journal of non-newtonian fluid mechanicsold-2017.metal.ntua.gr/uploads/3798/1054/a134.pdf ·...

Journal of Non-Newtonian Fluid Mechanics 166 (2011) 1262–1271

Contents lists available at SciVerse ScienceDirect

Journal of Non-Newtonian Fluid Mechanics

journal homepage: ht tp : / /www.elsevier .com/locate / jnnfm

Thixotropic flow of toothpaste through extrusion dies

Hesam A. Ardakani a, Evan Mitsoulis b, Savvas G. Hatzikiriakos a,⇑a Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, V6T 1Z3, Canadab School of Mining Engineering and Metallurgy, National Technical University of Athens, Zografou 157 80, Greece

a r t i c l e i n f o

Article history:Received 13 April 2011Received in revised form 30 May 2011Accepted 9 August 2011Available online 14 September 2011

Keywords:ToothpasteAxisymmetric contractionPaste extrusionThixotropySlip lawStructural parameter

0377-0257/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jnnfm.2011.08.004

⇑ Corresponding author. Tel.: +1 604 822 3107; faxE-mail addresses: mitsouli@metal.ntua.gr (E. Mits

ubc.ca (S.G. Hatzikiriakos).

a b s t r a c t

A commercial toothpaste is investigated in this work as a model paste system to study its processingcharacteristics in capillary flow using various dies. Its rheological behaviour has been determined as thatof a yield-stress, thixotropic material with a time-dependent behaviour, and severe slip at the wall. Therheological data obtained from a parallel-plate rheometer were used to formulate a constitutive equationwith a structural parameter which obeys a kinetic equation, typically used to model thixotropy. The pre-dictive capabilities of this model are tested against capillary data for a variety of capillary dies having dif-ferent length-to-diameter ratios (L/D), contraction angles (2a), and contraction ratios (Db/D)2, where Db isthe diameter of the barrel of the capillary rheometer. The major trends are well captured by the thixotro-pic model and show that slip is the essential parameter in predicting the flow behaviour of toothpaste.

1. Introduction

A paste can be defined as a mixture of a solid and a liquid phaseor as a dense suspension, which is shown to demonstrate proper-ties between liquid and solid [1]. Many pastes are able to retaintheir shapes against gravity like solids [2]. At high shear stressesthey start to flow, which implies the existence of a yield stress, thatis, a critical stress for transition from solid-like to fluid-like behav-iour [1,3–5]. Moreover most of these pasty materials have shownto exhibit thixotropic behaviour, i.e., their viscosity decreases withtime, which implies structure breakup [1]. Such a paste is commer-cial toothpaste, the subject of study of the present work. Knowingthe yield stress and the thixotropic behaviour of pastes is extre-mely important from the rheological point of view in order to de-velop a rheological constitutive equation capable of predictingcorrectly their flow behaviour in well defined flows.

A broad range of products in industry are in paste form or usedin paste form during processing, including bricks, tiles, catalyst pel-lets, drugs, dental materials, pencil leads, and poly-tetra-fluoro-ethylene (PTFE), among others [1]. Many fabricating processesare available to handle pastes with paste extrusion being dominantamong them. During extrusion, the paste is forced through the dieby a pressure difference. As the paste comes out of the die land ittakes the shape of the die [6]. It is important to be able to predictthe flow behaviour of pastes in such flows related to real process-

ll rights reserved.

: +1 604 822 6003.oulis), hatzikir@interchange.

ing by using constitutive equations developed from fundamentalrheological measurements. Such flow simulations provide addi-tional testing for the validation of these rheological constitutiveequations.

It is common to introduce toothpaste as an example for Bing-ham plastic behaviour [7]. Barnes [8] listed toothpaste as ashear-thinning material and assumed that its viscosity changes al-most instantaneously. However, toothpaste is usually consideredas a thixotropic material. There are various additives to improvethe rheological properties of toothpaste aiming at increasing itsshear-thinning and shorten its thixotropy [9]. Thixotropic behav-iour of toothpaste results in a delay between instantaneous rheo-logical behaviour and its equilibrium shear-thinning flow curve.The effects of toothpaste thixotropy on its flow regime in a large-gap Couette geometry have been studied in a recent paper by Pot-anin [10]. His simulations, which are verified by experimental datain simple shear, have shown that the calculated torque is greaterfor the thixotropic model compared to an equilibrium model. Hiswork on toothpaste and similar works on thixotropic materials[11] clearly show the importance of considering thixotropy in flowanalysis, and therefore the present work follows those guidelines.

In the present paper, we are interested in studying first the rhe-ological behaviour of commercial toothpaste using simple rheolog-ical flows. Based on these experimental results, a rheologicalconstitutive equation is developed. Furthermore, capillary extru-sion experiments are performed using capillary dies of differentgeometrical characteristics such as die diameter, D, length-to-diameter ratio, L/D, and contraction angle, 2a. Using the constitu-tive equation developed, capillary flow simulations are performed

H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 166 (2011) 1262–1271 1263

to predict the pressure drop in the various geometries and examinein detail the flow behaviour inside the dies with emphasis on struc-ture evolution.

2. Experimental

A commercial toothpaste has been chosen for this study (Col-gate). Because of the homogeneous structure of toothpaste, it is ex-pected that liquid migration becomes negligible compared to thecase of granular materials with a liquid medium, an assumptionof the present work [12].

