PDepartment of Mathematics and Statistics SAVVAS G ...

88 Applied RheologyMarch/April 2002

On numerical Simulations of Polymer Extrusion Instabilities

Evdokia Achilleos, Georgios C. Georgiou *

PDepartment of Mathematics and StatisticsUniversity of Cyprus

P.O. Box 20537, 1678 Nicosia, Cyprus

and

SAVVAS G. HATZIKIRIAKOS

Department of Chemical EngineeringThe University of British Columbia

2216 Main Mall, Vancouver, BC, V6T-1Z4, Canada

* E-mail: [email protected]: x357.2.339061

Received: 22.11.2001, Final version: 13.4.2002

Abstract:The objective of this study is mainly to review recent work concerning the numerical modeling of the stick-slipand gross melt fracture polymer extrusion instabilities. Three different mechanisms of instability are discussed:(a) combination of nonlinear slip with compressibility; (b) combination of nonlinear slip with elasticity; and (c)constitutive instabilities. Furthermore, preliminary numerical simulations of the time-dependent, compressibleextrudate-swell flow of a Carreau fluid with slip at the wall, using a realistic macroscopic slip equation that isbased on experimental data for a high-density polyethylene, are presented.

Zusammenfassung:Das Ziel dieses Übersichtsartikels ist es die neueste Arbeiten der numerischen Modellierung des “stick-slip” undder Extrusionsinstabilitäten bei Polymeren, die durch Schmelzebruch verursacht werden, vorzustellen. Es wer-den drei verschiedene Instabilitätsmechanismen besprochen: (a) die Kombination von nichtlinearem Gleitenmit Kompressibilität; (b) die Kombination von nichtlinearem Gleiten mit Elastizität; und (c) konstitutive Insta-bilitäten. Im weiteren werden vorläufige numerische Simulationen von zeitabhängiger, kompressibler Stran-gaufweitung einer Carreau-Flüssigkeit mit Wandgleiten vorgestellt, wobei für HDPE eine auf experimentellenDaten beruhende makroskopische Gleit-Gleichung verwendet wird.

Résumé:L’objectif de cette étude est principalement de passer en revue les travaux récents qui visent à modélisernumériquement les instabilités rencontrées lors de l’extrusion de fondus de polymères telles que le glissement-accrochage et le phénomène de fracture. Trois mécanismes différents d’instabilités sont discutés: (a) la combi-naison d’un glissement non linéaire avec la compressibilité; (b) la combinaison d’un glissement non linéaire avecl’élasticité; et (c) les instabilités constitutives. De plus, des simulations numériques préliminaires sont présen-tées. Elles simulent l’écoulement d’extrusion compressible avec gonflement d’un fluide de type Carreau avecglissement aux parois. Ce dernier est modélisé à l’aide d’une équation macroscopique réaliste, basée sur desdonnées expérimentales obtenues avec un polyéthylène haute densité.

Key Words: Extrusion instabilities, Melt fracture, Slip, Carreau model, Oldroyd-B model, Compressible flow, Con-stitutive instability, Viscoelastic flow

© Appl. Rheol. 12 (2002) 88-104

AR •Vol 12• Nr 2 • 04/02.qxd 08.05.2002 11:17 Uhr Seite 88

1 INTRODUCTIONPolymer extrusion instabilities appear frequent-ly in industrial practice, limiting the productionrates and influencing the appearance and quali-ty of polymer extrudate products. They have thusreceived considerable attention in the past fiftyyears in numerous experimental, theoretical,and numerical studies. The main goal of all theabove studies was to reveal the origins and themechanisms of the various forms of extrudatedistortion, in order to develop efficient tech-niques for suppressing or eliminating the insta-bilities. Reviews on extrusion instabilities includethose by Petrie and Denn [1], Denn [2, 3], Larson[4], Leonov and Prokunin [5], Piau et al. [6], El Kissiand Piau [7], and Wang [8]. In most experimen-tal studies on extrusion instabilities, capillaryrheometers are used, under either constant flowrate (i.e., constant piston speed) or constant pres-sure drop operation [9]. This work focuses oninstabilities observed at constant piston speed.

It is well known that during the extru-sion of polymers from capillary or slit dies underfixed piston speed, a variety of extrudate distor-tions are observed when the volumetric flow rate(or, equivalently, the apparent shear rate)exceeds a critical value [10 - 16]. As the volumet-ric flow rate increases, the size and the severityof the extrudate distortions increase. For mostlinear polymer melts, the flow curve (i.e., the plotof the wall shear stress versus the apparent shearrate at the wall, or, equivalently, the plot of thepressure drop versus the volumetric flow rate)consists of two positive-slope branches separat-ed by a zone where the pressure drop oscillates,although the flow rate is kept constant [13 - 17].A typical flow curve for a linear polyethylene, rou-tinely obtained using a capillary rheometerunder constant speed operation, is shown in Fig.1. Connecting the two branches with an imagi-nary line results in a non-monotonic flow curvewith a maximum and a minimum which definethe range of the pressure drop oscillations.

When the flow rate is controlled, fourdistinct flow regions are, in general, observed forlinear, narrow molecular weight distributedpolymers (e.g., LLDPE [11], HDPE [13, 18], and lin-ear polydimethylsiloxane [12]). These flowregions are illustrated in Fig. 1. At small shearrates, the extrudate has a smooth, glossy surfacefinish. This stable regime persists up to a criticalshear rate at which the extrudate starts losing

89Applied RheologyMarch/April 2002

gradually its glossiness, and small-amplitude,short-wavelength periodic distortions appear onits surface, even though the pressure remainsconstant. This surface defect is known as thesharkskin instability or surface melt fracture andis observed up to the endpoint of the left branchof the flow curve. Some researchers associatesharkskin with a small departure from the no-slipboundary condition [10, 11, 13, 18 - 20], which issometimes accompanied by a sharp change inthe slope of the flow curve. Some experimentsindicate that increased adhesion between thefluid and the wall reduces the sharkskin instabil-ity [10], while others show that enhancing wallslip suppresses the instability [6, 21, 22]. Howev-er, slip is not always present in the sharkskinregime; the experiments of El Kissi and Piau [14],Wang [8], and El Kissi et al. [15] showed that slipis negligible.

Beyond the sharkskin regime and with-in a certain range of apparent shear rates sepa-rating the two branches of the flow curve, nosteady-state solutions can be achieved. Althoughthe flow rate is kept constant, sustained pressureand mass flow rate oscillations are observed, andthe extrudate surface is characterized by alter-nating sharkskin and relatively smooth regions.This instability is known as the stick-slip instabil-ity or spurt phenomenon or oscillating melt frac-ture. As discussed below, the pressure drop oscil-lations are attributed, by some researchers, tothe combined effects of the finite compressibili-ty of the polymer and the periodic transitionsfrom a weak to a strong slip at the capillary walland vice versa. Piau et al. [6] have shown exper-imentally that the oscillations can be eliminated

Figure 1: Typical flow curvefor a linear polyethyleneand regions of instability.


by using slippery surfaces. Slip can be promotedeither by using dies constructed with appropri-ate materials, such as brass [22], or by means ofpolymer processing additives, such as fluo-ropolymers or Boron nitride [23 - 25]. It should benoted that in pressure-controlled experiments,an abrupt jump in the flow rate is observed in thespurt flow regime. Finally, at even higher wallshear rates corresponding to the right positive-slope branch of the flow curve, steady state canagain be achieved, although the distortion of theextrudate is more severe, with its surface beingrough and wavy. This instability is known as wavyor gross melt fracture. In this last regime, thevelocity profile in the capillary is nearly plug [26,27], which implies the occurrence of strong slip.

Melt fracture is a special phenomenon notexhibited by Newtonian fluids or by dilute to mod-erately concentrated polymer solutions. The vari-ous instabilities mentioned above may not allappear as the volumetric flow rate is increased,depending on the polymer used. Sharkskin is notalways present in experiments [3, 28]. On the otherhand, in certain cases before the appearance ofsharkskin, superficial scratching (or matting) mayappear in the form of stripes laid longitudinallyalong the axis and usually spaced equally aroundthe free surface of the extrudate [7, 12, 29]. The stick-slip instability is found only with linear polymers,such as high-density or linear low-density polyeth-ylenes [2, 27]. It has never been observed with long-branched polyethylenes [3, 27]. In some cases, a sec-ond distinct stick-slip region has been reported athigher shear rates [6, 7, 30]. The experiments ofRobert et al. [31] with linear high-density polyethyl-ene also showed that this secondary stick-slipregion may occur at flow rates just above, or with-in, the primary stick-slip region. Another strikingphenomenon reported recently by Fernández et al.[32], who carried out flow rate-controlled capillaryextrusion experiments with copolymers of etheneand propylene, is the split of the extrudate into twoseverely sharkskinned branches above a criticalflow rate.

