research article role of nonmonotonic constitutive curves in extrusion...

Research ArticleRole of Nonmonotonic Constitutive Curves inExtrusion Instabilities

Yu Cao123 Wen-Jing Yang12 Xiao-Wei Guo12 Xin-Hai Xu12 Juan Chen12

Xue-Jun Yang12 and Xue-Feng Yuan234

1College of Computer National University of Defense Technology Changsha Hunan 410073 China2State Key Laboratory of High Performance Computing National University of Defense Technology ChangshaHunan 410073 China3School of Chemical Engineering and Analytical Science Manchester Institute of Biotechnology University of ManchesterManchester M13 9PL UK4National Supercomputer Centre in Guangzhou and Research Institute on Application of High Performance ComputingSun Yat-Sen University Guangzhou Guangdong 510006 China

Correspondence should be addressed to Xue-Feng Yuan xuefengyuannscc-gzcn

Received 10 September 2015 Revised 14 December 2015 Accepted 15 December 2015

Academic Editor Angel Concheiro

Copyright copy 2015 Yu Cao et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Flow instabilities of non-Newtonian fluids severely hamper the quality of products during various chemical processes such asfibre spinning extrusion and film blowing The origin of extrusion instability has been studied over many decades Howeverno consensus has been reached among the research community so far In this paper the possible cause of extrusion instabilitiesis explored using the finitely extensible nonlinear elastic conformation-dependent (FENE-CD) model with a nonmonotonicconstitutive curve Many well-documented experimental phenomena are reproduced in our simulations and it could be concludedthat the nonmonotonic constitutive curve plays an essential role in extrusion instabilities In addition the results imply that the dieexit singularity may generate or magnify oscillations

1 Introduction

A great deal of effort has been devoted to investigatinginstabilities of non-Newtonian fluids over the past decades[1] The instabilities occurring during the extrusion of poly-meric fluids are called extrusion instabilities The existenceof extrusion instability would generally reduce the qualityof chemical products Therefore the researchers have triedhard to characterize the nature of such instabilities Extrusioninstability is firstly observed as small amplitude short-wavelength periodic distortions on the surface which isnamed as sharkskin With the increase of flow rate thesurface profile of the extrudatemight alternate between roughand smooth which is called spurt or slip-stick At a higherflow rate the extrude surface becomes relatively smoothand experiences a long-wavelength distortion A furtherincrease of the flow rate would result in a gross distortion ofextrudate surface The last feature is usually called wavy flow

or gross melt fracture A typical flow curve about extrusioninstabilities with different flow regions can be seen in Figure 1[2ndash4] Note that flow instabilities are sensitive to the typeof polymer materials For instance the spurt effect may beobserved in linear low density polyethylene (LLDPE) butit would never occur in low density polyethylene (LDPE)Nonetheless gross melt fracture exists in almost all unstableextrusion cases

The origin of the extrusion flow instability is widelystudied since its first observation however it is still con-troversial at present Among various theories developedso far ldquoslip on the surface of the dierdquo [1 3 5 6] andldquoeffects of constitutive instabilityrdquo [7ndash9] are two well-knownexplanations Slip theory attributes the observation thatthere is a sudden jump of fluid velocity near the wall tothe fact that the polymer molecules fail to adhere to thewall once the shear stress near the wall exceeds a criticalvalue Therefore special slip-law is integrated into numerical

Hindawi Publishing CorporationInternational Journal of Polymer ScienceVolume 2015 Article ID 312839 8 pageshttpdxdoiorg1011552015312839

2 International Journal of Polymer Science

WavySpurtor orSmooth

stick-slip fractureShar

kski

n

gross melt

Shea

r stre

ss

A

B

Shear rate

Figure 1 Typical flow curve with different regions of extrusion in-stability

simulations However these methods usually fail to simulateasymmetrical oscillations such as wavy flow Constitutiveinstability theory regards the instability as intrinsic naturesof the polymer itself and describes it with a nonmonotonicconstitutive equation This point of view could be validatedby successful numerical simulation of spurt flow or wavyflow without slip boundary conditions imposed on the wallAs a matter of fact constitutive instabilities are due to thenonexistence of steady state solutions for the flow in thenegative slope regime Several comparisons have been madebetween these two theories [4 10]The constitutive instabilitytheory is criticized by some researchers [10] for its failurein reproducing some of the experimental phenomena butprobably it is the inaccuracy of the constitutive equation thatshould be blamed Besides it has made great achievements inthe simulation of shear banding [11ndash14] which is another kindof polymeric instabilities

When the behaviour of viscoelastic fluids under complexconditions is simulated the constitutive model plays a vitalrole in reproducing experimental results The FENE-CDmodel is one of the dumbbell models which is widely usedin viscoelastic fluid simulations In order to obtain a betterdescription of the slipping effect the upper-convected timederivative in the FENE dumbbell model is replaced by theJohnson-Segalman derivative [15] which is named hereafteras FENE-CD-JS It is generally used to simulate the behaviourof dilute and semidilute polymer solutions

Previous researches about the extrusion instability wereusually based on mathematical analysis of the constitutiveequation [16] or relied on simulations without the extrudateregion [3] These two approaches successfully explained andverified the possible reason of extrusion instability to someextent However their results did not take the extrudate partinto account The challenge in extrusion flow simulationsis how to track the severely distorted flow surface Thoughsome researchers also showed the feature of the extrudateoscillation [17] they revealed neither the flow behaviours inthe extrudate part nor the irregular oscillations as wavy flowobserved in experiments

This paper aims at studying the original cause of extrusioninstabilities The relationship between the wavy extrudate

flow outside the die and the oscillation flow inside is analysedBoth symmetrical and asymmetrical oscillations in and out ofthe die are observed with the modified FENE-CD-JS modelwhich indicates that the constitutive instability might be theoriginal cause of extrusion instabilities Moreover the resultsimply that the singularity at the die exit may generate ormagnify oscillations