The rheological properties of toothpaste, such as yield stress andthixotropy, have been studied using a rotational rheometer (Kinex-us, Malvern) equipped with a cup-and-bob geometry. Most experi-ments were performed using the cup-and-bob geometry of largegap (2 mm). Although the handling of material is harder for thecup-and-bob geometry, the advantage of this geometry over othergeometries is that it provides constant shear rate, which is criticalfor maintaining uniform structure during the pre-shearing processand also during the steady and transient experiments. However,experiments with a cup-and-bob geometry having a small gap(1 mm) have been performed to check for wall slip effects (gapdependence of the flow curve). A few experiments were also per-formed with parallel plates (20 mm diameter and 1 mm gap) to ver-ify the consistency of measured rheological properties.

In order to obtain consistent and reproducible results, pastesamples are pre-sheared (a common practice for thixotropic fluids)for a certain period of time (15 min) under a certain shear rate(50 s�1) and left to rest for 2 h to recover their structure. All exper-iments have been carried out at room temperature (23 �C). Thedensity of this particular toothpaste has been measured and foundto be 1300 kg/m3.

2.1. Equilibrium flow curve

First, steady shear experiments were performed in order todetermine the equilibrium viscosity (steady-state). After pre-shearing, steady shear experiments were performed at specifiedshear rate values. After increasing the shear rate to a new value,a sudden increase in shear stress was observed due to thixotropy.However, due to continuous structural breakdown, the resistingshear stress slowly decreased to its equilibrium value. This equilib-rium shear stress value corresponds to a certain value of structurethat corresponds to the shear rate of pre-shearing. This procedurewas continued until enough points are obtained to define the com-plete equilibrium flow curve. Fig. 1 shows the steady values of the

Shear Rate, γ (s-1)0.001 0.01 0.1 1 10 100

r Stre

ss, τ

1000parallel plateslarge-gap cup-and-bobsmall-gap cup-and-bobstructural parameter modelcreep test results

Fig. 1. The equilibrium flow curve and the fitted thixotropic-structural model.

measured shear stress as a function of the shear rate. Different setsof steady-state experiments with different geometries (parallelplates, small- and large-gap cup-and-bob) have been carried outto check for slip effects. It was noted that for all these three geom-etries (including different gaps), the results fall on the same curveat least for high shear rates, which means that slip effects are neg-ligible, consistent with previous reports [10].

For shear rates beyond about 0.1 s�1, the experimental mea-surements were shown to be reproducible and consistent overthree decades of shear rate. Below 0.1 s�1, the data has shownsome inconsistency, which might be due to the occurrence of shearbanding. It has been reported in the literature that for many mate-rials at low shear rates, experimental measurements are not repro-ducible due to occurrence of shear banding, flow bifurcation andshear rejuvenation [13]. The flow behaviour of some materials withshear banding behaviour has been studied with flow visualisationtechniques. It has been shown that for the parallel-plate geometryunder some critical shear rate there is no homogeneous flow, andshear rate is localized in a plane between the plates, meanwhilethe rest of the material is in solid form [13,14]. Due to shear band-ing the flow curve exhibits a minimum and for each nominal shearstress there are at least two existing true shear rates. In simplewords, flow below a certain shear rate (minimum of the flowcurve) is always unstable. As a result of this instability it is impos-sible to report any equilibrium data below the critical shear rate.

To show the existence of shear banding, creep tests have beenperformed at different shear stress values at the low end of theflow curve, typically for rates below 0.1 s�1, where the data arenot reproducible. The same pre-shearing protocol discussed inthe experimental section has been applied to ascertain that allthe samples are at the same initial structural state. Fig. 2 showsthis flow bifurcation. For all shear stress values below a critical va-lue (in this case about 20 Pa), a continuous decrease of shear rate isobserved. Meanwhile, for shear stress beyond that critical stress,the shear rate attains its steady-state value. Since it takes timefor the flow to develop or to reach cessation, the stability analysisis affected by the duration of the experiment. Therefore, it is notsimple to define a precise value for critical stress. There is a win-dow of shear rate values (roughly below 0.1 s�1) that are shownnot to be reachable by any shear stress value. For example, for ashear stress value of 20 Pa, a steady shear rate 0.001 s�1 does notlast for more than 200 s, and flow terminates as the degree ofstructure is increased. For some other higher values of wall shearstress, such as 25 and 30 Pa (Fig. 2), apparent ‘‘steady-state’’ shearrates are obtained, which are plotted in Fig. 1 and labelled as creeptest results. Comparing these data with data obtained from

Time, t (s)

10 100 1000 10000

τ = 1 Pa51015202530 .

Fig. 2. Creep tests at small values of shear stress to show shear banding or flowbifurcation.

Shear Rate, γ (s-1)0 10 20 30 40

r Stre

ss, τ

700restγ = 2 s-1

γ = 5 s-1

γ = 10 s-1

γ = 30 s-1

structural model (rest)structural model (γ = 30 s-1)

Fig. 3. Iso-structure flow curves.

1264 H.A. Ardakani et al. / Journal of Non-Newtonian Fluid Mechanics 166 (2011) 1262–1271

shear-rate sweep experiments shows that the shear rates at thesame shear-stress values are not necessarily the same due to theoccurrence of shear banding.

