Entry and exit flows of Bingham fluids

Entry and exit flows of Bingham fluidsS. S. Abdali, Evan Mitsoulis, and N. C. Markatos

Citation: Journal of Rheology (1978-present) 36, 389 (1992); doi: 10.1122/1.550350 View online: http://dx.doi.org/10.1122/1.550350 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/36/2?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Non-modal stability in Hagen-Poiseuille flow of a Bingham fluid Phys. Fluids 26, 014102 (2014); 10.1063/1.4861025 Entrance and Exit Effects for Fluid Flow in Metal Foam AIP Conf. Proc. 1254, 299 (2010); 10.1063/1.3453828 On three-dimensional linear stability of Poiseuille flow of Bingham fluids Phys. Fluids 15, 2843 (2003); 10.1063/1.1602451 Extrudate swell behavior of polyethylenes: Capillary flow, wall slip, entry/exit effects and low-temperatureanomalies J. Rheol. 42, 1075 (1998); 10.1122/1.550919 Roll waves on a layer of a muddy fluid flowing down a gentle slope—A Bingham model Phys. Fluids 6, 2577 (1994); 10.1063/1.868148

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Entry and exit flows of Bingham fluids

S. S. Abdali and Evan Mitsoulis

Department of Chemical Engineering, University of Ottawa,Ottawa, Ontario KIN 9B4, Canada

N. C. Markatos

Computational Fluid Dynamics Unit, Department ofChemical Engineering, National Technical University of

Athens, Zographou; Athens 15773, Greece

(Received17 July 1991; accepted 18 November 1991)

Synopsis

Entry and exit flows through extrusion dies are studied numerically for Bingham fluids exhibiting a yield stress, A constitutive equation proposed by Papanastasiou is used, which applies everywhere in the flow field in both yielded andpractically unyielded regions. The emphasis is on determining the extent andshape of unyielded/yielded regions along with the extrudate swell (contraction)for planar and axisymmetric dies. The results for pressure are used to determinethe excess pressure losses that give rise to entrance, exit, and the total end (orBagley) correction.

INTRODUCTION

A plastic material exhibits little or no deformation up to a certainlevel of stress, called the yield stress. Above this yield stress the materialflows. These materials are often called Bingham plastics after Bingham(1922), who first described paint in this way in 1919. Paint, slurries,pastes, and food substances like margarine, mayonnaise, and ketchupare good examples of Bingham plastics. A list of several materials exhibiting yield was given in a seminal paper by Bird et al. (1983), whohave also provided an initial analysis of such materials in simple flowfields. Since then a renewed interest has developed among several researchers for the study of Bingham fluids as evidenced by a series ofpapers in the literature [see Lipscomb and Denn (1984), Gartling and

© 1992 by The Society of Rheology, Inc.J. Rheol. 36(2), February 1992 0148-6055/92/020389-19$04.00 389


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390 ABDALI, MITSOULlS, AND MARKATOS

Phan-Thien (1984), O'Donovan and Tanner (1984), Keentok et al.(1985), Dzuy and Boger (1985), Beris et al. (1985), Papanastasiou(1987), Beverly and Tanner (1989), and Ellwood et al (1990)].

To model the stress-deformation behavior, several constitutive relations have been proposed and different yield criteria have been used [see,e.g., discussion by Ellwood et al (1990)]. The existence of a true yieldstress has been questioned by several investigators. Quite recently, Papanastasiou (1987) proposed a novel constitutive equation for materialswith yield, where a material parameter controls the exponential growthof stress and which is valid for both yielded and unyielded areas. It wasshown by Papanastasiou (1987) and Ellwood et al. (1990) that thisequation closely approximates the ideal Bingham plastic and it providesa better approximation to real data of such materials.

In the present work, we shall use Papanastasiou's constitutive equation to examine the entry and exit flows of Bingham fluids throughextrusion dies. The emphasis will be on finding the extent and shape ofthe yielded/unyielded zones by using the criterion that the materialflows and deforms significantly only when the magnitude of the extrastress tensor exceeds the yield stress. The determination of the extrudateshape will also be carried out for planar and circular dies. Finally,excess pressure losses will be determined as a function of a dimensionless yield stress or Bingham number.