As for the sharkskin instability, there is ageneral agreement that it is initiated at or nearthe die exit and is due to the relaxation of strains[4], rupture of the polymer at the exit [33], or localstick-slip at the exit due to polymer disentangle-ment [8]. Several reported causes of sharkskinare reviewed by Wang [8], Denn [3] and Baroneand Wang [34]. The experiments of Piau et al. [6,

7, 12] indicate that melt fracture is initiated in thedie entry region where unstable vortices appearand symmetry is lost. Unstable upstream vorticeshave also been observed by Pérez-González et al.[35]. More recently, Piau et al. [36] have shownexperimentally that using porous media at theentrance of an extrusion die delays the occur-rence of melt fracture. Migler et al. [37] usinghigh-speed optical velocimetry and videomicroscopy have demonstrated that two distinctmaterial failures during each sharkskin cycle takeplace. These failures resemble the rupture mech-anism proposed by Cogswell [33].

The mechanisms of extrusion instabilityhave long been the subject of controversy [2, 4,28, 38, 39]. The proposed theories on the shark-skin instability (see the book of Leonov andProkunin [5] and the recent reviews by Inn et al.[21], Wang [8], Graham [40], Venet and Vergnes[41], Rutgers and Mackley [42], and Denn [3]) arebeyond the scope of the present work, since theyinvolve microscopic phenomena near the diewall and/or the die exit which are not easily incor-porated into the numerical dynamic simulationsof the macroscopic extrusion problem. Numeri-cal simulations for the sharkskin instability arerestricted to steady-state calculations aiming atcorrelating the magnitudes of the stress concen-tration at the exit of the die to the experimentaldata on the onset of instability (see [44] and ref-erences therein). The main objective of this paperis to review numerical simulations based on twobasic mechanisms proposed for the stick-slip andthe gross fracture instabilities. These are [2, 3, 4]:

(i) slip (or adhesive failure) at the die wall, and(ii) constitutive or material (or bulk failure) insta-bilities.

Both mechanisms require a non-monotonic lawwhich leads to a nonmonotonic flow curve, i.e., aflow curve with a maximum and a minimum. Thefirst is based on the non-monotonicity of the slipequation which relates the wall shear stress tothe slip velocity, while the second is based on thenon-monotonicity of the constitutive equation,i.e., on the non-monotonicity of the shearstress/shear rate curve in simple shear andPoiseuille flows.

The rest of the paper is organized as fol-lows: Section 2 reviews experimental observa-tions associating extrusion instabilities with wall



slip and discusses theories of slip. Numerical sim-ulations of extrusion instabilities based on thecombination of slip with compressibility arereviewed in Section 3. Section 4 presents newnumerical calculations for the time-dependent,compressible extrudate-swell flow of a shear-thinning fluid with slip at the wall. These havebeen obtained using the Carreau model and arealistic non-monotonic slip law which is basedon the slip equations proposed by Hatzikiriakosand Dealy for a high-density polyethylene [13, 18].Section 5 reviews numerical simulations basedon the combination of slip with elasticity. Section6 is devoted on the constitutive instability mech-anism and relevant numerical simulations. Final-ly, the limitations of the three mechanisms andthe related numerical simulations are discussedin Section 7.

2 SLIP AT THE WALL: EXPERIMENTS ANDTHEORYUnlike Newtonian fluids, for which the no-slipboundary condition is practically valid regardlessof the fluid and the material composition of theboundary, polymer melts and solutions slip oversolid surfaces when the wall shear stress exceedsa critical value, which ranges from 0.1-1 MPa inmelts, but it can be significantly lower in polymersolutions [43 - 45]. An extensive compilation ofreferences concerning experimental observa-tions on wall slip in polymer solutions, theoreti-cal analyses on slip for polymer solutions andmelts, and experimental observations of wall slipin extrusion of polymer melts is provided by Joshiet al. [46]. Recent reviews of slip and its relationto polymer processing are those by Piau et al. [6]and Denn [3]. The latter author points out thatapparent slip is observed with some highlyentangled linear polymers, but not withbranched polymers or linear polymers with a suf-ficient number of entanglements per chain.

Many groups have reported indirectmeasurements of slip velocities, based on themacroscopic characterization of flow curves incapillary [10, 12, 18, 47 - 50], sliding-plate [43, 51 -54], and torsional rheometers [55]. The experi-mental fluids in these studies included melts oflow- and high-density polyethylenes, of silicones,of polyisobutylene and polystyrene, as well asconcentrated polystyrene solutions.

Direct measurements of nonzero fluidvelocities at stationary solid surfaces have been

reported by many researchers. Migler et al. [56]and Durliat et al. [57] used an evanescent wavefluorescence technique to measure velocities ofa polydimethylsiloxane melt in plane Couetteflow in a zone within 0.1 mm from the wall. Sim-ilarly, Archer and co-workers [45, 58] visualizedthe motions of micron-sized tracer particles inplane Couette flow of entangled polystyrenesolutions using an optical microscope. Finally,direct velocity measurements in extrusion exper-iments through capillary and slit dies have beentaken by Legrand et al. [59], Münstedt et al. [27],and Migler et al. [24] (also see references there-in). Legrand et al. [59] used a fluorescence tech-nique, together with a theoretical model thataccounts for diffusion effects, to determine thevelocity fields at a microscopic scale close to thewall, during extrusion of a high molecular weightpolydimethylsiloxane through a rough slit die.Münstedt et al. [27] investigated the flow behav-ior of linear low- and high-density polyethylenemelts in a slit die using a laser-Doppler velocime-ter of high spatial and temporal resolution. Theydid not detect any indications of measurable wallslip for long-chain branched polyethylene. Forlinear high-density polyethylene, however, theyreported pronounced slip velocities at low appar-ent shear rates before the appearance of pres-sure oscillations. In the stick-slip regime, theymeasured velocity fluctuations of the same fre-quency as that of the pressure oscillations butwith different shapes of amplitude. Beyond thisregime, the observed velocity profiles are nearlyplug, which is an indication for strong slip at thewall. Migler et al. [24] used depth-resolved stro-boscopic optical microscopy to measure thevelocity profiles of a linear low-density polyeth-ylene with and without a fluoropolymer additivein a transparent tube located at the exit of a twin-screw extruder. In the absence of the additive,they found that no slippage occurs and sharkskinis observed. Their measurements indicated thatthe additive migrates to the capillary wall whereit sticks and induces slippage between itself andthe polymer, which results in the elimination ofsharkskin.

It should be noted that, in the capillaryexperiments of Piau and co-workers [6, 12], extru-date distortions accompanied with pressureoscillations are observed in the absence of slip,indicating that upstream instabilities in the dieentrance region are, indeed, important. Similar



observations have been made by Pérez-Gonzálezet al. [35], in their capillary extrusion experimentswith branched polyethylenes; they attribute theinstability to the effect of compressibility on theupstream vortices. The recent experiments ofPiau et al. [36] showed that the occurrence ofmelt fracture is delayed when porous media areplaced at the the die entrance. As already men-tioned, earlier experiments of Piau et al. [6]showed that oscillating flow regimes can also beeliminated using slippery surfaces, whileupstream instabilities are still observed.

For a given die (i.e., material of con-struction), the slip velocity of linear polymers is afunction of temperature, pressure, shear stresshistory, and molecular parameters [19, 60]. At themolecular level, de Gennes [61, 62] developed awidely accepted slip model for the case of a pas-sive interface (no interaction between the poly-mer and the solid interface) by introducing thenotion of the extrapolation length. Togetherwith Brochard and co-workers [63 - 65], heextended the theory to distinguish a passiveinterface (no polymer absorption) from anabsorbing one. The extended theory predicts thetransition from a weak to a strong slip, in agree-ment with experimental observations [10, 48, 56,66]. Weak slip occurs at shear rates sufficientlylow so that bulk-wall entanglements are main-tained, while strong slip occurs at higher shearrates when the bulk and wall chains are effec-tively disentangled. This and two other slip the-ories, based on the assumption that slip is theresult of adhesive failure of the polymer chainsat the solid surface or on the existence of a lubri-cated layer at the wall, are discussed by Denn [3].

Most macroscopic slip equations pro-posed in the literature predict a power-law rela-tion between the shear stress at the wall and theslip velocity (at constant temperature) [13, 58, 67- 69]. As mentioned above, of particular interestare equations which exhibit maxima and mini-ma, such as the equations proposed by El Kissiand Piau [70] and Leonov [71]. The non-monoto-ne slip equation derived by Leonov [71], from asimple stochastic model of interface moleculardynamics for cross-linked elastomers, hasrecently been modified for polymer melts byAdewale and Leonov [9]. Hatzikiriakos and Dealy[13] give two different slip equations corre-sponding to the two positive-slope branches oftheir experimental flow curve, with which the

slip velocity is again multi-valued for some rangeof the wall shear stress.