For all practical purposes, the low shear stress part of the flowcurve can be described by the existence of a yield stress, sy. Aspointed out by Papanastasiou [15], the downturn of the experi-mental data at low shear rates can be obtained by multiplyingthe yield stress with the following exponential function of theshear rate _c:

sy ! sy½1� expð�m _cÞ�; ð1Þ

where m is a stress-growth exponent with units of time. Choosing asuitable value for m makes a yield-stress curve pass through thelow-shear-rate data (see Fig. 1). Although, the experimental datafor shear rates below 0.1 s�1 are shown to be irreproducible dueto the occurrence of shear banding/flow bifurcation, the simpleyield stress model, modified with the Papanastasiou method, canbe used to roughly capture the ‘‘apparent’’ steady-state behaviourof the material even at these small shear rates.

Time, t (s)0.1 1 10 100 1000

r Stre

ss, τ

step shear rate γ = 0.1 s-1

γ = 1 s-1

γ = 5 s-1

structural model (from rest)

Fig. 4. Rheological response of toothpaste to different values of shear rate.

2.2. Time dependency

It is also necessary to characterise the time dependency andthixotropic behaviour of the toothpaste. To do so, various start-up experiments are carried out. In each experiment, the rheometeris first loaded with a sample. After pre-shearing and rest to recoverthe structure, the shear rate is suddenly increased from zero to acertain value. As a result the shear stress increases from zero to ahigh value rapidly and subsequently gradually decreases with timeuntil it reaches its steady value. Essentially a shear stress overshootis obtained in each case, also indicating time dependency.

Even though the thixotropic behaviour is amplified in the case ofsudden step shear rate from rest, similar, although milder, thixotro-pic behaviour is observed when the shear rate suddenly increasesfrom a lower to a higher value. Therefore, experiments have beencarried out using the same procedure with sudden shear-ratechanges from a lower to a higher value. The same procedure hasbeen followed in order to design a multistep shear-rate test formodel evaluation.

Two main methods have been suggested in the literature tostudy thixotropy. The first method, referred to as the iso-structureflow curve, is based on the idea of measuring the flow curve of thematerial at a certain level of its structure [16]. This method re-quires the determination of the flow curve with respect to a certainlevel of shear rate (reference rate). Each point in the flow curve isdetermined after the material is pre-sheared at the reference rate.For a purely thixotropic material in steady shear after obtainingsteady-state at the reference pre-shearing rate, application of anew higher shear rate will suddenly increase the shear-stress level.The first measured shear-stress value corresponds to the iso-struc-ture flow curve for the previous shear-rate value, i.e., time neededfor structural changes. However, as shear continues at this highershear rate, the structure gradually starts breaking down and theshear stress eventually reaches a new lower steady-state value,which represents the equilibrium structure for this shear-rate va-lue. Fig. 3 shows flow curves with various levels of the referenceshear rate. We observe that increasing the pre-shearing shear ratecauses the flow curve to shift to small shear stress values due tothe breakdown of the structure. The data also show that the sensi-tivity of the structure slightly increases in some cases with shearrate, which the model cannot capture adequately. Two possiblereasons are that (i) almost no material shows purely thixotropicbehaviour and typically thixotropic behaviour is combined withviscoelasticity. Therefore, the overshoot due to viscoelasticityinterferes with the sudden increase of the stress due to thixotropy

at the new shear-rate value, although these two effects can be dis-tinguished as discussed below; and (ii) technical limitations due tothe fact that it takes time for the rheometer to collect the first pointafter changing to the new shear rate. Our experiments are at rela-tively small shear-rate values, and the latter reason might not be aproblem.

Another method for the characterisation of thixotropy is theobservation of the material’s rheological response to any changein the flow regime like the shear rate. In this work, shear-rate steptests have been performed. It was observed that as the shear ratesuddenly increased, the shear stress started from zero and quicklyrose to a higher value. This initial part is related to the viscoelasticresponse of the material. After this viscoelastic response, thixotro-pic behaviour can be observed. Fig. 4 shows both viscoelastic andthixotropic effects. The viscoelastic effect can only be tracked atlow shear rates where the thixotropic effect is limited. Thereforefor model development, the elastic response is neglected and thematerial is assumed to be purely thixotropic.

Figs. 5 and 6 show two different sets of shear-rate jump tests;the first from a material at rest to a high shear rate and the secondfrom an equilibrium state of shear rate at 5 s�1 to a new highershear rate. As can be seen the overall description of the changesare satisfactory. Deviations at short times are related to the visco-elastic nature of the material; as such effects are neglected suchdeviations are expected.

Time, t (s)

0 20 40 60 80 100

r Stre

ss, τ

γ = 25 s-1

γ = 55 s-1

γ = 100 s-1

structural model

Fig. 5. Thixotropic behaviour of paste in three start-up tests at different shear rates.

Time, t (s)

0 20 40 60 80 100

r Stre

ss, τ

600γ = 40 s-1

γ = 30 s-1

γ = 20 s-1

γ = 10 s-1

structural model

Fig. 6. Thixotropic behaviour of toothpaste at various shear-rate jumps from aninitial shear rate of 5 s�1.

2.3. Slip at the wall

Experiments were performed with capillary dies havingdifferent diameters to detect effects of slip. Fig. 7 depicts theresults, where a diameter dependence of the flow curve is obvious.

Shear Rate, γΑ (s-1)

1 10 100 1000 10000

r Stre

ss, τ

D=0.85 mmD=1.27 mmD=2.23 mmlarge gap cup-and-bob

Fig. 7. The effect of die diameter, D, on the flow curve of toothpaste. Slip effectscause the diameter dependence.