MATHEMATICAL MODELING

The flowis governed by the usual conservation equations of mass andmomentum for an incompressible fluid under isothermal, creeping flowconditions (Re::::O) [Bird et al. (1983)].

The constitutive equation that relates T to the deformation is a modified Bingham equation proposed by Papanastasiou (1987) and is written as

( 1)

where fL is a constant viscosity, 7"y is the apparent yield stress, and m isa stress growth exponent. The magnitude Ir Iof the rate-of-strain tensorr = Vv + vifT is given by

• ~ 1 ........ 1/2Irl = V2II r = b{ r: r}] , (2)


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LAGRANGIAN-UNSTEADY BINGHAM FLOWS 391

3.0

2.5

.CJ)

~ 1.5a:::ICIl

a:::< 1.0wJ:CIl

0.5

0.00.0 2.0 4.0 6.0 8.0 10.0 12.0

SHEAR RATE, X (i')

FIG. 1. Shear stress vs shear rate according to the modified Bingham constitutive equation(3) proposed by Papanastasiou (1987) for several values of the exponent m.

where II t is the second invariant of r. Equation (1) approximates thevon Mises criterion for relatively big exponent m (m> 1(0), and holdsuniformly in yielded and unyie1ded regions. The one-dimensional analogof Eq. (1) in simple shear flow gives

(3)

where T12 and YI2 are the shear stress and shear rate, respectively.Figure I shows a graphical representation of Eq. (3) for different valuesof the exponent m. Clearly, for m » 100 the above equation mimics theideal Bingham plastic.

To track down yielded/unyielded regions, we shall employ the criterion that the material flows (yields) only when the magnitude of theextra stress tensor ITI exceeds the yield stress TY' i.e.,


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yielded: ITJ =~ = [H':T:T}]1/2 > TY'

unyielded: ITJ <Ty-

(4a)

(4b)

(5)

Note that previous work by Papanastasiou (1987) employed the criterion of the second invariant of half the rate-of-strain tensor i5 = ~rexceeding an arbitrarily small value of 0.001. This led to erroneousyielded/unyielded regions, as pointed out by Beverly and Tanner(1989), who, however, used very coarse grids and were not able tocapture well the extent and shape of these regions. In the present work,we try to resolve this matter adequately. We also believe (as was shownby numerical trials) that using as a criterion the value of yield stress,which is not arbitrary and is also a large number, reduces the uncertainty of using values very close to zero, which may be numerical noise.

For materials with yield stress, it is appropriate to introduce a dimensionless yield stress -1, defined by Papanastasiou (1987) as

T*- TyHY-fLVN'

where H is a characteristic length (half the channel width or radius R)and VN is a characteristic speed, taken by Papanastasiou (1987) as theaverage velocity of a corresponding Newtonian liquid with viscosityfLatthe same pressure gradient. However, this dimensionless number againintroduces dependence on a Newtonian counterpart. Bird et al (1983)suggest the Bingham number

(6)

(7a)

(7b)

where D is the diameter or channel width (2H) and VB is the averagevelocity of the Bingham fluid. In both cases, the Newtonian fluid corresponds to '1 = Bi = O. However, at the other extreme of anunyielded solid, Bi --+ 00, while T1 reaches a dimensionless pressure gradient defined by

(planar) !::J.P* = (!::J.P) H2

,!::J.L fL VN

(axisymmetric) !::J.p*=~(!::J.P) R2

.2 !::J.L fLVN


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In the present study both numbers will be used to provide comparisonswith the previous works by Papanastasiou (1987) and Ellwood et al(1990).

METHOD OF SOLUTION

The conservation and constitutive equations along with the appropriate boundary conditions [Papanastasiou (1987)] are solved by thefinite element method. The primary variables are the two velocities andpressure (u-v-p formulation). Streamlines are obtained a posteriori bysolving the Poisson equation for the stream function. Although aNewton-Raphson iterative scheme would be recommended for the solution of the nonlinear system of equations, here we have used a directsubstitution scheme (Picard method), sometimes with underrelaxationto accelerate convergence. This avoids calculating the Jacobian whichcan be time consuming, and it also enjoys a wider convergence range.The solution process starts from the Newtonian field (Ty=O), which isused to obtain a first approximation. Iterations are performed untilconvergence for the current yield stress is achieved (usually when thenorm-of-error < 0.01, although we have tried runs with tolerance< 10 - 4 with results virtually identical). The new solution is then usedas an initial estimate for the higher yield stress value (zero-order continuation on the 7"yparameter). The number of iterations increased substantially as Ty increased (departure from the Newtonian liquid andapproach to a solid plastic). From previous experience [Ellwood et al( 1990)] and our own runs the value of m in Eq. (I) was kept constantat 200, since higher values (m= 1000) had no effect on the results.

