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Measurements of the bulk and interfacial velocity profiles in oscillating

Newtonian and Maxwellian fluids

M. Torralba,1 J.R. Castrejon–Pita,2 A.A. Castrejon–Pita,2 G. Huelsz,2 J.A. del Rıo,2 and J. Ortın1

1Departament d’Estructura i Constituents de la Materia,Universitat de Barcelona, Av. Diagonal 647, E-08028 Barcelona, Spain

2Centro de Investigacion en Energıa, Universidad NacionalAutonoma de Mexico, A.P. 34, 62580 Temixco (Morelos), Mexico

(Dated: February 2, 2008)

We present the dynamic velocity profiles of a Newtonian fluid (glycerol) and a viscoelastic Maxwellfluid (CPyCl/NaSal in water) driven by an oscillating pressure gradient in a vertical cylindrical pipe.The frequency range explored has been chosen to include the first three resonance peaks of the dy-namic permeability of the viscoelastic fluid / pipe system. Three different optical measurementtechniques have been employed. Laser Doppler Anemometry has been used to measure the mag-nitude of the velocity at the centre of the liquid column. Particle Image Velocimetry and OpticalDeflectometry are used to determine the velocity profiles at the bulk of the liquid column and atthe liquid–air interface respectively. The velocity measurements in the bulk are in good agreementwith the theoretical predictions of a linear theory. The results, however, show dramatic differencesin the dynamic behaviour of Newtonian and viscoelastic fluids, and demonstrate the importance ofresonance phenomena in viscoelastic fluid flows, biofluids in particular, in confined geometries.

PACS numbers: 47.50.+d, 47.60.+i, 83.60.Bc

I. INTRODUCTION

Coupling between flow and liquid structuremakes the dynamic response of non–Newtonian(complex) fluids much richer than that of Newto-nian (simple) fluids [1, 2]. In particular, dependingon the relevant time scale of the flow, viscoelasticfluids exhibit the dissipative behaviour of ordinaryviscous liquids and the elastic response of solids.Due to their elastic properties, these fluids are po-tential candidates to exhibit interesting resonancephenomena under different flow conditions.

In this respect, the response of a viscoelastic fluidto an oscillatory pressure gradient has been anal-ysed theoretically in some detail. The response,measured in terms of the velocity for a given am-plitude of the pressure gradient, exceeds that of anordinary fluid by several orders of magnitude at anumber of resonant frequencies. The remarkableenhancement in the dynamic response of the vis-coelastic fluid is attributed to a resonant effect dueto the elastic behaviour of the fluid and the geom-etry of the container [3, 4, 5, 6].

This theoretical prediction has been recently con-firmed by Laser Doppler Anemometry (LDA) mea-surements of velocity at the centre of a fluid columndriven by an oscillating pressure gradient [7]. Theexperiments show that a Newtonian fluid exhibits asimple dissipative behaviour, while a Maxwell fluid(the simplest viscoelastic fluid) exhibits the reso-nant behaviour predicted by the linear theory atthe expected driving frequencies. While this anal-

ysis was performed in the central point of the fluidcolumn, an exploration of the whole velocity field isnecessary to ensure that the linear model capturesthe main features of the flow.

Resonance effects can only be observed when theelastic behaviour of the fluid is dominant. This isproperly characterised by a dimensionless numberDe≫ 1, where De (Deborah number) measures therelative importance of the relaxation time of thefluid to the typical time scale of the flow. More-over, at the resonant driving frequencies the systemresponse is nearly purely elastic and the dissipativeeffects are very small.

In the present paper we extend the previous ex-perimental investigation in several directions. Inthe first place we present new LDA measurementsfor two Maxwell fluids of different composition, andshow that the linear theory correctly predicts thelocation of the resonance frequencies in terms ofmaterial parameters. In addition, measurements atdifferent depths along the column centre show theattenuation of the velocity as the upper free inter-face is approached. The main result is the determi-nation of the velocity profiles in the radial directionof the cylindrical tube, at several time intervals anddriving frequencies. We have measured the velocityprofiles at two different locations: the bulk of theliquid column, and the upper liquid–air interface.Particle Image Velocimetry (PIV) has been used inthe former case, and an original technique basedon Optical Deflectometry (OD) in the latter. ThePIV results compare satisfactorily with the velocity

2

profiles predicted by the linear theory [8]. The ODresults show the attenuation of the velocity due tosurface tension.

