Experimental investigations and numerical simulations for an openchannel flow of a weak elastic polymer solution around a T-profile

C. Balan, V. Legat, A. Neagoe, D. Nistoran

Abstract The present paper is concerned with experi-mental and numerical investigations of planar complexflows of ‘‘weak’’ elastic polymer solutions (whose con-centration are below the critical overlap concentration),characterised by small relaxation times (k<0.1 s) and al-most constant shear viscosities for small and mediumshear rates. The main aim of the study is to detect to whatextent a very small amount of elasticity present in a vis-cous fluid can influence its behaviour in complex flows,without introducing major modifications of classical rhe-ological tests. The samples are polymer solutions of lowPIB molecular weight dissolved in highly viscous Newto-nian mineral oil. The analysed motion is steady, and takesplace in an open channel around a ‘‘T’’ profile. Maximumvalues of the characteristic parameters for the experi-ments, the Reynolds and Weissenberg numbers, were 45and 0.1, respectively. The experiments show a decrease ofthe wake length downstream the profile for weak elasticsolutions in comparison to the Newtonian solvent. Actu-ally, the same wake length as in the Newtonian case wasobtained for tested polymer solutions, but at higher Renumbers. Numerical simulations using the Giesekus modelpredict the same behaviour and are consistent with

experiments from both qualitative and quantitative pointof views. The results of research conclude that, even insmall amounts, the presence of elasticity in pure viscousliquids induces quantitative changes from Newtonian flowin complex dominant elongational flows, at elongationalrates for which the sudden thickening of extensional vis-cosity is remarkable. The study is important, since itshould enable better understanding and modelling of vis-coelastic flows that involve dilute polymer solutions, orfluids with similar rheology; biofluid mechanics being onearea of application of this research. Corroboration ofexperimental flow visualization with numerical simulationis currently a feasible method used to characterise weakelastic polymer solutions, since classical rheological tech-niques generally fail to obtain realistic values of relaxationtime for these particular viscoelastic fluids.

1IntroductionMuch of the recent research in the field of the rheology ofpolymer solutions has focused on establishing an effectiveprocedure that can be used to experimentally andnumerically test the validity of constitutive relations incomplex flows. A number of dissertations (see for exampleMcKinley 1991; Williams 1997; Ringenbach 1996;Schoonen 1998), and numerous papers have been dedi-cated to this subject, and have analysed the correlationbetween experimental data and numerical simulations forsome of the most common constitutive relations (Oldroyd-B, Giesekus, Phan Thien Tanner, K–BKZ and FENE-Pmodels).

The samples investigated are well-defined viscoelasticfluids (Boger, M1, S1, A1); their material constants areusually determined from classical viscometric tests, andoccaisonally from elongational experiments as well. Thestandard test flow geometries are either planar or axially-symmetrical contractions or expansions (Boger et al. 1992;Quinzani et al. 1994, 1995; Azaiez et al. 1996; Baaijens1993; Binding and Walters 1988; Purnode and Crochet1996; Yoo and Na 1991; Phillips and Williams 2002) andflows around cylinders (Baaijens et al. 1994; Baaijens 1995;Barakos and Mitsoulis 1995; Carew and Townsend 1991;Chen et al. 1995; Hu and Joseph 1990; Koniuta et al 1979;Dhahir and Walters 1989; McKinley et al. 1993). Usingdifferent visualization methods or birefringence and NMR-based techniques, the velocity and stress distributions inthe flow field are determined experimentally. Then, fol-lowing various numerical schemes, the combined equation

Experiments in Fluids 36 (2004) 408–418

DOI 10.1007/s00348-003-0727-5

408

Received: 14 May 2003 / Accepted: 26 September 2003Published online: 29 November 2003� Springer-Verlag 2003

C. Balan (&), A. Neagoe, D. NistoranHydraulics and Hydraulic Machinery Department – REOROM Group,University ‘‘Politehnica’’ of Bucharest, Splaiul Independentei 313,79590 Bucharest, RomaniaE-mail: balan@hydrop.pub.roTel.: +40-21402-9705Fax: +40-21402-9865

