13
J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 Convective linear stability analysis of two-layer coextrusion flow for molten polymers R. Valette a,b,, P. Laure b , Y. Demay b,a , J.-F. Agassant a a Centre de Mise Forme des Matériaux, Ecole Nationale Supérieure des Mines de Paris, UMR 7635 CNRS, 06904 Sophia Antipolis, France b Institut Non-Linéaire de Nice, Université de Nice Sophia Antipolis, UMR 6618 CNRS, 06560 Valbonne, France Received 30 December 2003; received in revised form 9 April 2004 Abstract The interface instability of the coextrusion flow of a polyethylene and a polystyrene is studied both experimentally and theoretically in a slit geometry. For prototype industrial conditions, we have found a stable/unstable transition which bounds the occurrence of stable/unstable sheets at die exit. By investigating a large range of processing conditions, we have shown that this transition is controlled by both temperature and flow rate ratios. Close to the transition, we used a transparent die to measure spatial amplification of different controlled perturbations at die inlet and pointed out the convective nature of the instability which exhibits a dominant mode (for which the instability is the most severe). We have then found that a convective stability analysis, using the White–Metzner constitutive equation, is able to account for the spatial amplification rate experimentally measured on controlled perturbation experiments. By considering that the instability is controlled by its dominant mode, we performed a convective stability analysis for all studied prototype industrial conditions and showed that such an analysis is able to forecast the occurrence of defects at die exit. © 2004 Elsevier B.V. All rights reserved. Keywords: Coextrusion; Interfacial stability; Viscoelastic fluids 1. Introduction This paper is devoted to the experimental and theoretical study of the stability of a coextrusion flow of a low den- sity polyethylene and a polystyrene. Coextrusion process is widely used for food packaging. Polymers are first separately melted in screw extruders and then flow simultaneously in the extrusion die. In some processing conditions wavy interfaces between the different polymer layers, due to a flow instability of the system in the die, are observed on the final product. These disturbances of the thickness regularity of each layer result in altered mechanical and optical properties. From a theoretical point of view, the occurrence of such instabilities has been largely studied by means of a linear sta- bility analysis of the two-layer plane Poiseuille flow. In such analysis, one studies the growth or decay rate of periodic and arbitrary small perturbations. In the past 10 years, sta- bility analysis has been carried out for different viscoelastic Corresponding author. E-mail address: [email protected] (R. Valette). constitutive equations and main contributions can be found in refs. [1–4]. This instability occurs at a very low Reynolds number and it is now accepted that the stability is governed by the viscosity ratio, elasticity ratio and layer thickness ra- tio (or flow rate ratio) of the two polymers. Experimental studies concerning coextrusion instabilities can be split into two methodologies. In the first one, the tran- sition between stable and unstable processing conditions is investigated by examining the extrudate obtained at die exit on industrial or prototype industrial devices. This approach has been followed by Han and Shetty [5–7], Antukar et al. [8] and Valette et al. [9,10] for commercial polymers. The authors have tried to point out a connection between obser- vations and thickness ratio, viscosity or first normal stress difference ratios at the interface. In the second approach, which has been followed by Khomami and co-workers [11–15], the instability is studied by looking at its development inside a partially transparent die. Evolution of the interface deviation along the die is ob- served through four optical windows for various layer thick- ness ratios (or flow rate ratios). The interfacial perturbations are generated by adding periodic oscillations of small am- 0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2004.04.002

Convective linear stability analysis of two-layer coextrusion flow for molten polymers

Embed Size (px)

Citation preview

Page 1: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

Convective linear stability analysis of two-layercoextrusion flow for molten polymers

R. Valettea,b,∗, P. Laureb, Y. Demayb,a, J.-F. Agassanta

a Centre de Mise Forme des Matériaux, Ecole Nationale Supérieure des Mines de Paris, UMR 7635 CNRS, 06904 Sophia Antipolis, Franceb Institut Non-Linéaire de Nice, Université de Nice Sophia Antipolis, UMR 6618 CNRS, 06560 Valbonne, France

Received 30 December 2003; received in revised form 9 April 2004

Abstract

The interface instability of the coextrusion flow of a polyethylene and a polystyrene is studied both experimentally and theoretically in aslit geometry. For prototype industrial conditions, we have found a stable/unstable transition which bounds the occurrence of stable/unstablesheets at die exit. By investigating a large range of processing conditions, we have shown that this transition is controlled by both temperatureand flow rate ratios. Close to the transition, we used a transparent die to measure spatial amplification of different controlled perturbationsat die inlet and pointed out the convective nature of the instability which exhibits a dominant mode (for which the instability is the mostsevere). We have then found that a convective stability analysis, using the White–Metzner constitutive equation, is able to account for thespatial amplification rate experimentally measured on controlled perturbation experiments. By considering that the instability is controlledby its dominant mode, we performed a convective stability analysis for all studied prototype industrial conditions and showed that such ananalysis is able to forecast the occurrence of defects at die exit.© 2004 Elsevier B.V. All rights reserved.

Keywords:Coextrusion; Interfacial stability; Viscoelastic fluids

1. Introduction

This paper is devoted to the experimental and theoreticalstudy of the stability of a coextrusion flow of a low den-sity polyethylene and a polystyrene. Coextrusion process iswidely used for food packaging. Polymers are first separatelymelted in screw extruders and then flow simultaneously inthe extrusion die.

In some processing conditions wavy interfaces betweenthe different polymer layers, due to a flow instability of thesystem in the die, are observed on the final product. Thesedisturbances of the thickness regularity of each layer resultin altered mechanical and optical properties.