II. LINEAR THEORY

In this section we recall the theoretical expressionof the velocity field in the bulk of a viscoelastic fluidcontained in a vertical tube, and subjected to an os-cillating pressure gradient [7, 8]. The tube is sup-posed to be infinite in the vertical direction. Thederivation is based on a linear approximation of thehydrodynamic equations and a linearized constitu-tive equation for a Maxwell fluid [3, 4, 5].

Following Ref. [7], the velocity field in Fourier’sfrequency domain reads:

V (r, ω) = − (1 − iωtm)

ηβ2

(

1 − J0(βr)

J0(βa)

)

dP

dz. (1)

In this expression r, z are the radial and verticalcylindrical coordinates in a reference system cen-tered with the tube axis, ω = 2πν, where ν isthe driving frequency, tm is the relaxation time ofthe Maxwell fluid, η its dynamic viscosity, β =√

ρ [(tmω)2 + iωtm] /(ηtm), a is the cylinder ra-dius, J0 is the cylindrical Bessel function of zerothorder, and P is the applied pressure in Fourier’sfrequency domain.

In our experiment, oscillations of the pressuregradient are induced by the harmonic motion of apiston at the bottom end of the fluid column, withamplitude z0 (Figure 1). Thus,

dp(t)

dz= ρz0ω

2 sin(ωt). (2)

The velocity profile in the radial direction of thetube, at the driving frequency ω, is then given bythe real part of the following expression:

v(r, t) = −i

(

1 − J0(βr)

J0(βa)

)

z0ωeiωt. (3)

For a Newtonian fluid (tm = 0) there is only onenode of the velocity profile, at r = a, which ac-counts for the non–slip condition at the wall. Re-markably, if tm 6= 0 (Maxwellian fluid) the velocityprofile may present several nodes. These nodes de-fine quiescent points of the flow. For given mate-rial and geometrical parameters, the location of thenodes depends on the driving frequency, as shownin Fig. 2.

As mentioned in the Introduction, the elasticproperties of the fluid dominate the dynamic re-sponse of the system at the resonant frequencies.

FIG. 1: Schematic view of the experimental device, in-cluding the setups for PIV and deflectometry measure-ments.

0.000 0.005 0.010 0.015 0.020 0.0250

4

8

12

16

20

ν (H

z)

r (m)

FIG. 2: Diagram showing the location of the quiescentflow points along the radial coordinate of the cylinder,r, as a function of driving frequency, ν, for a Maxwellfluid. The dashed horizontal lines separate the fre-quency axis in intervals of constant number of quies-cent flow points. The parameters ρ, tm, and η used tocompute the diagram are the ones listed in the text forthe 60/100 CPyCl/NaSal solution, together with thecylinder radius a = 25 mm.

3

In Eq. (3) this is manifest in that, at the resonantfrequencies, the oscillation in the centre of the tube(maximum velocity) is in phase with the driving.

III. EXPERIMENTAL DEVICE

The experimental device is shown in Fig. 1. Thecylindrical container (inner radius a = 25 mm,length 500 mm) is made of transparent acrylic.In order to avoid optical aberrations this cylin-der is placed inside a second recipient of squaresection, made also of transparent acrylic, which isfilled with glycerol to match the refractive indexof the acrylic walls. A Teflon piston driven by amotor of variable frequency produces harmonic os-cillations of the pressure gradient in the liquid col-umn. Although the motor controller can operatein the frequency range from 1.5 to 200 Hz, ourexperiments have been performed in the reducedfrequency range from 1.5 to 15 Hz. The piston os-cillation amplitude is set to 0.80±0.05 mm to keepthe Reynolds number low, within the accessible fre-quency range, for the two fluids studied. Additionaldetails of this part of the setup are described in Ref.[7].