V. LegatUniversity of Louvain-la-Neuve–Center for Systems Engineering Ap-plied Mechanics (CESAME), Batiment Euler, Av. Georges Lemaıtre 4,1348 Louvain-la-Neuve, Belgium

Corneliu Balan dedicates this paper to the anniversary of onehundred years from the birth of Academician Dumitru Dumit-rescu (1904–1983), charismatic personality of the Romanianschool of fluid mechanics.The authors wish to thank Professor Ken Walters for the helpfulsuggestions and advice regarding experiments, and for allowingMrs. D. Nistoran to use the laboratory facilities of the Universityof Wales, Aberystwyth. The authors also acknowledge the supportand useful remarks regarding the paper of Professor RolandKeunings from University of Louvain-la-Neuve and ProfessorValeriu Panaitescu, the head of the Boundary Layer and Turbu-lence Laboratory from the ‘‘Politehnica’’ University of Bucharest.

of motion and constitutive relation are solved for the sameconfigurations (usually the motion is consideredisochoric) and the results of the simulations are comparedwith experimental patterns of velocity distributions (orstreamlines) and stresses at the same Reynolds andWeissenberg (Deborah) numbers.

The majority of this work is a search for explanations ofthe differences observed between experiments and simu-lations, and inferences are suggested regarding the inad-equacy of the constitutive formulation for the problem athand. Limitations of the numerical procedures at highWeissenberg numbers is also evident, but in most of thepapers mentioned above, the influence of Reynolds num-ber is not discussed, since it is normally considered to bemuch less than one, so the convective acceleration in theequation of motion is neglected.

In this work, a constitutive relation (and its domain ofapplication) for modelling the rheological behaviour ofmaterial in various flow geometries is proposed. The re-sults are then extended to similar three-dimensional con-figurations which are normally encountered in industrialapplications.

From the analysis of the published papers on the sub-ject, one can summarize some important conclusions:

1. One single constitutive relation is not able to properlymodel the whole range of Weissenberg (Wi) numbersindependent of the Reynolds (Re) number. Criticalvalues of the controlling parameters (Wi and Re num-bers) always exist beyond which each model loses itsvalidity.

2. The fundamental requirement of the constitutive rela-tions theory -- that boundary conditions (the type ofmotion and flow geometry) do not affect the constitu-tive relation of the material -- is not always easy toenforce. For a particular constitutive relation, thematerial constants found in viscometric flows shouldhave the same values in elongational flows as well, in-deed for any complex flow. Unfortunately, for manypolymer solutions this is not true. For example, thevalue of relaxation time, k, predicted from viscometricdata (usually k<0.1 s) is much less than the value ob-tained from elongational measurements, especially fordilute or ‘‘weak’’ elastic polymer solutions1.

3. The first approximation of any steady viscoelasticcomplex flow of low or medium elasticity is given bythe Newtonian model (the Navier-Stokes solution); thesecond approximation is given by the coupling of theCauchy momentum balance equation with a linearviscoelastic constitutive relation, the most commonmodel being the Oldroyd-B relation.

Therefore, it is assumed that the symmetric extra-stressof the polymer solution, T, is a linear combination of the

Newtonian solvent contribution, Ts, and the polymeric

contribution, 2Tp ¼Pn

i¼1

2Tpi; characterized by n relaxation

modes:

T ¼ Ts þ Tp; ð1Þ

where

Ts ¼ 2gsD; ð2Þ

and

kiTr

piþ Tpi

¼ 2gpiD; ð3Þ

Here D is the stretching tensor, gs is the viscosity of thesolvent, gpi and ki (i= 1, 2, ..., n) are the viscosity coeffi-cients and relaxation times of the polymer, respectively.Equation 3 represents the differential Maxwell constitutiverelation, with upper convected objective tensor derivative

Tr

p ¼ T�

p � LTp � TpLT: ð4Þ

It is well known that the system (1) ‚ (3), with onesingle Maxwell model, is equivalent to the Oldroyd-Bconstitutive equation

k1Trþ T ¼ 2g0Dþ 2g0k2D

r; ð5Þ

where g0 ¼ gs þ gp is the zero shear viscosity coefficient,and k1 ¼ k and k2 ¼ gs

g0k are the relaxation and the retar-

dation time, respectively.In principle, any other constitutive model of differential

or integral type can be used instead of the Maxwell models(Eq. 3), for example the Giesekus model:

I þ akgp

Tp

" #

Tp þ kTr

p ¼ 2gpD ð6Þ

(here k is the relaxation time and a is the mobility factor;for more details see Barnes et al. 1989 and Larson 1999).