The rest of the paper is organized as follows Details of theproposed method are presented in Section 2 The numericalresults and discussion are reported in Section 3 Conclusionsare drawn in Section 4

2 Methodology

In this section the mathematical formulation of FENE-CD-JS model and numerical algorithms used in this paper will bebriefly introduced

21 Mathematical Model The dynamics of the isothermaland incompressible viscoelastic fluids can be governed by thecontinuity equation and the momentum equation

120588(120597u120597119905

+ (u minus w) sdot nablau) = minusnabla119901 + nabla sdot 120591119901 + 120578119904Δu

nabla sdot u = 0

(1)

where 120588 is the fluid density u is the fluid velocity w is thevelocity of the cell [18 19] 119901 is the pressure and 120578119904 is thesolvent viscosity 120591119901 is the polymer contribution to the stresstensor which can be calculated from a constitutive equation

FENE-CD-JS model is a constitutive equation whichconsidered the behaviour of finite extensible polymer chainsIt could reproduce a range of experimental phenomena butit still remains a relatively simple mathematical form It canbe written as

nabla

A + 120585 (D sdot A + A sdotD) = minus119891 (tr (A))

120582(A minus I) (2)

where 120585 is the slip parameter which reflects the nonaffineresponse of polymer chains when they are deformed 120582 is thecharacteristic relaxation time D is the rate of deformationtensor which can be expressed as D = (nablau + (nablau)119879)2 Iis the unit tensor A = ⟨Q Q⟩ is a conformation tensorin which Q is the end-to-end vector of a polymer moleculeand the symbol ⟨sdot⟩ represents an ensemble average The timeevolution of the conformation tensor is given by

nabla

A =120597A120597119905

+ (u minus w) sdot nablaA minus (nablau)119879 sdot A minus A sdot nablau (3)

119891(tr(A)) is a function of the trace of the tensor A and itdefines a nonlinear coupling between the applied macro-scopic flowfield and the evolution of the polymermicrostruc-ture By introducing the dumbbell extensibility parameter119872and a new model parameter 120581 it can be written as

119891 (tr (A)) = 1198722

(1198722 minus tr (A)) (1 minus 120581 + 120581radic(12) tr (A)) (4)

International Journal of Polymer Science 3

The generalised no-linear spring force may be expressedas F = 119891(tr(A))119867Q where 119867 is related to the shortestrelaxation time1205820 and a constant drag coefficient 1205890The term[1minus120581+120581radic(12)tr(A)] can be regarded as the conformational-dependent friction coefficient which is based on the simplescaling friction law by taking hydrodynamic interactionsbetween fluid and isolated chain molecule into account[20 21] 120581 varies from 0 to 1 which is used to tune thespring stiffness Moreover the increase of 120581 would result ina decrease of the force F under the same tr(A) [22] Thepolymeric stress tensor can be written as

120591119901 = 119873 (⟨F Q⟩ minus I) =120578119901

120582119891 (tr (A)) (A minus I) (5)

where119873 is the number of polymers per unit volume and 120578119901is the polymer viscosity

For the convenience of comparison dimensionless vari-able Reynolds number Re and Weissenberg number Wi areintroduced as

Re =120588119880119871

120578

Wi = 120582119880

119871

(6)

in which 119871 is the characteristic length (the half width of thedie in this work) and 119880 is the characteristic velocity (theaverage velocity in this work)

22 Numerical Algorithm

221 Implementation of Free Surface Boundary ConditionThe boundary conditions imposed on the free surface arecomprised of the kinematic boundary condition and thedynamic boundary condition The kinematic boundary con-dition can be expressed as

u sdot n = w sdot n (7)

It means the relative speed between the fluid and the grid per-pendicular to the interface is zeroThis has been implementedby Tukovic and Jasak [23]

In terms of the dynamic boundary condition it requiresthat the traction exerted from the fluid to the atmosphereshould be equal to the atmospheric forces imposed onthe fluid Supposing the atmosphere pressure is zero theboundary condition is given by

120590 sdot n = (minus119901I + 2120573D + 120591119901) sdot n = (minus119901I + T) sdot n = 0 (8)

where 120590 is the total stress tensor namely the Cauchy stresstensor across the free surface and T = 2120573D + 120591119901 is theextra stress tensor Unlike in FEM systems this boundarycondition is hard to satisfy directly in FVM systems Noticingthat velocity is stored in the cell center and the usage ofthe traction boundary condition here is only to calculatethe velocity of cells which own the free surface boundarywe could introduce a balance force 120590lowast = minus120590 on the freesurface and zero elsewhere (including the internal field and

other boundary fields) to solve this problem It is known thatminusnabla119901 + nabla sdot 120591119901 + 120578119904Δ119906 = nabla sdot 120590 and then by applying theoperation nabla sdot 120590

lowast in Gauss method and adding it into theoriginal momentum equation the equation is actually solvedas

int119881

(nabla sdot 120590 + nabla sdot 120590lowast) 119889119881 = int

119878

119889S sdot (120590 + 120590lowast)

= sum

119886

120590119886 sdot n119886119878119886 minus 120590119887 sdot n119887119878119887 = sum

119886 =119887

120590119886 sdot n119886119878119886(9)

where 119881 is the volume of the cell S is the face area vectorand n is the unit normal vector of face area 119886 representsfaces around a cell and 119887 represents free surface faces It canbe seen that the influence of the total stress imposed on thefree surface is offset in this way which means the tractionboundary condition is satisfied equivalentlyThemomentumequation is actually solved in the form

120588(120597u120597119905

+ (u minus w) sdot nablau)