RESULTS AND DISCUSSION

Entry flow of Bingham fluids

The numerical simulations were performed for Bingham fluids in a4:I abrupt contraction, as has been customarily done for entry flows indies. Calculations were performed with two grids, one having 336 elements, 741 nodes, and 1492 unknown degrees of freedom [Mitsoulis(l986a)] and another denser having 560 elements, 1197 nodes, and2504 degrees of freedom. The results were virtually identical and thusmesh independent. Figure 2 shows the dense finite element grid used(21 points in the r direction).

Calculations were carried out for different Ty values and using m

= 200 in the constitutive equation (I). Following Papanastasiou


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<;~:J-12.0 -B.O -4.0 ~ 0.0, 4.0 B.O, ,

,.'/ z/R "~ -,

/ ,~ ,

~ ,2.0 / ,

'"'- 1.0

0.0, I R

FIG. 2. Finite element grid used for the entry flow calculations in a 4:I abrupt contraction.

(1981), we have pursued the calculations for 0.;;;; r: .;;;; b.P* (or, respectively, O.;;;;Bi < (0), where the dimensionless pressure gradient b.P* isequal to 3 for planar and 4 for axisymmetric geometries in the downstream channel. The progressive growth of solid (unyielded) regions inthe flow field is shown in Fig. 3, for increasing values of r: (or Hi). Theshaded areas correspond to unyielded regions (solid). The fully developed conditions at entry and exit [analytical solutions available forPoiseuille flow of a Bingham fluid, Bird et ai. (1983)] give rise tohorizontal lines in the envelope for the solid region, which is muchlarger in the reservoir than the downstream tube, as expected due to amuch slower flow (hence stress level) in the reservoir. Also note thedead space near the comers, where the material remains unyielded. Asthe Bingham number increases, a bigger solid region appears closer tothe die entrance. Finally for very high Bi values only the high gradientareas near the entrance comers remain yielded. In a physical situationand for a given Bingham material, we expect the process to be reversedfrom the one shown, i.e., at very low extrusion rates (or shear rates) thematerial will be mostly unyielded; as the throughput increases and the


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YIELDED/UNYIELDED REGIONS

.;:1.0 (Bi:3.9)

2.0 ~.O 6.0 8.0 10.0

'.0 ,-,--,.----,---,-......__.---,.----,---,--,-----,

3.0

2.0

1.0

0.0

-1.0

-2,0

-3.0

• I H

~. 0 ,--,--,.----,---,---.__

3.0

2.0

1.0

0.0

-1.0

-J.O

- ..~~2l::.0-......L--l--J.---l-....-·

.;:2.0 (Bi:27.1l

0.0 2.0 ... 0 6.0 8.0 10.0

• I H

'.03.0

2.0

, .0

0.0

-1.0

-04.0-12.0

.;:2.5 (Bi:l27)

• I H

.;:2.7 (Bi:528)

2.0 ".0 6.0 B.O

c". -, ,.

~.O

3.0

2.0

1.0

0,0

-1.0

FIG. 3. Progressive growth of the unyielded zone (shaded) for entry flow of Binghamfluids in a 4:I planar abrupt contraction (11.1'" = 3).


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3.2 _-..,..-__.--__-...,.--,...--..,..-...,.---,

-+-axisym....... planar

3.01.0 1.~ 2.0 2.\YIELD STRESS, 'Ty

o.e0.0 ....- .....- ....--....- ....--.....---.....--

0.0 ••0

FIG. 4. Entrance correction vs dimensionless yield stress if for Bingham fluids flowing ina 4: I abrupt contraction (N corresponds to Newtonian result for if = 0).

shear rates get higher, the material behaves more like a fluid havingreduced solid regions until finally it flows as a Newtonian fluid (-r1= Bi = 0).