In some studies (Schoonen 1998) it has been observedthat some constitutive relations fit the complex kine-matics (velocity and streamline distributions) reasonablywell but fail to obtain the measured stress distribution.There is also experimental evidence that for some non-symmetric flow geometries, the visualised flow pattern isproperly simulated numerically for different values of thecontrolled parameters than the values found in the lab-oratory. In the authors� opinion, reasons this happenedmay include: (i) the constitutive relation was not properlychosen; (ii) the flow depends on an unknown combina-tion of Re and Wi numbers (or some additional con-trolled parameters); (iii) the scale factors associated withthe process are difficult to establish (for details, seeMcKinley et al. 1996).

Other experimental evidence is equally important forthe study of viscoelastic flows at small or medium Winumber: (i) the elasticity does not significantly influencethe streamline patterns, but does have a remarkable effectupon the stress distribution (Carew and Townsend 1991;Quinzani et al. 1994); (ii) the presence of local flowanomalies around sharp corners depends on elasticity

1In a weak polymer solution the polymer concentration c is belowthe critical overlap concentration c*. Harrison et al. (1998) cor-relate the presence of elasticity with the polymer concentrationand define ultradilute and dilute solutions as ones for whichc/c*<0.1 and 0.1<c/c*<O(1), respectively (in the case of a weakpolymer solution the molecular weight of the polymer is alsoassumed to be low).

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properties and may indicate ‘‘stress singularities’’ (Waltersand Webster 1982; Baaijens 1993; Quinzani et al. 1994);(iii) the elasticity influences the stability of the flow(Larson 1992; Shaqfeh 1996; McKinley et al. 1996).

The present study is focused on experimental investi-gation of the flow of a weak elastic polymer solution (inother words k<0.1 s and Wi<0.1) in complex planar flows.The goal of the work is to either validate or disprove thehypothesis that the presence of elasticity in planar com-plex flows of a weak elastic polymer solution does notqualitatively affect the stationary flow field in comparisonto that of the pure viscous flow, but does influence theevolution of the flow separation areas and vortex struc-tures.

For a given planar flow configuration, the streamlinepattern of a pure viscous fluid at a certain Reynoldsnumber, Re0

1[1, is very similar to the streamline pattern ofa viscoelastic fluid with low elasticity at a higher Reynoldsnumber Re1[Re0

1. This means that the presence of a smallamount of low molecular weight polymer in a viscousliquid directly influences the onset and evolution ofseparation areas and eddy structures in complex flows, inthe sense that the elasticity has a stabilising effect on theinertial instability of the planar flow (contrary to highelasticity solutions, in which the elasticity might have adestabilizing role, especially in flows with curved stream-lines, see Larson 1992 and McKinley et al. 1996).

This issue is also consistent with the phenomenon ofdrag reduction in pipes (Kulik 2001) or in rotationalgeometries (Koeltzsch et al. 2003), where a very smallamount of polymer is added to a viscous fluid in order toextend the laminar flow regime at higher Reynoldsnumbers (see also Nieuwstadt and den Toonder 2001;Cressman et al. 2001). In particular, we are looking forpossible applications of this work in the field of biofluidmechanics, especially in the modelling of blood flow inlarge branching vessels (in this flow geometry the bloodbehaves like a weak elastic fluid, see Vlastos et al. 1997;Balan et al. 2003).

2Experimental

2.1Rheometry of the polymer solutionsIn this study, elastic properties of the test liquids wereobtained by adding a non-toxic low molecular weightpolyisobutilene (PIB, Vistanex LM-MH, withMw=5.5·104 g/mol and a density of qp=910 kg/m3) to aNewtonian solvent (mineral oil with densityqs=900 kg/m3), in mass concentrations up to 10%.Vistanex has the advantage of possessing elastic proper-ties even at room temperature, so that experiments canbe performed without the need for complex heating ortemperature control devices. To prepare the solutions,small amounts of PIB were first dissolved in a smallvolume of oil via continuous mixing at a constanttemperature of approximately 70 �C. The concentratedsolutions obtained were diluted down to the desiredconcentrations of 1 to 10% and then mixed for more than20 days.

