Journal of Non-Newtonian Fluid Mechanics - TU/e

Journal of Non-Newtonian Fluid Mechanics 175-176 (2012) 25–37

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Journal of Non-Newtonian Fluid Mechanics

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Stability of fiber spinning under filament pull-out conditions

C. van der Walt a,b, M.A. Hulsen a,⇑, A.C.B. Bogaerds c, H.E.H. Meijer a, M.J.H. Bulters d

a Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsb Dutch Polymer Institute, P.O. Box 902, 5600 AX Eindhoven, The Netherlandsc Department of Biomedical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlandsd DSM Research, Materials Science Centre, P.O. Box 18, 6160 MD Geleen, The Netherlands

a r t i c l e i n f o

Article history:Received 31 October 2011Received in revised form 23 February 2012Accepted 11 March 2012Available online 24 March 2012

Keywords:Fiber spinningFilament pull-outStability analysisDraw resonance

0377-0257/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jnnfm.2012.03.005

⇑ Corresponding author. Tel.: +31 40 247 5081; faxE-mail address: [email protected] (M.A. Hulsen).

a b s t r a c t

Flow instabilities of wet-spun fibers in the form of draw resonance can result in radius fluctuations whichimpose limitations on either fiber quality or production rate. Also, at high winding velocities, if the fluidstrength is sufficiently high, a filament can be pulled out of the die. The force balance between the inte-grated normal stress that occurs during flow in the upstream region and the spinning force determinesthe position of the detachment point when the filament detaches from the spinneret wall. Filamentpull-out complicates the stability analysis in the sense that the upstream boundary conditions nowdepend on the position in the spinneret. In addition, the filament length varies in time. In this workwe extend the stability analysis on fiber spinning of a single isothermal filament by including theupstream pull-out condition in a one-dimensional fiber spin model using the eXtended Pom–Pom(XPP) constitutive model. Using a spectral method, our analysis incorporates the detachment point whichposition is allowed to vary according to the prescribed slope of the upstream integrated normal stress.Changing the slope S for a certain DR (draw ratio) and De (Deborah) number, the growth rate for eachS value can be determined. We compare the stability regions of our fiber spin model with pull-out, usingdifferent S values, to fiber spinning without pull-out. For low De values a finite value of S is destabilizingthe flow whereas for higher De values there is a range of S values that stabilizes the flow. For S = 0 weobtain fiber spinning with a constant force at pull-out, but the critical DR is greatly reduced. This is incontrast to fixed length fiber spinning of a Newtonian fluid at a constant force which is known to be sta-ble for any DR.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In wet-spinning, a polymeric fluid is pushed through a spin-neret die. After that, the extrudate is taken up downstream at ahigher velocity than the extrusion velocity and cooled in, e.g., awater bath to form a filament. By pulling harder on the filament,i.e., increasing the take-up velocity, the filament stretches andthe force in the filament increases. A periodic variation in filamentdiameter can occur beyond a critical draw ratio DR, the ratio be-tween the average extrusion and take-up velocity. This diameterfluctuation is a flow instability in the form of draw resonancewhich imposes limitations on the production rate. At high stretch-ing force, the filament can either break or pull out of the die, i.e.,the filament detaches from the die wall, if the fluid strength is suf-ficiently high (see Sridhar et al. [1] and Bulters et al. [2]). Pull-outrequires a fluid with high trouton ratio which is well known in iso-thermal solution spinning of spiders, for example Breslauer et al.[3], and high performance polymers and industrial wet-spinning

ll rights reserved.

: +31 40 244 7355.

of UHMwPE solutions as in Bulters et al. [2]. In wet-spinning theair gap is small and the residence time short hence, the area ofinterest, from the detachment point to the water bath, can be con-sidered isothermal. Fig. 1 illustrates this pull-out phenomenon.

The issue of draw resonance has been frequently addressed inliterature and mathematical models exist to describe the un-wanted instability of diameter fluctuations [6–13]. Draw reso-nance is a hydrodynamic instability and its onset can bepredicted by employing a linear stability analysis, see for exampleHyun et al. [14]. Determining the correct boundary conditions formathematical models is not always trivial. In Schultz et al. [15]the effect of take-up boundary conditions on the stability charac-teristics for a viscous dominated fiber in one-dimensional fiberspinning is discussed. According to Schultz et al. [15] a fiber dis-plays draw resonance instability if a constant take-up velocity con-dition is applied. However, this instability disappears if a constanttake-up force condition is imposed. The constant take-up forceboundary condition was also presented in Pearson et al. [7]. Renar-dy [16] revisited the above downstream boundary conditions forviscous flows. He additionally discussed the stability of prescribingthe cross-sectional filament area, with the aim of maintaining

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Fig. 1. Pull-out as induced by increasing the pulling force.

26 C. van der Walt et al. / Journal of Non-Newtonian Fluid Mechanics 175-176 (2012) 25–37

uniform filament thickness, at take-up as well as controlling thedrawing velocity in response to changes in the filament cross sec-tion or force. Different boundary conditions at the spinneret die forNewtonian and viscoelastic fiber spin models have also been inves-tigated. For example, Tsou et al. [17] studied the effect of die flowon the stability of isothermal melt spinning. They investigated theonset of draw resonance as a function of, among other things, thestresses in the die through different inlet stress boundary condi-tions. In Papanastasiou et al. [18] the application of an open bound-ary condition to the inlet stress is presented which avoidsspecifying an a priori unknown stress value at the spinneret exit.

Filament pull-out complicates the stability analysis in the sensethat the upstream boundary conditions are no longer constant butdepend on the deformation history of the polymeric fluid in thespinning die. Moreover, the upstream boundary conditions dependon the position of the detachment point as the filament length var-ies in time. The purpose of this paper is to implement the upstreampull-out condition in a one-dimensional fiber spin model for a vis-coelastic fluid. Our aim is to investigate to what extent the stabilityof pull-out on fiber spinning differs from the stability of a filamentwith a fixed length, i.e., with no pull-out included. Intuitively onewould expect that including filament pull-out will stabilize thespinning process since the change in filament length adjusts thespinning force to result in a process similar to spinning with a con-stant take-up force as in [7,15,16].