From a theoretical point of view, the occurrence of suchinstabilities has been largely studied by means of a linear sta-bility analysis of the two-layer plane Poiseuille flow. In suchanalysis, one studies the growth or decay rate of periodicand arbitrary small perturbations. In the past 10 years, sta-bility analysis has been carried out for different viscoelastic

∗ Corresponding author.E-mail address:[email protected] (R. Valette).

constitutive equations and main contributions can be foundin refs.[1–4]. This instability occurs at a very low Reynoldsnumber and it is now accepted that the stability is governedby the viscosity ratio, elasticity ratio and layer thickness ra-tio (or flow rate ratio) of the two polymers.

Experimental studies concerning coextrusion instabilitiescan be split into two methodologies. In the first one, the tran-sition between stable and unstable processing conditions isinvestigated by examining the extrudate obtained at die exiton industrial or prototype industrial devices. This approachhas been followed by Han and Shetty[5–7], Antukar et al.[8] and Valette et al.[9,10] for commercial polymers. Theauthors have tried to point out a connection between obser-vations and thickness ratio, viscosity or first normal stressdifference ratios at the interface.

In the second approach, which has been followed byKhomami and co-workers[11–15], the instability is studiedby looking at its development inside a partially transparentdie. Evolution of the interface deviation along the die is ob-served through four optical windows for various layer thick-ness ratios (or flow rate ratios). The interfacial perturbationsare generated by adding periodic oscillations of small am-

0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.jnnfm.2004.04.002

Page 2: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

42 R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

plitude on the flow rate at die entrance. The frequency ofthis forcing allows one to select the spatial periodicity ofthe interfacial wave. A polypropylene/polyethylene flow isstudied in[11–14] whereas[15] is concerned with modelfluid systems such as silicone oil/propylene glycol, siliconeoil/polybutene–kerosene or silicone oil/Boger fluid. Au-thors show that perturbations grow or decay as they traveldownstream, pointing out clearly the convective nature ofthe interfacial instability[16,17], which means that the flowbehaves like an amplifier on upstream disturbances. Growthor decay rates are measured and compared with results of aconvective linear stability analysis computed with Gaster’stransformation[18], for a 4-mode Giesekus constitutiveequation in[19,20]. A good quantitative agreement withexperimental observations is found both in terms of criti-cal flow rate ratio and spatial amplification of interfacialdisturbances.

In a recent paper, Valette et al.[9] have investigated theinterface stability of a polyethylene/polystyrene system andtested the capability of linear stability analysis to forecast on-set of wavy interface defects. Experiments were performedboth in industrial and laboratory conditions. The influenceof temperature and respective flow rates of polyethylene andpolystyrene was experimentally studied and compared withtheoretical results of the longwave stability analysis usingthe White–Metzner constitutive law. It was found both ex-perimentally and theoretically that the interface becomes un-stable as the thickness of the polystyrene layer decreases.However, stable sheets were obtained in a parameter rangethat was predicted unstable by linear stability analysis.

Consequently, we propose in this paper to check the abilityof a convective stability analysis to forecast the occurrence,in prototype industrial conditions, of stable/unstable sheetsat the die exit. This paper is organized as follows.Section 2describes our experimental approach; we show that thereexists a stable/unstable transition observed on the final filmby studying different coextrusion processing conditions withthe couple PE/PS[9]. Close to the transition, we measurespatial amplification of controlled interfacial waves inside atransparent die[10]. Section 3then describes our theoreti-cal approach; we show that a convective stability analysisallows one to retrieve the spatial amplification rate experi-mentally measured on controlled interfacial waves. Finally,by considering that the dominant mode of the instabilitycontrols the instability, we apply systematically a convec-tive stability analysis to all coextrusion experiments carriedout and compare to experimental results. InSection 4, wesummarize the results and draw conclusions.

2. The polymers and the experimental investigations

2.1. The polymers

Polymers used are a polyethylene 1003 FE 23 and apolystyrene 1240 from ATOFINA as described in a previous

study[9]. Dynamic and continuous shear rheology measure-ments have been performed on a parallel plate rheometer atdifferent temperatures. The activation energy is very differ-ent for the two polymers (62.5 kJ/mol K for the polyethy-lene and 123.16 kJ/mol K for the polystyrene) and it resultsin viscosity and elasticity ratios very sensitive to the flowtemperature.

Viscosity η∗ (Fig. 1) and relaxation timeλ (Fig. 2) asfunctions of the shear rate are obtained from dynamic mea-surements, assuming the validity of the Cox–Merz rule. Re-laxation time is assumed to be shear rate dependent as in theWhite–Metzner model (used in our theoretical approach). Itsexpression is given byλ = G′/ωG′′. Continuous shear datacomplete the experimental measurements at low shear rates.

Polystyrene presents a Newtonian plateau (Fig. 1b) at lowshear rate whereas viscosity of polyethylene is markedlyshear rate dependent in the same range (Fig. 1a). It resultsin a shear rate (i.e. flow rate) dependent viscosity ratio asshown onFig. 3a; at 180◦C polystyrene is more viscousthan polyethylene in all the processing ranges located be-tween 0.03 and 300 s−1. At 200◦C polystyrene remains

0.01 0.1 1 10 100Shear rate (/s)

1000

10000

100000

Vis

cosi

ty (

Pa.s

)

0.01 0.1 1 10 100Shear rate (/s)(a)

(b)

1000

10000

100000

Vis

cosi

ty (

Pa.s

)

Fig. 1. Plot of the shear viscosity at 200 ◦C vs. shear rate for the polyethy-lene (a) and the polystyrene (b); (�) master viscosity (η∗) curve deducedfrom linear rheometry data by using the Cox–Merz rule; (�) low shearrates continuous rheometry data; (—) White–Metzner model fit.