The variations of the elastic modulus G¢ and lossmodulus G¢¢ as a function of concentration and frequencyare shown in Fig. 1a, and in Fig. 1b the correspondingdynamic viscosity g¢ is shown.

The average value for critical (overlap) concentrationwas established at c*@2.9% for the analysed solutions (fordetails see Nistoran 1999, Schurtz 1991, and Bird et al.1980). One can observe that the shear-thinning characterof PIB solutions is remarkable only for shear rates (oroscillatory frequencies) greater than 10 s-1 and concen-trations beyond the overlap concentration, in other wordswhere c>c*. Using the Time-Temperature SuperpositionPrinciple, master curves of the material functions at thereference temperature h0=25 �C were obtained for thesolvent, solutions, and polymer in simple motions (vanGurp and Palmen 1988). From these curves the zero shearviscosity coefficient g0 and the relaxation time k wereidentified, according to the formulae

g0 ¼ limx!0

G00

x; k ¼ lim

x!0

G0

g0x2; ð7Þ

see Table 1 and Barnes et al. (1989) for details.

Fig. 1. Rheology of polymer solutions, as a function of polymerconcentration; dynamic test, cone and plate configuration: adynamic moduli (elastic modulus G¢, viscous modulus G¢¢); bdynamic viscosity g¢

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Considering the purpose of this paper, investigationswere focused on the flow behaviour of the 2% polymersolution in planar, complex motions. For this concentra-tion and analysed flow regime, the shear thinning char-acter is not relevant; the ratio between infinite shearviscosity and zero shear viscosity is over 0.85, the viscosityfunction being almost constant for shear rates less than100 s–1, see Fig. 1b (the same variation of viscosity withsteady shear rates was obtained in simple shear motion,see Nistoran et al. 2000).

Furthermore, even if the presence of elasticity wasproved through classic experiments like the Weissenbergeffect, reliable values for the first normal stress difference,N1, have not been measured except for concentrationswhere c‡30%, (Nistoran 1999). The same conclusion wasdrawn after performing the other usual viscometric tests inoscillatory and transitory motions (Quinzani et al. 1990;Ferguson and Hudson 1994): for concentrations less thanc*, commercial rheometers were not able to measure sig-nificant differences in the characteristic material functionsbetween Newtonian solvent behaviour and the weak elasticpolymer solutions (see also Kalashnikov 1994). At thisstage of the work we haven�t performed tests based onpure elongational flow. Extensional rheological tests basedon the filament technique seem to be the only ones withpotential to reveal differences in the flow behaviour be-tween pure viscous fluid and weak elastic fluid (McKinleyand Sridhar 2002). However, accurate measurements ofsimple extensional flows of dilute polymer solutions arevery difficult to obtain with usual experimental techniques(Jones 1989; Dontula 1997; Rothstein and McKinley 2001;Lindner et al. 2003). So far, only qualitative estimates ofelongational viscosity ge have been obtained for solutionsof PIB with higher molecular weights than the testedsamples in this study: Mw>106 g/mol (Hudson and Jones1993; Ferguson and Hudson 1994; Willenbacher andHingmann 1994; Park et al. 1994; Ooi and Shridhar 1994;Ng et al. 1996). These studies showed that ge valuesincrease sharply for deformation rates _e[_ecr, independentof the constituent type or the molecular weight of thepolymer (the critical value of the elongation rate, _ecr wasfound to be of O(10 s-1)). On this issue, Walters andWebster (1982) and Walters and Jones (1988) stated thatthe presence of elasticity has an important influence uponthe fluid behaviour in complex flows with a significantelongational component, at extensional rates exceeding thecritical value. A ‘‘hidden elasticity’’ of the weak elasticsolutions is also expected to be found in extensional flows,undetectable in viscometric flows. Therefore, the visco-elastic fluids under investigation might be characterised bya larger relaxation time in elongational-dominated

complex motions, than is detected in oscillatory visco-metric flows (but it should not exceed 0.1 s).