In this paper we present the one-dimensional stability analysisresults of an isothermal fiber spin model including filament pull-out. In Section 2 we describe the governing equations and bound-ary conditions for spinning with a fixed length as well as withfilament pull-out. In our model including filament pull-out, the fil-ament position at the take-up wheel is fixed but the filament ispermitted to move inside the die. We make use of the ArbitraryLagrangian Eulerian (ALE) moving-mesh formulation of Doneaet al. [19] where the moving axial filament coordinate is mappedonto a fixed spatial grid coordinate and the length perturbation,along with the state variables are solved as part of the eigenvalueanalysis. The ALE approach is different from the technique used inPearson et al. [20], Chang et al. [21], and Devereux at al. [22] to in-clude the movement of the solidification point in melt spinning,close to the take-up wheel, into their model. They define a pertur-bation in the freeze point on a fixed spatial regime where thefreeze point can be calculated from an explicit relation with thetemperature. Although the techniques differ, the ALE approachshould end in the same dynamics as the technique used in themoving solidification point problem. Next, in Section 3 the dimen-sionless system of equations is given, followed by the numericalmethod in Section 4 used to compute the eigenspectrum of thespinning model in order to study the stability. Results are pre-sented in Section 5 starting by determining the stability region ofa prescribed constant take-up velocity for the fixed length model.We also confirm that the fixed length model, using the eXtendedPom–Pom constitutive model of Verbeeten et al. [23], remainsstable as in the case of a Newtonian filament when prescribing aconstant take-up force and we refer to [15,16] for more details.

Finally, the differences between the fixed length and pull-out spin-ning models are discussed as related to their stability.

2. Model development

One-dimensional isothermal fiber spinning with the polymerfilament along the elongational direction z is considered here(see, for example, [6–10]). The velocity component uz(z, t) is as-sumed to be uniform over the cross-sectional filament area A(z,t) that is quantified by a varying function of the axial position zand time t. We assume the initial cross-sectional filament area A0

to be equal to the die hole. The initial velocity u0 of the filament de-pends on the throughput and the initial filament area. At a fixeddistance L0 from the spinneret, the filament is taken up at a con-stant end velocity uL, or spinning force F. The draw ratio DR, a de-gree of filament stretching in fiber spinning, is defined as the ratioof the velocities uL and u0. First, the one-dimensional model basedon the above defined geometry and boundary conditions is pre-sented before extending the model to include filament pull-out.

2.1. Fixed filament length

2.1.1. Governing equationsNeglecting inertia, gravity, and surface tension, the only force

considered to act on the filament segment is the drawing forcewhich reduces the equations for the conservation of mass andmomentum to (for a derivation see [6,24,25])

@A@tþ @

@zðAuzÞ ¼ 0; ð1Þ

@

@zðAðszz � srrÞÞ ¼ 0; ð2Þ

with srr and szz the extra stresses in the radial and filament direc-tion, respectively. In the momentum equation A(szz � srr) = Ffil isthe spinning force, which is independent of the axial position z,but can fluctuate in time. The problem is supplemented with a con-stitutive equation that relates the extra stresses, srr and szz, to thedeformation history of the polymer filament.

There are many possibilities for the constitutive relation, eachwith different properties depending on the material used, whichin turn will also influence the results of the stability analysis. How-ever, we believe that the results vary only quantitatively betweendifferent constitutive relations and therefore we choose theapproximate differential form of the eXtended Pom–Pom (XPP)model of Verbeeten et al. [23]. This choice is based on the quanti-tative agreement of the Pom–Pom constitutive equations and thedynamical experimental data in describing real viscoelastic melts(see [26,27]).

The XPP equations for the 1D fiber spin model are given by

@szz

@t� 2 szz þ

gkb

� �@uz

@zþ uz

@szz

@zþ 1

kbFszz þ

gkbðF� 1Þ

� �¼ 0; ð3Þ

@srr

@tþ srr þ

gkb

� �@uz

@zþ uz

@srr

@zþ 1

kbF srr þ

gkbðF� 1Þ

� �¼ 0; ð4Þ

with an auxiliary scalar valued function

F ¼ 2r emðK�1Þ 1� 1K

� �þ 1

K2 ; ð5Þ

and tube stretch

K ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ kbðszz þ 2srrÞ

3g

s: ð6Þ

Here we define r as the ratio of the relaxation times, i.e., r = kb/ks,with kb the relaxation time of the backbone tube orientation andks the relaxation time of the backbone tube stretch. The zero shear

(a)

(b)

Fig. 2. A schematic drawing of the two extreme cases from the mixed boundarycondition at take-up distance L0 for a single polymer filament in fiber spinning. Thefilament is either wound up at (a) with constant velocity uL or (b) with constantspinning force F.

Fig. 3. Schematic view of the pull-out of a single polymer filament with the totalfilament length defined as L(t) = L0 � d(t). The assumed initial steady state pull-outfilament length is L0.

C. van der Walt et al. / Journal of Non-Newtonian Fluid Mechanics 175-176 (2012) 25–37 27

rate material viscosity is indicated by g. The parameter m in Eq. (5) istaken as m = 2/q with q the number of dangling arms at both ends ofthe backbone. For the simulations described in this paper, we setr = 2 and q = 9 based on the work of Blackwell et al. [28].

The steady state spinning solution is obtained from the timeindependent model in Eqs. (1)–(6). We employ a linear stabilityanalysis method, described in Section 4, to study draw resonancein fiber spinning with a fixed filament length.

2.1.2. Boundary conditionsThe flow domain along the axial direction z for fiber spinning

with a fixed filament length is defined from z = 0 until z = L0. Evi-dently, the set of Eqs. (1)–(6) should be supplemented with suit-able conditions on the boundaries of the flow domain. We startwith the steady state boundary conditions. At z = 0 we prescribethe cross-sectional filament area, A(0) = A0. Further, we also pre-scribe the axial filament velocity, uz(0) = u0 and radial stress,srr(0) = srr0 at z = 0. The value assigned to srr0 is discussed in Sec-tion 5.1. For the downstream boundary condition, i.e., at z = L0,we impose a constant take-up velocity uz(L0) = uL = DR u0 to obtainthe steady state solution. In the stability analysis of the spinningprocess we impose a linear combination (mixed boundary condi-tion) of the velocity and force due to slip between the take-upwheel and the fiber (for more details we refer to Schultz et al.[15] and Renardy [16])

uzðL0; tÞ þ e FðtÞ ¼ constant ! ~uL þ eeF 0 ¼ 0: ð7Þ

with ~uL and eF 0 the linearly perturbed velocity and force at L0,respectively (see Section 4). Here e is an arbitrary value wheree = 0 represents spinning at a constant velocity and e =1 spinningat a constant force as the two extreme cases depicted in Fig. 2.