Multivalued microstructural slip modelshave more recently been proposed by Mhetar andArcher [45], Yarin and Graham [72], and Wang andco-workers [8, 73], who developed theories similarto that of Brochard and de Gennes, and by Hill [74],who developed a continuum slip model based onadhesive failure. Slip models in which the slipvelocity depends not only on the shear stress butalso on the pressure or the normal stress have alsobeen proposed [13, 17, 18, 69, 74 - 77].

Linear stability analyses of simple shearand/or Poiseuille flows with slip along the wallshow that steady-state solutions correspondingto the negative-slope part of the slip equation arelinearly unstable [78 - 84]. As discussed in subse-quent sections, the combination of the nonmo-notonicity of the slip equation with either com-pressibility or elasticity leads to oscillatorysolutions at fixed volumetric flow rate, providedthat the latter is in the negative slope regime ofthe flow curve.

3 COMPRESSIBILITY COMBINED WITHNONLINEAR SLIPBased on the linear stability analysis of incom-pressible Newtonian Poiseuille flow with slip atthe wall, Pearson [85] pointed out that the com-bination of a nonmonotonic slip equation withmelt compressibility can lead to self-sustainedoscillations of the pressure drop and of the massflow rate, similar to those observed experimen-tally with the stick-slip instability. Georgiou andCrochet [86, 87] verified numerically the ideas ofPearson [85], by solving the time-dependent com-pressible Newtonian Poiseuille and extrudate-swell flows with slip along the wall. Their calcula-tions showed that steady-state solutionscorresponding to the negative-slope regime of theslip equation are unstable in agreement with thelinear stability analysis of Pearson and Petrie [78].Compressibility acts as the storage of elastic ener-gy that sustains the oscillations of the pressuredrop and the mass flow rate and generates waveson the extrudate surface, in the case of the extru-date-swell problem. These oscillations are similarto those observed with the stick-slip instability.The amplitude and the wavelength of the free-sur-face waves increase with compressibility. Bymeans of finite-element calculations, Georgiou[88] demonstrated that imposing periodic volu-




(1)

where vw is the relative velocity of the fluid withrespect to the wall, sw is the shear stress on thewall, vc2 is the maximum slip velocity at sc2, vminis the minimum slip velocity at smin, and m1, m2,and m3, are power-law parameters. The thirdbranch is the power-law slip equation suggestedby Hatzikiriakos and Dealy [13] for the rightbranch of their flow curve. The first branch resultsfrom the slip equation they propose for the leftbranch of their slope curve after substituting allparameters for resin A at 180˚C and taking thenormal stress as infinite. The intermediate, neg-ative-slope branch, which corresponds to theunstable region of the flow curve, is simply theline connecting the other two branches. The val-ues of all the slip equation parameters are givenin Tab. 1.

va v va v v va v v

w

wm

w c

wm

c w

wm

w

=

£ £

£ £

≥

Ï

Ì

ÔÔÔÔÔ

Ó

ÔÔÔÔÔ

21

3 2

2

1

3

2

0s

s

s

,

,

,min

min

metric flow rate at the inlet of the die leads to moreinvolved periodic responses and to free surfaceoscillations similar to those observed experimen-tally with the stick-slip instability.

The combination of compressibility andnon-linear slip represents the underlying mech-anism in various one-dimensional phenomeno-logical relaxation/oscillation models describingthe oscillations of the pressure and the volumet-ric flow rate in the stick-slip instability regime(see Refs. [89] and [90] and references therein).These models, include the reservoir region andtake into account the compressibility of the fluid.However, they require the calculation of certainparameters from experimental data and arebased on the assumption that the time-depen-dent solution follows the hysteresis loop result-ing from the nonmonotonic steady-state flowcurve. One-dimensional phenomenologicalmodels cannot predict the onset of instabilityand the wall slip. Moreover, they do not includethe extrudate region, and, therefore, they cannotrelate the distortions of the extrudate to thepressure oscillations. Kumar and Graham [91]used a one-dimensional relaxation model andmodified the slip equation used by Georgiou andCrochet [86], in order to include the pressuredependence of the slip velocity. They consideredboth generalized Newtonian and viscoelastic flu-ids. An interesting consequence of the depen-dence of slip velocity on pressure, which may beused for explaining the sharkskin instability, isthat the flow curve might not be multivaluedeven if the slip model is.

4 COMPRESSIBLE EXTRUDATE-SWELLFLOW OF A CARREAU FLUID WITH SLIPTwo simplifications in the simulations of Geor-giou and Crochet [86, 87] were the use of a sim-ple, arbitrary nonmonotonic slip equation andthe employment of the Newtonian constitutiveequation. To overcome these limitations, we con-sider the time-dependent, compressible axisym-metric extrudate-swell flow of a Carreau (i.e.shear thinning) fluid using a realistic slip equa-tion based on the experimental data of Hatzikiri-akos and Dealy [13, 18], and present preliminaryfinite-element calculations in the unstable flowregime.

We employ the non-monotonic slip lawshown in Fig. 2. This is a multi-valued slip modelgiven by

Table 1: Values of the slipmodel parameters.

Figure 2: The nonmonotonicslip law based on theexperimental data ofHatzikiriakos and Dealywith a high-density poly-ethylene at 180˚C [13, 18]

0.01

0.1

1

10

100

1000

0.1 1sw [MPa]

v w [c

m/s

]

smin sc2

Parameter Value

a1, (MPa)-m1 cm/s 125.09m1 3.23a2, (MPa)-m1 cm/s 1000m2 2.86a3, (MPa)-m1 cm/s 5.484*10-3

m3 -4.434

sc2, (MPa) 0.27smin, (MPa) 0.19vc2, cm/s 1.82vmin, cm/s 8.65



Hatzikiriakos and Dealy [13] measuredthe power-law parameters and the isothermalcompressibility, b, of the experimental polymermelt (HDPE), assuming that the density varies lin-early with pressure. Here we use the Carreaumodel with zero infinite-shear-viscosity whichgeneralizes the power-law model as follows:

(2)

where h is the viscosity, h0 =0.03 MPas is the zero-shear-rate viscosity, l is a time constant, n = 0.44is the power-law constant, and IId is the secondinvariant of the rate-of-deformation tensor d.The latter is defined as:

(3)

where v is the velocity vector, and the supersciptT denotes the transpose.

To non-dimensionalize the governingequations, we scale the lengths by the radius R,the velocity by the mean velocity V in the capil-lary, the pressure, p, and the stress tensor, ss, byh0 ln-1 Vn / Rn, the density, r, by the density atatmospheric pressure r0, and the time by R/V.With this scaling, the continuity and momentumequations for time-dependent, compressible,isothermal flow become

(4)

and

(5)

where all variables are now dimensionless, andRe is the Reynolds number defined as

(6).Re∫

- -r lh

02 1

0

R Vn n n

Ret

r∂∂

+ ◊—Ê

ËÁÁ

ˆ

¯˜̃ = —◊

v v v s

∂∂

+—◊ =r

rt

v 0

d v v= — —[ ]12

( ) + ( )T

h h l= +[ ] -

02 2 1 2

1 2( )( )

IIdn

The dimensionless stress tensor for the Carreaufluid is

(7)

where I is the unit tensor, and

(8).

Another dimensionless number is the compress-ibility number,

(9)

which appears in the equation of state,

(10)

The dimensionless form of the slip equation is

(11)

where

(12).

The volumetric flow rate is scaled by pR2V.The geometry and the boundary condi-

tions of the time-dependent, compressible,axisymmetric extrudate-swell flow are shown inFig. 3. Along the axis of symmetry we have theusual symmetry conditions. Along the wall, theradial velocity vanishes whereas the axial veloc-ity satisfies the slip Eq. 11. At the inlet plane, weassume that the radial velocity component, vr,vanishes, and that vz is given by

A a VR

iim m n

m n m n

i i

i i∫ =

-

-

hl

01

1 1 2 3( ) , ,

vA v vA v v vA v v

w

wm

w c

wm

c w

wm

w

=

£ £

£ £

≥

Ï

Ì

ÔÔÔÔÔ

Ó

ÔÔÔÔÔ

1

3 2

2

1

3

2

0s

s

s

,

,

,min

min

2

r= +1 Bp

B VR

n

n n∫ -

bhl

01

L∫lVR

s =- + +[ ] - —◊Ê

ËÁÁÁÁ

ˆ

¯˜̃̃˜

-p IId

nI d I v1 2 2 2

32 2 1 2

L ( )( )


(13)

where Q is the volumetric flow rate, and v*w is theslip velocity calculated by means of the slip equa-tion. At the outlet plane, we assume that srz andthe total normal stress, szz, are zero. Finally, onthe free surface, we assume that surface tensionis zero and impose vanishing normal and tangen-tial stresses. The unknown position h(z, t) of thefree surface satisfies the kinematic condition.

The steady-state solution of the stick-slip flow (i.e. for flat free surface) is taken as theinitial condition, and the free surface is releasedat t = 0. The governing equations along with theabove boundary conditions are solved usingstandard finite elements in space and the stan-dard fully-implicit scheme in time [87].