= minusnabla119901 + nabla sdot 120591119901 + 120578119904Δu + ⟨nabla sdot 120590lowast⟩Γ119891

(10)

where Γ119891 indicates the term is only applied on the freesurface boundary A proper pressure boundary condition onthe free surface is required when coupling the velocity andthe pressure in SIMPLE algorithm According to the forcebalance in the normal direction on the interface it can bederived as

119901 = n sdot T sdot n (11)

222 Preserving the Physical Soundness of Molecular Con-formation In the original FENE-CD-JS model there onlyexists a maximum extension of polymer conformation astr(A) could not exceed the maximum extensional length 119871However there should be a minimum length (or volume)as well to prevent tr(A) from becoming negative Thereforebefore solving the constitutive equations the state of currentpolymer conformation should be checked Modification oftr(A) by setting it back to the initial value would be done ifnumerical underflowoccurredThis kind ofmodificationwasproposed by Yang [24] and had achieved a great success in thesimulation of viscoelastic flows in complex situations

223 Other General Numerical Schemes Used in the Simu-lation In addition to the approaches described above thediscrete elastic split stress (DEVSS) numerical strategy whichis proposed by Matallah et al [25] is used to enforce thestability of numerical simulation Moreover underrelaxationalgorithm is also used with the relaxation parameters 03 and05 for 119901 and 119880 respectively

Second-order backward Euler method is used for timederivatives The discretized linear systems are solved byPCG (Preconditioned Conjugate Gradient method) withAMG (Algebraic Multigrid method) preconditioning usedfor pressure and BiCGstab (Biconjugate Gradient Stabilizedmethod) with Diagonal Incomplete Lower-Upper (DILU)preconditioning for velocity and stress [26 27]


120581 = 1 120585 = 007 120578s120578 = 19

120581 = 01 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 05 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

Extensional

Shear

10minus110minus2 102101100

120582 120576 or 120582

10minus4

10minus2

100

102

120578E120578

or120578s120578

Figure 2 Shear and extensional viscosities for FENE-CD-JS model

3 Results and Discussion

31 Model Analysis under Rheometric Flow Figure 2 showsthe rheological response in simple shear and uniaxial exten-sional flows for the FENE-CD-JS model Note that theviscosity is normalised by the zero-shear viscosity The upperlines are the uniaxial extensional viscosities while the lowerones are the shear viscosities

The shear viscosity of the FENE-CD-JS model undergoesa shear thickening or shear thinning In terms of the elon-gational viscosity it is bounded and would reach a plateauvalue at last It means the relationship between the strainstress and the strain rate is linear and the coefficient isconstant after a specified strain rate The solid lines andthe dashed lines in Figure 2 illustrate the parameter 120581 hasa significant influence on both the shear viscosity and theextensional viscosity Furthermore larger 120581 would show ahigher extensional viscosity In the case with a large 120581 (120581 = 1)the shear viscosity experiences a shear thickening at first andthen shear thinning The shear-thickening effect would beeliminated gradually as 120581 decreases and the curve of 120581 = 01

only shows shear thinningThe parameter 120585 is mainly used tocontrol the shear thinning A larger 120585 would result in a morepronounced shear thinning From the solid and the dottedlines in Figure 2 it can be seen that the shear-thickeningeffect is weakened or even disappeared with the increase of120585 The effect of the viscosity ratio on the extensional viscosityis almost negligible as shown by the solid lines and the dash-and-dot lines in Figure 2 The shear viscosity with a largerviscosity ratio is decreased due to relatively smaller solventviscosity

The FENE-CD-JSmodel reproduces various flow regimesas the rheological flow curves shown in Figure 3 The solidlines in Figure 3 depict the first normal stress and the shearstress of the FENE-CD-JS model with 120581 = 01 120585 = 01 and120578119904120578 = 19 under simple shear flow The first normal stressreaches a plateau and the shear stress increasesmonotonicallyas shear rate increases With a different model parameter

120581 = 01 120585 = 01 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

N1

Txy

10minus4

10minus2

100

102

104

N1

orTxy

120582

10minus110minus2 102100 101

Figure 3The normalised first normal stress and shear stress for theFENE-CD-JS model in a simple shear flow with various parameters

setting that is 120581 = 1 120585 = 007 and 120578119904120578 = 19 theprediction of the first normal stress is almost the same butthe absolute value is a little bit higher The shear stress variesnonmonotonically with the increase of the shear rate Withfurther decrease of the viscosity ratio 120578119904120578 = 150 theshear stress of the FENE-CD-JS model exhibits even morepronounced nonmonotonical behaviour as shown by thedotted line in Figure 3 Such a nonmonotonical behaviourof constitutive model could give three shear rate states underone shear stress hence it is capable of generating constitutiveinstabilities The constitutive instability in both Poiseuilleflow and die swell flow is studied in the following sectionsso as to shed light on the origin of extrusion instabilities

32 Basic Geometric Shape and Boundary Conditions Asketch of the geometry and the boundary conditions is shownin Figure 4 The half width of the die 119871 is chosen as thecharacteristic length The length of the upstream and thedownstream is set as 1198711 and 1198712 respectively They should belong enough to ensure the flow before the entry of the dieexit and further downstream at the outlet is fully developedIn this paper 119871 = 1 1198711119871 = 1546 and 1198712119871 = 16 Auniform inflow is imposed at the inlet No-slip boundarycondition is set at the wall and zero derivative is set at theoutlet and free surface for velocity The pressure at the outletis set as zero while on the free surface it is initially set aszero and would change dynamically during the simulationThe conformation tensor (A) at the input is set as the unittensor and the corresponding polymer tensor is zero