Moreover the flow curves determined from capillary flow deviatesignificantly from the one obtained from the parallel-plate rheom-eter (data of Fig. 1).

The Mooney analysis has been used to determine the slip veloc-ity of toothpaste as a function of the wall shear stress [17–19]. Thisanalysis gives:

_cA ¼ _cA;s þ8Vs

D; ð2Þ

where _cA is the apparent shear rate, _cA;s is the slip-corrected appar-ent shear rate, which corresponds to the parallel-plate flow curve ofFig. 7, and Vs is the slip velocity. Fig. 8 plots the slip velocity versusthe wall shear stress, sw. A linear relationship for slip is obtained,which can be written as:

Vs ¼ bsw; ð3Þ

where b is the slip coefficient. Its value was found to be 8 � 10�5 m/(Pa s). For the flow conditions used in the capillary experiments andthe rheological properties at hand, this value amounts to massiveslip, as will become evident in the simulations further down. Itshould be noted that our previous papers on PTFE paste extrusion[20–22] also showed severe slip occurring at the die walls, whichis the norm rather than the exception for pastes in die flows.

2.4. Capillary extrusion results

Capillary extrusion experiments were also performed using apiston-driven constant-speed capillary rheometer (Bohlin RH2000). Various dies were used in order to study the effects of geo-metrical parameters of the die on the capillary flow (essentiallypressure drop) of toothpaste. Important parameters include thedie diameter, D, the length-to-diameter ratio, L/D, the reductionratio, RR � ðD2

b=D2Þ, where Db is the barrel diameter, and theentrance contraction angle, 2a. Table 1 lists all capillary dies usedin this study along with their geometrical characteristics. Thebarrel diameter, Db, is 15 mm. These results will be presented alongwith the flow simulations below.

3. Constitutive modeling

A constitutive model for a typical paste should possess at leasttwo elements. First, a yield stress which is the stress below no flowis observed. Then, thixotropy should be included through anappropriate kinetic equation involving a structure parameter. Ithas been claimed by many authors that the existence of yield stress

Wall Shear Stress, τw (Pa)0 500 1000 1500 2000 2500

, V s (

RR=311RR=140RR=45slip model

Vs = 8x10-5τw (m/s)

Fig. 8. Slip velocity of toothpaste as a function of wall shear stress at 23 �C.

Table 1List of capillary dies used in the present work. The barrel diameter, Db, is 15 mm.

Die number RR L/D 2a D# (–) (–) (�) (mm)

1 277 18 15 0.92 311 20 30 0.853 311 20 45 0.854 311 20 60 0.855 311 20 90 0.856 311 20 180 0.857 45 20 45 2.2258 139 20 45 1.279 311 0 45 0.85

10 311 10 45 0.85

Table 2Fitted parameters for the structural model.

k1 (–) k2 (s�1) sy (Pa) g1(Pa s) m (s)

0.002 0.1 83.73 4.89 50

indicates the existence of structure that requires initial energy todecay and let the material flow, which implies that flow mightalter the structure and thus thixotropy is expected [16,23,24].

To represent toothpaste rheology, the proposed model includestwo terms and can be written as follows:

s ¼ syð1� expð�m _cÞÞ þ ð1þ nÞg1 _c: ð4Þ

The first term represents the yield stress behaviour of the material.It is often assumed that the yield stress is related to the structure ofthe material through a structural parameter, n. However, in thepresent case the apparent yield stress shows a weak dependenceon n, and we have neglected this dependence. Papanastasiou’s expo-nential modification (Eq. (1)) is applied on this term for fitting thedata of Fig. 1 at the lower shear-rate range.

The second term includes the viscous part of the response, andit is a function of the structural parameter n, which can bedescribed by a kinetic equation. The rate of change in structurecan be described by [11]:

dndt¼ �k1 _cnþ k2ð1� nÞ: ð5Þ

This way the structural parameter n is a normalized quantity thatvaries between 0 and 1 and indicates the integrity of the network(n = 0: no network or structure; n = 1: fully developed network or

ur= n =τA

un=0,D E

LcLres

uz=f(r)ur=0 ξ=1

Fig. 9. Field domain and boundary condit

structure). The first term on the RHS of Eq. (5) indicates breakdownof the network due to material deformation; the second term isresponsible for build-up of the network with a time constant 1/k2

associated to it (here found to be equal to 10 s).Structure formation occurs due to Brownian motion and is par-

tially due to imposition of shear rate [16]. According to the aboveEq. (5), the shear contribution in structure build-up is neglected,and the rate of formation is set proportional to (1 � n) [24,25]. Ithas been assumed that the shear rate may break down structure.The rate of structural break-down is proportional to the shear rateand also to the degree of structure.