In Fig. 4, preliminary calculations in theunstable negative-slope regime of the flow curveare given for Q = 1.5, Re = 0.01, B = 1.541*10-4, L =349.2, A1 = 0.0583, A2 = 0.929 and A3 = 4.04. Allthe above dimensionless numbers correspond toexperimental conditions, the only exceptionbeing the Reynolds number which is rather high.Fig. 4a depicts the evolution and the develop-ment of self-sustained oscillations of the pres-sure gradient, - — P, at constant volumetric flowrate. In Fig. 4b, the evolution of the solution onthe flow curve plane is illustrated (M·

0 is the massflow rate at the outlet plane). The solid curve isthe non-monotonic, steady-state flow curve;after the initial oscillations, a limit cycle is even-tually reached. Finally, in Fig. 4c, a representativefree surface profile is illustrated. Small-ampli-tude, high frequency waves are obtained, whichstresses the need for adequately refined finite-element mesh and sufficiently small time step.The numerical simulations are thus verydemanding in CPU time. As indicated by thenumerical simulations, both the amplitude andthe wavelength of the free surface waves are

v v r v Qn

n r n v

nn

r Q

z z wn

w

n

= =+

−

+

++

−

+

+

( , , )* 11

3 1 2

3 11

1

1 1

1 1

( + )

*


reduced and the time-dependentflow problem becomes stiffer, asRe is decreased. Therefore,extremely refined meshes and

very small time steps must be employed in orderto simulate the experiments in which Re is about10-5. This work is currently under progress.

5 ELASTICITY COMBINED WITH NONLIN-EAR SLIPIn some recent papers [26, 79, 80, 81], Georgiouand co-workers demonstrated that viscoelastic-

Figure 3 (left): Geometryand boundary conditions forthe compressible extrudate-swell flow of a Carreau fluidwith slip at the wall.

Figure 4 (right): Transientextrudate-swell flow of aCarreau fluid with n = 0.44,Re = 0.01, B = 1.541*10-4 andQ =1.5: (a) Evolution of thepressure drop; (b) Trajectoryof the solution on the flowcurve plane; (c) Oscillationsof the free surface.


ity may replace compressibility and, when com-bined with nonlinear slip, can act as a storage ofelastic energy generating oscillations of the pres-sure drop similar to those observed experimen-tally in extrusion instabilities. They employed theOldroyd-B constitutive model which exhibits amonotonic steady-shear response in the absenceof slip. Therefore, their approach is fundamen-tally different from the constitutive instabilitymechanism in which the constitutive modelsexhibit a nonmonotonic (i.e. double-valued)steady shear response (see Section 6). They firststudied the time-dependent one-dimensionalshear flow and Poiseuille flow of an Oldroyd-Bfluid assuming that slip occurs along the fixedwall, following an arbitrary nonmonotonic slipequation which relates the shear stress to thevelocity at the wall. The stability of the steady-state solutions has been investigated by meansof a one-dimensional linear stability analysis andtime-dependent calculations, which revealedthe existence of unstable solutions in the nega-tive-slope regime of the flow curve, even thoughthe volumetric flow rate, Q, is kept constant. Fig-

ure 5a shows the neutral stability curves for dif-ferent values of the dimensionless Newtonianviscosity h2 on the plane of the elasticity numbere and the derivative F’ of the slip function F (sw)

= -F(vw). The elasticity number is defined as

(14)

where We = l V/R and Re = r VR/(h1 + h2) are theWeissenberg and Reynolds numbers, respective-ly, l, h1 and h2 are the usual material parametersof the Oldroyd-B model [81], and h1 + h2 is the zeroshear-rate viscosity. The slip function F is the arbi-trary non-monotonic function employed byGeorgiou and Crochet [86, 87]. This is subject tothe constraint F’(v–w) > -4, which ensures that thevolumetric flow rate for fully-developed roundPoiseuille flow is a monotonic function of thesteady-state slip velocity v–w. If F’(v–w) > 0, thesolution is stable independent of the elasticitynumber e. If F’(v–w) <0, the solution is unstable forvalues above the corresponding marginal stabil-ity curve. Hence, the Newtonian solutions arestable everywhere, and the interval of instabilitygrows as one moves from the Newtonian to theupper-convected Maxwell model. When anunstable steady-state is perturbed, periodic solu-tions are obtained, while the volumetric flow rateis kept fixed (as in the experiments). The ampli-tude and the period of the oscillations areincreasing functions of the elasticity [79, 80].

In [81], the numerical calculations havebeen extended to the time-dependent one- andtwo-dimensional axisymmetric Poiseuille andextrudate-swell flows using the elastic-viscoussplit stress (EVSS) method for the integration ofthe constitutive equation. The two-dimensionalcalculations showed that the pressure oscillationsobtained in the unstable regime are of smalleramplitude and period than in the one-dimension-al flow (Fig. 5b). They also revealed the existenceof a second mode of periodicity in space, i.e. in theaxial direction (Fig. 6a). The oscillations in the dieresult in small-amplitude oscillations of the extru-date surface, as shown in Fig. 6b.

More recently, Shore and co-workers[82 - 84] presented a phenomenological hydro-dynamic model describing the Poiseuille flow of

e =WeRe


Figure 5: (a) Linear stabilitydiagram for the Poiseuilleflow of an Oldroyd-B fluid

with slip at the wall andconstant volumetric flowrate; solutions above thecorresponding marginal

stability curve up to -F’(v–w) = 4 are unstable;

(b) Pressure-drop oscillations in two-dimen-sional Poiseuille flow and

comparisons with theone-dimensional calcula-

tions (broken line); Q = 0.45,Re = 1, We = 1, and h2 = 0.1.


a viscoelastic Maxwell fluid. They assumed thatthe polymer near the surface undergoes a first-order transition in conformation as the wallshear stress increases, which produces stick-slipbehavior and leads to a multivalued shearstress/slip velocity curve. Using linear stabilityanalysis and carrying out numerical calculationsof the linearized, incompressible, two-dimen-sional momentum equations under the assump-tion of periodic conditions in the direction of theflow, they demonstrated the existence of self-sustained oscillations, and related the latter tothe sharkskin instability [82 - 84]. Nevertheless,Denn [3] points out that the linearized stress-con-stitutive equation used by Shore and co-workersis not properly invariant.

Black and Graham [75 - 77] demonstratedthat the combination of elasticity with a monoto-nic slip model that takes into account the depen-dence of the slip velocity on the normal stressleads to short wavelength shear flow instabilitiesat sufficiently high shear rates, which is qualita-tively consistent with experimental observationsof the sharkskin instability. (If the slip velocitydepends only on shear stress, then the flow isalways stable.) Black and Graham employed a sim-ple phenomenological evolution equation for theslip process and carried out a linear stability analy-sis for the creeping, plane, incompressible Couetteflow of the upper convected Maxwell and thePhan-Thien-Tanner fluids.

6 CONSTITUTIVE INSTABILITIESWhile slip-induced instability is based on thenonmonotonicity of the slip equation, most ofthe proposed studies of constitutive instabilityare caused by the nonmonotonicity of the shearstress/shear rate curve. It should be pointed outthat, in both mechanisms, the flow curves forPoiseuille flow have similar forms, i.e. they exhib-

it a maximum fol-lowed by a minimum.Nonmonotone consti-tutive models includethe three-constantOldroyd model [92],the Doi-Edwardsmodel with a Rouserelaxation mode [93,94], the Johnson-Segalman model withan added Newtonian

(solvent) viscosity [95], the Giesekus model [95,96], and the KBKZ model with an extra viscousterm [97]. The nonmonotonicity of the constitu-tive equation is predicted by reptation theoriesfor highly entangled polymer melts and can beviewed physically as the separation of flow intotwo dynamic regimes with histories at low andhigh deformation rates, away from or near to thesolid boundary, respectively [93], which impliesthe existence of two characteristic times of theviscoelastic fluids. There is experimental evi-dence that wormlike surfactant solutions obey anonmonotone constitutive law [98 - 100]. More-over, Callaghan et al. [101] and Britton et al. [100]observed the spurt (stick-slip) instability inwormlike micelles using nuclear magnetic reso-nance velocimetry techniques.

A consequence of the nonmonotonicityof the constitutive models is that, in viscometric(one-dimensional) flows, such as the simpleshear and Poiseuille flows, multiple steady-statesolutions are allowed, within a certain range ofshear stress. In addition to the standard steady-state solution, there exist uncountably infiniteweak solutions which are characterized by anarbitrary number of shear rate discontinuities. InPoiseuille flow, these discontinuities are locatedover a very thin layer of the fluid close to the wallwhere the shear rate can be high. The thin high-shear-rate layer near the wall leads to an appar-ent slip, whereas the flow in the bulk is almostplug [95, 102, 103]. The apparent slip may practi-cally be considered as wall slip, and, thus, anoth-er possible physical interpretation of the stick-slip instability is provided through constitutiveinstabilities. The appearance of zones of differ-ent shear rates, also referred to as shear banding[104], has been shown experimentally for worm-like surfactant semidilute solutions which exhib-it a narrow spectrum of relaxation times [98, 99].