The computational mesh is shown in Figure 5 in which itcan be seen that uniform grids were regenerated Simulationspossess 81600 (680 lowast 120) cells in total with minimum gridrelative interval spaces at Δ119909min119871 = 0025 and Δ119910min119871 =

0017 Flow behaviours in a strait pipe flow are also studiedwith minimum relative interval spaces at Δ119909min119871 = 0025

and Δ119910min119871 = 00032


Inlet Outlet

No-slip Free surface

2L

L1 L2

u = w = 0

u middot n = w middot n120590 middot n = 0

p = 0120591p

uin

Figure 4 Die swell flow configurations

Figure 5 Typical computational grid

33 Results and Discussion In this section the relationshipbetween wavy flow and constitutive instability is investigatedby using the FENE-CD-JS model The rheometric behaviourof FENE-CD-JS model with different parameters has beenillustrated in Section 31 The model is firstly studied underPoiseuille flow to reveal possible flow instabilities After-wards die swell simulations are implemented to study therelationship between wavy flow and constitutive instabilitySince flows with monotonic shear stress (the solid line inFigure 3) are extremely stable in both Poiseuille flow andextrusion flow only the behaviour of flows with nonmono-tonic shear stress is analysed in the following sections

331 Simulations of the FENE-CD-JS Model in PoiseuilleFlow The effects of the constitutive instability are firstlyinvestigated in Poiseuille flow For the model parametersgiven as the dashed curves in Figure 3 the extrusion flowsunder nominal flow rates Wi = 01 Wi = 05 Wi = 1and Wi = 3 are investigated The actual local Wi numberat the wall which is evaluated by multiplying the shear rateof the cell most close to the wall and the relaxation timeafter simulation equals 03 77 89 and 136 respectivelyTherefore the first case is located in the smooth region whilethe latter three cases are located in the wavy region as shownin Figure 1 (the Wi of local bottom point B is correspondingto 72) The flow of the first case remains stable But underthe latter three conditions they begin to oscillate For thecase of Wi = 05 there are only very small oscillationsoccurring near the wall which almost does not affect thecentral flow while for Wi = 1 the flow changes graduallyfrom stable to asymmetrical oscillating The intensity imagesof tr(A) in time sequence are shown in Figure 6 It can be seenclearly that the oscillation occurs near thewall where polymermolecules are stretched most Numerous experimental andnumerical simulations have proved that elastic stress tendsto destabilize the fluid For example instabilities of pipe flowwere observed at a lower Re in viscoelastic fluids (about 2230or even lower) than those in Newtonian fluids (about 5800)[16 28]

Pipe flows under model parameters of the dotted lines inFigure 3 are studied under nominal flow rate of Wi = 01Wi = 04 Wi = 05 and Wi = 3 By calculating the shear

rates after the simulations the local Wi numbers at the wallare around 03 12 158 and 318 respectively The first twocases are located in the smooth region and display completelystable status as expected In contrast the latter two casesare located in the spurt and wavy flow regime and both ofthem exhibit oscillations Note that at Wi = 04 the shearrate near the wall has been very close to the local top (pointA in Figure 1) of the shear stress curve Thus after a slightincrease of the nominal flow rate the flownear thewall wouldrun into the spurt regime and experience a sudden changeof the flow rate Figure 7 illustrates that there is a suddenvelocity jump near the wall at Wi = 05 in contrast to thecase of Wi = 04 It is trustworthy that the sudden jumpof the shear rate is the direct cause of extrusion instabilityFurthermore the position of the discontinuity is very close tothe wall which has also been discovered in experiments thatthe polymer could form a thin layer near the wall with largevelocity gradient called the spurt layer [7] Similar simulationresults are also observed by Fyrillas et al [8] with Johnson-Segalman model By drawing all cells in the middle regionof the pipe and averaging over the period of last 5 relaxationtimes of each cell point a comparison of the relationshipbetween shear rate and shear stress is shown in Figure 8 Theerror bars in the horizontal direction and vertical directionare the standard deviation of shear rate and shear stressrespectivelyThemarkers represent the average values of eachcell during this period It can be seen clearly that the shearstress linearly increases with respect to the shear rate for thecase of Wi = 04 as it is located in the smooth region Incontrast for the case of Wi = 05 the shear rate near thewall might depart from the smooth flow regime and jumpto a much higher value at nearly the same shear stress Thisis due to the existence of negative region in its shear stressflow curve by which the spurt effect happens In additionthe length of the error bar for points with higher shear rateis much longer than the ones with lower shear rates whichmeans the spurt layer is particularly unstable For the casewithWi = 3 the flow is totally in chaos as shown in Figure 9This might explain why the polymer would distort severelyin gross melt fracture The figure exhibited by the diamondsin Figure 8 shows that the error bars for Wi = 3 (red bars)are much longer than that for Wi = 05 (blue bars) whichobviously indicates the oscillation is more serious

332 Simulations of the FENE-CD-JS Model in Die SwellThe effects of constitutive instability in free surface flow arefurther studied to reveal the possible cause of extrusion insta-bilities For flowswith nonmonotonic constitutivemodel (thedashed curves in Figure 3) flow behaviours of die swellprocess with different flow rates are given in Figure 10 Thelocal Wi number at the wall for the nominal flow rate ofWi = 08 is about 91 At Wi = 01 both flows in thedie and in the plug domain are stable With the increase offlow rate to Wi = 05 instabilities emerge and the extrudateoscillates with a small amplitude however the flow in the dieis still almost stable By increasing the flow rate further toWi = 08 periodical and symmetry slight oscillations beginto appear in the die and spurt appears near the wall of thedie In terms of the extrudate part it oscillates in the same


2 4 8 12 16 179

Trace A

(a) 119905 = 5

2 4 8 12 16 179

Trace A

(b) 119905 = 30

Figure 6 Intensity image of tr(A) in time sequence for FENE-CD-JS model with 120581 = 1 120585 = 007 and 120578119904120578 = 19 at Wi = 1