2.2. Filament pull-out

2.2.1. Governing equationsFor a relatively large spinning force, it is possible to create pull-

out when the filament detaches from the wall inside the die. Inpractice, the die shape consists of a contraction followed by a cap-illary region; we study the detachment of the filament inside thecapillary region. We assume a steady spinning solution having afixed length L0, starting from a point inside the die. Thus, we setz = 0 at the detachment point of the steady state filament.

In order to study the stability of the steady state solution, we as-sume that the detachment point can change in time, where we de-note the position of the detachment point with d(t), i.e., for d > 0the point will move downstream in Fig. 3. The filament lengthchanges as a result of the movement of the detachment pointand, therefore, the domain of the fiber along the fiber axis is de-fined on

dðtÞ 6 z 6 L0: ð8Þ

Hence, the total length of the filament is

LðtÞ ¼ L0 � dðtÞ; ð9Þ

with L0 the assumed initial steady state filament length as depictedin Fig. 3. Notice that the coordinate origin is taken at the positionwhere the steady state filament pulls away from the capillary walland that d(t) can be positive or negative.

To have a well defined computational system we make use of adeforming grid for the filament by employing the ArbitraryLagrangian Eulerian (ALE) moving-mesh formulation of Doneaet al. [19]. The z coordinate is mapped onto a fixed grid coordinaten 2 [0,1] by writing z in terms of the grid coordinate as follows

z ¼ nðL0 � dðtÞÞ þ dðtÞ or n ¼ z� dðtÞL0 � dðtÞ : ð10Þ

From the above projection we can write small variations in the zdirection as

dz ¼ ðL0 � dðtÞÞdn: ð11Þ

In order to obtain the ALE form of our model, the material velocityuz in the convective terms in Eqs. (1)–(4) has to be replaced with thevelocity relative to the grid. Hence, for example, the mass balanceequation after applying the product rule changes as follows:

@A@tþ @

@zðA uzÞ )

@A@t

��n

þ ðuz � ugridÞ@A@zþ A

@uz

@z: ð12Þ

Notice that @A/@tjn on the right-hand side of the double arrow is thegrid time derivative.

As can be seen in Eq. (12), a grid velocity is needed which wefind by differentiating Eq. (10) in time

ugrid ¼ _zjn ¼ ð1� nÞ _dðtÞ: ð13Þ

Here the dot refers to the derivative with respect to time. Also notethat

ugridðn ¼ 0Þ ¼ _dðtÞ and ugridðn ¼ 1Þ ¼ 0: ð14Þ

Lastly, by applying the product rule to the momentum equation, themodel given by Eqs. (1)–(4) can be rewritten in terms of the gridcoordinate n in the ALE description through the steps mentionedabove as:

(a)

(b)

Fig. 4. The steady state spinning force as a function of the detachment point for afixed draw ratio, DR.

Fig. 5. Development of the integrated normal stress as function of the capillarylength.


ðL0 � dÞ @A@tþ ½uz � _dð1� nÞ� @A

@nþ A

@uz

@n¼ 0; ð15Þ

ðszz � srrÞ@A@nþ A

@szz

@n� A

@srr

@n¼ 0; ð16Þ

ðL0 � dÞ @szz

@t� 2 szz þ

gkb

� �@uz

@nþ ½uz � _dð1� nÞ� @szz

@n

þ L0 � dkb

Fszz þgkbðF� 1Þ

� �¼ 0; ð17Þ

ðL0 � dÞ @srr

@tþ srr þ

gkb

� �@uz

@nþ ½uz � _dð1� nÞ� @srr

@n

þ L0 � dkb

Fsrr þgkbðF� 1Þ

� �¼ 0: ð18Þ

The auxiliary function F, together with the tube stretch, remainsthe same as in Eqs. (5) and (6).

2.2.2. Boundary conditionsIn fiber spinning with a fixed filament length, we focus on two

choices for the downstream boundary condition: a constant take-up velocity or constant force. However, for the pull-out modelwe only apply a constant velocity at the take-up wheel, uL. Thesame upstream boundary conditions are applied in the detachmentpoint as mentioned at the end of Section 2.1 for the fixed length fi-ber spinning model. Hence, in the detachment point we prescribethe cross-sectional filament area A0, velocity u0, and radial stresssrr0. The final boundary condition comes from the filament beingpulled out of the capillary. The detachment of the filament fromthe capillary is determined by balancing in the detachment posi-tion the normal stress in the capillary, generated by the upstreamextensional and shear flow (steady contribution), pushing the fluidtowards the wall, and the spinning force needed to obtain a pre-scribed draw ratio pulling the fluid away from the wall (see Sridharet al. [1] and Bulters et al. [2]). We will now discuss the boundarycondition for pull-out.

The steady state spinning force depends on the stretchinglength and therefore, on the position of the detachment point.The force becomes smaller if the point moves upstream, for a con-stant uL, as depicted in Fig. 4. From an integral momentum balanceincluding the spinning force, the force balance at the pull-out pointz = 0 is given by (see [2])

Fc ¼ 2pZ R0

0N1 þ

N2

2

� �r dr; ð19Þ

with R0 the capillary radius and N1 and N2 the first and second nor-mal stress difference from the upstream conditions, respectively.The normal stress N averaged over the cross-section of the capillarycan be written as

N ¼ Fc

A0: ð20Þ

Fig. 5 depicts schematically the development of the integrated nor-mal stress in the die from the contraction point. The intersection ofthe integrated normal stress and spinning force curves defines thesteady state pull-out position, see Fig. 6.

When studying the stability of the steady state we assume thatthe upstream profile will not change but the spinning force is fullydetermined by the dynamical state of the fiber. The force balancegiven by Eq. (19) now becomes

A0½szz � srr �ðn ¼ 0; tÞ ¼ Fcðz ¼ dÞ: ð21Þ

The linearization of the integrated normal stress at the capillarywall is as follows:

Fcðz ¼ dÞ ¼ A0dNdz

��0

dþ N0

!; ð22Þ

with A0N0 the integrated normal stress in the detachment pointfrom the assumed steady state, d the fluctuation in the position ofthe detachment point, and dN=dzj0 the averaged stress slope atz = 0. Substitution of the linearized normal stress and the prescribedradial steady state stress srr0 into Eq. (21) gives us a boundary con-dition for the axial stress

szzðn ¼ 0; tÞ ¼ dNdz

��0

dðtÞ þ szz0: ð23Þ

Here szz0 is the axial stress at z = 0 from the assumed steady state.The intersection angle between the two curves in Fig. 6 is of

considerable importance since we are interested in the sensitivityof the filament stability depending on the upstream stress profilewhen pull-out occurs in fiber spinning. Refer to the beginning ofSection 5.3 for more details about the intersection angle.