Figure 6: (a) Cells of radialvelocity contours inPoiseuille flow of anOldroyd-B fluid with slip atthe wall; (b) Detail of thefree-surface oscillations inextrudate-swell flow for Q =0.45, Re = 1, We = 1, and h2 =0.1 (the broken line corre-sponds to the steady-stateposition of the free surface).


Linear stability and/or time-dependentnumerical analyses of the shear and Poiseuilleflows of three-constant Oldroyd [92], Doi-Edwards [93, 94], Johnson-Segalman andGiesekus [95] fluids show that steady-state solu-tions with segments corresponding to thedecreasing portion of the shear stress/shear ratecurve may be unstable and that a flow curve hys-teresis is obtained between the two stablebranches. Various investigators considered thisinherent constitutive instability in order toexplain the spurt effects observed during thepressure-controlled capillary extrusion of certainpolymer melts. Kolkka et al. [95] solved the pres-sure gradient-controlled plane Poiseuille flow ofa Johnson-Segalman fluid with an added New-tonian viscosity and showed that the time-dependent solution jumps to one of the two sta-ble positive-slope branches of the steady shearstress/shear rate curve and thus the negative-slope branch is unattainable. Malkus et al. [105]obtained numerical time-dependent results forthe startup of axisymmetric Poiseuille flow (i.e.,at fixed volumetric flow rate) and found that thepressure drop becomes periodic for very smallvalues of the Newtonian viscosity. The persistentoscillations arise as a Hopf bifurcation to period-ic orbits as the volumetric flow rate is increasedbeyond a critical value. Similar results have beenobtained more recently by Aarts and van de Ven[106]. Solutions of oscillatory character have alsobeen reported by Yuan et al. [107] who solved thecreeping two-dimensional plane Poiseuille flowof a Johnson-Segalman fluid using aLagrangian/Eulerian simulation technique andperiodic boundary conditions at the inlet andoutlet planes of the flow domain. They did notmention, however, whether oscillations persistor decay.

Georgiou and co-workers [103, 108]solved numerically the one-dimensional time-dependent simple shear and plane Poiseuilleflows of a Johnson-Segalman fluid with addedNewtonian viscosity, and investigated the sta-bility of the steady-state solutions focusing onthe case where the steady-state shearstress/shear rate curve is not monotonic. Theirnumerical simulations showed that the steady-state solutions are unstable only if a part of thevelocity profile corresponds to the negative-slope regime of the standard steady-state shearstress/shear rate curve, in agreement with linear

stability analysis. Their time-dependent solu-tions are always bounded and converge to dif-ferent stable steady states, depending on the ini-tial perturbation. No regimes of self-sustainedoscillations (similar to those observed in thestick-slip extrusion instability) have been found,for values of the dimensionless solvent viscosityas low as 0.01. Oscillatory solutions are observedin certain cases but the oscillations are alwaysdecaying, in contrast to the numerical findings ofMalkus et al. [105, 109], and Aarts [106]. Similarresults demonstrating that the selection of thefinal steady-state is history dependent have alsobeen reported by Olmsted et al. [110]. The addi-tion of a diffusion (nonlocal) term to the John-son-Segalman model allows, under most condi-tions, for unique stress selection andhistory-independent banding profiles in theunstable regime [111, 112]. Note that, for very smallvalues of the solvent viscosity, the negative sloperegime of the constitutive equation becomeslarge and extends to very high shear rates, whichis not expected physically, since short-relaxationtime processes become important causing thestress to increase [104].

We believe that the osci l lationsobtained with non-monotonic constitutive mod-els are due to the inability of the numericalschemes to capture the shear rate discontinuitywhich tends towards the wall as the dimension-less solvent viscosity goes to zero. Malkus et al.[109] note that the oscillations might be an arti-fact of the numerical algorithm, i.e., a conse-quence of the system being slightly damped.Aarts [106] also admits that the oscillations of thevelocity gradient and of the stress componentshave a larger amplitude and ‘tend to become dis-continuous’ near the wall. Similarly, Malkus [105]reports that the velocity rises and falls nearlyperiodically at the thin layer near the boundary.Note also that neither Español et al. [113], whosolved the creeping one- and two-dimensionalplane shear flows of a Johnson-Segalman fluidwith added Newtonian viscosity using aLagrangian/Eulerian method, nor Greco and Ball[114], who solved the creeping circular Couetteflow of a Johnson-Segalman fluid using a varia-tional principle, have reported the appearance ofself-sustained oscillations in the negative-sloperegime of the flow curve. They have showninstead, that a steady-state solution is alwaysreached according to a selection mechanism for



the position of the discontinuity and for the totalshear stress. Other works in which no periodicsolutions have been found are those of DenDoelder et al. [89] and Ashrafi and Khayat [115].Den Doelder et al. [89] adjusted the one-dimen-sional relaxation-oscillation model of Molenaarand Koopmans [116] for a Johnson-Segalmanfluid with a no-slip boundary condition. Ashrafiand Khayat [115] examined the nonlinear stabili-ty of the one-dimensional plane Couette flow ofa Johnson-Segalman fluid and obtained a six-dimensional dynamical system using theGalerkin projection method. They found thattime-dependent solutions always converge tothe corresponding steady states either monoto-nically or oscillatorily, at low or high Reynoldsnumbers, respectively. Thus, the fact that non-monotonic constitutive equations allow the exis-tence of unstable steady states in some range ofthe shear rate cannot alone be used for explain-ing the stick-slip instability.

7 DISCUSSIONRecent work concerning the numerical modelingof the stick-slip and gross melt fracture instabil-ities in polymer melt extrusion has beenreviewed. The following three mechanisms ofinstability have been considered:

(i) combination of nonlinear slip with compress-ibility;(ii) combination of nonlinear slip with elasticity;and(iii) constitutive instabilities.

It is clear that only the first two mechanisms, i.e.those involving slip, lead to periodic solutions in(time-dependent) Poiseuille and extrudate-swellflows. Both compressibility and elasticity, in com-bination with a nonmonotonic slip law, act as astorage of elastic energy generating self-sus-tained oscillations of the pressure-drop inPoiseuille flow and waves on the extrudate sur-face in extrudate-swell flow, similar to thoseobserved in extrusion experiments. There is ageneral agreement that the compressibility-slipinstability mechanism is present in the stick-slipand gross melt fracture regimes. The elasticity-slip mechanism may also be present and appearsto generate small-amplitude high-frequencyoscillations which are qualitatively consistentwith the sharkskin instability [75 - 77], and are

superimposed to the much larger oscillationscaused by the combined effect of compressibili-ty and slip.

We believe that the osci l lationsobtained with non-monotonic constitutive mod-els are of numerical nature, caused by the inabil-ity of the numerical schemes to capture the shearrate discontinuities admitted by such models.This is not the only criticism to the constitutive-instability mechanism. Even though Larson [4]notes that gross fracture is most probably causedby constitutive instabilities and not by slip, sincethe onset of gross melt fracture is not signifi-cantly affected by changes in the die material, hepoints out that the main drawback of the consti-tutive instability approach is that it cannotaccount for the dependence of the oscillations onthe capillary die effects [4, 117]. The flow curvesobtained from a capillary rheometer are diame-ter-dependent in the sharkskin regime, whichindicates the crucial role of slip [96]. Hence theconstitutive instability mechanism may only beused for molten polymers exhibiting no-slip inthe sharkskin regime, such as nearly monodis-perse polybutadienes [16, 118]. The strong depen-dence of the upper branch of the apparent flowcurve on the capillary radius and the importanceof slip to the spurt instability are also noted byAdewale and Leonov [9]. The interfacial nature ofthe stick-slip transition, which depends on thesurface roughness and the surface energy of thecapillary die, is supported by the experiments ofof Piau et al. [6] and Wang and Drda [50]. On theother hand, Aarts and Van de Ven [106] note thatthe critical volumetric flow rate for the Johnson-Segalman fluid does scale with the radius, andthis is consistent with the experimental data ofEl Kissi and Piau [119].

Other criticisms on the use of the con-stitutive instability mechanism for explainingthe spurt instability for polymer melts or solu-tions have also been reported. Denn [2] remarkedthat the oscillations obtained by Malkus et al.[109] are not of the type of the slip induced per-sistent oscillations between the two stablebranches of the flow curve that are observedexperimentally with polydisperse LLDPE in thestick-slip instability regime. Adewale and Leonovhave shown that the Johnson-Segalman modelis unable to match simultaneously the experi-mental data for narrow-distributed polyethylenein the critical regime of shear flow leading to the



spurt phenomenon and in the critical regime ofelongational flow [120], and underline the factthat non-monotone flow curves have beenreported only for some materials with yield stressand not for common polymer melts or elas-tomers [9]. De Kee and Wissbrun [16] note thatthe constitutive instability theory does notaccount for the lack of spurt flow for most poly-mers. Finally, Wang et al. [117] note that theobserved hysteresis in spurt flow cannot beattributed to the constitutive properties of thepolymer that is being continuously extruded outand therefore does not re-visit itself during theexperiment, and that stick-slip oscillations canbe eliminated in the same stress range using anidentical die with its wall chemically treated toweaken polymer adsorption.