At the end of this section we conclude that the 2%polymer solution is characterised by the following materialconstants at 25 �C: density q=900 kg/m3, viscosityg0=1.2 Pa s (solvent viscosity gs=1.2 Pa s), and the valueof relaxation time is somewhere in the interval0.0015 s<k<0.1 s.

2.2Experimental set-upFor the present work, a complex planar motion with adominant elongational component was chosen: the motionaround a ‘‘T’’ profile, with a contraction ratio of 2:1 in thechannel. The test profile was also characterized by sharpcorners, so (according to Hinch 1993, and Davies andDevlin 1993) unusual effects in the vicinity of the sharpedges of the profile are to be expected. The experimentalset-up consists of an open channel with a visualisationzone of approximately 1 m length, width of 0.23 m anddepth of 0.35 m, see Fig. 2. The total volume of viscoelasticfluid re-circulated in the channel is approximately 0.4 m3.

Table 1. Material constants for PIB solutions, at referencetemperature h0=25 �C and different polymer concentrations,obtained by fitting the oscillatory shear data

Concentration (%) g0 (Pa s) k (s)

0 0.5 01 0.7 0.0012 1.2 0.00155 3.5 0.0077

Fig. 2. Geometry of the channel and flow configuration: a generalview of the channel; b flow configuration (the channel dimensionswere B=0.23 m, H=0.35 m). L is the length of the wake (in otherwords separation area) downstream the ‘‘T’’ profile (the openchannel is located at the Boundary Layer and TurbulenceLaboratory of the ‘‘Politehnica’’ University of Bucharest)

411

The re-circulation of the fluid is obtained from rotationof an open transversal impeller with fixed geometry. Thetranslated flow rate depends only on the rotational speedof the impeller, and is not influenced by the presence andposition of the ‘‘T’’ profile. Channel geometry and flowcontrol devices ensure a parabolic velocity profile acrossthe channel width in a cross section sufficiently upstreamof the visualisation zone, see Fig. 3.

Only steady-state flow conditions were investigated.Velocity field measurements were performed by the PhotoTracer Method (PTM), an experimental procedure basedmainly on Particle Image Velocimetry (Gilbert and John-son 2003) and Particle Tracking Velocimetry (Devasena-thipathy et al. 2003), without making use of expensiveequipment. The individual small tracer particles (locatedin the flow field, on the liquid surface) were photographedfor a fixed exposure time. The length of the trace dividedby the exposure time is the local velocity of the steadyisothermal motion (care was taken to continuously mon-itor and control the temperature during the photographingsessions; differences of less than 0.5 �C were observed

during an experimental session). Photographs of the flowfield in laminar motion were taken at different flow ratesfor the Newtonian solvent and for the 0%, 1%, 2% and 5%polymer solutions. The exposure time texp was adjusted tothe flow rates and the average velocities of the vortexes(separation flow areas) formed in the wake of the profile.Therefore, if Lf is the length of the particle trace on the filmfor exposure time texp, at any distance y from the channelaxis Ox, the local velocity is

vxðyÞ ¼Lf

texp; ð8Þ

(see Dumitrescu and Cazacu 1970 for details).As Baaijens et al. (1994) and Quinzani et al. (1995)

predicted, upstream of the obstructed cross-section thevelocity distributions of the polymer solution in a channeldo not differ significantly from the Newtonian parabolicones (in our case, the deviations near the wall from theparabolic shape are assumed to be due to measurementerrors, and the influence of the downstream channelgeometry, see Fig. 3). Subsequently, the mean velocity Vm

at the surface has been calculated using the well-knownrelationship for Newtonian fluid laminar motion in planarchannels (Oroveanu 1967):

Vm ¼2

3Vmax ð9Þ

where Vmax is the maximum velocity, measured in thedirection of the 0x axis of the channel for y=0, see Fig. 4.The characteristic Reynolds and Weissenberg numbers ofthe flow are defined as