Summarized, the boundary conditions for filament pull-out are

n ¼ 0 : uzð0; tÞ ¼ u0; Að0; tÞ ¼ A0;

srrð0; tÞ ¼ srr0; szzð0; tÞ ¼dNdz

��0

dðtÞ þ szz0; ð24Þ

n ¼ 1 : uzð1; tÞ ¼ DR u0: ð25Þ

Fig. 6. Force balance between the integrated normal stress (red line) and the steady state spinning force (blue line) as defined in Fig. 4.


3. Dimensionless equations

The following dimensionless variables are defined

bA ¼ AA0; uz ¼

uz

u0; t ¼ u0

L0t;

szz ¼L0

g u0szz; srr ¼

L0

g u0srr; d ¼ d

L0:

ð26Þ

Substitution of the above variables into the pull-out model gi-ven by Eqs. (15)–(18) yields after omitting the hats

ð1� dÞ @A@tþ ½uz � _dð1� nÞ� @A

@nþ A

@uz

@n¼ 0; ð27Þ

ðszz � srrÞ@A@nþ A

@szz

@n� A

@srr

@n¼ 0; ð28Þ

ð1� dÞ @szz

@t� 2 szz þ

1De

� �@uz

@nþ ½uz � _dð1� nÞ� @szz

@n

þ 1� dDe

Fszz þ1

DeðF� 1Þ

� �¼ 0; ð29Þ

ð1� dÞ @srr

@tþ srr þ

1De

� �@uz

@nþ ½uz � _d ð1� nÞ� @srr

@n

þ 1� dDe

Fsrr þ1

DeðF� 1Þ

� �¼ 0; ð30Þ

with the auxiliary scalar valued function

F ¼ 2 r emðK�1Þ 1� 1K

� �þ 1

K2 ; ð31Þ

and tube stretch

K ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ De

3ðszz þ 2srrÞ

r: ð32Þ

The same set of dimensionless variables given in Eq. (26) can beused to obtain a dimensionless set of equations for the fixed lengthfiber spinning model in Eqs. (1)–(4). The model is similar to the onegiven above but with d and _d equal to zero since there is no pull-out.

The dimensionless boundary conditions for both models at n = 0are

uzð0; tÞ ¼ 1; Að0; tÞ ¼ 1; and srrð0; tÞ ¼ srr0: ð33Þ

Additionally, the pull-out model has a boundary condition for thedetachment point in terms of the axial stress

szzð0; tÞ ¼ SdðtÞ þ szz0: ð34Þ

At n = 1 the dimensionless boundary condition for the pull-outmodel is

uzð1; tÞ ¼ DR; ð35Þ

whereas, for the fixed length fiber spin model

uð1; tÞ þ bFðtÞ ¼ constant ! ~uL þ beF 0 ¼ 0: ð36Þ

From the dimensionless equations and supplemented boundaryconditions, we obtain four dimensionless groups. First, the Deborahnumber that is the ratio of the characteristic material time kb, to thecharacteristic process time L0/u0

De ¼ kbu0

L0: ð37Þ

The constant take-up velocity boundary condition at the end of thefilament gives us the draw ratio, a degree of filament stretching de-fined as

DR ¼ uL

u0: ð38Þ

The mixed velocity and force boundary condition at the take-upwheel used in the fixed length fiber spin model gives us the thirddimensionless group

b ¼ gA0

L0e: ð39Þ

Finally, from the integrated normal stress slope at the detachmentpoint inside the capillary for the pull-out condition

S ¼ dN0

dzL2

0

gu0: ð40Þ

4. Numerical method

In order to study the influence of the boundary conditions andpull-out on the stability of fiber spinning, we consider a linear sta-bility analysis of the derived model in Eqs. (27)–(32). First a steadystate analysis for the fixed length model is carried out since thisrepresents also the steady state we assume for the pull-out model.The stability analysis requires an expansion of the governing equa-tions around the computational fixed length steady state solutionin which only the first order terms of the perturbation variablesare used. Thus, neglecting higher order terms, we may expressthe variables

yðn; tÞ ¼ ½A;uz; srr; szz; L�ðn; tÞ; ð41Þ

as the sum of the steady state and the perturbed values

yðn; tÞ ¼ �yðnÞ þ �yðn; tÞ: ð42Þ

0 0.25 0.5 0.75 1−60

−50

−40

−30

−20

−10

0

z

τ rr

0 0.00125 0.0025 0.00375 0.005

−3

−2

−1

0

boundary layer

De = 10−4 DR = 20

Fig. 7. A small boundary layer appears at the beginning of the steady state radial stress profile when imposing a zero radial stress condition at z = 0. The number of spectralmodes used were N = 300.

0 4 8 12 16 20 24−4

−2

0

2

4

DR

τ rr(0

)

De = 1De = 0.55De = 0.1

Fig. 8. The steady state stress value srr at z = 0 obtained from an open boundarycondition simulation for increasing DR using three different De numbers. A positivesolution is produced when increasing the DR for De = 1.


Here, �y denotes the steady state values and �y denotes the perturba-tion values of the variables in Eq. (41) which we write as

�yðn; tÞ ¼ ~yðnÞert; ð43Þ

with r the complex eigenvalue.A discrete approximation of the eigenspectrum of the polymer

filament is obtained by discretizing the governing equations usinga Chebyshev-tau method as described in Gottlieb et al. [29]. Eachvariable in y is expanded into coefficients of a truncated Chebyshevpolynomial expansion of order N. Values for the fixed length steadystate are obtained by neglecting the time derivatives in Eqs. (27)–(32) with d and _d set to zero and solving the unknown coefficientsusing a Newton iteration scheme. If the perturbation variables aredefined and the necessary boundary conditions are imposed forboth the fixed length and pull-out model, a generalized eigenvalueproblem is obtained

K~y ¼ rM~y; ð44Þ

with ~y representing the unknown coefficients, K the (real) steadydifferential operator, and M the mass matrix. The problem is solvedusing a QZ-algorithm in Matlab. While our numerical scheme yieldsall the eigenvalues with the use of the QZ algorithm, it leads toindefinite eigenvalues. We remove these eigenvalues using themethod described by Goussis et al. [30]. Notice that r = a + ib isthe complex eigenvalue. If the real part a of the eigenvalue is posi-tive, the perturbations will grow in time indicating unstable flow,

whereas the imaginary part b determines if the instability is oscilla-tory. Furthermore, if b is non-zero we have a Hopf bifurcation.