Page 3: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 43

0.01 0.1 1 10 100Shear rate (/s)

0.01

0.1

1

10

Rel

axat

ion

time

(s)

0.01 0.1 1 10 100Shear rate (/s)(a)

(b)

0.01

0.1

1

10

Rel

axat

ion

time

(s)

Fig. 2. Plot of the relaxation time at 200◦C vs. shear rate for the polyethy-lene (a) and the polystyrene (b); (�) master relaxation time (λ = G′/ωG′′)curve deduced from linear rheometry data by using the Cox–Merz rule;(�) low shear rates continuous rheometry data; (—) White–Metznermodel fit.

more viscous than polyethylene between 1 and 100 s−1

whereas at 220 ◦C polyethylene is always more viscous thanpolystyrene. As shown in Fig. 3b, the situation is quite dif-ferent when considering the relaxation time λ. Polystyreneis less elastic than polyethylene at low shear rate and itis the opposite at high shear rate. Relaxation times matchat shear rate values increasing with temperature (1 s−1 at180 ◦C and 70 s−1 at 220 ◦C). In previous experimentalstudies viscosity curves did not cross each other and theviscosity ratio was more or less uniform [14]. This is not thecase in our study where both viscosity ratio and elasticityratio change significantly with flow rate and temperature.

2.2. Analysis of extruded samples

Coextrusion experiments are performed with laboratoryapparatus (Fig. 4) already described in [9]. This appara-tus consists of a Kaufmann extruder used for polyethy-lene and a Haake-Rheocord extruder for polystyrene. Thetwo polymers are coextruded in a temperature-controlledtwo-manifold die which has a flat final part with 100:1 and

Fig. 3. Plot of viscosity ratio ηPS/ηPE (a) and relaxation time rationλPS/λPE (b) vs. shear rate, deduced from the White–Metzner model fit,for various temperatures.

Page 4: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

44 R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

Fig. 4. Schematic of experiment apparatus: “ laboratory” device with a multi-manifold die.

40:1 width and length aspect ratios, respectively. It has beenshown in [10] that intefacial instabilities develop in this finalpart of the die. The large width aspect ratio ensures that theflow motion can be approximated by a two-layer Poiseuilleflow. After extrusion the film is slightly stretched in air (aminimum draw ratio is necessary) by a chill roll imposinga controlled velocity and finally quenched in a water bath.

The first experiments (setup 1) are made for three differ-ent temperatures, namely 180, 200, 220 ◦C and various flowrates. At each time, a film sample was taken for stabilizedprocessing conditions and its optical properties were exam-ined by using an overhead projector. Flow rate of polyethy-lene and polystyrene denoted in the following as QPE andQPS was measured by weighing.

Fig. 5. Samples at die exit: (a) stable; (b) “slightly” unstable with aperiodic wave.

It was found that, in the experimental range of flow ratesand temperatures, there exists a transition between stableand unstable processing conditions. The transition can bedescribed as follows:

• For similar values of flow rates (QPE ∼ QPS) the interfacebetween polymers is regular. A sample of the obtainedextrudate is presented in Fig. 5a. We called “stable” suchprocessing conditions.

• For larger values of the flow rate ratio QPE/QPS a wavyinterface between the two polymers is observed at die exitand on the solidified film as shown in Fig. 5b. We called“unstable” such processing conditions.

The observations made for all processing conditions stud-ied are summarized in Fig. 6. It appears also that the criticalflow rate ratio QPE/QPS giving the transition between sta-ble/unstable sheets increases with total flow rate QPE +QPS.The temperature influences the transition by shifting thisratio to a higher one when increased. Moreover, no unsta-ble processing conditions were observed at 220 ◦C and thatcould mean that the transition moves outside the experimen-tally reachable range of flow rates.

2.3. Analysis of interfacial waves

To study developments of defects inside the die, we havemodified our device by replacing the lateral walls with trans-parent glass windows (setup 2). Moreover, we have modifiedthe generator which controls rotation speed of the Kaufmannscrew extruder in order to be able to impose a periodic forc-ing on flow rate (see [10] for more details). One polymer iscolored in black (PS layer) and a CCD camera records theinterfacial fluctuation.

When no external forcing is imposed in “spontaneously”unstable conditions, one observes an unstable interface as de-picted in Fig. 7, which shows a sequence of pictures recordedthrough the transparent wall, in the final part of the die (be-

Page 5: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 45

Fig. 6. Summary of experimental, longwave analysis and spatial stability analysis in the (QPE,QPS) plane. From top to bottom: T = 180, 200 and220 ◦C. The samples are either stable � or unstable �; the longwave stability analysis (—) bounds stable (symbol ) and unstable (symbol ) areas;the (convective) linear stability computation gives stable flow (symbol S) or unstable flow (symbol Unst), where −ki (mm−1) is the spatial amplificationrate of the perturbation of wavenumber kr (mm−1) obtained using Gaster’s relation.

Page 6: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

46 R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

Fig. 7. Interfacial wave in the last part of the die (32 mm ≤ x ≤ 40 mm): sequence recorded with a time interval of 0.08 s.

tween positions x = 32 and 40 mm). That indicates that theinterface position varies as a function of time and stream-wise coordinate x. For example, a defect appearing at posi-tion x = 32 mm and time t = 0 s is advected and reaches dieexit at time t = 0.96 s. One also observes that the amplitudeof this defect increases as it is advected. Such a sequenceclearly proves the convective nature of the instability.