In the numerical simulations of Section4, some progress has been made with respect toour previous work on the compressibility-slipmechanism [86 - 88], by using the Carreau modeland employing a macroscopic slip equation sug-gested by the experimental data of Hatzikiriakosand Dealy for a HDPE [13, 18]. It should be noted,however, that many other important factorsmust be taken into account in order to achievemeaningful macroscopic simulations of extru-sion instabilities that will allow the prediction ofthe critical conditions for the onset of slip and/orunstable flow and will establish the relation ofthe extrudate characteristics, such as amplitudeand wavelength of the surface waves, to the flowconditions. These factors include [31, 81]:

(i) The use of more sophisticated dynamic slipequations describing the polymer/wall adhesionphysics and involving the dependence of slip onnormal stress (pressure) and history. These equa-tions can be tested by micro-macro or multiscalesimulations describing accurately the entangle-ment dynamics and the conformational changesnear the wall.

(ii) The use of appropriate viscoelastic constitu-tive equations for the polymer melts under study.

(iii) The inclusion of the reservoir region. In addi-tion to the fact that the compressibility effect ismore important in this region, the die-entry flowis crucial in the development (or the elimination)of melt fracture, as shown by Piau et al. [7, 12, 36]and Pérez-González et al. [35].

(iv) The development of special numerical methodsthat will overcome the convergence difficultiesencountered in the neighborhood of the stress sin-gularity at the exit of the die, as emphasized byDenn [3]. Such singular methods, however, requirethe knowledge of the strength and the angulardependence of the singularity [121], which areunknown for all viscoelastic constitutive equationsof interest. Convergence difficulties also occur inthe neighborhood of the re-entrant corner, whenthe reservoir region is included.

ACKNOWLEDGMENTSWe are grateful to Mr. Eric Brasseur (UniversitéCatholique de Louvain, Belgium), Prof. MarcelCrochet (Université Catholique de Louvain, Bel-gium), Dr. Marios Fyrillas (Hyperion SystemsEngineering, Cyprus), and Dr. Dimitris Vlas-sopoulos (Foundation for Research and Technol-ogy, Greece), for their contributions and com-ments. This research was partially supported bythe Cyprus Foundation for the Promotion ofResearch (PENEK Program 02/2000).

REFERENCES[1] Petrie CJS, Denn MM: Instabilities in Polymer

Processing, AIChE J. 22 (1976) 209-236.[2] Denn MM: Issues in Viscoelastic Fluid Mechan-

ics, Annu. Rev. Fluid Mech. 22 (1990) 13-34.[3] Denn MM: Extrusion Instabilities and Wall Slip,

Annu. Rev. Fluid Mech. 33 (2001) 265-287.[4] Larson RG: Instabilities in Viscoelastic Flows,

Rheol. Acta 31 (1992) 213-263.[5] Leonov AI, Prokunin AN: Nonlinear Phenomena

in Flows of Viscoelastic Polymer Fluids, Chapmanand Hall, London (1994).

[6] Piau JM, El Kissi N, Toussaint F, Mezghani A: Dis-tortions of Polymer Melt Extrudates and theirElimination using Slippery Surfaces, Rheol. Acta34 (1995) 40-57.

[7] El Kissi N, Piau JM: Stability Phenomena duringPolymer Melt Extrusion, in Rheology for PolymerMelt Processing, Piau JM, Agassant JF (Eds.), Else-vier Science, New York (1996) 389-420.

[8] Wang SQ: Molecular Transitions and Dynamicsat Polymer/Wall Interfaces: Origins of FlowInstabilities and Wall Slip, Adv. Polym. Sci. 138(1999) 227-275.

[9] Adewale KEP, Leonov AI: Modeling Spurt andStress Oscillations in Flows of Molten Polymers,Rheol. Acta 36 (1997) 110-127.

[10] Ramamurthy AV: Wall Slip in Viscous Fluids andInfluence of Materials of Construction, J. Rheol.30 (1986) 337-357.



[11] Kalika DS, Denn MM: Wall Slip and ExtrudateDistortion in Linear Low-Density Polyethylene, J.Rheol. 31 (1987) 815-834.

[12] Piau JM, El Kissi N, Tremblay B: Influence ofUpstream Instabilities and Wall Slip on MeltFracture and Sharkskin Phenomena during Sili-cones Extrusion through Orifice Dies, J. Non-Newtonian Fluid Mech. 34 (1990) 145-180.

[13] Hatzikiriakos SG, Dealy JM: Role of Slip and Frac-ture in the Oscillating Flow of HDPE in a Capil-lary, J. Rheol. 36 (1992) 845-884.

[14] El Kissi N, Piau JM: Stability Phenomena duringPolymer Melt Extrusion, J. Rheol. 38 (1994) 1447-1463.

[15] El Kissi N, Piau JM, Toussaint F: Sharkskin andCracking of Polymer Melt Extrudates, J. Non-Newtonian Fluid Mech. 68 (1997) 271-290.

[16] De Kee D, Wissbrun KF: Polymer Rheology,Physics Today 51 (1998) 24-29.

[17] Hill DA, Hasegawa T, Denn MM: On the Appar-ent Relation between Adhesive Failure and MeltFracture, J. Rheol. 34 (1990) 891-918.

[18] Hatzikiriakos SG, Dealy JM: Wall Slip of MoltenHigh Density Polyethylenes. II. CapillaryRheometer Studies, J. Rheol. 36 (1992) 703-741.

[19] Mackay ME, Henson DJ: The Effect of MolecularMass and Temperature on the Slip of PolystyreneMelts at Low Stress Levels, J. Rheol. 42 (1998)1505-1517.

[20] Wise GM, Denn MM, Bell AT, Mays JW, Hong K,Iatrou H: Surface Mobility and Slip of Polybuta-diene Melts in Shear Flow, J. Rheol. 44 (2000)549-567.

[21] Inn YW, Fischer RJ, Shaw MT: Visual Observationof Development of Sharkskin Melt Fracture inPolybutadiene Extrusion, Rheol. Acta 37 (1998)573-582.

[22] Ghanta VG, Riise BL, Denn MM: Disappearanceof Extrusion Instabilities in Brass Capillary Dies,J. Rheol. 43 (1999) 435-442.

[23] Amos SE, Giacoletto GM, Horns JH, Lavallée C,Woods SS: Polymer Processing Aids (PPA), in Plas-tic Additives, Hanser, New York (2001) 553-584.

[24] Migler KB, Lavallée C, Dillon MP, Woods SS, Get-tinger CL: Visualizing the Elimination of Shark-skin through Fluoropolymer Additives: Coatingand Polymer-Polymer Slippage, J. Rheol. 45(2001) 565-581.

[25] Achilleos E, Georgiou G, Hatzikiriakos SG: TheRole of Processing Aids in the Extrusion of Poly-mer Melts, J. Vinyl & Additive Technology 8(2000) 7-24.

[26] Fyrillas MM, Georgiou GC, Vlassopoulos D,Hatzikiriakos SG: A Mechanism for ExtrusionInstabilities in Polymer Melts, Polymer Eng. Sci.39 (1999) 2498-2504.

[27] Münstedt H, Schmidt M, Wassner E: Stick andSlip Phenomena during Extrusion of Polyethyl-

ene Melts as Investigated by Laser-DopplerVelocimetry, J. Rheol. 44 (2000) 413-427.

[28] Tanner RI, Walters K: Rheology: An HistoricalPerspective, Elsevier, Amsterdam (1998).

[29] Liang Y, Oztekin A, Neti S: Dynamics of Vis-coelastic Jets of Polymeric Liquid Extrudate, J.Non-Newtonian Fluid Mech. 81 (1999) 105-132.

[30] Li H, Hürlimann HP, Meissner J: Two SeparateRanges for Shear Flow Instabilities with PressureOscillations in Capillary Extrusion of HDPE andLLDPE, Polym. Bull. 15 (1986) 83-88.

[31] Robert L, Vergnes B, Demay Y: Complex Tran-sients in the Capillary Flow of Linear Polyethyl-ene, J. Rheol. 44 (2001) 1183-1187.

[32] Fernández M, Santamaria A, Muñoz-Escalona A,Méndez L: A Striking Hydrodynamic Phenome-non: Split of a Polymer Melt in Capillary Flow, J.Rheol. 45 (2001) 595-602.

[33] Cogswell FN: Stretching Flow Instabilities at theExits of Extrusion Dies, J. Non-Newtonian FluidMech. 2 (1977) 37-57.

