋅+⋅+⋅++−∇⋅−=

∂∂

µλαξλλµλλ }{)(2 †

1 ddLLdu (3)

)(1 ii

iiptt

i trf ττττµ

λε+= , 0≥i

pttε and 20 ≤≤ iξ ,

∑=

=M

ii

1

ττττττττ .

Above, ( ipttε , iξ ) represent the LPTT non-dimensional parameters. To recover the LPTT

model, the mobility factor (αi) should vanish. This factor is related to the Giesekus model, and is associated with anisotropic material response, with range specification 0.10 ≤≤ α for physical meaning. In contrast, the Giesekus model is recovered by setting (fi=1 and ξι=0). When fi=1, ξi=0 and αi=0, the Oldroyd-B model is recovered. To non-dimensionalise the governing set of equations, the following non-dimensional quantities are employed for distance, time, velocity, pressure, stress and viscosity

L*xx = , *tU

Lt = , U*uu = , *

0 pL

Up µ= , *

0 ττττττττL

Uµ= , *0 ii µµµ = . (4)

This yields a set of non-dimensional group numbers of Reynolds, Weissenberg (for each mode i), Bond and Capillary (function of surface-tension coefficient χ), defined viz.

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0µρUL

Re = , L

UWe i

i

λ= , gL

Boχ

ρ 2

= , χ

µ UCa 0= . (5)

After discarding asterisk for notation clarity, the resulting non-dimensional system emerges,

gspRet

Re Fduuu ++⋅∇+∇−∇⋅−=

∂∂

)2( ττττµ , (6)

0=⋅∇ u , (7) †2 ( )i

i i i i i i i i iWe We f Wet

µ∂ = − ⋅∇ − + + ⋅ + ⋅∂ττττ τ τ τ ττ τ τ ττ τ τ ττ τ τ τu d L L +

{ } i ii i i i i i

i

WeWe

αξµ

⋅ + ⋅ − ⋅τ τ τ ττ τ τ ττ τ τ ττ τ τ τd d . (8)

Here, ∑=

=M

ii

1

ττττττττ , )(1 ii

iiptt

i trWe

f ττττµ

ε+= ,where Fg represents the ratio of gravitational to viscous

forces given by

†

0

2† ,0(),0( g)

U

LFgg µ

ρ==F ,

yielding the axial coefficient, CaBoFg /= .

3. Numerical discretisation and problem specification

In this paper, a semi-implicit transient decoupled hybrid finite volume/element scheme is employed to discretise and solve the related non-linear system of field Eqs. (6-8), with appropriate initial and boundary conditions. The scheme base is that of a two-step Lax-Wendroff within a Taylor series approximated up to O(∆t2). The pressure is incremented through a Pressure-correction strategy (TGPC) [19]. Hence, a three-stage scheme emerges. The first stage is related to updating both stress and non-solenoidal velocity fields. At the second stage, the pressure is then updated. Finally, the velocity field is updated at a third stage. Concerning the discretisation of the full system in space, a finite-element approach is performed on a triangular tessellation with piecewise-continuous quadratic interpolation for velocity, and linear for pressure. The stress equations are discretised through a sub-cell (within the velocity element) finite volume method of cell-vertex form. The velocity/pressure element is sub-divided into four sub-cells (for stress). To solve the resulting algebraic system, a Jacobi-iterative method is employed for velocity (using mass-iterations of number three to five), and a direct Choleski decomposition scheme for pressure. The stress is directly evaluated through the resulting diagonalised fv-stencil, yielding an explicit simple nodal evaluation. The numerical scheme may be found in full detail in [15]. Here, the problem of relevance is that of a viscoelastic filament contained between two coaxial discs, and extended in exponential or linear-rate. For the single-mode approximation, both linear and exponential-stretching are considered, and the effects of gravity and surface tension are also taken into account. In contrast, under multi-mode (three-mode) prediction, we consider only exponential-stretching and gravity/surface tension effects are omitted. Throughout this study, inertia, represented through the Reynolds number (O(10-3)), has practically negligible effect.

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Mesh refinement is performed to address scheme accuracy under the multi-mode case. Accuracy in time is also dealt with through a particular implementation. For the multi-mode case, mesh-refinement is tackled in two directions, axially and radially. For the axial direction, three initial meshes of generalised rectangular (triangular element-pairs) element description (20x100), (20x150), (20x200) are employed, whilst two are taken radially, meshes of (15x150 and 20x150). Under single-mode simulations, the single (20x100) mesh is employed as in our earlier work (see [15,20]). Initial conditions are taken as quiescent. Consistent boundary conditions are displayed in Fig. 1. On the moving-plates, velocity and length control are imposed as follows:

a) exponential-stretching:

),exp()2/()( 000 tLtVz

••

±= εε 0)( =tVr , )exp()2/()( 00 tLtLp

•

±= ε .

b) linear-stretching:

),2/()( 00 LtVz

•

±= ε 0)( =tVr , tLtLp 00 )2/()(•

±= ε ,

where 0

•

ε , L0 represent an imposed initial stretch-rate and the initial filament length, which

provide an appropriate characteristic velocity scale, 00 0U V Lε•

= = .

The Trouton ratio may be defined as

0

Trµµ

−+

= , where 2

( )z

mid mid

FO F

E R E R

χµπ

−+ = − +⋅ ⋅ . (9)

Above, ( E⋅

) is an extension rate, and Fz represents the exerted force on the area (A) of the moving-plates given through the surface integral,

{ }2z s zz zz

A

F p d dAµ τ= + +∫ .