With regard to the well-known small Newtonian vortex in the reservoir, it was found that this space always remains about the same sizeand becomes an unyielded region even for small Bi numbers.

The overall pressure drop aP in the system obtained from each runcan be used to evaluate the entrance correction nen defined by

(8)fJ.p - (fJ.Pres + fJ.Po)

2'Tw

where aPres is the pressure drop obtained for fully developed Poiseuilleflow in the reservoir, fJ.Po the corresponding value obtained for thecapillary (slit), and 1"w the wall shear stress at the capillary (slit) wall.The Newtonian values were found to be 0.579 (capillary) and 0.381(slit) and compare favorably with previous results [Mitsoulis (1986a)].

The results for the two geometries are shown in Fig. 4 as a function


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3.2

c -+- axisym...~planar~

:i 2.•0

E~ 1.8U

~ 0.8

N

N

0.0H1"1 Hf 1d 1d 1d Ht

BINGHAM NUMBER, B i

FIG. 5. Entrance correction vs Bi for Bingham fluids flowing in a 4:. abrupt contraction(N corresponds to Newtonian result for Bi = 0).

of T1 and in Fig. 5 as a function of Bi. Note the virtually linear relationship with T1 and sigmoidal behavior with Bi. The linear relationshipfollows the equations

(planar) nen=0.552rj + 0.381, (9a)

(axisymmetric) nen=0.562rj + 0.579. (9b)

These results indicate that increasing the yield stress increases substantially the excess pressure losses and thus the entrance correction.Similar results have been found for power-law fluids for decreasingpower-law index [Boger et al. (1978), Mitsoulis et al. (1984a»).

Exit flow of Bingham fluids

The numerical simulations were performed for Bingham fluids forboth planar and axisymmetric geometries (slit and capillary dies, respectively). Previous results for this case have appeared in the literature[Papanastasiou (1987), Ellwood et al. (1990)] and a comparison is thuspossible, especially for extrudate swell calculations. Our results extend


12:47:48


further to calculations of excess pressure losses and a more accuratedepiction of yielded/unyielded regions.

Calculations were performed with two grids, one having 276 elements, 611 nodes, and 1274 unknown degrees of freedom [Mitsoulis(1986a)) and another denser having 460 elements, 987 nodes, and 2106degrees of freedom. Good agreement was found between the resultsfrom the two grids. Figure 6 shows the dense finite element grid used(21 nodes in the r direction) .

Calculations were carried out for different Ty values. FollowingPapanastasiou (1987), we have pursued the calculations for 0 <;;; T:< aP* (or, respectively, D<;;;Bi < 00), where the dimensionless pressuregradient AP* is equal to 3 for planar and 4 for axisymmetric geometriesin the upstream channel. Comparisons, whenever possible, with theresults of Ellwood et al. (1990) for the free surface profiles showed avery good agreement between the two works [Abdali (1991)]. The extrudate swell (or contraction in this case) as a function of T: is given inFig. 7 and as a function of Bi in Fig. 8 for both geometries (slits andcapillaries). It is seen that increasing the yield stress reduces the Newtonian value of swelling from the well-established values of 19% forplanar and 13% for axisymmetric dies [Mitsoulis et al. (1984b)], to 0%as expected for extrusion of solids (plug-flow profile throughout theflow field, hence no rearrangement near the exit from parabolic to pluglike profile). The results of Papanastasiou (1987) for planar geometriesare also shown in Fig. 7. It is interesting to note that our results reveala slight contraction below I for a certain range of yield stress. Thiscontraction reaches about 5% for planar and 2% for axisymmetric dies.Extensive numerical experiments (different grids, lower tolerance forthe norm-of-the-error, increase of m parameter in the constitutive equation from 200 to 1000) showed that these contractions were alwayspresent.