[34] Barone JR, Wang SQ: Rheo-optical Observationsof Sharkskin Formation in Slit-Die Extrusion, J.Rheol. 45 (2001) 49-60.

[35] Pérez-González J, Pérez-Trejo L, de Vargas L,Manero O: Inlet Instabilities in the Capillary Flowof Polyethylene Melts, Rheol. Acta 36 (1997) 677-685.

[36] Piau JM, Nigen S, El Kissi N: Effect of Die EntranceFiltering on Mitigation of Upstream Instabilityduring Extrusion of Polymer Melts, J. Non-New-tonian Fluid Mech. 91 (2000) 37-57.

[37] Migler KB, Son Y, Qiao F, Flynn K: ExtensionalDeformation, Cohesive Failure, and BoundaryConditions During Sharkskin Melt Fracture, J.Rheol. 46 (2002) 383-400.

[38] Joseph DD, Joe Liu Y: Letter to the Editor: SteepWave Fronts on Extrudates of Polymer Melts andSolutions, J. Rheol. 40 (1996) 317-319.

[39] Kurtz SJ: Comment to the Editor, J. Rheol. 40(1996) 319-320.

[40] Graham MD: The Sharkskin Instability of Poly-mer Melt Flows, Chaos 9 (1999) 154-163.

[41] Venet C, Vergnes B: Stress Distribution aroundCapillary Die Exit: an Interpretation of the Onsetof Sharkskin Effect, J. Non-Newtonian FluidMech. 93 (2000) 117-132.

[42] Rutgers R., Mackley M: The Correlation of Exper-imental Surface Extrusion Instabilities withNumerically Predicted Exit Surface Stress Con-centrations and Melt Strength for Linear LowDensity Polyethylene, J. Rheol. 44 (2000) 1319-1334.

[43] Demarquette NR, Dealy JM: Nonlinear Vis-coelasticity of Concentrated Polystyrene Solu-tions: Sliding Plate Rheometer Studies, J. Rheol.36 (1992) 1007-1032.



[44] Chen YL, Larson RG, Patel SS: Shear Fracture ofPolystyrene Melts and Solutions, Rheol. Acta 33(1994) 243-257.

[45] Mhetar V, Archer LA: Slip in Entangled PolymerSolutions, Macromolecules 31 (1998) 6639-6649.

[46] Joshi YM, Lele AK, Mashelkar RA: Slipping Fluids:A Unified Transient Network Model, J. Non-New-tonian Fluid Mech. 89 (2000) 303-355.

[47] Barone JR, Plucktaveesak N, Wang SQ: InterfacialMolecular Instability Mechanism for SharkskinPhenomenon in Capillary Extrusion of LinearPolyethylenes, J. Rheol. 42 (1992) 813-832.

[48] Wang SQ, Drda PA: Superfluid-like Stick-SlipTransition in Capillary Flow of Linear Polyethyl-ene. 1. General Features, Macromolecules 29(1996) 2627-2632.

[49] Wang SQ, Drda PA: Superfluid-like Stick-SlipTransition in Capillary Flow of Linear Polyethyl-ene. 2. Molecular Weight and Low-TemperatureAnomaly, Macromolecules 29 (1996) 4115-4119.

[50] Wang SQ, Drda PA: Stick-Slip Transition in Cap-illary Flow of Linear Polyethylene. 3. Surface Con-ditions, Rheol. Acta 36 (1997) 128-134.

[51] Hatzikiriakos SG, Dealy JM: Wall Slip of MoltenHigh Density Polyethylenes. I. Sliding PlateRheometer Studies, J. Rheol. 35 (1991) 495-523.

[52] Reimers MJ, Dealy JM: Sliding Plate RheometerStudies of Concentrated Polystyrene Solutions:Large Amplitude Oscillatory Shear of a Very HighMolecular Weight Polymer in Diethyl Phthalate,J. Rheol. 40 (1996) 167-186.

[53] Reimers MJ, Dealy JM: Sliding Plate RheometerStudies of Concentrated Polystyrene Solutions:Nonlinear Viscoelasticity and Wall Slip of twoHigh Molecular Weight Polymers in TricresylPhosphate,’’ J. Rheol. 42 (1998) 527-548.

[54] Koran F, Dealy JM: Wall Slip of Polyisobutylene:Interfacial and Pressure Effects, J. Rheol. 43(1999) 1291-1306.

[55] Awati KM, Park Y, Wisser E, Mackay ME: Wall Slipand Shear Stresses of Polymer Melts at HighShear Rates without Pressure and Viscous Heat-ing Effects, J. Non-Newtonian Fluid Mech. 89(2000) 117-131.

[56] Migler KB, Hervet H, Leger L: Slip Transition of aPolymer Melt under Shear Stress, Phys. Rev. Lett.70 (1993) 287-290.

[57] Durliat E, Hervet H, Leger L: Influence of GraftingDensity on Wall Slip of a Polymer Melt on a Poly-mer Brush, Europhys. Lett. 38 (1997) 383-388.

[58] Archer LA, Larson RG, Chen YL: Direct Measure-ments of Slip in Sheared Polymer Solutions, J.Fluid Mech. 301 (1995) 133-151.

[59] Legrand F, Piau JM: Wall Slip of a Polydimethyl-siloxane Extruded through a Slit Die with RoughSteel Surfaces: Micrometric Measurement at theWall with Fluorescent-Labeled Chains, J. Rheol.42 (2000) 1389-1402.

[60] Hatzikiriakos SG: A Slip Model for Linear Poly-mers Based on Adhesive Failure, Intern. PolymerProcessing VIII(2) (1993) 135-142.

[61] De Gennes PG: “Mécaniques des Fluides: Écoule-ments Viscométriques de PolymèresEnchevêtrés”, C.R. Acad. Sci. Paris 288 (1979) 219-220.

[62] De Gennes PG: Wetting: Statics and Dynamics,Rev. Modern Phys. 57 (1985) 827-863.

[63] Brochard F, de Gennes PG: Shear-DependentSlippage at a Polymer/Solid Interface, Langmuir8 (1992) 3033-3037.

[64] Ajdari A, Brochard-Wyart F, de Gennes PG, LeiblerL, Viory LL, Rubinstein M: Slippage of an Entan-gled Polymer Melt on a Grafted Surface, PhysicaA 204 (1994) 17-39.

[65] Brochard-Wyart F, Gay C, de Gennes PG: Slippageof Polymer Melts on Grafted Surfaces, Macro-molecules 29 (1996) 377-382.

[66] Drda PA, Wang SQ: Stick-Slip Transition at Poly-mer Melt/Solid Interfaces, Phys. Rev. Let. 75(1995) 2698-2701.

67] Lau HC, Schowalter WR: A Model for AdhesiveFailure of Viscoelastic Fluids during Flow, J.Rheol. 30 (1986) 193-206.

[68] Denn MM: Surface-Induced Effects in PolymerMelt Flow, in Theoretical and Applied Rheology,Moldenaers P, Keunings R (Eds.), Elsevier SciencePublishers (1992) 45-49.

[69] Stewart CW: Wall Slip in the Extrusion of LinearPolyolefins, J. Rheol. 37 (1993) 499-512.

[70] El Kissi N, Piau JM: Écoulement de FluidesPolymères Enchevêtrés dans un Capillaire. Mod-élisation du Glissement Macroscopique á laParoi, C. R. Acad. Sci. Paris 309, Série II (1989) 7-9.

[71] Leonov AI: On the Dependence of Friction Forceon Sliding Velocity in the Theory of Adhesive Fric-tion of Elastomers, Wear 141 (1990) 137-145.

[72] Yarin AL, Graham MD: A Model for Slip at Poly-mer/Solid Interfaces, J. Rheol. 42 (1998) 1491-1503.

[73] Wang SQ, Drda PA, Inn YW: Exploring MolecularOrigins of Sharkskin, Partial Slip, and SlopeChange in Flow Curves of Linear Density Poly-ethylene, J. Rheol. 40 (1996) 875-897.

[74 Hill DA: Wall-Slip in Polymer Melts: A Pseudo-chemical Model, J. Rheol. 42 (1998) 581-601.

[75] Black WB, Graham MD: Wall-Slip and PolymerMelt Flow Instability, Phys. Rev. Lett. 77 (1996)956-959.

[76] Black WB, Graham MD: Effect of Wall Slip on theStability of Viscoelastic Plane Shear Flow, Phys.Fluids 11 (1999) 1749-1756.

[77] Black WB: Wall Slip and Boundary Effects in Poly-mer Shear Flows, Ph.D. Thesis, Department ofChemical Engineering, University of Wisconsin -Madison (2000).



[78] Pearson JRA, Petrie CJS: On the Melt-Flow Insta-bility of Extruded Polymers, Proc. 4th Int. Rheo-logical Congress 3 (1965) 265-282.

[79] Georgiou GC: On the Stability of the Shear Flowof a Viscoelastic Fluid with Slip along the FixedWall, Rheol. Acta 35 (1996) 39-47.