Wi = 04

Wi = 05

0

02

04

06

08

1

12

14

16

0 103 04 05 06 07 080201 09

y998400 = y2L

u998400=uU

Figure 7 Normalised velocity along normalised 119910 axis in pipe flowfor FENE-CD-JS model with 120581 = 1 120585 = 007 and 120578119904120578 = 150 atWi = 04 and Wi = 05

Wi = 04

Wi = 05

Wi = 3

minus8

minus6

minus4

minus2

0

2

4

Shea

r stre

ss

minus200 0 200 400minus400

Share rate

Figure 8 Shear stress against shear rate in pipe flow for FENE-CD-JSmodelwith 120581 = 1 120585 = 007 and 120578119904120578 = 150 atWi = 04Wi = 05and Wi = 3

manner as the flow in the die with a short wavelength Itseems that the wavelength of the periodical oscillation in andout of the die preserves the same order of magnitude whichindicates that there might exist a close relationship betweenthe instabilities in the inside and outside of the die Onepossible explanation is that the extrudate instability comesfrom the unstable flow in the die and the other one is that

0 0002 0004 0006

U magnitude

Figure 9 Simulation results (velocity profile) of pipe flow for FENE-CD-JS model with Wi = 3

the instability is caused by the singularity of the die exit andpropagates towards both sides To increase the flow rate toWi = 1 more severe distortions are observed Moreover theflow is no longer oscillating symmetrically but forms wavesThe shear rates at the wall vary drastically approximatelybetween 70 and 20 when approaching the die exit for bothconditions at Wi = 08 and Wi = 1 primarily because theyare located in the wavy or melt fracture region Concerningthe extrudate domain it is still smooth at the surface butdistorts with a larger amplitude and the oscillation is alsono longer symmetric compared with the previous resultsTheoscillation of flow in the die impels the movement of theextrudate Thus in this case oscillations in the die could beconsidered as the driving cause of the extrudate instabilityPalza et al [29] also agreed with the point that spurt beganat the entrance of the die and propagated from upstream todownstream along the die by analysis of physical experimentswith a long die Combined with the performance in thestraight pipe described in Section 331 instability in the diewhich is caused by constitutive instability is more likely tobe the origin of wavy flow The difference between straightpipe flow and extrusion flow is that extrusion flow is morelikely to be unstableThis could be attributed to the influencebrought by the die exit singularity and the extrudate partThe existence of singularity would enhance the shear rate tospurt region quickly Moreover once the extrudate begins tooscillate it would influence the upstream flow in return andprevent the flow from going back to the stable status

For the case with a deeper negative region (the dottedcurves in Figure 3) a specific simulation of die swell atWi = 05 is shown in Figure 11(c) It is found that thevelocity profile is significantly different from casesmentionedabove Although the plug flow region is extremely uniformand stable the flow in the die oscillates symmetrically Spurtoccurs near the wall periodically which might suggest thatspurt effect is the initial cause of instability and is reproducedby constitutive equations More interestingly the visual effect


(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1

Figure 10 Simulation results (velocity profile) for FENE-CD-JS model with different Wi 120581 = 1 120585 = 007 and 120578119904120578 = 19

(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269

Figure 11 Process of flow instability for FENE-CD-JS model (velocity profile) with 120581 = 1 120585 = 007 and 120578119904120578 = 150 at Wi = 05

with the time passing by looks like the position of the spurtis moving backward as displayed in Figures 11(c)ndash11(f) Fromthese time serial figures it could be seen clearly that the blueregion near the wall and the red region in the middle ofthe die are both moving backward from the die exit to theinput region which is a counterintuitive finding In order todemonstrate that this is not an illusion caused by frame ratethe generation of the first spurt appearance at the wall andfirst ellipsoid-like phenomena of the central flow are given inFigures 11(a) and 11(b) respectively From these two figuresit could be seen clearly that the instability firstly occurrednear the wall at the die exit and then affected the runningof the central flow Instabilities were apparently propagatingbackward It is reasonable that spurt firstly happened near thedie exit since polymers are always stretchedmost severely andthe shear rate is also the highest near the singularities Timeand mesh density convergence tests were implemented andthe results were the sameThis counterintuitive phenomenonmight suggest that one kind of instability is related with thesingularity at the die exit Considering the fact that slightoscillation was also observed in pipe flow it is reasonableto believe that the singularity generated or magnified theinstability in extrusion flows This point may be supportedby the experimental effect that periodical oscillations closeto the exit were found [30] and it owned higher relativepressure fluctuations than other upstream positions of the die[29] Although spurt phenomenon is seen in the simulationno obvious evidence has been proved that slip effects inexperiments are also an intrinsic property of constitutiveinstability and could be integrated into a constitutive equa-tion Moreover sharkskin was not seen in the extrudate

region in the simulation Therefore this result could onlybe used as an implication that sudden jump of velocity atwall could be reproduced by a proper constitutive equationbut not an obvious evidence that sharkskin is also originallycaused by constitutive instabilities

4 Conclusion

In this work the original cause of extrusion instabilitywas discussed systematically with the modified FENE-CD-JS model By carefully studying different flow regimes theconclusion may be drawn that nonmonotonic constitutivemodel reproduced wavy flow which supported the theorythat the constitutive instability may be the origin of theextrusion instabilities in spurt and wavy flow Some well-documented experimental phenomena (symmetrical andasymmetrical oscillations in and out of the die) were repro-duced in this paper which provided numerical evidences tosupport the constitutive instability theory Detailed analysisshowed that wavy flow may be attributed to the instabilityemerging in the die Moreover experiments have shownthat spurt instabilities may be generated or magnified atthe die exit and propagate to both sides In future workmore constitutive equations with worm-like rheological flowcurves can be tested so as to further investigate the influenceof nonmonotonic shear stress on extrusion instabilities