First the linearized set of equations for the pull-out model ispresented. Substitution of Eqs. (42) and (43) into the governingEqs. (27)–(32) and retaining only the first order terms of theperturbation variables yields

reA þ �uzdeAdnþ dA

dn~uz � rð1� nÞ dA

dn~dþ A

d~uz

dnþ d�uz

dneA ¼ 0; ð45Þ

ð�szz � �srrÞdeAdnþ dA

dnð~szz � ~srrÞ þ

d�szz

dn� d�srr

dn

� �eAþ A

d~szz

dn� d~srr

dn

� �¼ 0; ð46Þ

r~szz þd�szz

dn½~uz � rð1� nÞ~d� þ �uz

d~szz

dn� 2 �szz þ

1De

� �d~uz

dn

� 2d�uz

dn~szz þ

F

De~szz þ 2

F0

De�szz þ

1De

� �~szz þ

F0

De�szz þ

1De

� �~srr

� F �szz þ1

De

� �� 1

De

� � ~dDe¼ 0; ð47Þ

r~srr þd�srr

dn½~uz � rð1� nÞ~d� þ �uz

d~srr

dnþ �srr þ

1De

� �d~uz

dn

þ d�uz

dn~srr þ

F

De~srr þ 2

F0

De�srr þ

1De

� �~srr þ

F0De

�srr þ1

De

� �~szz

� F �srr þ1

De

� �� 1

De

� � ~dDe¼ 0; ð48Þ

~szzð0; tÞ � S~dðtÞ ¼ 0; ð49Þ

with

F ¼ 2r emðK�1Þ 1� 1K

� �þ 1

K2 ; ð50Þ

F0 ¼m DeðK� 1Þ þ De

K

3K2 r emðK�1Þ � De3K4 ; ð51Þ

K ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Deð�szz þ 2�srrÞ

3

r: ð52Þ

We arrive at the same set of linearized equations for the fixedlength fiber spinning model as for the pull-out model above bysetting ~d ¼ 0 and replacing Eq. (49) with

~uzð1; tÞ þ b½Að1Þð~szzð1; tÞ � ~srrð1; tÞÞ þ eAð1; tÞð�szzð1Þ � �srrð1ÞÞ� ¼ 0:

ð53Þ

0 0.1 0.2 0.3 0.4−4

−3

−2

−1

0

z

τ rr

τrr(0) = 0

τrr(0) = −1/(6De)

τrr(0) = −1/(4De)

no τrr(0) prescribed

−0.4 −0.3 −0.2 −0.1 0−4

−2

0

2

4De = 0.1 DR = 10

real (σ)

imag

( σ)

Fig. 9. (Left) The steady state stress profile close to the spinneret exit by imposing different boundary condition values at the spinneret for the radial stress. (Right) Thecomputed eigenspectrum for the corresponding boundary condition values used in the left-hand side figure.


5. Results

In this section results are presented for the linear stability of thefiber spin model with a fixed length and filament pull-out. First, wediscuss the choice made for the stress boundary condition, srr0, atthe spinneret exit using the fixed length model. For the fixed lengthmodel, we focus on the effect different downstream boundary con-ditions have on the stability of the model. Lastly, the stability re-gion of the fiber spin model with filament pull-out is presentedand compared with the stability region of the fixed length model.Notice that all the results presented in this section are dimension-less unless indicated otherwise.

5.1. Stress boundary condition

In one-dimensional modeling, it is not obvious how the stressboundary condition at z = 0 has to be determined. The inlet stressvalues of srr and szz depend on the upstream processing conditionsin the spinneret as well as the tension imposed on the spinning lineby the take-up velocity. This boundary condition problem does notoccur when using Newtonian or generalized Newtonian fluids be-cause the viscous stresses are eliminated from the governing equa-tions by being explicitly related to the velocity gradient (seePapanastasiou et al. [18]). In literature, different initial stress con-ditions have been used in one-dimensional viscoelastic fiber spin-ning models. We refer to Tsou et al. [17] where the effect of dieflow on the stability of isothermal melt spinning is studied. Theyinvestigated the onset of draw resonance as a function of, amongother things, the stresses in the die through different inlet stressboundary conditions. We will now briefly state how different entryvalues for the radial stress boundary condition affect the stabilityof our fixed length fiber spin model.

� Prescribed radial stressIn Fisher et al. [8], Beris et al. [11], and Forest et al. [31],srr(0) = 0 has been used for the inlet stress condition. The originof the spinning distance coordinate z was chosen at the maxi-mum extrudate swell position. Setting the inlet radial stressequal to zero means that the pressure at the point of maximumextrudate swell is zero. When imposing this boundary conditionin our model we found that for De < 10�2 a discretizationdependent wiggly stress solution is generated. The wigglesextend to the take-up point of the filament. Increasing the num-ber of spectral modes until the solution appears more smooth,we see a small boundary layer at the beginning of the radial

stress profile where the stress immediately jumps to a solutionof the system and not the imposed inlet boundary conditionvalue as illustrated in Fig. 7. This boundary layer scales withthe De number. Fluids with De < 10�2 approaches the Newto-nian fluid regime for which the stress memory vanishes. Thus,adding a stress boundary condition over-constrains the model.� Open boundary condition

Papanastasiou et al. [18] presented the application of an openboundary condition to the inlet stress which avoids specifyingan a priori unknown stress value at the spinneret exit. For theirstudy they have used the upper-convected Maxwell and Gies-ekus constitutive equations. Due to the filament being stretchedwe expect negative radial stress values. However, applying anopen boundary condition, i.e., prescribing no inlet value, tothe stresses at the spinneret exit we were only able to obtaina negative and decreasing inlet stress srr0 for increasing DR atsmall De numbers. This seems to be an anomaly of the openboundary condition combined with our model. The inlet steadystate stress value srr0 for increasing DR is given in Fig. 8.

Comparing the radial stress solution profiles close to the spin-neret when imposing different inlet stress srr0 values and usingan open boundary condition, we found that the open boundaryimplementation tends to find the most smooth solution profile.For large De numbers (while fixing DR = 10) the initial stress srr0

obtained by the open boundary implementation seems to scale be-tween 1/(5De) for De = 0.1 and 1/De for De = 10. Increasing Detends to increase the inlet stress srr0 to zero, thus, it makes senseto prescribe srr0 = 0 for De P 10�2. For De < 10�2 we use the openboundary condition for the inlet stresses. The stability of the XPPmodel, when imposing the different steady state srr0 values, furthershowed to be the most stable for srr0 = 0. An example of the steadystate radial stress profiles and the corresponding eigenspectra canbe seen in Fig. 9. Based on the results above, we apply a prescribedsteady state inlet radial stress srr0 = 0 if De P 10�2, and an openboundary condition for the inlet stresses if De < 10�2.