The term O(F) represents a correction arising from imposed inertial and gravitational forces (F). Procedure through time The scheme employed for single-mode simulations has been validated through a number of studies, both steady and transient [20,15,7]. In the present study, this scheme is extended to accommodate for the multi-mode context. There are two important new issues involved here. The first lies in dealing with small alongside large relaxation time-scales, which requires improved time-implicitness treatment for the constitutive equation. The second aspect relates to handling the near-Maxwellian form of the momentum equation, as solvent viscosity is largely absorbed within the individual modes. The outcome is to arrive at equitable time-steps for both multi-mode as well as single-mode implementations. The multi-mode scheme is similar to that of the single-mode, adopting some minor modifications, described below. For any single Hencky-strain step ( Henckyt∆ ), we appeal to

the following sequence of algorithmic steps:

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Step 1. Update Hencky-strain, Henckynn ttt ∆++++====++++1

Step 2. Fix plate-boundary conditions at 1++++nt

Step 3. Shift plate locations through a single step (Henckyt∆ ).

Step 4. Shift free-surface nodes to updated 1++++nt position.

Readjust position of interior/domain nodes governed by ALE-scheme [15]. Step 5. First, we update kinematics through single-mode modelling, gathering current

velocity and pressure field on the shifted domain, solving fractional-stages, with dynamic boundary conditions.

Step 6. Next, we synchronously evolve the stress state forward for each mode to coincide with the kinematics.

Step 7. Finally, we correct the pressure/kinematics, to be consistent with the updated total multi-mode stress.

Step 8. Stopping criteria: determine local time of stretch; if not at terminating time, goto Step 1.

At the outset and for the initial mesh, we use elliptic-mapping mesh distribution for interior nodes in the axial direction according to a height function method. Some further detail on specific steps is in order. At Step 5, velocity and pressure field solutions are determined, with frozen stress and heavy-side solvent viscosity weighting ( 10 µµµ −=s ), equivalent to a

single-mode representation. This is performed over each Hencky-strain step to a specified iterative tolerance (normally 10-7). The pseudo-sub-steps demanded at this stage are termed inner-steps, usually O(103) in number (N). In Step 6, we evaluate primary variables (u, p, ττττ ) in a synchronous manner, so that, the total number of steps employed (M) corresponds to the ratio of the Hencky-strain step Henckyt∆ to the local inner time-step innert∆ . Here, each stress

mode (i) is solved subject to its own polymeric viscosity ( iµ ). Similarly, in Step 7, we

calculate (u, p) with frozen multi-mode stress ττττ, employing the actual solvent viscosity ( ∑−=

i is µµµ 0 ).

4. Results and discussion Results are reported under two sections: the first deals with single-mode modelling with a choice of rheological models. This work includes consideration of surface tension and body force effects. Various fluids are modelled by adjusting the LPTT and Giesekus parameters (εptt, ξ) and α, respectively. The single-mode relaxation time is evaluated by discarding the influence of the shorter modes, retaining only the largest mode. Hence, all polymeric viscosities of the discarded modes are taken to contribute to the solvent viscosity. This aids the numerical simulation by stabilising the numerical scheme, as it improves the semi-implicitness of the momentum balance equation. In the second section, concentration is given to multi-mode modelling with three-modes, ignoring the effects of surface tension and body force. Consistency through mesh refinement is confirmed with the Giesekus model choice through four meshes, as described above, at a specific Hencky-strain of ε=1.8 units.

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4.1 Single-mode results: exponential-stretching

We begin by reporting on results for the single-mode approximation with (µs=0.262), where

greater detail may be found in our earlier work of Ref. [15].

Effect of body forces and surface tension (LPTT-ξξξξ=0.13) To evaluate the effect of both body force and surface tension, we show in Fig. 2, axial stress contours for the LPTT model (ξ=0.13) at a Hencky-strain level of ε=3 and 3.2 units. The axial stress reaches its maximum at the filament mid-plane, decreasing towards the moving end-plates, nearly doubling in value from 55 units at ε=3 to 112 units at ε=3.2. At ε=3 units, there are some signs of the onset of bead generation, slightly off-set from the mid-plane. The beads are clearly shown at ε=3.2 units, in a symmetrical manner with a dip at the mid-plane region. The combined influence of body force and surface tension is translated to greater extension experienced at the filament mid-plane, increasing the axial stress to 55 units, compared to its counterpart without body/surface tension forces that yields about 35 units. In the case where these forces are discarded, the axial stress maxima are described depending on the range of the Hencky-strain. At ε < 3 units, the position of axial stress maxima lie at the filament mid-plane, which is no different to its counterpart when body/surface tension forces are present. In contrast, at ε=3 units and beyond, these stress maxima lie slightly off-centre, yet symmetry is maintained. This leads one to appreciate that greater body/surface tension forces, aggravate enhanced fluid extension. Shear-thinning effects: LPTT and Giesekus To analyse the bead-generation process somewhat further, we focus on alternative model choices to LPTT(ξ=0.13), and discard the body force and surface tension effects. Here, we concentrate on results for LPTT(ξ=0) and Giesekus models. These two models have similar shear-thinning behaviour, yet dissimilar to that for LPTT(ξ=0.13), of premature shear-thinning form. In contrast, both LPTT(ξ=0.13) and LPTT(ξ=0) display exaggerated extensional properties compared to those for the Giesekus model. The presence of the ξ-parameter for the LPTT model generates non-symmetrical behaviour under ε>3 units. This is clarified through cross-checking LPTT results under vanishing ξ. The LPTT (ξ=0) model yields a similar trend to its Giesekus counterpart, yet noting differences in the level of stress values. The Giesekus model with a critical Hencky-strain of εcrit=4.0 units, and τzz-maxima on the central-plane of 12 units, is devoid of such bead-like or asymmetrical features. Similarly, the LPTT model (ξ=0) shows no sign of emerging beads, reaching Hencky-strain levels of εcrit=4.4 units, whilst maintaining symmetry (see Fig. 3 and Fig. 4c). Stress maxima achieved for this LPTT model are about 81 units at ε=4 units, compared to 13 units for the Giesekus model. For ξ=0.13, the axial stress-maxima position lies off-centre at ε≥3 units (see on for multi-mode penetration in contrast to that for the single-mode case at ε=3 units). Strain-hardening effects: Giesekus model To study the strain-hardening behaviour of the Giesekus model, it is necessary to adjust the mobility factor throughout an appropriate range. When this factor vanishes, the Oldroyd-B model is recovered, which represents the limiting situation in maximum strain-hardening under Giesekus modelling. In contrast, by increasing the mobility factor, the model strain-softens and one expects less axial stress to be developed through the filamentation process. Hence, the problem becomes more tractable to numerical solution. One notes that the solution loses physical meaning beyond the limit of α of unity. The corresponding critical Hencky-