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


Acknowledgments

The authors would like to thank the University ofManchesterand N8 High Performance Computing for providing compu-tational resources and partial funding from the National Nat-ural Science Foundation of China under grant no 61221491no 61303071 and no 61120106005 and Open fund (no201503-01 no 201503-02) from State Key Laboratory of HighPerformance Computing

References

[1] R G Larson ldquoInstabilities in viscoelastic flowsrdquo RheologicaActa vol 31 no 3 pp 213ndash263 1992

[2] D S Kalika andMMDenn ldquoWall slip and extrudate distortionin linear lowdensity polyethylenerdquo Journal of Rheology vol 31no 8 pp 815ndash834 1987

[3] J D Shore D Ronis L Piche and M Grant ldquoTheory of meltfracture instabilities in the capillary flow of polymer meltsrdquoPhysical Review E vol 55 no 3 pp 2976ndash2992 1997

[4] K P Adewale and A I Leonov ldquoModeling spurt and stressoscillations in flows of molten polymersrdquo Rheologica Acta vol36 no 2 pp 110ndash127 1997

[5] W B BlackWall slip and boundary effects in polymer shear flows[PhD thesis] 2000

[6] M M Denn ldquoExtrusion instabilities and wall sliprdquo AnnualReview of Fluid Mechanics vol 33 pp 265ndash287 2001

[7] A Aarts Analysis of the flow instabilities in the extrusion ofpolymeric melts [PhD thesis] 1997

[8] M M Fyrillas G C Georgiou and D Vlassopoulos ldquoTime-dependent plane Poiseuille flow of a Johnson-Segalman fluidrdquoJournal of Non-Newtonian Fluid Mechanics vol 82 no 1 pp105ndash123 1999

[9] D S Malkus J A Nohel and B J Plohr ldquoDynamics ofshear flow of a non-Newtonian fluidrdquo Journal of ComputationalPhysics vol 87 no 2 pp 464ndash487 1990

[10] E Achilleos G Georgiou S G Hatzikiriakos and E AchilleosldquoOn numerical simulations of polymer extrusion instabilitiesrdquoApplied Rheology vol 12 no 2 pp 88ndash104 2002

[11] C Chung T Uneyama Y Masubuchi and H WatanabeldquoNumerical study of chain conformation on shear bandingusing diffusive rolie-poly modelrdquo Rheologica Acta vol 50 no9-10 pp 753ndash766 2011

[12] JM Adams SM Fielding and P D Olmsted ldquoTransient shearbanding in entangled polymers a study using the rolie-polymodelrdquo Journal of Rheology vol 55 no 5 pp 1007ndash1032 2011

[13] X-WGuo S Zou X Yang X-F Yuan andMWang ldquoInterfaceinstabilities and chaotic rheological responses in binary poly-mer mixtures under shear flowrdquo RSC Advances vol 4 no 105pp 61167ndash61177 2014

[14] X-W Guo W-J Yang X-H Xu Y Cao and X-J Yang ldquoNon-equilibrium steady states of entangled polymer mixtures undershear flowrdquoAdvances inMechanical Engineering vol 7 no 6 pp1ndash10 2015

[15] M W Johnson Jr and D Segalman ldquoA model for viscoelasticfluid behavior which allows non-affine deformationrdquo Journal ofNon-Newtonian FluidMechanics vol 2 no 3 pp 255ndash270 1977

[16] T C Ho and M M Denn ldquoStability of plane poiseuille flowof a highly elastic liquidrdquo Journal of Non-Newtonian FluidMechanics vol 3 no 2 pp 179ndash195 1977

[17] E Brasseur M M Fyrillas G C Georgiou and M J CrochetldquoThe time-dependent extrudate-swell problem of anOldroyd-Bfluid with slip along the wallrdquo Journal of Rheology vol 42 no 3pp 549ndash566 1998

[18] H Jasak ldquoDynamicmesh handling in openfoam inrdquo in Proceed-ings of the 47th AIAA Aerospace Sciences Meeting Including TheNew Horizons Forum and Aerospace Exposition Orlando FlaUSA January 2009

[19] H Jasak and Z Tukovic ldquoAutomatic mesh motion for theunstructured finite volume methodrdquo Transactions of Famenavol 30 no 2 pp 1ndash20 2006

[20] E J Hinch ldquoMechanical models of dilute polymer solutions instrong flowsrdquo Physics of Fluids vol 20 no 10 pp S22ndashS30 1977

[21] P G De Gennes ldquoCoil-stretch transition of dilute flexiblepolymers under ultrahigh velocity gradientsrdquo The Journal ofChemical Physics vol 60 no 12 pp 5030ndash5042 1974

[22] S C Omowunmi and X-F Yuan ldquoTime-dependent non-lineardynamics of polymer solutions in microfluidic contractionflow-a numerical study on the role of elongational viscosityrdquoRheologica Acta vol 52 no 4 pp 337ndash354 2013

[23] Z Tukovic andH Jasak ldquoAmovingmesh finite volume interfacetracking method for surface tension dominated interfacial fluidflowrdquo Computers amp Fluids vol 55 pp 70ndash84 2012

[24] W Yang Parallel computer simulation of highly nonlineardynamics of polymer solutions in benchemark flow problems[PhD thesis] 2014

[25] H Matallah P Townsend and M F Webster ldquoRecovery andstress-splitting schemes for viscoelastic flowsrdquo Journal of Non-Newtonian Fluid Mechanics vol 75 no 2-3 pp 139ndash166 1998