In our stability analysis we put the perturbation on the inletstress srr(0,t) equal to zero. The critical DR obtained for De < 10�2

corresponds to DR = 20.21 for Newtonian fluids, see Section 5.2.1.Summarized, the initial stress boundary condition for our mod-

el is as follows:

� Newtonian regime (De < 10�2) : open stress boundarycondition,� Viscoelastic regime (De P 10�2) : radial stress equal to zero.

−40 0 40 80 120 160 200 240−240

−160

−80

0

80

160

240DR = 15 De = 1

real (σ)

imag

(σ)

stress : N modesstress : N−2 modes

spuriousmodes

2.3.

1.

0 0.25 0.5 0.75 1−2−1

012

x 1084

u z

0 0.25 0.5 0.75 1−2−1

012

x 1084

A

0 0.25 0.5 0.75 1−2−1

012

x 1086 σr = 197.81

z

τ zz

0 0.25 0.5 0.75 1−1

−0.50

0.51

x 1084 σi = 226.65

z

τ rr

N = 50 Chebyshev modes1.

0 0.25 0.5 0.75 1−1.4−0.7

00.71.4

u z

0 0.25 0.5 0.75 1−0.09

−0.0450

0.0450.09

A

0 0.25 0.5 0.75 1−140

−700

70140

σr = 3.7526

z

τ zz

0 0.25 0.5 0.75 1−0.1

−0.050

0.050.1

σi = 16.732

z

τ rr

2.

0 0.25 0.5 0.75 1−0.1

−0.050

0.050.1

u z

0 0.25 0.5 0.75 1−5

−2.50

2.55

x 10−3

A

0 0.25 0.5 0.75 1−8−4

048

σr = 0.81976

z

τ zz

0 0.25 0.5 0.75 1−4−2

024

x 10−3 σi = 36.562

z

τ rr

3.

Fig. 10. The computed eigenspectrum of the XPP fixed length fiber spin model for a DR = 15 and De = 1. The spurious modes in the eigenspectrum are removed when thestress variables srr and szz are expanded into N � 2 and the rest of the governing variables into N Chebyshev coeffients as indicated by the circles. The subgraphs show theamplitude of the mode of the governing variables for the modes numbered 1–3 with 1 being a spurious mode.

1 For interpretation of color in Figs. 1–17, the reader is referred to the web versionof this article.


5.2. Fiber spinning with a fixed length

In this section we present the stability regions for fiber spinningwith a fixed length using a mixed boundary condition at the take-up wheel. We focus the stability analysis on spinning with a con-stant take-up velocity (b = 0) and constant take-up force (b =1);see Eqs. (36) and (39).

5.2.1. Prescribed take-up velocityBefore we present the stability results for fiber spinning with

a constant take-up velocity we have to make sure that ourspectral code is able to resolve the eigenspectra. According toDawkins et al. [32] the Chebyshev-tau spectral method forapproximating eigenvalues of boundary value problems may pro-duce spurious eigenvalues with large positive real parts, evenwhen all true eigenvalues of the problem are known to have neg-ative real parts. By this we mean an eigenvalue which is in thecollection of eigenvalues but is not a solution to the non-approx-imated differential equations and thus an unwanted mode ofoperation. Possible reasons for the occurrence of these spuriousmodes are, for example, the way the boundary conditions are im-posed and incompatible truncation orders for the perturbationvariables.

If we change the number of Chebyshev modes N, we observethat there is a value of r that oscillates from a very large negativevalue to a very large positive one and vice versa. This indicatesthat next to the regular resolved eigenspectra, there are also spu-rious modes present. We found that by changing the expansion

order of the stress variables srr and szz to N � 2 and expandingthe rest of the governing variables to N Chebyshev coeffients,we were able to eliminate the spurious modes in our calculationswithout affecting the real eigenspectrum as can be seen inFig. 10. The subgraphs show the amplitude of the mode of thegoverning variables for the modes numbered 1–3. Only modes2 and 3 have converged and a complete period using 10 timeintervals of the solutions are presented while for mode 1, aspurious mode, just the solution in the first time interval ispresented. The solution in the first time interval for modes 2and 3 is given in red.1

Finally, we present the stability diagram resulting from thefixed length spinning model with a prescribed take-up velocityby setting b = 0 in Eq. (36) as shown in Fig. 11. For De < 10�2 thestability threshold is DR = 20.21, i.e., the real part of the first eigen-value passes from negative to positive at a critical draw ratio of20.21. Below this value all disturbances decay in time. Above thisvalue the first mode in the eigenfunction expansion grows andthe system is unstable to arbitrarily small disturbances. This criti-cal DR corresponds to the critical draw DR of draw resonance for aNewtonian fluid in isothermal and uniform tension spinning (see,for example, [12,8,16]). For De > 10�2 there is a slight increase inthe stability threshold before it goes down again and at De = 0.2the critical draw ratio decreases from DR = 19.8 to DR = 4.1 atDe = 10.

10−4 10−3 10−2 10−1 100 1010

5

10

15

20

25

De

DR

STABLE

UNSTABLE

Fig. 11. Stability diagram resulting from the fixed length fiber spin model with a prescribed velocity uL at the end of the filament. All DR and De numbers below the line arestable whereas all numbers above the line are unstable.

10−4 10−3 10−2 10−1 100 1010

10

20

30

40

50

60

70

De

DR

β = 0β = 0.5082β = 0.9650β = 1.8056

Fig. 12. Stability diagram resulting from the fixed length (XPP) fiber spinning model with a mixed boundary condition at the end of the filament. For b = 0 the stability curveof the fixed length spinning model with a constant end velocity, as in Fig. 11, is obtained.


5.2.2. Prescribed forceWe now consider the case where b ?1 which represents the

constant force boundary condition for fixed length fiber spinning.The value of b is positive since the fiber velocity at the take-upwheel decreases as the resistance of the fiber to elongate increases.In Schultz et al. [15] the mixed boundary condition, ~uzð1; tÞþbeF 0 ¼ 0, was used to investigate spinning with a constant take-up force for viscous filaments. Superposing small disturbances tothe steady state in the form of

yðz; tÞ ¼ �yðzÞ½1þ ~yðzÞ ert �; ð54Þ

where y represents the area and velocity, they found that the criti-cal DR increases linearly with b. Critical DR’s of up to 8000 could beobtained with their scaling of b showing that the critical constantforce value approaches infinity and, thus, stabilizing the filament.