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strain decreases as α declines: so, εcrit is {2.69, 4.20, 4.40, 4.80} for α of {0.0, 0.32, 0.5, 1.0}. The trend in critical Hencky-strain is consistent across the range 0.0≤α≤1.0, but not so beyond. As displayed in Fig. 5, shear is dominant near the region of the moving-plates, and as the mobility factor α increases this shearing effect decreases. In contrast, extension prevails in the filament mid-plane zone, and this also declines with increase of α. The shear/axial-stress maxima are: (8.11;80.0), (0.51;9.77), (0.37;6.69) and (0.28;3.65) at α=0.0, 0.32, 0.50 and 1.0, respectively. When shearing is large, the fluid tends to adhere more closely to the moving-plates. In contrast, less extension at the filament mid-plane leads to greater decrease in filament mid-plane radius. The foot is pinched, whilst the mid-plane thickens. So, as rV

increases in absolute value and reaches its maximum near the plates, the fluid there is pushed further towards the centre of the plates. This is explained through changes in max

rV , with

-0.28≤ maxrV ≤-0.23 for 0.0≤α≤1.0 (see Table 1 below).

4.2 Linear versus exponential-stretching: Oldroyd-B

The motivation for this part of the study is to compare and contrast predictions for Oldroyd-B fluids under the linear-stretching mode against those for the corresponding, more common, exponential configuration, whilst adopting similar discrete approximations. The benefit of exponential-stretching is that the stretch-rate remains constant and this aids in the direct study of such physical properties as Trouton ratio and extensional viscosity. The advantage of linear-stretching, with variable stretch-rates but constant plate-retraction rate, is that longer filament lengths and stretch times may be accommodated. This also has some impact on temporal adjustment of min-max element aspect-ratios, under such a compression-extension filamentation process. Comparison between these two stretching configurations can be taken both at the same times and equal lengths (equivalent Hencky-strain levels), to analyse axial stress, normal force, Trouton ratio and radial velocity. This can provide insight into longer duration times under linear-stretching and corresponding physical features, in contrast to its exponential counterpart. Comparison at equivalent times As displayed in Fig. 6, stress maxima are observed as contours at the filament mid-plane under both stretching modes. At the earlier time of t=0.2 units, axial stress values are comparable between the two stretching instances. As time progresses, around unity and beyond, the differential in stress broadens. For example, at t=2.6 units, exponential-stretching produces a maximum axial stress (τzz) of some 80 units, which is almost sixteen times greater than that generated by its linear counterpart (5 units). Such stress maxima lie at the filament mid-plane section, a zone of minimum cross-sectional area, whilst the applied force through the whole filament is constant. Hence, as stress is proportional to the inverse of area, the smaller area tends to attract larger stress under both stretching configurations. We also look to the extra-stress along the mid-plane axis as depicted in Fig. 7, under both stretching modes. The exponential form axial stress grows functionally as {τzz= 0.0129exp (4.0959t)}. Alternatively, under linear-stretching such stress rises mildly until a time between 5 <t< 6 units, and drops gradually thereafter, until it almost vanishes. As time proceeds the difference between extra-stress maxima for the two stretching instances grows increasingly wider. This is in agreement with other findings concerning Trouton ratios and normal force on the moving plates, which are decreasing functions under linear-stretching and ascending functions under exponential-stretching.

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Concerning the radial velocity rV in Fig. 8, the main difference between the stretching

instances, is the location of the radial velocity minima. For the linear case, rV minima lie at

the filament mid-plane at all times. In contrast, under exponential-stretching and beyond early times of agreement, rV -minima are located near the moving-plates at times t ≥1.8 units. This