Re ¼ qVmB

g0

; Wi ¼ kVm

Bð10Þ

where B=0.23 m is the channel width.The maximum measured velocity at 2% concentration

was Vmax@0.43 m/s, which corresponds to Re@45. Photo-graphs were taken in perfect stationary laminar motion atdifferent Re numbers and corresponding Wi numbers lessthan 0.1. Special attention was given to observing the flowin the vicinity of the channel bottom. In this region, the

Fig. 3. Non-dimensional velocity distribution, measured at adistance of 0.3 m upstream of the ‘‘T’’ profile (2% polymerconcentration in Newtonian solvent)

Fig. 4. Experimental velocity distributionaround the profile at Re@45 and 2%polymer concentration in Newtonian sol-vent

412

motion is three-dimensional, but the flow at the free sur-face remains planar for the whole range of tested veloci-ties; no influence of three-dimensional flow structures wasobserved at the free surface of the channel for ratioH/B>0.5 (here H is the height of the fluid in the channel;the experiments were performed at H/B>0.8, see Fig. 2b).At relatively high Re numbers, Re>30, the planar freesurface was locally curved (due to inertia) in the vicinity ofthe profile�s corners; see also Boger (1987).

Following the described method, we obtained thestreamline pictures of the flow around the profile for afixed Reynolds number. Assuming a parabolic profile ofvelocity upstream from this profile, the values of 10streamlines were computed (for the case of symmetric flowgeometry). By overlaying a grid on the picture, see Fig. 4,the velocities in the nodes of the network were computedusing a finite difference scheme.

Using this graphic procedure the shear rates _c andelongational rates _e in the direction of the channel axiswere calculated; as expected, their maximum values oc-curred in the cross-sections near the edges of the ‘‘T’’profile, namely ‘‘S–1’’ and ‘‘S+1’’ (in these regions we alsoregistered an increased local circulation of the viscoelastic

flows, in comparison with the Newtonian flow at the sameReynolds numbers). As one can see from Table 2, themaximum extensional rates have the same order as theassumed critical value _ecr at which the elongational vis-cosity of the PIB solutions has a sudden increase and,consequently, the elongational effects dominate the shearcomponent of complex motion.

2.3Experimental results: visualisationsThe photographed stream lines of the Newtonian and 2%PIB solution are shown at various Re numbers in Fig. 5,around the ‘‘T’’ profile placed at 0� inclination with respectto the flow axis. In Fig. 6 we show the experimentalstreamlines at various profile inclination angles.

Numerical simulation for the Newtonian fluid (Navier-Stokes equation of motion at Re=45), performed usingFLUENT code, was compared with the experimental pic-ture in Fig. 7; the streamline patterns and the length of thewake are very similar in both cases. The obtainednumerical results for the Newtonian fluid are consistentwith experiments for all tested Re numbers and planargeometries (see also Nistoran 1999; Nistoran et al. 2000).

Table 2. Maximum experimental shear and elongational rates for the Newtonian solvent and the 2% PIB solution, at two different flowrates corresponding to Re @ 30 and Re @ 45, respectively (symmetric distance upstream and downstream of the profile, see Fig. 4)

Newtonian mineral oil 2% PIB in mineral oil

_c _e _c _e

upstream downstream upstream downstream upstream downstream upstream downstream

35 s-1 46 s-1 25 s-1 27 s-1 35 s-1 40 s-1 15 s-1 17 s-1

40 s-1 52 s-1 20 s-1 20 s-1 39 s-1 43 s-1 10 s-1 13 s-1

Fig. 5. Comparison betweenexperimental streamlines pat-terns around the symmetric‘‘T’’ profile for Newtonian fluidand weak elastic solution (2%polymer concentration inNewtonian solvent) at differentReynolds numbers

413

Comparison between the visualised flow fields fromFig. 5 shows the influence of elasticity on the flowseparation area and vortex geometry downstream of the‘‘T’’ profile. One can see that a small amount of elas-ticity in a Newtonian solvent delays the onset of flowseparation downstream of the profile and limits the rateof increase of the vortex length L with Reynoldsnumber. Therefore, the vortex geometry and streamlinedistributions are preserved in weak elastic polymersolutions at higher Reynolds numbers than in aNewtonian fluid.