According to this kinetic equation, the structural parameter ap-proaches a steady-state value, neq, at a given value of shear rate, _c,that is:

neq ¼k2

k1 _cþ k2: ð6Þ

Therefore the equilibrium flow curve is given by:

seq ¼ syð1� expð�m _cÞÞ þ ð1þ neqÞg1 _c: ð7Þ

Then the equilibrium apparent viscosity geq is given by:

geq ¼sy

_cð1� expð�m _cÞÞ þ ð1þ neqÞg1: ð8Þ

Optimized values for the proposed models are summarized in Table2. The ratio k1/k2 and m were calculated by fitting Eq. (7) to the dataof Fig. 1 by writing neq ¼ 1=½1þ ðk1=k2Þ _c� from Eq. (6). Subsequently,using the data of Figs. 5 and 6, the individual values of k1 and k2

were calculated. Each flow curve at a given level of the structuralparameter (or apparent viscosity) should look like a Bingham modeland cross the equilibrium flow curve at its corresponding initialshear rate (see Fig. 3). Figs. 5 and 6 show two different sets ofshear-rate jump tests; the first from a material at rest and the sec-ond from an equilibrium state of shear rate at 5 s�1. The thixotropicbehaviour of the material has been adequately captured by the pro-posed model.

4. Governing equations

We consider the conservation equations of mass and momen-tum for incompressible fluids under isothermal, creeping, steadyflow conditions. These are written as [7]:

r � �u ¼ 0; ð9Þ

0 ¼ �rpþr � ��s; ð10Þ

ut=βτw, ξn=0

S Fz=0 ur=ξn=0

ions for capillary flow of toothpaste.

where �u is the velocity vector, p is the pressure and ��s is the extrastress tensor.

The viscous stresses are given for inelastic incompressible fluidswith structure by the relation [7]:

��s ¼ ngðj _cjÞ��_c; ð11Þ

where gðj _cjÞ is the apparent viscosity of Eq. (8), in which the shearrate _c is replaced by the magnitude j _cj of the rate-of-strain tensor��_c ¼ r�uþr�uT , which is given by:

j _cj ¼ffiffiffiffiffiffiffiffiffi12

2ð _c :

��_cÞ� �1=2

; ð12Þ

where II _c is the second invariant of ��_c

II _c ¼ ð _c : _cÞ ¼X

_cij _cij; ð13Þ

Thus, the apparent viscosity is written as:

Fig. 10. Yielded/unyielded (shaded) regions in toothpaste extrusion at 23 �C for three diD = 20, 2a = 45�, RR = 311.

gðj _cj; nÞ ¼ sy

j _cj ð1� expð�mj _cjÞÞ þ ð1þ nÞg1: ð14Þ

In the above, the apparent viscosity is a function of n, which obeysthe kinetic Eq. (5). In general flows, Eq. (5) becomes the convective-transport equation [21,22]:

@n@tþ �u � rn ¼ �k1j _cjnþ k2ð1� nÞ: ð15Þ

For steady-state conditions, on/ot = 0.The above rheological model (Eqs. (11) and (14)) is introduced

into the conservation of momentum (Eq. (10)) and closes the sys-tem of equations. Boundary conditions are necessary for the solu-tion of the above system of equations. Fig. 9 shows the solutiondomain and boundary conditions for the tapered contractiongeometry of capillary flow. It should be noted that for the struc-tural parameter, n, the entry boundary condition is n = 1 (i.e., it isassumed that the network is fully established) and @n=@�n ¼ 0 atthe walls, where �n is the unit outward normal vector. The last

fferent apparent shear rates. Incompressible flow with slip at the wall for Die #3, L/

boundary condition means that the structural parameter is free totake its values at the wall such that normal to the wall there are nochanges in n. Simulations with different initial n-values haveshown that the results are not affected much, a finding whichhas been observed experimentally for thixotropic fluids [26,27].Values of n = 0 at entry affected only the n-values in the reservoir,not in the die, and only at the upper end of apparent shear rates( _cA > 2000 s�1). Because of symmetry only one half of the flowdomain is considered, as was done in our previous works [21,22].

All lengths are scaled by the die radius R, all velocities by theaverage velocity U at the die exit, all pressures and stresses byg1(U/R).

5. Method of solution

The numerical solution is obtained with the Finite ElementMethod (FEM), employing as primary variables the two velocities,pressure, and structural parameter (u–v–p–n formulation). Notingthe similarity of the kinetic equation with the energy equation,we have substituted in our previous formulation the temperature,T, with the structural parameter, n [21,22]. We use Lagrangian

Fig. 11. Contours of the structural parameter n in toothpaste extrusion at 23 �C for three dD = 20, 2a = 45�, RR = 311.

quadrilateral elements with biquadratic interpolation for thevelocities and the structural parameter, and bilinear interpolationfor the pressures. The solution process is similar to that employedby Mitsoulis and Hatzikiriakos [21,22], using the same grids asbefore. As in the previous results with PTFE paste modeling, thesolution process was based on incrementing the apparent shearrate from low to high values. Due to the massive slip at the wallpresent in both cases (PTFE and toothpaste extrusion), convergencewas fast and easy and could be obtained within few iterations tosatisfy criteria for the norm-of-the-error and the norm-of-the-residuals below 10�5.

6. Flow simulations

6.1. Flow field

The flow simulations have been performed with the above con-servation, constitutive and convective-transport equations, theirboundary conditions, and the parameters of Table 2. Referenceresults are given for the highest apparent shear rate ( _cA ¼4Q=pR3 ¼ 4U=R = 2560 s�1) and one die design (Die #3, 2a = 45�,

ifferent apparent shear rates. Incompressible flow with slip at the wall for Die #3, L/

L/D = 20). The purpose is to find out the effect of the structuralparameter n on the results, and how it affects the relative impor-tance of the various forces at play in toothpaste extrusion.