has some corresponding impact upon filament-shape, particularly at the filament feet and mid-plane. Comparison at equivalent lengths (equitable Hencky-strains) Switching instead to consider comparison of configurations at equivalent lengths, we plot in Fig. 10b the evolution of the mid-plane radius, Rmid (ε), depicted as a function of Hencky-strain (ε). Numerical predictions under constant plate-retraction rate lie in close agreement with those for the exponential alternative, up to Hencky-strains close to 1.6 units, around which point departure is detectable. So, noticeably, with the constant Vz-plate choice, there is a slight variation in Rmid beyond ε >2.0 units to be observed between exponential and linear-stretching. Alternatively, necking in Rmid is observed to progress more slowly under exponential-stretching. This response goes hand-in-hand with the much longer elongation times permitted under linear-stretching. For example, at ε=2.4 units the time for linear-stretching is t≈10 units, whilst under exponential-stretching this is only around t=2.4 units. This is slightly more than a four-fold difference, which well compensates for the radial velocity that is typically three-times larger under exponential-stretching (see Fig. 8). As a result, this causes thicker mid-plane filament radii under exponential-stretching. The development of extra-stress under both stretching modes is depicted in Fig. 9. Under exponential-stretching, extra-stress increases continuously from ε=0.2 to ε=2.6 units. Whilst, under linear-stretching, τzz displays a rising trend up to ε=1.8 units; it declines thereafter, so that τzz at ε=2.6 closely approximates its value at ε=0.2 units. As above, the discrepancy between solutions under linear and exponential motion widens as the stretching process proceeds further. At ε=0.2 units, there are barely any differences in τzz; yet at ε=2.6 units, exponential-stretching predictions are O (102) times greater than their counterparts under the linear-mode. This discrepancy allows for further stretching under the linear-case. For linear-stretching, maxima in pressure magnitude arise on the plate at small Hencky-strains and these maxima are maintained up to t=5.0 units, causing rise in the Trouton ratio and an indication of rise in τzz. Thereafter, the location of pressure maxima switches toward the vertical axis of symmetry on the end-plates, alongside a sharp decrease in magnitude associated with a drop in Trouton ratio, that itself corresponds to a decrease in τzz. In contrast, under exponential-stretching the pressure continues to rise at the plates, leading to an increase in Trouton ratio and normal force. Along with this, there is continuous rise in τzz under the exponential-stretching mode. Trouton ratio (Tr) development versus Hencky-strain is displayed in Fig. 10a, where the trend is one of rising uniformly under exponential-stretching. With linear-stretching, Trouton ratio is insubstantial (close to zero), decreasing around ε≈2.0 units and remaining almost constant thereafter. Normal force for such strain-hardening (Oldroyd-B) fluids under exponential-stretching drops until ε≈1.0, and subsequently rises sharply corresponding to an exponential rise in Trouton ratio. The trend in normal force under the linear-mode lies below that for the exponential form, corresponding to far lower Trouton ratio settings. We neglect the inertial force term in the Trouton ratio definition of Eq. 9 and consider the fact that mid-plane filament radius is larger under exponential-stretching at the same levels of

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Hencky-strain (see Fig. 10b). Here, the surface tension coefficient employed is the same under both modes of stretching. It is the rising normal force which renders the first term significant of definition Eq. 9. Consequently under exponential-stretching, a rise is observed in extensional viscosity and Trouton ratio (see Fig. 10a), compared to under its linear counterpart. This rise in normal force, extensional viscosity and Trouton ratio stimulates axial stress to rise also under exponential-stretching, and yet decays under linear-stretching. 4.3 Multi-mode results: exponential-stretching

Next, we turn to multi-mode representation for Giesekus and LPTT models and only consider the exponential-stretching configuration. Here, 3-mode representation is selected for these strain-hardening models. The relevant selected parameter values of (α, εptt and ξ) are displayed in Table 2 for each model. First, we consider mesh refinement with restriction to a single model, chosen for convenience as the Giesekus model. This analysis is restricted to a moderate Hencky-strain value of ε=1.8 units. Refinement is performed in both axial and radial directions. Three meshes (100x20, 150x20 and 200x20) cover axial refinement (z-axis), and two meshes (150x20 and 150x15) govern the radial refinement (r-axis). Once we have confirmed consistency, we shift attention to address the rheological consequences of the multiple modes involved. The influence of inclusion of the shorter modes is also investigated in contrast to solutions based on a single mode (the longest). Along the way, comparison is made to the literature to confirm our findings. Time-discretisation: Giesekus model: To analyse transient behaviour in the multi-mode context, it is necessary to establish more robust numerical procedures to discretise the constitutive equations of the relevant viscoelastic models. In this respect, we may apply either the standard explicit time-discretisation, developed earlier (see Ref. [15] ), or invoke an alternative semi-implicit approach to tackle both the requirement for smaller time-steps, and simultaneously, the occurrence of low to high relaxation times. This semi-implicit scheme adopts a Crank-Nicolson time-splitting stencil for the τ-term and provides a product factor (2Wei+∆t), with dependency upon both the time-step (∆t) and the Weissenberg number (Wei). This accommodates for the Wei→0 scenario, in the context of other larger non-trivial Wei-values. In this fashion, we solve for the multi-mode Giesekus model, using a single mesh for convenience, mesh (15x150), at ε=2 units under conventional explicit and semi-implicit procedures. In Fig. 11, the corresponding axial velocity zV and axial stress zzτ contour fields

are displayed. Excellent agreement is established between the two solutions generated in both components throughout the whole field (graphically no difference), at this and other comparable Hencky-strain levels tested. Mesh refinement for Giesekus model First, axial refinement is illustrated through stress fields of τzz plotted at ε=1.8 units in Fig. 12a. These data show complete agreement between the solutions on the three axially refined meshes (100x20{blue-solid}, 150x20{green-dashed} and 200x20{red-dashdot}). The contours are displayed from 0.0 to 8.0, in steps of 0.5 units. Consistent with above, the axial stress achieves its maxima at the filament mid-plane region, spreading and reducing in magnitude towards the plates, where it vanishes. This trend was also observed in the single-mode instance. In Fig. 12b, zV is plotted from -3.0 to +3.0 units in steps of 0.5 units. Once

more, the axial velocity is visibly overlapping across the three meshes considered. Symmetry is apparent around the filament mid-plane. There, axial velocity vanishes with its maxima

11

located towards the moving-plates. Likewise, agreement in profiles may be gathered from Fig. 14c,e, covering axial velocity and stress (zV and τzz) along the filament centreline.