[26] M A Ajiz and A Jennings ldquoA robust incomplete choleski-conjugate gradient algorithmrdquo International Journal for Numer-ical Methods in Engineering vol 20 no 5 pp 949ndash966 1984

[27] J Lee J Zhang and C-C Lu ldquoIncomplete LU preconditioningfor large scale dense complex linear systems from electro-magnetic wave scattering problemsrdquo Journal of ComputationalPhysics vol 185 no 1 pp 158ndash175 2003

[28] K C Porteous and M M Denn ldquoLinear stability of planepoiseuille flow of viscoelastic liquidsrdquo Transactions of TheSociety of Rheology vol 16 no 2 pp 295ndash308 1972

[29] H Palza S Filipe I F C Naue and M Wilhelm ldquoCorrela-tion between polyethylene topology and melt flow instabili-ties by determining in-situ pressure fluctuations and applyingadvanced data analysisrdquo Polymer vol 51 no 2 pp 522ndash5342010

[30] J Barone and S-Q Wang ldquoFlow birefringence study of shark-skin and stress relaxation in polybutadiene meltsrdquo RheologicaActa vol 38 no 5 pp 404ndash414 1999

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


WavySpurtor orSmooth

stick-slip fractureShar

kski

n

gross melt

Shea

r stre

ss

A

B

Shear rate

Figure 1 Typical flow curve with different regions of extrusion in-stability

simulations However these methods usually fail to simulateasymmetrical oscillations such as wavy flow Constitutiveinstability theory regards the instability as intrinsic naturesof the polymer itself and describes it with a nonmonotonicconstitutive equation This point of view could be validatedby successful numerical simulation of spurt flow or wavyflow without slip boundary conditions imposed on the wallAs a matter of fact constitutive instabilities are due to thenonexistence of steady state solutions for the flow in thenegative slope regime Several comparisons have been madebetween these two theories [4 10]The constitutive instabilitytheory is criticized by some researchers [10] for its failurein reproducing some of the experimental phenomena butprobably it is the inaccuracy of the constitutive equation thatshould be blamed Besides it has made great achievements inthe simulation of shear banding [11ndash14] which is another kindof polymeric instabilities

When the behaviour of viscoelastic fluids under complexconditions is simulated the constitutive model plays a vitalrole in reproducing experimental results The FENE-CDmodel is one of the dumbbell models which is widely usedin viscoelastic fluid simulations In order to obtain a betterdescription of the slipping effect the upper-convected timederivative in the FENE dumbbell model is replaced by theJohnson-Segalman derivative [15] which is named hereafteras FENE-CD-JS It is generally used to simulate the behaviourof dilute and semidilute polymer solutions

Previous researches about the extrusion instability wereusually based on mathematical analysis of the constitutiveequation [16] or relied on simulations without the extrudateregion [3] These two approaches successfully explained andverified the possible reason of extrusion instability to someextent However their results did not take the extrudate partinto account The challenge in extrusion flow simulationsis how to track the severely distorted flow surface Thoughsome researchers also showed the feature of the extrudateoscillation [17] they revealed neither the flow behaviours inthe extrudate part nor the irregular oscillations as wavy flowobserved in experiments

This paper aims at studying the original cause of extrusioninstabilities The relationship between the wavy extrudate

flow outside the die and the oscillation flow inside is analysedBoth symmetrical and asymmetrical oscillations in and out ofthe die are observed with the modified FENE-CD-JS modelwhich indicates that the constitutive instability might be theoriginal cause of extrusion instabilities Moreover the resultsimply that the singularity at the die exit may generate ormagnify oscillations

The rest of the paper is organized as follows Details of theproposed method are presented in Section 2 The numericalresults and discussion are reported in Section 3 Conclusionsare drawn in Section 4

2 Methodology

In this section the mathematical formulation of FENE-CD-JS model and numerical algorithms used in this paper will bebriefly introduced

21 Mathematical Model The dynamics of the isothermaland incompressible viscoelastic fluids can be governed by thecontinuity equation and the momentum equation

120588(120597u120597119905

+ (u minus w) sdot nablau) = minusnabla119901 + nabla sdot 120591119901 + 120578119904Δu

nabla sdot u = 0

(1)

where 120588 is the fluid density u is the fluid velocity w is thevelocity of the cell [18 19] 119901 is the pressure and 120578119904 is thesolvent viscosity 120591119901 is the polymer contribution to the stresstensor which can be calculated from a constitutive equation

FENE-CD-JS model is a constitutive equation whichconsidered the behaviour of finite extensible polymer chainsIt could reproduce a range of experimental phenomena butit still remains a relatively simple mathematical form It canbe written as

nabla

A + 120585 (D sdot A + A sdotD) = minus119891 (tr (A))

120582(A minus I) (2)

where 120585 is the slip parameter which reflects the nonaffineresponse of polymer chains when they are deformed 120582 is thecharacteristic relaxation time D is the rate of deformationtensor which can be expressed as D = (nablau + (nablau)119879)2 Iis the unit tensor A = ⟨Q Q⟩ is a conformation tensorin which Q is the end-to-end vector of a polymer moleculeand the symbol ⟨sdot⟩ represents an ensemble average The timeevolution of the conformation tensor is given by

nabla

A =120597A120597119905

+ (u minus w) sdot nablaA minus (nablau)119879 sdot A minus A sdot nablau (3)

119891(tr(A)) is a function of the trace of the tensor A and itdefines a nonlinear coupling between the applied macro-scopic flowfield and the evolution of the polymermicrostruc-ture By introducing the dumbbell extensibility parameter119872and a new model parameter 120581 it can be written as

119891 (tr (A)) = 1198722

(1198722 minus tr (A)) (1 minus 120581 + 120581radic(12) tr (A)) (4)



120591119901 = 119873 (⟨F Q⟩ minus I) =120578119901

120582119891 (tr (A)) (A minus I) (5)