The minimum in the swelling as the Bi number increases may be atfirst sight quite puzzling since one would expect a monotonic decreaseto no swelling. To explain the appearance of the minimum it is instructive to consider the two regions, a plug-flow region (unyielded) occupying the core and the surrounding fluid region (yielded) between thecore and the wall. This is equivalent to having two fluids with differentviscosities, a very high-viscosity core and a low-viscosityouter layer. Asshown by Tanner (1980) in his inelastic theory of extrudate swell andby Mitsoulis (1986b) in extrudate swell studies for double-layer flows,a less viscous outer layer results in a reduced swelling that can go belowI, depending on the viscosity ratio and also on the feed ratio (relative


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free surface

I6 754

---

2

-4

o

.-6

T1

1

wall e~il

~==I~•-5 -3 -2-----1---- 0------8 -7

FIG. 6. Finite element grid for the die exit flow calculations and the determination ofextrudate swell.

1.20N

-+ axisym.1.J.5 ....... planar

~ N8

~1.10

!!<.. 1.06

~i

1.00

0.86

1.0 US 2.0 2.5 • 3.0YJELD STRESS. Ty

FIG. 7. Extrudate swell vs dimensionless yield stress l' for Bingham fluids extrudedthrough planar and circular dies (N corresponds to Newtonian result for l' = 0). Thedotted line corresponds to planar results of Papanastasiou (1987).


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1.20N

, .-+-axisym.,

~ planar,1.15

,

N

~......1.10s

~

~1.05

;0.85

l~ ld ld ldBINGHAM NUMBER. Bi

let

FIG. 8. Extrudate swell vs Hi for Hingham fluids extruded through planar and circulardies (N corresponds to Newtonian result for Hi= 0).

height of the core with respect to the cross section of the die). It hasbeen established [Mitsoulis (1986b)] that outer layers near the wallshrink rather than swell. When the core occupies progressively more ofthe cross section of the die (say 50% or 70% or 90%), it has beenshown that for the same viscosity ratio there is less reduction in theswelling [Mitsoulis (l986b)]. In the present case, as Bi is increased andthe plug-flow core occupies more of the flow domain, as this plugemerges the surrounding thin layers of fluid shrink and are simplyaccelerated to the (higher) velocity of the core, giving a swell ratio less


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FIG. 9. Progressive growth of the unyielded zone (shaded) in planar extrusion ftowofBingham ftuids (aJ-=3).


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0.32 _-__- .....--_-_-_-_~-_-..

••0

-e- planar

--+- axisym,

1.0 1.G 2.0 2.G... 3.0YlBLD STRESS. Ty

0.00 '-_....._-""'__...._ ....._...L...;;;;;:-........_.&..._.::III

0.0 O.G

FIG. 10. Exit correction vs dimensionless yield stress '1for Bingham fluids extruded fromcapillary and slit dies (N corresponds to Newtonian result for '1 = 0).

than unity. Thus, as Bi is increased, the swell ratio always approachesunity from below.

This behavior is in agreement with statements presented by Beverlyand Tanner (1989), who argue that at high yield stress values (orcorrespondingly very low shear rates) "the only feasible kinematic pattern is that the unyielded core continues straight through near the axisand all velocity rearrangements take place near the die wall. In this case,it is clear that no swelling can take place and, from mass conservation,a slight reduction in diameter must result." However, their insufficientlydense grid could not capture this phenomenon. Our highly dense gridscapture these changes very well as will be shown in what follows.

The corresponding progressive growth of solid (unyielded) regionsin the flow field is shown in Fig. 9 for increasing values of r: (or Bi).The shaded areas correspond to unyielded regions (solid). The fullydeveloped conditions at entry give rise to horizontal lines for the envelope of the solid region.

The case of r: = 1.6, liP* = 3 has also been shown by Papanastasiou


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0.32

-e- planar

-+- axisym.It N --'

I::l O.a.

~

I 0.18N +

~ 0.08

0.001et ul 1et 1(110-· ul 1d

BINGHAM NUMBER. B i

FIG. 11. Exit correction vs Bi for Bingham fluids extruded from capillary and slit dies (Ncorresponds to Newtonian result for Bi = 0).