Unfortunately, the defects do not usually exhibit atwo-dimensional shape [10] in the spanwise direction and

0 10Time (s)1.651.7

1.751.8

1.85

h (m

m)

0 10Time (s)1.651.7

1.751.8

1.85

h (m

m)

0 10Time (s)1.651.7

1.751.8

1.85

h (m

m)

0 10Time (s)1.651.7

1.751.8

1.85

h (m

m)

0 10Time (s)1.651.7

1.751.8

1.85

h (m

m)

x = 31mm

x = 32mm

x = 33mm

x = 34mm

x = 35mm

Fig. 8. h(t) diagrams for a forcing frequency 0.49 Hz in successive positions between x = 31 and 35 mm.

a superposition of different waves is observed through thetransparent walls.

To prevent the appearance of a nonlinear regime and todeal with uniform waves, we then control the upstream per-turbation by imposing a periodic forcing in slightly unstableconditions. In fact, due to large width aspect ratio, interfacialwaves remain two dimensional in a small parameter range.The die thickness has also been increased from 1 to 2 mm inorder to ensure better accuracy of interface measurements.

Page 7: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 47

Table 1Summary of defect features as a function of forcing frequency

External forcing N (Hz) 0.26 0.49 0.64 0.85 1.26Pulsation ωr (rad/s) 1.63 3.08 4.02 5.34 7.92Time periodicity T (s) 3.92 2.06 1.56 1.17 0.80Velocity Vφ (mm/s) 5.68 6.22 5.40 5.08 4.97Wavenumber kr (rad/mm) 0.28 0.50 0.74 1.05 1.59Spatial periodicity λ (mm) 22.27 12.69 8.44 5.98 3.95

With this die gap, one obtains “slightly” unstable con-ditions for QPE = 87.4 g/min and QPS = 7.4 g/min. Onethen observes that, without external forcing, the interfaceis flat but, when introducing controlled perturbations, atwo-dimensional defect is detectable on extrudates. Oneobserves that controlled perturbations are also spatiallyamplified. In order to analyze the influence of the forcingfrequency on the spatial growth rate, we record the interfacedeviation at various streamwise coordinates. In this way,one gets h(t) diagrams (h denotes the position of the inter-face), which allows one to measure the spatial and temporalfeatures of interfacial waves (see Fig. 8). The methodologyused to deduce this information from the pictures recordedby the CCD camera is explained in detail in [10].

The results are summarized in Table 1 and one can seethat a specific wavenumber is selected by each forcing fre-quency. The spatial amplification rate is determined, for agiven wavenumber, by the slope of the graph of ln(h(t, x))as a linear regression of x(h = h − 〈h〉 is the interfacedeviation). An example is given in Fig. 9 for the forcing fre-quency of 0.49 Hz. One then plots on Fig. 10, for all experi-mental results, the spatial amplification rate as a function ofthe forcing wavenumber. One observes that the growth ratealso depends on the forcing frequency. The amplification issmall for low wavenumbers, then it increases and reaches a

30.0 31.0 32.0 33.0 34.0 35.0 36.0Position in the die (mm)

3.0

3.2

3.4

3.6

3.8

log

(Am

plitu

de)

Fig. 9. Spatial growth rate of the perturbation for forcing frequency0.49 Hz: the linear regression on a (x, ln(h(x))) diagram (logarithm ofinterface deviation h = h − 〈h〉 as function of position) gives a slope of0.1544/mm.

maximum value for moderate waves (wavenumber 1 rad/mm). Finally it decreases for larger wavenumbers.These results are in agreement with Wilson and Khomami’sobservations [11].

For frequency 0.26 Hz the spatial amplification rate ofthe perturbation is quite small, and for frequency 1.26 Hzthe initial perturbation is small due to the limitation of theapparatus in imposing a forcing of large amplitude andfrequency. Consequently, for these two frequencies, the in-terface deviations are rather small and the correspondingerror bars are relatively large.

This figure will be used to check the capability of theWhite–Metzner constitutive law (used in our theoretical ap-proach) to quantitatively predict the amplification rates ofthe defects.

3. Linear stability analysis

The purpose of this section is to perform a linear stabilityanalysis of coextrusion flows described in previous sections.In the sequel, the following assumptions are made:

• The flow is isothermal and two-dimensional (invariant inthe spanwise direction).

• The two polymers are incompressible and immiscible,gravity and surface tension are neglected.

• The basic flow located in the flat part of the die is assumedto be a Poiseuille flow.

That means that the velocity uuu, stress tensor σσσ and interfaceposition h are not functions of time t and are invariant inthe streamwise direction x.

3.1. Basic flow

Modelling requires the choice of a constitutive equationdescribing the behaviour of polystyrene and polyethylene.Furthermore, as derivatives of viscometric functions appearin the “ linearized” equations, a smooth behaviour of theirderivatives is necessary. It is the main reason for which wehave preferred the White–Metzner model to a less regularmultimode model. Moreover, Ganpule and Khomami [20]have shown that many models, including the White–Metznermodel, can predict accurately the critical layer depth ratioand the most dangerous wavenumber of the instability, pro-vided that they can describe accurately the steady shear vis-coelastic behaviour of the polymers. They also claim thatdifference in the elongational behavior of the models is notexpected to significantly influence the interfacial stability.Actually, when comparing the White–Metzner model withthe Giesekus model, Ganpule and Khomami have shownthat quantitative differences in linear stability predictions ofperturbations growth rates arise due to differences in thetransient behaviour of the two models. Nevertheless, as thedifferences are only significant far from the stable/unstabletransition, one can expect the White–Metzner model to give

Page 8: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

48 R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

0 0.5 1 1.5 2Wavenumber (/mm)

0

0.1

0.2

0.3

0.4

Spat

ial g

row

th r

ate

(/m

m)

Fig. 10. Spatial amplification rate of the perturbations function of forcing wavenumber for a die gap of 2 mm, T = 180 ◦C, QPE = 87.4 g/min andQPS = 7.4 g/min: (×) experimental data; (— ) theoretical result computed with Gaster’s relation for the White–Metzner constitutive law.

satisfactory qualitative results, and eventually a quantitativedescription of the instability at least close to the transition.