Re =120588119880119871

120578

Wi = 120582119880

119871

(6)











int119881


119878

119889S sdot (120590 + 120590lowast)

= sum

119886


119886 =119887

120590119886 sdot n119886119878119886(9)


120588(120597u120597119905



(10)


119901 = n sdot T sdot n (11)





120581 = 1 120585 = 007 120578s120578 = 19

120581 = 01 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 05 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

Extensional

Shear

10minus110minus2 102101100

120582 120576 or 120582

10minus4

10minus2

100

102

120578E120578

or120578s120578







120581 = 01 120585 = 01 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

N1

Txy

10minus4

10minus2

100

102

104

N1

orTxy

120582

10minus110minus2 102100 101






and Δ119910min119871 = 00032


Inlet Outlet


2L

L1 L2

u = w = 0


p = 0120591p

uin









2 4 8 12 16 179

Trace A

(a) 119905 = 5

2 4 8 12 16 179

Trace A

(b) 119905 = 30


Wi = 04

Wi = 05

0

02

04

06

08

1

12

14

16

0 103 04 05 06 07 080201 09

y998400 = y2L

u998400=uU


Wi = 04

Wi = 05

Wi = 3

minus8

minus6

minus4

minus2

0

2

4

Shea

r stre

ss

minus200 0 200 400minus400

Share rate



0 0002 0004 0006

U magnitude





(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1


(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269




4 Conclusion





Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





120591119901 = 119873 (⟨F Q⟩ minus I) =120578119901

120582119891 (tr (A)) (A minus I) (5)



Re =120588119880119871

120578

Wi = 120582119880

119871

(6)











int119881


119878

119889S sdot (120590 + 120590lowast)

= sum

119886


119886 =119887

120590119886 sdot n119886119878119886(9)


120588(120597u120597119905



(10)


119901 = n sdot T sdot n (11)





120581 = 1 120585 = 007 120578s120578 = 19

120581 = 01 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 05 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

Extensional

Shear

10minus110minus2 102101100

120582 120576 or 120582

10minus4

10minus2

100

102

120578E120578

or120578s120578







120581 = 01 120585 = 01 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

N1

Txy

10minus4

10minus2

100

102

104

N1

orTxy

120582

10minus110minus2 102100 101






and Δ119910min119871 = 00032


Inlet Outlet


2L

L1 L2

u = w = 0


p = 0120591p

uin









2 4 8 12 16 179

Trace A

(a) 119905 = 5

2 4 8 12 16 179

Trace A

(b) 119905 = 30


Wi = 04

Wi = 05

0

02

04

06

08

1

12

14

16

0 103 04 05 06 07 080201 09

y998400 = y2L

u998400=uU


Wi = 04

Wi = 05

Wi = 3

minus8

minus6

minus4

minus2

0

2

4

Shea

r stre

ss

minus200 0 200 400minus400

Share rate



0 0002 0004 0006

U magnitude





(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1


(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269




4 Conclusion





Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




120581 = 1 120585 = 007 120578s120578 = 19

120581 = 01 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 05 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

Extensional

Shear

10minus110minus2 102101100

120582 120576 or 120582

10minus4

10minus2

100

102

120578E120578

or120578s120578







120581 = 01 120585 = 01 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 19

120581 = 1 120585 = 007 120578s120578 = 150

N1

Txy

10minus4

10minus2

100

102

104

N1

orTxy

120582

10minus110minus2 102100 101






and Δ119910min119871 = 00032


Inlet Outlet


2L

L1 L2

u = w = 0


p = 0120591p

uin









2 4 8 12 16 179

Trace A

(a) 119905 = 5

2 4 8 12 16 179

Trace A

(b) 119905 = 30


Wi = 04

Wi = 05

0

02

04

06

08

1

12

14

16

0 103 04 05 06 07 080201 09

y998400 = y2L

u998400=uU


Wi = 04

Wi = 05

Wi = 3

minus8

minus6

minus4

minus2

0

2

4

Shea

r stre

ss

minus200 0 200 400minus400

Share rate



0 0002 0004 0006

U magnitude





(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1


(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269




4 Conclusion





Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Inlet Outlet


2L

L1 L2

u = w = 0


p = 0120591p

uin









2 4 8 12 16 179

Trace A

(a) 119905 = 5

2 4 8 12 16 179

Trace A

(b) 119905 = 30


Wi = 04

Wi = 05

0

02

04

06

08

1

12

14

16

0 103 04 05 06 07 080201 09

y998400 = y2L

u998400=uU


Wi = 04

Wi = 05

Wi = 3

minus8

minus6

minus4

minus2

0

2

4

Shea

r stre

ss

minus200 0 200 400minus400

Share rate



0 0002 0004 0006

U magnitude





(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1


(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269




4 Conclusion





Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




2 4 8 12 16 179

Trace A

(a) 119905 = 5

2 4 8 12 16 179

Trace A

(b) 119905 = 30


Wi = 04

Wi = 05

0

02

04

06

08

1

12

14

16

0 103 04 05 06 07 080201 09

y998400 = y2L

u998400=uU


Wi = 04

Wi = 05

Wi = 3

minus8

minus6

minus4

minus2

0

2

4

Shea

r stre

ss

minus200 0 200 400minus400

Share rate



0 0002 0004 0006

U magnitude





(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1


(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269




4 Conclusion





Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a) Wi = 01 (b) Wi = 05

(c) Wi = 08 (d) Wi = 1


(a) 119905 = 10 (b) 119905 = 12

(c) 119905 = 265 (d) 119905 = 267

(e) 119905 = 268 (f) 119905 = 269




4 Conclusion





Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




Acknowledgments


References






































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



research article role of nonmonotonic constitutive curves in extrusion...

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