(1987) in his Fig. 12(a). A point to be noticed is the difference in theyielded/unyielded regions with his previous results. There, theunyielded (shaded) region extended all the way along the centerlinefrom entry to exit. As was pointed out by Beverly and Tanner (1989),at low r1 values where the fully developed profile gives a height ofunyielded/yielded region much less than the die gap (or radius), thiscannot be the case because the rearrangement near the exit will causegradients and the material will yield. Papanastasiou's results were obtained by applying an arbitrary criterion for ti.eldin~ i.e., that the second invariant of half the rate-of-strain tensor D = ! r exceed the valueof 0.001. This also caused the determination of an incorrect entry heightfor the unyielded region, which is in disagreement with the analyticalsolution for this r1 value. Here, we have used the criterion that yieldingoccurs when the magnitude Irl of the extra-stress tensor "7 exceeds thevalue 7 y of the yield stress, a number which is not arbitrary and also notvery small or close to zero. Our results are then in agreement with thoseof Beverly and Tanner (1989), showing that yielding occurs near andafter the die exit for small 7: or Bi values. The yielded region becomessmaller as 7: (or Bi) increases and disappears only for high enoughvalues of 71, where most of the material behaves as solid in the die (see


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~.

3.2 ....-"'""'--r--"""T-.....,--,.....-"'""'--r--~

...-..- axisym.~planar

3.1I1.0 1.1I 2.0 2.\ 3.0YIELD STRESS, Ty

0.0 --....- ...........---......----.....----0.0 0.1I •.0

FIG. 12. End (Bagley) correction vs dimensionless yield stress l' in extrusion of Bingham fluids (N corresponds to Newtonian result for l' = 0).

Fig. 9). In a physical situation and for a given Bingham material, weexpect the process to be reversed from the one shown, i.e., at very lowextrusion rates (or shear rates) the material will be mostly unyielded; asthe throughput increases and the shear rates get higher, the materialbehaves more like a fluid having reduced solid regions, until finally itflows inside the die as a Newtonian fluid and only in the extrudate,where the gradients are zero, it behaves like a solid. This behavior wasshown by Beverly and Tanner ( 1989) for increasing shear rates r in therange 1<;;; r<;;; 100 s - I (or equivalently 4.4 < Hi< 0.44) for a certain viscoplastic material.

The overall pressure drop ap in the system obtained from each runcan be used to evaluate the exit correction nex defined by

(10)

where aPois the pressure drop for fully developed Poiseuille flow in thedie channel and T w the wall shear stress. The Newtonian values were


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3.2

• ......... axisym.l:l

I__ planar

2••

f;lIS

UI0

~Ng 0.8

~N

1'&1

0.010'"1 Id Id Id Id let

BINGHAM NUMBER, Bi

FIG. 13. End (Bagley) correction vs Bi in extrusion of Bingham fluids (N corresponds toNewtonian result for Bi = 0).

found to be 0.248 (capillary) and 0.154 (slit) and compare favorablywith previous results [Mitsoulis (1986a), Mitsoulis et a/. (1984b)].

The results for the two geometries are plotted in Fig. 10 as a functionof '1 and in Fig. II as a function of Bi. It is interesting to see that forsmall values of '1 or Bi, there is a slight increase of nex' but afterwardsthe extra pressure losses decrease to finally reach zero values as Bi- 00,

as expected since a plug-like profile would generate no extra pressurelosses. This is also the case for power-law fluids as the power-law indexn approaches zero [Mitsoulis et 01. (1984a)]. However, the overshootfor low values of Bi is unexpected and reminds us of the reduction of theNewtonian extrudate swell for low values of the Weissenberg numberwhen using the Maxwell model [Tanner (1985)].

The sum of entry and exit pressure losses gives rise to the end (orBagley) correction [Boger et 01. (1978)], i.e.,

nB=nen + nex' (11)

The results for Bingham fluids as a function of '1are given in Fig. 12and as a function of Bi in Fig. 13. Note the virtually linear relationship


12:47:48


with '1and sigmoidal behavior with Bi, The linear relationship followsthe equations:

(planar) nB=0.4767; + 0.535,

(axisymmetric) nB=0.5877; + 0.827.

(12a)

(12b)

Thus, Bingham fluids exhibit an increase in the end (Bagley) correctionproportional to the yield stress, which is responsible for substantiallyincreasing the excess pressure losses in extrusion through dies.

CONCLUSIONS

Finite element simulations have been undertaken for the entry andexit flow of Bingham fluids through capillary and slit dies. The constitutive equation used has been proposed earlier by Papanastasiou (1987)and is valid uniformly in yielded and unyielded regions, thus eliminatingtracking the location of yield surfaces. The present results confirm earlier studies about reduction of extrudate swell as the Bingham numberincreases. Furthermore, the extent and shape of yielded/unyielded regions has been accurately captured using the criterion of the magnitudeof the extra stress tensor exceeding the yield stress.