The White–Metzner constitutive equation is:{σσσ = −pIII + τττ,

τττ + λ(γ)(∂tτττ + uuu∇τττ − ∇ uτuτuτ − τττT∇uuu) = 2η(γ)εεε,(1)

where εεε, τττ are, respectively, the strain rate tensor and theextra stress tensor. The viscosity η and the elasticity λ arefunctions of generalized shear rate γ and they follow aCarreau–Yasuda type law:

η(γ) = η0[1 + (kγ)a](m−1)/a,

λ(γ) = λ0[1 + (lγ)b](n−1)/b. (2)

The constant values η0, k, a, m, λ0, l, b, n are identifiedfrom rheological curves (Figs. 1 and 2) and are reported inTable 2.

In the following, the symbol · labels the basic or Poiseuilleflow. The basic velocity uuu = (u(y), 0) is given by the fol-lowing equation ([[.]]h denotes the jump at the interface):

η = η( ˙γ),∂y(η ˙γ) = ζ,

[[u]]h = 0,

[[η ˙γ]]h = 0,

(3)

Table 2White–Metzner parameters at 180 ◦C for both polymers

Viscosity η(γ) = η0[1 + (kγ)a](m−1)/a Relaxation time λ(γ) = λ0[1 + (lγ)b](n−1)/b

Parameter Polyethylene Polystyrene Parameter Polyethylene Polystyrene

η0 (Pa s) 3.188 × 105 1.252 × 105 λ0 (s) 1.618 × 102 3.489 × 102

k (s) 2.718 × 102 2.407 × 100 l (s) 2.649 × 102 1.598 × 102

a 3.817 × 10−1 5.506 × 10−1 b 6.871 × 10−1 2.323 × 10−1

m 2.944 × 10−1 1.731 × 10−1 n 1.108 × 10−1 6.429 × 10−2

where ˙γ is the shear rate, ∂yu, ζ the constant pressure gra-dient, ∂xp and h the position of flat interface. This sys-tem is completed by non-slip boundary conditions on diewalls.

The extra stress tensor is then given by:

τττ =(σ τ

τ 0

)=(

2λη ˙γ2η ˙γ

η ˙γ 0

), (4)

where λ = λ( ˙γ).

3.2. Temporal linear stability analysis

The methodology used for temporal linear stability anal-ysis is rather well known [21]. It consists in studying thebehaviour of small perturbations of the basic flow. Assum-ing a large or infinite channel length, one can introduce in-finitesimal perturbations which are spatially periodic in thestreamwise direction x. They have the form:

(uuu, τττ, p, h)(x, y, t) = (uuu,τττ, p, h)(y) ei(kx−ωt), (5)

where 2π/k is the spatial period, ωi (the imaginary part ofω) is either the growth or the damping rate and 2π/ωr (ωr isthe real part of ω) the temporal period. By linearizing equa-tions around the Poiseuille solution, one gets generalized

Page 9: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 49

Orr–Sommerfeld equations in each layer:

(1 + ikλu)σ − 2λu′τ − 2ik(λσ + η)u

− 2

(λ + ∂λ

∂u′ u′)τDu + λσ′v − 2ik

∂λ

∂u′ u′τv = iωλσ,

(6)

(1 + ikλu)τ − λu′γ −(η + ∂η

∂u′ u′)

Du

− ik

(λσ +

(η + ∂η

∂u′ u′))

v + λτ′v = iωλτ, (7)

(1 + ikλu)γ + 2ikηu − 2ikλτv = iωλγ, (8)

ikσ + Dτ − ikp = 0, (9)

ikτ + Dγ − Dp = 0, (10)

iku + Dv = 0, (11)

v(h) − iku(h)h = −iω, (12)

where Eqs. (6)–(8) are deduced from constitutive equation,Eqs. (9) and (10) correspond to conservation momentumequation, Eq. (11) is the mass conservation equation andEq. (12) reads the immiscibility between the two fluids. Thisset of equations is achieved by non-slip condition on wallsand the continuity of velocity (13) and (14) and stress (15)and (16) at interface:

[[v]]h = 0, (13)

[[u]]h + h[[u′]]h = 0, (14)

[[τ]]h − ikh[[σ]]h = 0, (15)

[[p]]h + [[γ]]h = 0. (16)

The details of algebraic calculations can be found in [22].Each variable (uuu,τττ, p) is discretized in terms of Chebyshevpolynomial and boundary conditions are imposed by usinga Tau method [21] and one obtains a generalized eigenvalueproblem for ω:

AAA(k)X = ωBBB(k)X, (17)

where AAA, BBB are two matrix (BBB is singular) and X the vectorformed by Chebyshev coefficients of all the variables.

If all the parameters are fixed, the basic flow (uuu, τττ) isdetermined and the Eq. (17) expresses the link between thewavenumber k and the pulsation ω, which is also calledthe dispersion relation, D(k, ω) = 0. Finally, the temporalstability analysis is given by the ω-roots ofD(k, ω) = 0 for kreal and the temporal growth rate is given by the eigenvaluehaving the largest imaginary part. If this value is positive,the two-layer Poiseuille flow is considered as unstable. Thatcorresponds to classical temporal stability analysis and it hasalready been investigated for the two-layer Poiseuille flowusing several constitutive laws [23–25,4].