3Numerical simulationsThe purpose of the simulations was to investigate thebehaviour of various shear thinning generalised Newto-nian fluids and viscoelastic models in the experimentallytested symmetric ‘‘T’’ profile flow geometry, up to Re=50.The aim of the numerical simulation was to prove thatanomalous phenomena observed in the flow of weakelastic polymer solutions are consequences of elasticityand not due to the shear thinning character. We were alsolooking to compare the models to experimental data, todetermine which viscoelastic model fits reality best; inother words, which one best simulates the decreasinglength of vortexes downstream of the ‘‘T’’ profile as elas-ticity in the fluid is increased, while maintaining very lowrelaxation times, k<0.1 s. Also, using the numerical com-putations we were looking to determine a more realistic

value for the relaxation time than the one calculated fromviscometric oscillatory flows.

Numerical simulations were performed in steady, 2-Dflow using the commercial software POLYFLOW, whichsimultaneously solves the equation of motion in Cauchyform and the constitutive relation for the polymer extra-stress, Eqs. 3 and 5, respectively. Standard mesh refine-ment analyses were performed in order to check theaccuracy of the computational grid, especially in thevicinity of the corners where the numerical algorithms forviscoelastic models are very sensitive. The two methodsused to compute the planar velocity field were the EVSSmethod (Elastic-Viscous Stress-Splitting) and the SUPGmethod (Streamline Upwind Petrov-Galerkin); for moredetails refer to Sun et al. (1999), Matallah et al. (1998), andWarichet and Legat (1997).

The simulation performed with Bird-Carreau and Crossgeneralised Newtonian models proved that for a solutionof 2% concentration the differences in comparison toNewtonian flows are insignificant (the values of thematerial constants from the generalised Newtonian modelswere determined by fitting the shear viscosity function, seeTudor et al. 2002). Same qualitatively results have beenobtained for the viscoelastic Oldroyd-B model (Eq. 5), aslong as the relaxation time is set to an upper limit of 0.1 s,and the Weissemberg number is less than 0.1.

The best modelling of experimental data, from both aqualitative and a quantitative point of view, was obtainedwith one Giesekus model, see Eqs. 2 and 6. This result wasexpected, since the Giesekus model predicts finite butremarkable thickening of elongational viscosity, even atsmall extensional rates; therefore Giesekus simulations of

Fig. 6. Experimental streamlines distribution at different orien-tation angles of the ‘‘T’’ profile (Re@ 45 and 2% polymerconcentration in Newtonian solvent). The flow configurationbehind the profile is highly dependent on the angle (the anglemeasured relative to the channel axis decreases from the top; the‘‘0’’ angle is the symmetric profile, see Fig. 5)

Fig. 7. Comparison between experiment and numerical simula-tion for the Newtonian fluid (symmetric ‘‘T’’ profile, Re=45,FLUENT code)

414

complex flows with dominant extensional character werefound consistent for many experimental cases (Schoonen1998).

Numerical simulations using the Giesekus model withadded Newtonian (solvent) viscosity were performed withthe following constant values of material parameters:a=0.0001 (chosen to be a very low value because of theslightly shear thinning character), gp=0.7 Pa s, andgs=0.5 Pa s for the Newtonian solvent, and various valuesof relaxation time k.

In Fig. 8 and Fig. 9 one can observe that at constantReynolds number (Re=30, respectively Re=10) the wakelength downstream of the profile decreases with increasingrelaxation time, see also Table 3.

The best fit of the experimental data presented in Fig. 5was obtained with the value of relaxation time k=0.06 s;for comparison between experiments and numerical sim-ulations see Fig. 10 and Fig. 11 (note that the Weissenbergnumber that corresponds to Re=45 is Wi=0.07).