New results include the determination of the entrance, exit, and end(or Bagley) correction as a function of a dimensionless yield stress 71 orBingham number (Bi). The entrance correction increases as well as theBagley correction, while the exit correction goes to zero as Bi -+ 00.

Linear equations are given for the Bagley correction as a function of

'f1·

ACKNOWLEDGMENTS

Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged. Mr.Abdali is sponsored by the Ras Lanuf Oil Company. Most of the workwas done while one of the authors (E. Mitsoulis) was visiting professorat NTUA in Greece.


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ReferencesAbdali, S. S., "Numerical Simulation of Viscoplastic Material Flow through Extrusion

Dies," M.A.Sc. Thesis, Dept. Chern. Eng., Univ. of Ottawa, Canada, 1991.Beris, A. N., J. A. Tsamopoulos, R. C. Armstrong, and R. A. Brown, "Creeping Motion

of a Sphere through a Bingham Plastic," J. Fluid Mech. 158,219-244 (1985).Beverly, C. R. and R. I. Tanner, "Numerical Analysis of Extrudate Swell in Viscoelastic

Materials with Yield Stress," J. Rheo\. 33, 989-1009 (1989).Bingham, E. C., Fluidity and Plasticity (McGraw-Hill, New York, 1922).Bird, R. 8., G. C, Doi, and B. J. Yarusso, "The Rheology and Flow of Viscoplastic

Materials," Rev. Chern. Eng. I, 1-70 (1983).Boger, D. V., R. Gupta, and R. I. Tanner, "The End Correction for Power-Law Fluids in

the Capillary Rheometer," J. Non-Newt. Fluid Mech. 4, 239-248 (1978).Dzuy, N. Q. and D. V. Boger, "Direct Yield Stress Measurement with the Vane Method,"

J. Rheo\. 29, 335-347 (1985).Ellwood, K. R. J., G. C. Georgiou, T. C. Papanastasiou, and J. O. Wilkes, "Laminar Jets

of Bingham-Plastic Liquids," J. Rheo\. 34, 787-812 (1990).Gartling, D. K. and N. Phan-Thien, "A Numerical Simulation of a Plastic Fluid in a

Parallel-Plate Plastometer,' J. Non-Newt. Fluid Mech. 14, 347 (1984).Keentok, M., J. F. Milthorpe, and E. J. O'Donovan, "On the Shearing Zone around

Rotating Vanes in Plastic Liquids: Theory and Experiment," J. Non-Newt. FluidMech. 17,23 (1985).

Lipscomb, G. G. and M. M. Denn, "Flow of Bingham Fluids in Complex Geometries,"J. Non-Newt. Fluid Mech. 14, 337 (1984).

Mitsoulis, E., "The Numerical Simulation of Boger Fluids: A Viscometric ApproximationApproach," Polym. Eng. Sci. 26, 1552-1562 (1986a).

Mitsoulis, E., "Extrudate Swell in Double-Layer Flows," J. Rheol, 30, S23-S44 (1986b).Mitsoulis, E., J. Vlachopoulos, and F. A. Mirza, "Numerical Simulation of Entry and Exit

Flows in Slit Dies," Polym. Eng. Sci. 24, 707-715 (1984a).Mitsoulis, E., J. Vlachopoulos, and F. A. Mirza, "Simulation of Extrudate Swell from

Long Slit and Capillary Dies," Polym. Proc. Eng. 2, 153-177 (1984b).O'Donovan, E. J. and R. I. Tanner, "Numerical Study of the Bingham Squeeze Film

Problem," J. Non-Newt. Fluid Mech. 15, 75 (1984).Papanastasiou, T. c, "Flow of Materials with Yield," J. Rheo\. 31, 385-404 (1987).Tanner, R. I., "A New Inelastic Theory of Extrudate Swell,' J. Non-Newt. Fluid Mech.

6, 289-302 (1980).Tanner, R. I., Engineering Rheology (Clarendon, Oxford, 1985).


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Entry and exit flows of Bingham fluids

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