Results are shown as contours of two important variables of themodel, namely, the yield stress, sy, and the structural parameter, n.First, Fig. 10 shows the yielded/unyielded (shaded) zones for differ-ent apparent shear rates. The dividing line between the tworegions is the contour of the magnitude of the stress tensor|s| = sy. For the lowest apparent shear rate of 25.6 s�1, the yieldedregion is contained in the tapered region just before the die entry.At intermediate shear rates, this region is extended inside the die,leaving a small unyielded plug in the die core. At the highest appar-ent shear rate, the yielded region is further extended both up-stream and downstream to encompass the whole tapered regionand die.

The corresponding results for contours of the structural param-eter n are given in Fig. 11. It becomes apparent that structurechanges are small, and occur mainly in the die at high levels ofshear rate. This is not surprising because the flow is fast as evi-denced by the inverse of the apparent shear rates, and no time isgiven for the material to break down its structure. It should benoted that n starts with 1 at entry (n not being 1 or 0 at the wallwhen slip is allowed), and then most of the changes occur in a conenear the die entry and inside the die.

6.2. Effect of flow and design parameters

The effect of the die entrance angle on the structural parametern is depicted in Fig. 12, where the axial distribution is presentedalong the centreline and the wall for the highest apparent shearrate _cA = 2560 s�1. We observe that for all three die designs thestructural parameter decreases continuously from the inlet to thedie exit on the die wall, while it remains constant within the diealong the centreline where there is no shearing. The continuouslinear decrease at the die wall is expected from the kinetic Eq.(15). However, all the values are close to 1, meaning that the struc-ture is mildly affected by the flow kinematics.

The angles have a small effect on n. These effects originate fromdifferences in the extensional components of the flow conditions atthe entry, which are functions of the entrance angle.

Axial Distance, z / R

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40

, ξ (-

2α = 30o

2α = 45o

2α = 60o

γA=2560 s-1, L/D=20, RR=311.

c-line

die wall

Fig. 12. The effect of die entrance angle, 2a, on the structural parameter, n. Axialdistribution of n along the centreline and the wall for toothpaste extrusion at 23 �C.Incompressible flow with slip at the wall for _cA = 2560 s�1, L/D = 20, RR = 311.

The effect of L/D on the structural parameter for a given geom-etry (2a = 45�) and the highest apparent flow rate ( _cA = 2560 s�1) isshown in Fig. 13. A longer die allows the material more time tobreak down its network and to reach lower values of n.

The effect of the apparent shear rate for a given geometry(2a = 45�) is shown in Fig. 14. Increasing the apparent shear ratedecreases the n-parameter, hence it leads to network breakdown,but again these changes are mild, hardly reaching n = 0.88. Againthe effects are more significant at high shear rates and at the diewalls.

At this point it is instructive to examine what causes this smallchange in n. The way to study this is to keep the ratio k1/k2 = 0.02 s�1 = constant (since this is dictated by the steady-stateflow curve) and change the individual values of k1 and k2. This ef-fect on the structural parameter for a given geometry (2a = 45�)and apparent flow rate ( _cA = 2560 s�1) is shown in Fig. 15. A largerk2 parameter (k2=1 s�1, hence a time constant 1/k2 = 1 s) bringsdown the structural parameter on the die wall to almost zero,while a smaller k2 parameter (k2 = 0.01 s�1, hence a time constant1/k2 = 100 s) keeps the structural parameter everywhere close to 1.Thus, thixotropic effects are well captured by the model but for thepresent toothpaste and its value of k2 = 0.1 s�1 the effect on thebreakdown of the network is small and the values of n are between1 and 0.8 in the range of experiments.

6.3. Comparison with experiments

As mentioned in the experimental part, experiments were per-formed in capillary dies of different designs (see Table 1). The sim-ulations have been carried out for all dies and shear rates andbelow they are compared with experimental results.

The first geometrical property studied is L/D, while 2a = 45� andRR = 311 are kept constant. Because of the smaller diameter of thedie land compared to the die entrance, it is assumed that the mainpressure drop occurs in the die land. Fig. 16 shows the extrusionpressure for three different L/D ratios of 0, 10, and 20. For the caseof L/D = 0, we have used in the simulations a small value L/D = 0.2.For the other L/D ratios, a linear relation between L/D and pressure

Axial Distance, z / R-50 -40 -30 -20 -10 0 10 20 30 40

, ξ (-

L/D = 5L/D = 10L/D = 20

γA=2560 s-1, 2α=45o, RR=311.

tapered wall

die wall

c-line

Fig. 13. The effect of die length, L/D, on the structural parameter, n. Axialdistribution of n along the centreline and the wall for toothpaste extrusion at23 �C. Incompressible flow with slip at the wall for _cA = 2560 s�1, 2a = 45�, RR = 311.

Axial Distance, z / R-60 -50 -40 -30 -20 -10 0 10 20 30 40

, ξ (-

γA = 25.6 s-1

γA = 102 s-1

γA = 310 s-1

γA = 1027 s-1

γA = 2560 s-1

Die #3, 2α=45o, L/D=20, RR=311

tapered wall

die wall....

c-linec-line

Fig. 14. The effect of apparent shear rate, _cA; on the structural parameter, n. Axialdistribution of n along the centreline and the wall for toothpaste extrusion at 23 �C.Incompressible flow with slip at the wall for 2a = 45�, L/D = 20, RR = 311.