Next, we turn to radial refinement and ε=1.8 units, in both axial stress (Fig. 13a) and velocity

zV (Fig. 13b). This provides graphically identical results across the two meshes considered

(150x20 and 150x15). Concentrating on the filament mid-plane region where much of the extension occurs, we plot profiles of τrr and τzz in Fig. 13c,d along the radial-axis. The overall pattern in τzz reflects a constant form across the filament mid-plane, with a slight increase near the free-surface. Axial stress profiles along the filament axis are identical across these mesh-solutions. Likewise, the two mesh-solutions lead to similar results in the radial τrr stress component. Lastly, we consider comparison with refinement in both coordinate directions, axially and radially. In Fig. 14b,d, we plot the radial and axial stress component profiles along the radial-axis at the filament mid-plane. There is barely any disparity detected between the different cases. Similarly, both stress components plotted along the filament centreline axis, Fig. 14a,c, show that there is little to no difference between the various mesh solutions generated here. This provides us with satisfactory mesh resolution extracted under the multi-mode context, both axially and radially. Acceptable level of agreement has been established between these mesh-solutions, leading to the consideration of a single-mesh to analyse rheological variations across different models. Hence, predictions on the (100x20)-mesh are considered as satisfactory, and are employed subsequently below. Giesekus v LPTT (ξξξξ=0.13 and ξξξξ=0) solution comparison In Fig. 15, we plot axial stress profiles for the Giesekus model, along the filament centerline at different Hencky-strain levels of ε=0.2, 1.0 and 2.0 units. The difference between both single and multi-mode solutions may be gathered, starting from a low Hencky-strain level, such as ε=0.2 units. The contribution of the shorter modes to the stress is felt through the larger value of the multi-mode solution, when compared to its single-mode counterpart. At ε=0.2 and the filament mid-plane where the maxima are observed, these contributions are, respectively, 0.89 units for the multi-mode case {composed of 0.3 (largest mode), 0.47, 0.12 (shortest mode)} and 0.3 units for the single-mode form. Similar, comments apply to the axial stress along the free-surface. From single-mode results of Fig. 3, the Giesekus model displays symmetry in axial stress around the filament mid-plane, being maintained across all Hencky-strains up to a critical level of εcrit=4.0 units. The multi-mode representation confirms these findings. In Fig. 16, axial stress contours are displayed up to ε=4.2 units. Beyond a Hencky-strain value of ε=3.0 units, the influence of the shorter modes starts to become apparent in the axial penetration of stress through the filament. In the radial direction, this penetration is uniform. Once more, neither single nor multi-mode solutions show signs of bead-like structure formation. The main discrepency lies in the trend associated with the development of axial stress-maxima. Across Hencky-strains of ε=(3.0, 3.2, 4.0), the single-mode representation yields a slight increase in the axial stress-maxima as (11.9, 12.0, 13.2) units. Equivalently, the multi-mode representation at ε=(3.0, 3.6, 4.0) gives axial stress-maxima of (12.0, 13.6, 12.0). This slight change in axial stress-maxima, around an average of 12.5 units may be interpreted as a constant trend with Hencky-strain variation beyond a value of ε=3 units.

12

Similarly, the LPTT model under ξ=0, (see Fig. 17a) generates symmetrical solutions in the axial stress around the filament mid-plane axis for Hencky-strain levels up to ε=3.6 units. Again, no bead-like structure is present at any Hencky-strain level shown. In contrast, as displayed in Fig. 17b, the LPTT(ξ=0.13), yields a bead-like structure but only at Hencky-strain levels beyond ε>3.4 units, as opposed to ε=3.2 units for the single-mode case. In the single-mode solution and at ε=3 units, an off-centre double stress-maxima arises in the axial stress component. This phenomena of off-centre double stress-maxima delays its appearance in the multi-mode case to Hencky-strains around and beyond ε=3.4 units. Such a time-lag is attributed to the direct impact of the shorter modes involved. Comparison across relaxation-modes The axial stress distribution pattern is similar across the modes at each Hencky-strain, with distinction in value differences, where the largest mode represents the main contributing factor. This is particularly true for large values of ε≥0.8 units. For example, with the LPTT (ξ=0) model at ε=3.6 units, τzz reaches 62.10 units for the largest mode, compared to 2.83 and 0.16 units with the second and third shorter modes, respectively (see Fig. 18). In contrast, at the early Hencky-strain of ε=0.2 units, the axial stress is dominated by the second mode. Maxima-τzz equate to (0.59, 0.92 and 0.20) units, ranging from the largest to the shortest mode, respectively. In the range, 0.2<ε<0.8 units, the first two modes provide relatively equal contributions, being larger than for their shortest-mode counterpart. 5. Conclusion In summary, the filament stretching problem has been studied under both a single-mode approximation and a multiple-relaxation spectrum. Predictions have been presented for the strain-hardening rheology of Oldroyd-B, Giesekus and LPTT models. Under multi-mode modelling, the influence of the shortest modes is apparent in the axial penetration of stress. In the absence of body/surface tension forces, both the Giesekus and LPTT(ξ=0) fluids yield comparable trends with symmetrical filament-shapes and distribution patterns for the axial stress component. There is no sign of bead-like structure here, in contrast to their LPTT(ξ=0.13) counterpart that reflects earlier shear-thinning properties. Linear versus exponential-stretching instances have also been compared under the single-mode approximation, where their differing kinematic and elastic response has been identified. Under the linear-stretching configuration, the axial stress relaxes during the stretching process, in contrast to under its exponential-counterpart, where the axial stress grows in an exponential manner.

13

References [1] R. Sizaire, V. Legat, Finite element simulation of a filament stretching extensional rheometer, J. Non-

Newtonian Fluid Mech. 71 (1997) 89-107. [2] A. Bach, H.K. Rasmussen, P.-Y. Longin, O. Hassager, Growth of non-axisymmetric disturbances of the

free surface in the filament stretching rheometer: experiments and simulation, J. Non-Newtonian Fluid Mech. 108 (2002) 163-186.