An example of evolution of the most critical eigenvalueωi with the wavenumber k is plotted in Fig. 11a, whereasFig. 11b shows the evolution of the pulsation ωr with thewavenumber. One can remark in this figure that in the long-wave limit (k → 0), there is always a zero eigenvalue (whichis a classical feature of interfacial instabilities) and the slopeof the eigenvalue, ∂kωi(k = 0), at this point is zero. Thevalue of the second derivative ∂2

k2ωi(k = 0) gives an infor-mation on the sign of ωi for larger values of k. The com-putation of this particular coefficient is called “ longwavestability analysis” and it allows one to give rapidly an indi-cation about the influence of parameters [6]. However, thismethod does not allow one to give a reliable criterion forstable flows as the flow might become unstable for shorterwaves (in this case ωi becomes positive for moderate orlarge values of the wavenumber k). This method has never-theless been used for comparison to our experiments in [22]and the “ longwave” stable and unstable boundaries in the(QPE,QPS) plane obtained in this way are also reported inFig. 6.

Finally comparison between experimental data, longwavestability analysis and the full temporal stability analysisgives the following results:

• At T = 180 ◦C, there is a reasonable agreement be-tween longwave stability analysis and experimental ob-servations. The transition from stable to unstable whenincreasing the polyethylene flow rate is well captured.The stability with respect to perturbations having a mod-erate wavenumber gives no more additional informati-ons.

• At T = 200 ◦C, the theoretical boundary given by long-wave stability analysis is moved to larger values of theflow rate ratio QPE/QPS. Comparison to experiments isthen rather good except for two experimental conditions:at (QPE,QPS) = (1281, 183 mm3/s) the flow is theoreti-cally stable for longwaves but experimentally unstable; at(QPE,QPS) = (117 mm3/s, 81 mm3/s) the flow is theo-retically unstable but experimentally stable. The first dis-crepancy is dropped when computing the full temporalstability analysis as it appears that the flow becomes theo-retically unstable for moderate wavelength. However, thesecond discrepancy (where the flow is experimentally sta-ble) is more intricate as the theoretical analysis gives asufficient condition for instability.

• At T = 220 ◦C, there is a drastic change in the theoreti-cal boundaries coming from longwave stability analysis.Most of the experimental conditions become theoreticallyunstable with respect to both longwave and full temporalstability analysis (except for two processing conditions).However, one observes that all these processing condi-tions give raise to a stable extrudate. Hence, temporal sta-bility analysis fails to predict the stability in a same wayas observed for (QPE,QPS) = (117, 81 mm3/s) at T =200 ◦C.

Page 10: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

50 R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

0 0.5 1 1.5 2 2.5 3 3.5Wavenumber (/mm)

0

0.5

1

1.5

2

2.5

Tem

pora

l gro

wth

rat

e (/

s)

0 0.5 1 1.5 2 2.5 3 3.5Wavenumber (/mm)

0

10

20

30

Puls

atio

n (/

s)

(a)

(b)

Fig. 11. Example of evolution the most critical eigenvalue for a die gap of 2 mm, T = 180 ◦C, QPE = 87.4 g/min and QPS = 7.4 g/min, obtained byusing the White–Metzner model: (a) temporal growth rate ωi vs. wavenumber k; (b) critical pulsation ωr vs. wavenumber k.

3.3. Gaster’s relation for convective stability analysis

However, since the experiments have shown that interfa-cial instability is of convective nature, the latter approach isno longer suitable. We are rather interested in linear dynam-ics triggered off by an initial localized and time-periodicdisturbance and its evolution in time and space. In thisway, one has to deal with a perturbed initial-value prob-lem [16] which can be solved by using a Fourier–Laplacetransform. Therefore the spatial growing waves can alwaysbe written in the latter form (5), but ω is now real and k

is complex. Finally, the spatial stability problem consistsin finding k-root of D(k, ω) = 0; ω being the forcing pul-sation and the largest −ki (the imaginary part of k) gives

the spatial amplification rate associated with the pulsationω.

In the case of small temporal and spatial growth rates,the following transformation linking approximately thesegrowth rates was proposed by Gaster [18]:

−ki(S) = ωi(T)

∂ωr/∂kr, (18)

where S and T mean that values are obtained with spatial andtemporal stability analysis, respectively. Gaster’s transfor-mation is widely used for computing spatial growth rates byusing a preliminary computation of temporal growth rates.It allows one to save computation time as spatial computa-tions are much more tricky than temporal stability ones.

Page 11: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 51

In a previous study on a film flow down an inclined plane[26], it is found that this relation holds perfectly as the nu-merical values given by both approaches are practically in-distinguishable for the entire range of unstable frequencies.Concerning the two-layer Poiseuille flow, one deals with thesame kind of interfacial instability (the liquid-air interfaceis replaced by a liquid-liquid interface) and we have foundthat the growth rate ωi(T) is rather small (at least close to astable/unstable transition). Therefore, one can assume thatthis relation would again give a good approximation of spa-tial growth rates.

Note also that Vg = ∂ωr/∂kr is the group velocity of theperturbation having wavenumber kr and, in a frame moving

0 0.5 1 1.5 2 2.5 3 3.5

Frequency (Hz)

0

0.1

0.2

0.3

0.4

Spat

ial g

row

th r

ate

(/m

m)

0 0.5 1 1.5 2 2.5 3 3.5

Frequency (Hz)

0

2

4

6

8

10

12

Gro

up V

eloc

ity (

mm

/s)

(a)

(b)

Fig. 12. Example of evolution critical values for a die gap of 2 mm, T = 180 ◦C, QPE = 87.4 g/min and QPS = 7.4 g/min deduced from Fig. 11 thanksto Gaster’ relation: (a) Gaster coefficient −ki = ωi/Vg vs. frequency ωr/2π; (b) group velocity Vg = ∂kωr vs. frequency ωr/2π.

with this group velocity, the perturbation would be tempo-rally unstable. The group velocity Vg is different from theinterface velocity and, if it is small enough, the perturbationstays a long time inside the die, then it is strongly amplified,leading to a pronounced defect at die exit.