[80] Fyrillas M, Georgiou GC: Linear Stability Dia-grams of the Shear Flow of an Oldroyd-B Fluidwith Slip along the Fixed Wall, Rheol. Acta 37(1998) 61-67.

[81] Brasseur E, Fyrillas MM, Georgiou GC, CrochetMJ: The Time-Dependent Extrudate-Swell Prob-lem of an Oldroyd-B Fluid with Slip along theWall, J. Rheol. 42 (1998) 549-566.

[82] Shore JD, Ronis D, Piché L, Grant M: Model forMelt Fracture Instabilities in the Capillary Flowof Polymer Melts, Phys. Rev. Let. 77 (1996) 655-658.

[83] Shore JD, Ronis D, Piché L, Grant M: Theory ofMelt Fracture Instabilities in the Capillary Flowof Polymer Melts, Phys. Rev. E 55 (1997) 2976-2992.

[84] Shore JD, Ronis D, Piché L, Grant M: SharkskinTexturing Instabilities in the Flow of PolymerMelts, Physica A 239 (1997) 350-357.

[85] Pearson JRA: Mechanics of Polymer Processing,Elsevier, London (1985).

[86] Georgiou GC, Crochet MJ: Compressible ViscousFlow in Slits, with Slip at the Wall, J. Rheol. 38(1994) 639-654.

[87] Georgiou GC, Crochet MJ: Time-DependentCompressible Extrudate-Swell Problem with Slipat the Wall, J. Rheol. 38 (1994) 1745-1755.

[88] Georgiou GC: Extrusion of a Compressible New-tonian Fluid with Periodic Inflow and Slip at theWall, Rheol. Acta 35 (1996) 531-544.

[89] Den Doelder CFJ, Koopmans RJ, Molenaar J, Vande Ven AAF: Comparing the Wall Slip and theConstitutive Approach for Modelling SpurtInstabilities in Polymer Melt Flows, J. Non-New-tonian Fluid Mech. 75 (1998) 25-41.

[90] Ranganathan M, Mackley MR, Spitteler PHJ: TheApplication of the Multipass Rheometer to Time-Dependent Capillary Flow Measurements of aPolyethylene Melt, J. Rheol. 43 (1999) 443-451.

[91] Kumar KA, Graham MD: The Effect of Pressure-Dependent Slip on Flow Curve Multiplicity,Rheol. Acta 37 (1998) 245-255.

[92] Yerushalmi J, Katz S, Shinnar R: The Stability ofSteady Shear Flows of some Viscoelastic Fluids,Chem. Eng. Sci. 25 (1970) 1891-1902.

[93] McLeish TCB, Ball RC: A Molecular Approach tothe Spurt Effect in Polymer Melt Flow, J. Polym.Sci. B24 (1986) 1735-1745.

[94] Lin YH: Explanation for Stick-Slip Melt Fracturein Terms of Molecular Dynamics in PolymerMelts, J. Rheol. 29 (1985) 605-637.

[95] Kolkka RW, Malkus DS, Hansen MG, Ierley GR,Worthing RA: Spurt Phenomena of the Johnson-Segalman Fluid and Related Models, J. Non-Newtonian Fluid Mech. 29 (1988) 303-335.

[96] Vlassopoulos D, Hatzikiriakos SG: A GeneralizedGiesekus Constitutive Model with RetardationTime and its Association to the Spurt Effect, J.Non-Newtonian Fluid Mech. 57 (1995) 119-136.

[97] Aarts K, Van de Ven AAF: Transient Behaviourand Stability Points of the Poiseuille Flow of aKBKZ-Fluid, J. Eng. Maths. 29 (1995) 371-392.

[98] Decruppe JP, Cressely R, Makhloufi R, CappelaereE: Flow Birefringence Experiments Showing aShear-Banding Structure in a CTAB Solution, Col-loid Polym. Sci. 273 (1995) 346-351.

[99] Mair RW, Callaghan PT: Shear Flow of WormlikeMicelles in Pipe and Cylindrical Couette Geome-tries as Studied by Nuclear Magnetic ResonanceMicroscopy, J. Rheol. 41 (1997) 901-924.

[100] Britton MM, Mair RW, Lambert RK, Callaghan PT:Transition to Shear Banding in Pipe and CouetteFlow of Wormlike Micellar Solutions, J. Rheol. 43(1999) 897-909.

[101] Callaghan PT, Cates ME, Rofe CJ, Smeulders JBAF:The Spurt Effect Observed in Wormlike MicellesUsing Nuclear Magnetic Resonance Microscopy,J. Phys. II France 6 (1996) 375-393.

[102] Hunter JK, Slemrod M: Viscoelastic Fluid FlowExhibiting Hysteretic Phase Changes, Phys. Flu-ids 26 (1983) 2345-2351.

[103] Fyrillas M, Georgiou GC, Vlassopoulos D: Time-Dependent Plane Poiseuille Flow of a Johnson-Segalman Fluid, J. Non-Newtonian Fluid Mech.82 (1999) 105-123.

[104] Spenley NA, Yuan XF, Cates ME: NonmonotonicConstitutive Laws and the Formation of Shear-Banded flows, J. Phys. II France 6 (1996) 551-571.

[105] Malkus DS: Numerical Simulation of Shear-FlowDynamics of Three Simple Flows using the John-son-Segalman Model, RRC144, University of Wis-consin, Madison (1997).

[106] Aarts ACT: Analysis of the Flow Instabilities in theExtrusion of Polymeric Melts, Ph.D. Thesis, Eind-hoven University of Technology, The Nether-lands (1997).

[107] Yuan XF, Ball RC, Edwards SF: A New Approachto Modelling Viscoelastic Flow, J. Non-Newton-ian Fluid Mech. 46 (1993) 331-350.

[108] Georgiou GC, Vlassopoulos D: On the Stability ofthe Simple Shear Flow of a Johnson-SegalmanFluid, J. Non-Newtonian Fluid Mech. 75 (1998) 77-97.

[109] Malkus DS, Nohel JA, Blohr BJ: Oscillation in Pis-ton-Driven Shear Flow of a Non-NewtonianFluid, in Numerical Simulation of Non-Isother-mal Flow of Viscoelastic Liquids, Dijksamn JF,Kuiken GD (Eds.), Kluwer, Dordrecht (1993) 57-71.



[110] Olmsted PD, Radulescu O: Johnson-SegalmanModel with a Diffusion Term in Cylindrical Cou-ette Flow, J. Rheol. 44 (2000) 257-275.

[111] Yuan XF: Dynamics of Mechanical Interface inShear-Banded Flow, Europhys. Lett. 46 (1999)542-548.

[112] Lu CYD, Olmsted PD, Ball RC: Effects of NonlocalStress on the Determination of Shear BandingFlow, Phys. Rev. Lett. 84 (2000) 642-645.

[113] Español P, Yuan XF, Ball RC: Shear Banding Flowin a Johnson-Segalman Fluid, J. Non-NewtonianFluid Mech. 65 (1996) 93-109.

[114] Greco F, Ball RC: Shear-Band Formation in a Non-Newtonian Fluid Model with a ConstitutiveInstability, J. Non-Newtonian Fluid Mech. 69(1997) 195-206.

[115] Ashrafi N, Khayat RE: A Low-DimensionalApproach to Nonlinear Plane-Couette Flow ofViscoelastic Fluids, Physics of Fluids 12 (2000)345-365.

[116] Molenaar J, Koopmans RJ: Modelling PolymerMelt-Flow Instabilities, J. Rheol. 38 (1994) 99-109.

[117] Wang SQ, Barone JR, Yang X, Deeprasertkul C,Plucktaveesak N, Chai CK, Capaccio G, Hope PS:Flow Instabilities in Polymer Processing: Past

Controversies, Current Understanding, andFuture Challenges, in Proceedings of the XIIIthInternational Congress on Rheology, BindingDM, Hudson NE, Mewis J, Piau JM, Petrie CJS,Townsend P, Wagner MH, Walters K (Eds.), Vol.3 (2000) 164-166.

[118] Lim FJ, Schowalter WR: Wall Slip of Narrow Mol-ecular Weight Distribution Polybutadienes, J.Rheol. 33 (1989) 1359-1382.

[119] El Kissi N, Piau JM: The Different Capillary FlowRegimes of Entangled PolydimethylsiloxanePolymers: Macroscopic Slip at the Wall, Hystere-sis and Cork Flow, J. Non-Newtonian Fluid Mech.37 (1990) 55-94.

[120] Adewale KEP, Leonov AI: On Modeling SpurtFlows of Polymers, J. Non-Newtonian FluidMech. 49 (1993) 133-138.

[121] Georgiou GC, Schultz WW, Olson LO: SingularFinite Elements for the Sudden-Expansion andthe Die-Swell Problems, Int. J. Numer. MethodsFluids 10 (1990) 357-372.



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