Axial Distance, z / R

-60 -50 -40 -30 -20 -10 0 10 20 30 40

, ξ (-

k2 = 1 s-1

k2 = 0.1 s-1

k2 = 0.01 s-1

Die #3, 2α=45o, L/D=20, RR=311

tapered wall

die wall

c-line

die wall

c-line

k1/k2=0.02 s

Fig. 15. The effect of k1 and k2 on the structural parameter, n, keeping the sameratio k1/k2 = 0.02 s. Axial distribution of n along the centreline and the wall fortoothpaste extrusion at 23 �C. Incompressible flow with slip at the wall for 2a = 45�,L/D = 20, RR = 311.

.Apparent Shear Rate, γα (s-1)

10000100000110

L/D = 20L/D = 10L/D = 0L/D = 20 (sim.)L/D = 10 (sim.)L/D = 0.2 (sim.)

2α = 45o

RR = 311

Fig. 16. Effect of L/D on extrusion pressure.

Apparent Shear Rate, γα (s-1)

100010010

RR = 45RR = 139RR = 311RR = 45 (sim.)RR = 139 (sim.) RR = 311 (sim.)

L/D = 20

2α = 45o

Fig. 17. Effect of reduction ratio RR on extrusion pressure.

Apparent Shear Rate, γΑ (s-1)

10000100010010

2α = 30o

2α = 45o

2α = 60o

2α = 180o

2α = 30o (sim.)2α = 45o (sim.)2α = 60o (sim.)

RR = 311L/D = 20

Fig. 18. Effect of contraction angle 2a on extrusion pressure.

drop is expected. However, as illustrated in Fig. 16, the pressuredrop for L/D = 20 is much greater than the pressure drop for L/D = 10. The origin of this phenomenon could be the formation ofan unyielded area inside the die land, which grows because ofstructural formation as the material relaxes through the die land.The growth of the unyielded area increases the shear rate insidethe yielded area. Therefore, the shear rate increases along the dieland wall and causes higher shear stresses.

The numerical simulations with the structural parameter modelshow that the general trends are captured by the model, althoughquantitatively the L/D = 10 results are over-predicted. The simula-tions show that doubling the die length leads to doubling thepressure in the system, while experimentally there is a quadru-pling effect!

The second geometrical property studied is RR, while 2a = 45�and L/D = 20 are kept constant. Typically the material wouldtolerate higher shear rates through the conical zone when thegeometry has a high reduction ratio. As it can be seen in Fig. 17,

the pressure drop is almost the same for the two lower reductionratios of 45 and 139, and slightly lower for the highest reductionratio of 311. This can be explained by a greater structure break-down for this case, which results in a lower apparent viscosity.The numerical simulations are in general agreement with thesefindings but they distinguish more clearly among the three cases.Thus the model can adequately take into account the effect of RRon toothpaste extrusion.

The third geometrical property studied is 2a, while RR = 311and L/D = 20 are kept constant. The extrusion pressure could beaffected by the die entrance angle. Since most of the pressure dropoccurs in the die land, and the pressure drop in the conical zone isalmost negligible, it is hard to detect the effect of die contractionangle on the pressure drop. However, Fig. 18 shows that for lowercontraction angles, the extrusion pressure is slightly higher due toan increased effect of shear. In the case of polymer melts, it hasbeen observed that the extrusion pressure is independent of thedie entrance angle for angles greater than about 30�. The same con-clusion can be made for thixotropic toothpaste.

The numerical simulations, in general, corroborate these find-ings as they show that for all cases the pressure drop is more orless the same, except for the lower angle of 30�, where the resultsare slightly higher. Thus the model underestimates the pressuresfor the lower angles, but can differentiate the effect of the contrac-tion angle 2a on toothpaste extrusion.

7. Conclusions

Experiments and simulations were performed in this work forcommercial toothpaste. The experiments showed the existence ofa yield stress and time-dependent phenomena associated withthixotropy, hence structure. Severe slip at the wall was found tooccur in capillary flow with different die designs. A simple thixo-tropic model was formulated and the parameters of the modelwere found by matching experimental data. The model accountsfor yield stress and structure through a kinetic-type convective-transport equation.

Simulations based on the proposed model for capillary flowshowed that the model essentially captures the pressure drop inthe system for the majority of the cases investigated. Slip in mas-sive and responsible for the lower-than-expected pressures at ele-vated shear rates reaching 2500 s�1. The simulations also providedetails about structure evolution in the die and show that due toslip and fast flows, structure breakdown is not strong. Some dis-crepancies between theory and experiments may be attributed topossible viscoelastic effects not accounted for in the model. Suchwork to include viscoelasticity together with thixotropy is cur-rently under way by the authors.

Acknowledgements

Financial assistance from the Natural Sciences and EngineeringResearch Council (NSERC) of Canada and the programme ‘‘PEBE

2009–2011’’ for basic research from NTUA are gratefullyacknowledged.

References

[1] P. Coussot, Rheometry of Pastes, Suspensions, and Granular Materials:Applications in Industry and Environment, Wiley, New York, 2005.

[2] H.A. Barnes, A Handbook of Elementary Rheology, University of Wales Instituteof Non-Newtonian Fluid Mechanics, Aberystwyth, Wales, 2000.

[3] E.C. Bingham, Plasticity and elasticity, J. Franklin Inst. 197 (1924) 99–115.[4] A.W. Blair, A.L. Prince, Influence of organic matter on crop yield and on

carbon–nitrogen ratio and nitrate formation in the soil, Soil Sci. 35 (1933)209–219.