3.4. Application of Gaster’s transformation toexperimentally forced flows (setup 2)

The aim of this section is to validate both the use ofGaster’s relation and the choice of the White–Metzner con-stitutive equation thanks to data coming from experimen-tally forced flow (setup 2). As explained above, the spatial

Page 12: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

52 R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53

amplification rate −ki is numerically computed from tem-poral stability analysis by computing ωi and Vg for a fixedwavenumber kr. For example Fig. 12a and b are deducedfrom Fig. 11 by using Gaster’s relation. In these experiments,die gap is 2 mm and flow rates are QPE = 87.4 g/min andQPS = 7.4 g/min. The numerical solution with the same pa-rameters gives an interface position for the Poiseuille flowh = 1.70 mm, which is close to the experimental measure-ment h = 1.75 ± 0.01 mm.

The results of this computation are plotted in Fig. 10and compared with experimental data. As shown by thisfigure, theoretical predictions are in quite good agreementwith experimental data. More precisely, theoretical resultspredict that the spatial amplification rate reaches a maximumand then decreases for large values of the wavenumber. Nearthis maximum, that is to say for values of the wavenumberequal to 0.74, 1.05 and 1.59 rad/mm, there is a quantitativeagreement with experiments. For values of the wavenumberequal to 0.28 and 0.50 rad/mm, one observes a discrepancythat could be explained as follows. For these values, theperturbations exhibit a spatial periodicity of, respectively,22.27 and 12.69 mm. When comparing these values to thedie length (40 mm), one can object that the assumption of alarge length of the channel is no more satisfied. Thus, thetheoretical approach is less accurate (or eventually no longervalid) if the wavelength perturbation is close to die length.

3.5. Comparison between convective stability analysis andcoextrusion experiments results (setup 1)

In the previous section, we have found that a convectivestability analysis can quantitatively predict the results ob-tained for experimentally forced flow. One found that thereexists a “dominant” mode for which the spatial amplifica-tion rate reaches its maximum. The goal is now to use thistheoretical tool to check if it is possible to forecast the ap-pearance of instabilities in coextrusion flows (without forcedperturbations) described in Section 2. For each processingcondition, we have computed the spatial growth rate andchecked if, for theoretically unstable processing conditions,there exists such a dominant mode. The associated valueof the spatial amplification rate is then a good candidate toquantify the “ intensity” of the instability.

Following this methodology, theoretical results are com-pared to experimental results in correlating the appearanceof stable/unstable extrudates to the largest value of the spa-tial growth rate. These values are also reported in Fig. 6 foreach processing condition and one can show that:

• At temperature 180 ◦C, one observes that the maximalspatial growth rate increases with the polyethylene flowrate. It is in agreement with experimental observationswhere film distortions become more and more importantby increasing QPE.

• At temperature 200 ◦C and for large polyethylene flowrates, one gets the same correlation between spatial growth

rates and experimental observations as found for T =180 ◦C. Nevertheless, for low polyethylene flow rates, oneobserves that, for a theoretical spatial amplification ratesmaller than 0.22 mm−1, the experimental coextrusionflow remains stable, whereas it becomes unstable for a the-oretical spatial amplification rate larger than 0.33 mm−1.This means that, for “slightly” unstable flows, perturba-tions are not sufficiently amplified to give rise to an ob-servable defect at die exit. The critical amplification ratebeyond which experimental flows become unstable wouldthen be in the range ]0.22 mm−1, 0.33 mm−1[.

• This assumption is confirmed by looking at the Fig. 6for temperature 220 ◦C: all processing conditions areeither theoretically stable or theoretically unstable witha corresponding spatial amplification rate smaller than0.04 mm−1. Thus, perturbations do not give rise to ob-servable defects, as it is experimentally observed.

4. Conclusion

The convective nature of interfacial instabilities in co-extrusion flows had been pointed out experimentally byKhomami and coworkers [11–15] and numerically by Valetteet. al. [17]. In this paper, we have performed a convectivestability analysis for a large range of prototype industrialexperiments and shown that it is possible to forecast the ap-pearance of interfacial defects on the extrudate.

This general approach has been validated by using a trans-parent device and a theoretical analysis based on the use ofGaster’s transformation for White–Metzner fluids, close tothe stable/unstable transition. Main results are:

• The use of the White–Metzner constitutive equation al-lowed us to describe the basic flow and the dynamics ofthe perturbed flow.

• Gaster’s transformation is sufficient to obtain a qualita-tive agreement with experiments. In particular, we havepointed out the existence of a dominant mode.

For prototype industrial experiments, a convective stabil-ity analysis is able to forecast the appearance of extrudatedefects much better than classical approaches:

• The temporal asymptotic study fails when the instabilityis only activated for moderate wavelength and/or whenthe spatial amplification is not sufficient enough.

• The temporal study doesn’ t take account of defects resi-dence time inside the die.

• The spatial study shows that there exists a critical valueof the spatial amplification rate which corresponds to theappearance of defects on the extrudate. Such a study thenshows a very good agreement with experimental results.

This critical value, which is dependent on rheologicalproperties of the polymers and on processing conditions, isalso strongly dependent on the die length and on the originof the perturbations. If the die length and the amplitude of

Page 13: Convective linear stability analysis of two-layer coextrusion flow for molten polymers

R. Valette et al. / J. Non-Newtonian Fluid Mech. 121 (2004) 41–53 53

upstream perturbations are small enough, one can expect thedefects to be weakly amplified at die exit.