[3] O. Hassager, M.I. Kolte, M. Renardy, Failure and nonfailure of fluid filaments in extension, J. Non-Newtonian Fluid Mech. 76 (1998) 137-151.

[4] H.K. Rasmussen, O. Hassager, Three-dimensional simulations of viscoelastic instability in polymeric filaments, J. Non-Newtonian Fluid Mech. 82 (1999) 189-202.

[5] M. Yao, G.H. McKinley, B. Debbaut, Extensional deformation, stress relaxation and necking failure of viscoelastic filaments, J. Non-Newtonian Fluid Mech. 79 (1998) 469-501.

[6] F. Langouche, B. Debbaut, Rheological characterisation of high density polyethylene with a multi-mode differential viscoelastic model and numerical simulation of transient elongational recovery experiments, Rheol. Acta 38 (1999) 48-64.

[7] M.F. Webster, H. Matallah, K.S. Sujatha, Sub-cell approximations for viscoelastic flows-filament stretching, J. Non-Newtonian Fluid Mech. 126 (2005) 187-205.

[8] W.F. Noh, A time dependent two space dimensional coupled Eulerian Lagrangian code, Meth. Comput. Phys. 3 (1964).

[9] C.W. Hirt, A.A. Amsden, J.L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys. 14 (1974) 227-253.

[10] T.J.R. Hughes, W.K. Liu, T.K. Zimmermann, Lagrangian-Eulerian finite element formulation for incompressible viscous flows, Comput. Methods Applied Mech. Eng. 29 (1981) 329-349.

[11] H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility, J. Non-Newtonian Fluid Mech. 11 (1982) 69-109.

[12] J.-M. Li, W.R. Burghardt, B. Yang, B. Khomami, Flow birefringence and computational studies of a shear thinning polymer solution in axisymmetric stagnation flow, J. Non-Newtonian Fluid Mech. 74 (1998) 151-193.

[13] K. Foteinopoulou, V.G. Mavrantzas, J. Tsamopoulos, Numerical simulation of bubble growth in Newtonian and viscoelastic filaments undergoing stretching, 122 (2004) 177-200.

[14] J. Koplik, J.R. Banavar, Extensional rupture of model non-Newtonian fluid filaments, (2006). [15] H. Matallah, M.J. Banaai, K.S. Sujatha, M.F. Webster, Modelling filament stretching flows with strain-

hardening models and sub-cell approximations, 134 (2006) 77-104. [16] M.F. Webster, H.R. Tamaddon-Jahromi, M. Aboubacar, Transient viscoelastic flows in planar

contractions, J. Non-Newtonian Fluid Mech. 118 (2004) 83-101. [17] N. Phan-Thien, R.I. Tanner, A new constitutive equation derived from network theory, J. Non-

Newtonian Fluid Mech. 2 (1977) 353-365. [18] N. Phan-Thien, A non-linear network viscoelastic model, J. Rheol. 22 (1978) 259-283. [19] H. Matallah, P. Townsend, M.F. Webster, Recovery and stress-splitting schemes for viscoelastic flows,

J. Non-Newtonian Fluid Mech. 75 (1998) 139-166. [20] K.S. Sujatha, H. Matallah, J. Banaai, M.F. Webster, Computational predictions for viscoelastic filament

stretching flows: ALE methods and free-surface techniques (CM and VOF), (in press) J. Non-Newtonian Fluid Mech. (2006).

14

List of Tables Table 1: Effect of mobility factor (α) on filament behaviour Table 2: Material properties for single and multi-mode models List of Figures Fig. 1: Simulation domain with boundary conditions Fig. 2: Effect of body and surface tension forces on axial stress component τzz, single-mode,

LPTT (ξ=0.13), a) Ca-1=0.1, Fg=0.317, b) Ca-1=0, Fg=0.317, c) Ca-1=0.1, Fg=0. Fig. 3: Effect of shear-thinning on axial stress τzz, single-mode, Giesekus model, Ca-1=0, Fg=0 Fig. 4: τzz contours, comparison between LPTT models (ξ=0.13 and ξ=0), single-mode, effect

of early shear-thinning; a) Ca-1=0.1, Fg=0.317, ξ=0.13, b) Ca-1=0, Fg=0, ξ=0.13, c) Ca-1=0, Fg=0, ξ=0.

Fig. 5: Shear and axial stress contours, effect of the mobility factor, Giesekus single-mode: τrz (a-e), a) α=0, b) α=0.32, c) α=0.5, d) α=1.0, e) α=1.5; τzz (f-k), f) α=0, g) α=0.32, h) α=0.5, j) α=1.0, k) α=1.5

Fig. 6: τzz field contours at equivalent times Fig. 7: Axial stress vs. time along mid-plane axis Fig. 8: Vr field contours at equivalent times Fig. 9: Axial stress, τzz at equal lengths Fig. 10: (a) Trouton ratio ( Tr) development vs. Hencky-strain (b) Mid-plane filament radius,

Rmid(ε) vs. Hencky-strain Fig. 11: Effect of time-implicitness on axial velocity and stress, 3-mode Giesekus, ε=2.0,

a) τzz, b) zV

Fig. 12: Axial mesh refinement, multi-mode Giesekus model, ε=1.8, meshes 200x20, 150x20 and 100x20: a) τzz, b) zV

Fig. 13: Radial mesh refinement, multi-mode Giesekus model, ε=1.8, meshes 150x20, 150x15: contours a) τzz, b) zV , profiles along filament mid-plane c) τrr, d) τzz

Fig. 14: Mesh refinement (axially and radially), multi-mode Giesekus model, ε=1.8, meshes 200x20, 150x20, 100x20 and 150x15; profiles along the filament centreline a) τrr and c) τzz, d) zV , along mid-plane b) τrr and d) τzz

Fig. 15: Comparison between single and multi-mode Giesekus, axial stress τzz profiles along filament centreline, increasing ε, 0.2, 1 and 2 units: --- single mode, ___ multi-mode.