[5] H.A. Barnes, The myth of yield stress fluids, Prog. Trends Rheol. V (1998) 201–202.

[6] J. Benbow, J. Bridgwater, Paste Flow and Extrusion, Clarendon Press, OxfordUniversity Press, Oxford, 1993.

[7] R.I. Tanner, Engineering Rheology, second ed., Oxford University Press, Oxford,2000.

[8] H.A. Barnes, The yield stress – a review or ‘panta rei’ – everything flows?J. Non-Newtonian Fluid Mech. 81 (1999) 133–178.

[9] A.O. Barel, M. Paye, H.I. Maibach, Handbook of Cosmetic Science andTechnology, CRC Press, 2001.

[10] A. Potanin, 3D simulations of the flow of thixotropic fluids, in large-gapCouette and vane-cup geometries, J. Non-Newtonian Fluid Mech. 165 (2010)299–312.

[11] J.J. Derksen, Prashant, simulations of complex flow of thixotropic liquids, J.Non-Newtonian Fluid Mech. 160 (2009) 65–75.

[12] A.B. Budi, S. Ebnesajjad, S.G. Hatzikiriakos, Preforming behavior ofpolytetrafluoroethylene paste, Powder Technol. 121 (2001) 249–258.

[13] P. Coussot, L. Tocquer, C. Lanos, G. Ovarlez, Macroscopic vs. local rheology ofyield stress fluids, J. Non-Newtonian Fluid Mech. 158 (2009) 85–90.

[14] F. Pignon, A. Magnin, J.M. Piau, Thixotropic colloidal suspensions and flowcurves with minimum: identification of flow regimes and rheometricconsequences, J. Rheol. 40 (1996) 573–587.

[15] T.C. Papanastasiou, Flows of materials with yield, J. Rheol. 31 (1987) 385–404.[16] J. Mewis, N.J. Wagner, Thixotropy, Adv. Colloid Interf. Sci. 48 (2009) 147–148

(214–227).[17] M. Mooney, Explicit relations for slip, J. Rheol. 2 (1931) 210–221.[18] A.V. Ramamurthy, Wall slip in viscous fluids and influence of materials of

construction, J. Rheol. 30 (1986) 337–357.[19] S.G. Hatzikiriakos, J.M. Dealy, Wall slip of molten high density polyethylenes.

II. Capillary rheometer studies, J. Rheol. 26 (1992) 703–741.[20] P.D. Patil, J.J. Feng, S.G. Hatzikiriakos, Constitutive modeling and flow

simulation of polytetrafluoroethylene (PTFE) paste extrusion, J. Non-Newtonian Fluid Mech. 139 (2006) 44–53.

[21] E. Mitsoulis, S.G. Hatzikiriakos, Steady flow simulations of compressible PTFEpaste extrusion under severe wall slip, J. Non-Newtonian Fluid Mech. 157(2009) 26–33.

[22] E. Mitsoulis, S.G. Hatzikiriakos, Modelling PTFE extrusion: the effect of anobjective flow type parameter, J. Non-Newtonian Fluid Mech. 159 (2009) 41–49.

[23] I.M. Krieger, The role of instrument inertia in controlled-stress rheometers, J.Rheol. 34 (1990) 471–483.

[24] C. Zhu, J.E. Smay, Thixotropic rheology of concentrated aluminum colloidalgels for solid freeform fabrication, J. Rheol. 55 (2011) 655–672.

[25] A. Mujumdar, A.N. Beris, A.B. Metzner, Transient phenomena in thixotropicsystems, J. Non-Newtonian Fluid Mech. 102 (2002) 157–178.

[26] J. Götz, K. Zick, W. Kreibich, Possible optimisation of pastes and the accordingapparatus in process engineering by MRI flow experiments, Chem. Eng. Proc.42 (2003) 517–534.

[27] S. Rodts, J. Boujlel, B. Rabideau, G. Ovarlez, N. Roussel, P. Moucheront, C. Lanos,F. Bertrand, P. Coussot, Solid–liquid transition and rejuvenation similarities incomplex flows of thixotropic materials studied by NMR and MRI, Phys. Rev. E81 (2010) 021402.

journal of non-newtonian fluid mechanicsold-2017.metal.ntua.gr/uploads/3798/1054/a134.pdf ·...

Documents

ad-a134 317 mechanical properties of longleaf pine …

in the united states court of appeals for the sixth …...

ad-a134 950 development of an organizational effectiveness...

modelling of microbial metabolism stoichiometry...

1.1 Τι είναι το άτομο -...

Κεφάλαιο 1....

lower road - images1.loopnet.com · the a1092 intersects...

installation instructions instructions front disc brake...

photocatalytic treatment of olive mill wastewaters (omw...

suffolk park bury st edmunds | ip32 7qb...ip32 7qb: suffolk...

chapter 1: introduction 1.0...

simulation of viscoplastic ﬂow past cylinders in...

2020 mock-up pages - church farm stow bardolph · us or...

some i a f d p c...

microbiology fundamentals -...

tubing extrusion of a fluoropolymer...

numerical simulation of viscoelastic effects in blade...

dc resistivity definition -...

a1066 diss a140 hoxne a143 a134 a143 yaxley eye...

annular extrudate swell of a fluoropolymer...