Table 1Comparison of the critical DR’s from the viscous model of Renardy [16] with the XPPmodel for De = 10�6 when increasing b in the mixed velocity and force boundarycondition at L0.

Renardy [16] XPP: De = 10�6

�R DRcrit b DRcrit

0 20.218 0 20.217985 26.561 0.5082 26.56069

10 31.632 0.9650 31.6323620 40.137 1.8056 40.1374750 60.37 4.0646 60.37052

100 87.468 7.4550 87.46781

Renardy [16] also investigated the constant force boundary condi-tion for viscous fluids using the same form of adding small distur-bances to the steady state as in Eqs. (42) and (43). Our results ofthe XPP model for De = 10�6 compare well with the Newtonian re-sults in Renardy [16] where the constant force stability thresholdvalue increases. A comparison of the critical DR’s is given in Table 1where b scales as follows with the parameter �R used in the mixedboundary condition of Renardy [16]

b ¼ �R

3 ln DR: ð55Þ

The derivation of how we arrived at the above relation is given inAppendix A.

We also applied the mixed boundary condition with b P 0 to aviscoelastic filament. For the range of De numbers considered inthe XPP model we see that the critical DR increases for increasingb, as shown in Fig. 12, which has a stabilizing effect on the fiber.Thus, the same stability trend as in the viscous filament case wasfound for a viscoelastic filament when using the mixed boundarycondition with increasing b (infinite b gives a constant forceboundary condition).

5.3. Filament pull-out

In our pull-out model the detachment position of the filamentfrom the capillary is determined by balancing the normal forcefrom the capillary and the pulling force in the filament. Since wedo not model the polymer flow in the capillary we have to usean approximation for the unknown stresses in the spinneret. A

−0.03 −0.02 −0.01 0 0.01 0.02 0.039

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8De = 0.1 DR = 15

δ

simulationτ* = 9.3782 δ + 9.3743

−600 −400 −200 0 200 400 600−0.2

0

0.2

0.4

0.6

0.80.8

S

S = 9.3782

S = −9.3782

S = −120 damping

growing

real

(σ)

τ * = [τ

zz−τ

rr](δ

)

Fig. 13. (Left) Normal stress difference as function of the pull-out length d obtained from the steady state of the fixed length fiber spin model by changing the fiber lengthfrom the original length for each d in the figure. (Right) The growth rate r for De = 0.1 and DR = 15 while varying S; the optimum growth rate is reached at S = �120. De = 0.1and DR = 15 is stable for the fixed length filament.

−0.03 −0.02 −0.01 0 0.01 0.02 0.0320.4

20.6

20.8

21

21.2

21.4

21.6

21.8De = 1 DR = 10

δ

τ * = [τ

zz−τ

rr](δ

)

simulationτ* = 21.036 δ + 21.087

−400 −200 0 200 400−0.2

0

0.2

0.4

0.6

0.8

S

real

(σ)

S = 21.036S = −21.036

S = −100

growing

damping

Fig. 14. (Left) Normal stress difference as function of the pull-out length d obtained from the steady state of the fixed length fiber spin model by changing the fiber lengthfrom the original length for each d in the figure. (Right) The growth rate r for De = 1 and DR = 10 while varying S; the optimum growth rate is reached at S = � 100.De = 1 andDR = 10 is stable for the fixed length filament.


qualitatively averaged normal stress profile after the contraction inthe die is shown in Fig. 5 (see Bulters et al. [2] for details). As part ofthe stability analysis we use a linear profile for the integratednormal stress with a slope S as given by Eq. (40). If we look at

10−4 10−3 10−2 10−1 100 1010

5

10

15

20

25

30

De

DR

S = −40

S = −3000

S = −200

no pull−out

S = 0

Fig. 15. Stability diagram resulting from the fiber spin model without pull-out(solid black line) and with pull-out for different negative S values (dashed lines)using an open boundary condition for srr0 if De < 10�2 and setting srr0 = 0 ifDe P 10�2. For a chosen negative slope S all DR and De numbers below that line arestable, correspondingly all numbers above that line are unstable.

Fig. 5 we see that S is negative and its value increases uponapproaching the contraction point in the capillary. Changing theslope S for a certain DR and De number, the growth rate (real partof r) for each S value can be determined. In the right hand side ofFigs. 13 and 14 we observe that the stability depends on S. An opti-mum S value, �120 and �100 in Figs. 13 and 14, respectively, isreached which yields the most stable decay rate for the De andDR considered here. We further observe that the most unstablegrowth rate is at S = 0, with a converged critical DR. This meansit is not a singular point as might appear in the figures. Eventhough a positive stress slope is not found in practice duringpull-out, we have included positive S values in the figures to showhow the stability is affected.

We have used a wide range of S values to see how they influ-ence the stability of fiber spinning under the pull-out condition.The stability is of course determined around a steady state witha certain spinning force. From the intersection of the integratednormal stress and dynamical spinning force curves in Fig. 6 (seealso the boundary condition in Eq. (21)), we intuitively expect thatthe smaller the intersection angle, i.e., the closer slope S is to thespinning force slope, the more freedom of movement the pull-out length would have which could stabilize the process. In the lefthand side graph of Figs. 13 and 14, the normal stress difference asfunction of the pull-out length in the detachment point can be

10−4 10−3 10−2 10−1 100 1010

5

10

15

20

25

30

De

DR

no pull−out

S = 3000

S = 200

S = 0

S = 40

Fig. 16. Stability diagram using the same conditions as in Fig. 15 but with positivenormal stress slope values S (dashed lines).

−3000 −2000 −1000 0 1000 2000 30000

5

10

15

20

25

30

S

DR

De = 10−2

De = 10−1

De = 100

Fig. 17. Stability diagram of the pull-out model with DR vs. S for three differentdifferent De numbers. For a chosen De number all DR and S values below that lineare stable, correspondingly all values above that line are unstable.


seen. The stress difference was determined from the steady state ofthe fixed length fiber spin model. A filament length L0 was chosenand then for each d in the figure, L0 was changed accordingly beforerunning the steady state simulation again. Using the slope ob-tained from these two figures as input value for S in the linear sta-bility analysis, we observed that the spinning process remainedunstable for the chosen operating conditions. Thus, using a slopeS with the same value as the steady state spinning force slope,i.e., a zero intersection angle, in the pull-out model does not stabi-lize the model as intuitively expected.