Acknowledgements

The PhD works of Rudy Valette has been supported byATOFINA, Cerdato Research Center, France. ATOFINA isacknowledged for financial support, technical assistance andadvice.

References

[1] K.P. Chen, Interfacial instability due to elastic stratification in con-centric coextrusion of two viscoelastic fluids, J. Non-Newtonian FluidMech. 40 (1991) 155–175.

[2] Y.Y. Su, B. Khomami, Interfacial stability of multilayer viscoelasticfluids in slit and converging channel dies geometries, J. Rheol. 36(1992) 357–387.

[3] Y.Y. Su, B. Khomami, Purely elastic interfacial instabilities in su-perposed flow of polymeric fluids, Rheol. Acta 31 (1992) 413–420.

[4] P. Laure, H. Le Meur, Y. Demay, J.C. Saut, S. Scotto, Linear sta-bility of multilayer plane Poiseuille flows of Oldroyd-B fluid, J.Non-Newtonian Fluid Mech. 71 (1997) 1–23.

[5] C.D. Han, R. Shetty, Studies on multilayer film coextrusion. I. Therheology of flat film coextrusion, Polym. Eng. Sci. 16 (10) (1976)697–705.

[6] C.D. Han, R. Shetty, Studies on multilayer film coextrusion. II.Interfacial instability in flat film coextrusion, Polym. Eng. Sci. 18 (3)(1978) 180–186.

[7] C.D. Han, Y.J. Kim, H.B. Chin, Rheological investigation of interfa-cial instability in two layer–layer flat-film coextrusion, Polym. Eng.Rev. 4 (3) (1984) 177–217.

[8] N.R. Anturkar, T.C. Papanastasiou, J.O. Wilkes, Estimation of criticalstability parameters by asymptotic analysis in multilayer extrusion,Polym. Eng. Sci. 33 (1993) 1532–1539.

[9] R. Valette, P. Laure, Y. Demay, J.-F. Agassant, Investigation of theinterfacial instabilities in the coextrusion flow of polyethylene andpolystyrene, Int. Polym. Process. 18 (2) (2003) 171–178.

[10] R. Valette, P. Laure, Y. Demay, J.-F. Agassant, Experimental inves-tigation of the development of interfacial instabilities in two layercoextrusion dies, Int. Polym. Process. 19 (2) (2004).

[11] G.M. Wilson, B. Khomami, An experimental investigation of inter-facial instabilities in multilayer flow of viscoelastic fluids. Part I.

Incompatible polymer systems, J. Non-Newtonian Fluid Mech. 45(1992) 355–384.

[12] G.M. Wilson, B. Khomami, An experimental investigation of inter-facial instabilities in multilayer flow of viscoelastic fluids. Part II.Elastic and nonlinear effects in incompatible polymer systems, J.Rheol. 37 (1993) 315–339.

[13] G.M. Wilson, B. Khomami, An experimental investigation of inter-facial instabilities in multilayer flow of viscoelastic fluids. Part III.Compatible polymer systems, J. Rheol. 37 (1993) 341–354.

[14] B. Khomami, M.M. Ranjbaran, Experimental studies of interfacialinstabilities in multilayer pressure-driven flow of polymeric melts,Rheol. Acta 36 (1997) 345–366.

[15] B. Khomami, K.C. Su, An experimental/theoretical investigation ofinterfacial instabilities in superposed pressure-driven channel flow ofNewtonian and well characterized viscoelastic fluids. Part I. Linearstability and encapsulation effects, J. Non-Newtonian Fluid Mech.91 (2000) 59–84.

[16] P. Huerre, A. Monkewitz, Local and global instabilities in spatiallydeveloping flows, Annu. Rev. Fluid Mech. 22 (1990) 473–537.

[17] R. Valette, P. Laure, Y. Demay, A. Fortin, Convective instabilities incoextrusion process, Int. Polym. Process. 16 (2) (2001) 192–197.

[18] M. Gaster, A note on the relation between temporally-increasing andspatially-increasing disturbances in hydrodynamic stability, J. FluidMech. 14 (1962) 222–224.

[19] H.K. Ganpule, B. Khomami, An investigation of interfacial insta-bilities in the superposed channel flow of viscoelastic fluids, J.Non-Newtonian Fluid Mech. 81 (1999) 27–69.

[20] H.K. Ganpule, B. Khomami, The effect of transient viscoelasticproperties on interfacial instabilities in superposed pressure drivenchannel flows, J. Non-Newtonian Fluid Mech. 80 (1999) 217–249.

[21] S. Orszag, Accurate solution of the Orr–Sommerfeld equation, J.Fluid Mech. 50 (4) (1971) 689–703.

[22] R. Valette, Etude de la stabilité de l’écoulement de Poiseuille defluides viscoélastiques, Application au procédé de coextrusion despolymères, Ph.D. Thesis, Ecole Nationale Supérieure des Mines deParis, 2001.

[23] B. Khomami, Interfacial stability and deformation of two stratifiedpower law fluids in plane Poiseuille flow. Part I. Stability analysis,J. Non-Newtonian Fluid Mech. 36 (1990) 289–303.

[24] B. Khomami, Interfacial stability and deformation of two stratifiedpower law fluids in plane Poiseuille flow. Part II. Interface deforma-tion, J. Non-Newtonian Fluid Mech. 37 (1990) 19–36.

[25] A. Pinarbasi, A. Liakopoulos, Stability of two-layer Poiseuille flowof Carreau–Yasuda and Bingham-like fluids, J. Non-Newtonian FluidMech. 57 (1995) 227–241.

[26] L. Brevdo, P. Laure, F. Dias, T.J. Bridges, Linear pulse and signallingin a film flow on an inclined plane, J. Fluid Mech. 396 (1999) 37–71.