4ConclusionsIn this work we have shown that it is impossible to getevidence of viscoelastic behaviour at low values of elas-ticity in stationary simple, viscometric motions, repro-duced by rotational and capillary rheometers. For polymersolutions with concentration below the critical value thefirst normal stress difference, N1, is not detectable and themeasurements of the elastic modulus G¢ are not relevantdue to the limit of the rheometer sensitivity. As stationarypure elongational motions are very difficult to obtainexperimentally, the only possible way to test constitutiverelations is through complex flows with strong elonga-tional components (where convective acceleration be-comes important). The paper brings a relevantcontribution to the study of weak elastic complex flows inan open channel around a ‘‘T’’ profile. This is an exampleof a sudden contraction with sharp edges, which causeselasticity to show its influence on flow separation andweak structure distinctively, even for very diluted polymersolutions. At the same time, an important decrease in thewake length is observed for weak elastic solutions whencompared with Newtonian solvent. Actually, the samewake length as in the Newtonian case was obtained forpolymer solutions, but at higher Re numbers.

Therefore, the presence of elasticity in viscous fluidshas a stabilizing hydrodynamic effect if polymer concen-tration is below the critical one.

The results of the study are found to be in excellentagreement with the affirmations of others researchers(Walters and Webster 1982; Walters and Jones1988; Har-rison 1998) concerning differences appearing in thebehaviour of viscoelastic extension-dominated (contrac-tion) flows compared to those exhibiting Newtonianbehaviour. As the shear viscosity is almost constant for the

Fig. 8. Numerical simulations of viscoelastic flows in the channelat Re=30 for different values of relaxation time k (Giesekusmodel)

Fig. 9. Numerical calculation at Re=10for the velocity component Vx in themiddle of the channel, behind the sym-metric profile, as a function of therelaxation time k and distance (x=0defines the profile). For k>0.09 s there isno re-circulated area downstream of theprofile (in other words Vx is positive atany x>0)

415

whole range of shear rates, and therefore shear-thinningeffects in the contraction are negligible, the only elasticeffect that can produce these differences has to be elon-gational viscosity, whose behaviour is sudden extensionalthickening at rates greater than a critical rate.

Very recently, similar conclusions to those from thepresent work have been obtained by Oliveira (2003).Numerical simulations with the FENE-CR model for dilutepolymer solutions in sudden 1:3 expansion flow show apostponement of the bifurcation point in the flow diagramat higher Re numbers, in comparison with the Newtonianflow.

The investigated complex flow can also be used to de-tect the ‘‘real value’’ of relaxation time, which is notapparent in simple shear motions. In the case underinvestigation, the relaxation time found in complex mo-tion with a strong extensional component is more than anorder of magnitude higher than the value found fromoscillatory shear flows (k=0.06 s, in comparison tok=0.0015 s).

We can therefore conclude that using visualizationtechniques in conjunction with numerical simulations ofcomplex flows is the best method to more realisticallycharacterise the rheology of weak elastic polymer solu-tions, since rheometry based on viscometric or elonga-tional motions cannot produce reliable values forrelaxation time and extensional viscosity, respectively.

The rheology and modelling of weak elastic flows isstill a challenge for scientists, but an important subjectto study, due to its multiple applications in industrialprocesses and daily life. Almost all bio-fluids flowinginside living bodies or plants can be considered to be, tothe first rheological approximation, dilute polymersolutions or ‘‘weak’’ elastic solutions. This is the reasonwhy the present and future work of the group is focusedon the investigation of the flows of similar polymersolutions in branching architecture and bifurcations, asthese configurations are the most common for the hu-man body and plants (for more on this, see Broboanaet al. 2003).

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Table 3. Variation of the relative wake lengthL/B as a function ofRe and relaxation time k; numerical simulations with POLYFLOWcode, Giesekus model used (a=0.0001, gp=0.7 Pa s,andgs=0.5 Pa s)

Re k=0 k=0.02 s k=0.03 s k=0.04 s

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Fig. 10. Comparison between experiments (2% polymer concen-tration in Newtonian solvent) and numerical simulations forviscoelastic flow around the symmetric ‘‘T’’ profile; Re=45 andWi=0.07 (POLYFLOW code, Giesekus model with k=0.06 s,a=0.0001, gp=0.7 Pa s, and solvent viscosity gs=0.5 Pa s)

Fig. 11. Relative length of the wake downstream of the ‘‘T’’profile, as a function of Re number, for Newtonian solvent and2% concentration of polymer in Newtonian solvent. The solidlines are the results of the numerical simulations

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