Repeating the two examples in the right hand side of Figs. 13and 14 for different DR and De numbers, we can compare the sta-bility regions of the fiber spin model without and with pull-out fordifferent negative and positive slopes of S as shown in Figs. 15 and16, respectively. Note that for large positive, as well as negative Svalues, the freedom of movement for the pull-out length is reducedand the fixed length stability result with a constant take-up veloc-ity is approached. For relatively low De numbers we see that a fi-nite value of S is destabilizing the flow whereas for higher Denumbers there is a range of intermediate S values that stabilizesthe flow. We found the optimum S value to be around S = �200in the region between De = 0.1 and De = 0.2. Further, for S = 0 weobtain fiber spinning with a constant force at pull-out where thecritical DR is greatly reduced. This is in contrast to fixed length fi-ber spinning of a viscous fluid at a constant force which is knownto be stable for any DR; see Section 5.2.2. That S = 0 is the mostunstable slope can also be seen in Fig. 17 where the stability region

of the pull-out model is given with the DR as function of S for threedifferent De numbers.

6. Conclusions

The stability of fiber spinning with filament pull-out has beeninvestigated. First we determined the stability regions of what weconsidered to be the limiting cases for filament pull-out: fixedlength fiber spinning with a prescribed take-up velocity and force.We were able to confirm that for the constant force condition, thestability threshold value for our viscoelastic fiber spin model in-creases considerably which stabilizes the fiber as reported in Schultzet al. [15] and Renardy [16] for viscous filaments. By setting the inte-grated normal stress slope S in the pull-out model equal to zero, weobtain fiber spinning with a constant force at varying filamentlength with the lowest critical DR. This is in contrast to spinning aviscous fluid at a fixed length with a constant force which is stablefor any DR. Further, from our filament pull-out stability results, a fi-nite value of the stress slope S destabilizes the flow in fiber spinningfor relatively low De numbers. However, for larger De numbersthere is a region of negative intermediate values of S where pull-out improves the stability. For large S values, the freedom of move-ment for the pull-out length is reduced and the fixed length stabilityresult with a constant take-up velocity is approached.

Acknowledgement

This work is part of the Research Programme of the Dutch Poly-mer Institute (DPI), Eindhoven, The Netherlands, Project No. 694.

Appendix A. Linear velocity and force boundary condition

We start by recapitulating the derivation of the dimensionlesslinear velocity and force boundary condition used in Renardy[16]. A one-dimensional fiber spin model with cross-sectionalaveraging from the spinneret z = 0 until filament take-up z = L0 isconsidered. The force in the filament is assumed to be due toviscous effects and is given by

3gA@u@z: ðA:1Þ

Therefore, the requirement of constant force in the filament leads to

@

@zA@u@z

� �¼ 0: ðA:2Þ

The boundary conditions are the given filament area and velocity atz = 0

Að0; tÞ ¼ A0 and uð0; tÞ ¼ u0; ðA:3Þ

and a prescribed velocity at z = L0

uðL; tÞ ¼ uL: ðA:4Þ

Using the constant force requirement together with the mass bal-ance and the boundary conditions given above, the steady statesolution is as follows:

usðzÞ ¼ u0ekz and AsðzÞ ¼ A0e�kz; ðA:5Þ

where

ekL ¼ uL

u0¼ DR; ðA:6Þ

is the draw ratio.For the stability analysis, exponentially varying perturbations

on the state variables are considered

uðz; tÞ ¼ �uðzÞ þ ~uðzÞekt and Aðz; tÞ ¼ AðzÞ þ eAðzÞekt : ðA:7Þ


Here the tilde represents the perturbation on the steady state vari-ables indicated by the bar. Linearizing the force in Eq. (A.2) aroundthe steady state yields

ddz

Ad~udzþ eA d�u

dz

� �¼ 0: ðA:8Þ

Using the spacial transformation x = ekz the steady state solutionbecomes

�uðxÞ ¼ u0x and AðxÞ ¼ A0

x: ðA:9Þ

The linearized force then takes the form

ddx

A0d~udxþ u0

eAx� �

¼ 0; ðA:10Þ

which can also be rewritten as

1u0

d~udxþ x

eAA0¼ C1; ðA:11Þ

with C1 an integration constant. Scaling the variables as in Eq. (26)and omitting the hats, the force equation can be madedimensionless

dudxþ xA ¼ C1: ðA:12Þ

Using the dimensionless form of the velocity and force we arrive atthe mixed boundary condition at x = ekL = DR as given in [16]

uðDRÞ þ �RdudxðDRÞ þ DR AðDRÞ

� �¼ 0; ðA:13Þ

with �R an arbitrary value.For small Deborah numbers the XPP fiber spin model reduces to

the Newtonian fiber spin model. Therefore, it is possible to com-pare the mixed boundary condition in Eq. (A.13) with Eq. (53).However, the spatial scaling used in these dimensionless equationsare not the same and should first be rescaled.

We will now rescale the spatial variable x = [1;DR] in the vis-cous model of Renardy [16], to the dimensionless variablez ¼ ½0; 1� of the XPP fiber spin model. From the transformation var-iable x = ekz we know that

dx ¼ kx dz: ðA:14Þ

Since lnx = k z, we can write the derivative of x at take-up in termsof z as

dxjDR ¼DR ln DR

L0dzjL0

: ðA:15Þ

Scaling the filament distance z with the filament length L0 as in Eq.(26) we arrive at

dzjDR ¼ DR ln DR dzj1: ðA:16Þ

Substitution of Eq. (A.16) into the mixed boundary condition Eq.(A.13) yields

uð1Þ þ �R1

DR ln DRdudzð1Þ þ DR Að1Þ

� �¼ 0; ðA:17Þ

which we can rewrite as

uð1Þ þ �R

DR ln DRdudzð1Þ þ DR2 ln DR Að1Þ

� �¼ 0: ðA:18Þ

By rewriting the mixed boundary condition of the XPP model givenby Eq. (53) in the same form as the above equation leads to

uð1Þ þ 3bbALdudzð1Þ þ sL

zz � sLrr

3bAL

" #Að1Þ

!¼ 0; ðA:19Þ

where bAL; sLzz, and sL

rr are the dimensionless steady values at z ¼ 1.Comparing Eqs. (A.18) and (A.19) we finally arrive at a relation be-tween the boundary condition parameters

b ¼ �R

3DR bAL ln DR: ðA:20Þ

Due to DR bAL being equal to one, the parameter relation can be sim-plified to

b ¼ �R

3 ln DRor e ¼ L�R

3gA0 ln DR: ðA:21Þ

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Journal of Non-Newtonian Fluid Mechanics - TU/e