Fig. 16: Axial stress τzz contours, multi-mode Giesekus, increasing ε up to ε=4.2. Fig. 17: Axial stress τzz contours, multi-mode LPTT, increasing ε up to ε=3.6: a) LPTT (ξ=0),

b) ξ=0.13 Fig. 18: Axial stress τzz contours, multi-mode LPTT (ξ=0), ε=3.6, contribution of separate

relaxation modes

15

α εcrit maxrzτ max

zzτ maxrV

0.0 2.69 8.11 80.0 -0.28 0.32 4.20 0.51 9.77 -0.26 0.5 4.40 0.37 6.69 -0.25 1.0 4.80 0.28 3.65 -0.23 1.5 3.10 0.34 2.68 -0.23

Table 1: Effect of mobility factor (α) on filament behaviour

Mode 1 Mode 2 Mode 3 λi (s) 0.421 0.0563 0.00306 µi (Pa s) 25.8 7.71 1.37 αi (Giesekus) 0.3162 0.2422 0.0993 εptt (LPTT) 0.035 0.035 0.035 ξ 0.13 0.13 0.13 Wei 1.886 0.252 0.014 µs (solvent viscosity) (Pa s)

0.069

Single-mode We 1.886 µs (solvent viscosity) (Pa s)

9.149

Table 2: Material properties for single and multi-mode models

16

Fig. 1: Simulation domain with boundary conditions

free-surface : b.i p = p0 (ambient pressure)

bottom plate Vr = 0 Vz = -Vz(t)

Vr = 0 Vz = Vz(t) top-plate

L(t) Sym Vr = 0 τrz = 0

17

a) body force & surface tension b) body force c) surface tension Fig. 2: Effect of body and surface tension forces on axial stress component τzz, single-mode, LPTT (ξ=0.13), a) Ca-1=0.1, Fg=0.317, b) Ca-1=0, Fg=0.317, c) Ca-1=0.1, Fg=0.

εεεε=3.0

εεεε=3.2

εεεε=3.4

εεεε=3.0

εεεε=3.2

εεεε=3.4

εεεε=3.0

εεεε=3.2

18

Fig. 3: Effect of shear-thinning on axial stress τzz, single-mode, Giesekus model, Ca-1=0, Fg=0

19

b) ξ=0.13 a) body force & surface tension, ξ=0.13 c) ξ=0 Fig. 4: τzz contours, comparison between LPTT models (ξ=0.13 and ξ=0), single-mode, effect of early shear-thinning; a) Ca-1=0.1, Fg=0.317, ξ=0.13, b) Ca-1=0, Fg=0, ξ=0.13, c) Ca-1=0, Fg=0, ξ=0.

εεεε=3.0

εεεε=3.2

20

a) α=0 b) α=0.32 c) α=0.5 d) α=1.0 e) α=1.5

f) α=0 g) α=0.32 h) α=0.5 j) α=1.0 k) α=1.5

Fig. 5: Shear and axial stress contours, effect of the mobility factor, Giesekus single-mode: τrz (a-e), a) α=0, b) α=0.32, c) α=0.5, d) α=1.0, e) α=1.5; τzz (f-k), f) α=0, g) α=0.32, h) α=0.5, j) α=1.0, k) α=1.5

τrz

τzz

21

(g) (a)

(c) (e)

τmax=80.0

τmax=23.2

τmax=5.4

τmax=2.0 τmax=3.9 τmax=5.0

τmax=0.6

τmax=0.6

linear exp

Fig. 6: τzz field contours at equivalent times

t =0.2 t=1.0 t =1.8 t =2.6 (a) (c ) (e) (g) (b) (d) (f) (h)

(b) (d) (f) (h)

22

Fig. 7: Axial stress vs. time along mid-plane axis

Time

τ zz(

p)

0 2 4 6 8 10 12 140

20

40

60

80

exp

linear

τzz= 0.0129 exp(4.0959t)

23

Fig. 8: Vr field contours at equivalent times

t =0.2 t=1.0 t =1.8 t =2.6 (a) (c ) (e) (g) (b) (d) (f) (h) Vrmax= - 0.3

Vrmax= - 0.3

Vrmax= - 0.4

Vrmax= - 0.4 Vrmax= - 0.6

Vrmax= - 1.2 exp

linear

Vrmax= - 0.2 Vrmax= - 0.3

Vrmax= - 0.1

exp

linear

(a) (c) (e) (g)

Vrmax= - 1.0

(b) (d) (f) (h)

24

Fig. 9: Axial stress, τzz at equal lengths

(g) (a) (c) (e)

(b) (d) (f) (h)

exp

linear τmax=0.6

τmax=3.9

τmax=5.0

τmax=0.9

τmax=0.6 τmax=5.4

τmax=23.2

τmax=80.0

linear (h) (f) (d) (b) (g) (e) (c ) (a) ε =2.6 ε =1.8 ε=1.0 ε =0.2

exp

25

(a)

(b) Fig. 10: (a) Trouton ratio ( Tr) development vs. Hencky-strain (b) Mid-plane filament radius, Rmid(ε) vs. Hencky-strain

Hencky-strain

Tr

1 2 3 4

0

40

80

120

160

exp

linear

26

Level Tzz

18 8.517 816 7.515 714 6.513 612 5.511 510 4.59 48 3.57 36 2.55 24 1.53 12 0.51 0

Level Vz

19 3.6930918 3.6682417 3.516 315 2.514 213 1.512 111 0.510 09 -0.58 -17 -1.56 -25 -2.54 -33 -3.52 -3.634421 -3.68678

a) τzz b) zV

Fig. 11: Effect of time-implicitness on axial velocity and stress, 3-mode Giesekus, ε=2.0, a) τzz, b) zV

27

0.00.5

0.5

2.0

2.0

2.5

3.0

4.0

6.5

7.5

0.51.0

1.5

2.5

3.0

3.0

4.0

8.0

0.0

0.0

0.5

2.0

3.0

3.5

4.5

4.5

6.5

7.0

7.0

Level Tzz

17 8.016 7.515 7.014 6.513 6.012 5.511 5.010 4.59 4.08 3.57 3.06 2.55 2.04 1.53 1.02 0.51 0.0

3.0

2.5

2.0

1.5

1.0

0.0

-1.0

-1.5

-2.0

-2.5

-3.0

Level

1110987654321

a) τzz b) Vz

Fig. 12: Axial mesh refinement, multi-mode Giesekus model, ε=1.8, meshes 200x20, 150x20 and 100x20: a) τzz, b) zV

28

Level Tzz

19 8.018 7.517 7.016 6.515 6.014 5.513 5.012 4.511 4.010 3.59 3.08 2.57 2.06 1.55 1.04 0.53 0.02 -0.51 -1.0

3.0

2.9

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

-2.0

-2.5

-2.8

-3.0

Level Vz

15 3.014 2.913 2.512 2.011 1.510 1.09 0.58 0.07 -0.56 -1.05 -1.54 -2.03 -2.52 -2.81 -3.0

a) τzz b) Vz

R/Rmid

τ rr

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

150x20150x15

R/Rmid

τ zz

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

150x20150x15

c) τrr d) τzz

Fig. 13: Radial mesh refinement, multi-mode Giesekus model, ε=1.8, meshes 150x20, 150x15: contours a) τzz, b) zV , profiles along filament mid-plane c) τrr, d) τzz

29

a) τrr along filament centreline b) τrr along filament mid-plane

c) τzz along filament centreline d) τzz along filament mid-plane

??? e) Vz along filament centreline f) Vr along filament mid-plane

Fig. 14: Mesh refinement (axially and radially), multi-mode Giesekus model, ε=1.8, meshes 200x20, 150x20, 100x20 and 150x15; profiles along the filament centreline a) τrr and c) τzz, d) zV , along mid-plane b) τrr and d) τzz

0.8 1

0005

R/Rmid

τ zz

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

200x20150x20100x20150x15

8 10

τrr

z/L

p

-1 -0.8 -0.6 -0.4 -0.2 0

-0.4

-0.2

0

0.2

0.4

200x20150x20100x20150x15

R/Rmid

τ zz

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

200x20150x20100x20150x15

τ zz

-0.6

-0.4

-0.2

0

e=

τzz

z/L p

0 2 4 6 8 10

-0.4

-0.2

0

0.2

0.4

200x20150x20100x20150x15

8 10

Vz

z/L

p

-3 -2 -1 0 1 2 3

-0.4

-0.2

0

0.2

0.4

200x20150x20100x20150x15

a) b)

c) d)

e)

τrr

τzz

30

a) ε=0.2 b) ε=1 c) ε=2 Fig. 15: Comparison between single and multi-mode Giesekus, axial stress τzz profiles along filament centreline, increasing ε, 0.2, 1 and 2 units: --- single mode, ___ multi-mode.

0.2

0.4

corrected multi-mode (New), e=2.0

τzz

z/L p

0 2 4 6 8 10

-0.4

-0.2

0

0.2

0.4

single-modemulti-mode

0.2

0.4

corrected multi-mode (New), e=4.0

τzz

z/L

p

0 1 2 3 4 5 6 7 8 9 10

-0.4

-0.2

0

0.2

0.4

single-modemulti-mode τzz

0 2 4 6 8 10

-0.4

τzz

z/L

p

0 1 2 3 4 5 6 7 8 9 10

-0.4

-0.2

0

0.2

0.4

single-modemulti-mode

--- single-mode ___ multi-mode

31

Fig. 16: axial stress τzz contours, multi-mode Giesekus, increasing ε up to ε=4.2.

ε=4.2 max 11.0

32

a) ξ=0

Fig. 17a): Axial stress τzz contours, multi-mode LPTT, increasing ε up to ε=3.6: a) LPTT (ξ=0)

ε=3.6 max 65.1

ε=3.0 max 48.6

ε=2.0 max 24.5 ε=1.0

max 7.1 ε=0.2 max 1.7

33

εεεε=0.2max 1.54

εεεε=1.0max 5.75

εεεε=2.0max 18.1

εεεε=3.0max 36.6

εεεε=3.6max 75.4

εεεε=0.2max 1.54

εεεε=1.0max 5.75

εεεε=2.0max 18.1

εεεε=3.0max 36.6

εεεε=3.6max 75.4

εεεε=0.2max 1.54

εεεε=1.0max 5.75

εεεε=2.0max 18.1

εεεε=3.0max 36.6

εεεε=3.6max 75.4

b) ξ=0.13

Fig. 17b): Axial stress τzz contours, multi-mode LPTT, increasing ε up to ε=3.6: b) ξ=0.13

34

Fig. 18: axial stress τzz contours, multi-mode LPTT (ξ=0), ε=3.6, contribution of separate relaxation modes