The viscoelastic fluid used in these experimentsis an aqueous solution of cetylpyridinium chlorideand sodium salicylate (CPyCl/NaSal). This sur-factant solution is known to exhibit the rheologi-cal behaviour of a linear Maxwell fluid in a rangeof concentrations [9, 10]. In most experiments wehave used a 60/100 concentration, for which the dy-namic viscosity η = 60 Pa·s, the density ρ = 1050kg/m3, and the relaxation time tm = 1.9 s [11]. InDLA measurements we have also used a 40/40 con-centration, for which η = 30 Pa·s, ρ = 1005 kg/m3,and relaxation time around tm = 1.25 s. Our rhe-ological characterisation of this last fluid, however,suggests that it cannot be fully described by a sin-gle relaxation time. As Newtonian fluid we haveused commercial glycerol, with nominal dynamicviscosity η = 1 Pa·s and density ρ = 1250 kg/m3.These values have been determined at the workingtemperature of (25 ± 0.5)◦C.

All sets of measurements performed on the samefluid have been carried out in order of increasingfrequency. The cylinder has been emptied and re-filled with fresh fluid every time that a series ofmeasurements at three different driving frequencieswas complete, i.e. approximately every day. Mea-surements in the bulk have been carried out at twodifferent heights, corresponding to distances of 6and 10 cm from the upper free interface.

The LDA technique used in the measurements

presented in the next Section has already been de-scribed in detail in Ref. [7]. We present now a briefdescription of the other two techniques (PIV anddeflectometry).

A. Particle Image Velocimetry

The PIV technique provides instant measures ofthe velocity maps in a plane of the flow [12]. Tothis purpose, the fluid is seeded with small parti-cles. The velocity maps are obtained by a measureof the statistical correlation of the displacement ofthe seeding particles in the fluid in a known timeinterval, in this case the time between two consec-utive laser pulses.

Our PIV system contains a two pulsed Nd-YAGlasers unit, that includes an optical array to pro-duce a laser light sheet in a vertical plane of theacrylic cylinder (Fig. 1). A high resolution cam-era (Kodak E1.0), perpendicular to the laser lightsheet, is used to record the digital images. Thecamera records two consecutive frames, one corre-sponding to each laser light pulse. The acquisi-tion rate is limited by the camera to three pairsof images every two seconds (1.5 Hz). A Dan-

tec FlowMap 1100 processor takes care of the syn-chronization between the laser pulses and the cam-era trigger. Post–processing of the data, to deter-mine velocity maps, is carried out by the Dantec

FlowMap v5.1 software. Dantec 20–µm polyamidspheres were used as seeding particles in the presentexperiments. These particles are small enough tofollow the flow with minimal drag, but sufficientlylarge to scatter enough light to obtain good particleimages.

B. Optical Deflectometry

Deflectometry is a technique used to study rela-tively small deformations in transparent fluid sur-faces [13]. In our case the liquid column serves asa variable–thickness lens. A thin transparent plas-tic sheet with a regular array (grid spacing d) ofblack dots (0.5 mm diameter) is placed on top ofthe Teflon piston, at the bottom of the acrylic cylin-der (Fig. 1). The grid is imaged through the liquidcolumn using a CCD camera interfaced to a com-puter. Light passing through the liquid column isrefracted at the liquid–air interface, distorting theimage of the grid as the interface deforms. Figure3 presents an example of the recorded images.

The displacement of an imaged grid point, δ, isrelated to the local slope of the interface, φ, in the

4

FIG. 3: A typical picture recorded in Optical Deflec-tometry measurements. The image of a regular arrayof black dots (grid spacing d = 5.0 mm) is distorted bythe interface deformation.

f

qf-q

d

n1

n2

h

FIG. 4: Sketch of the path followed by a light ray inthe system.

following way (Fig. 4). According to Snells’ refrac-tion law, n1θ = n2φ. On the other hand, for acolumn height h the figure shows that tan(φ−θ) =δ/h. Therefore, the relation between φ and δ reads:

φ =1

1 − n2/n1

arctan

(

δ

h

)

. (4)

In our setup the height of the liquid column(H = 240 mm) is much larger than the local heightvariations produced by the deformation of the in-terface (of the order of mm), and we can safely takeh ≃ H in Eq. (4).

OD is usually used in thin films. The deflectionof the points of the grid increases with the depthof the layer, making this technique useless for themeasurement of highly deformed interfaces of thicklayers. However, the small amplitude of the per-turbation in our experiments generates relatively

-20 -10 0 10 20

-0.4

-0.2

0.0

0.2

0.4

ν=6.5 Hz, t=0.106 s ν=2.0 Hz, t=0.423 s

φ

r (mm)

FIG. 5: Theoretical deformation profiles of the Maxwellfluid driven at 2 Hz (dashed line) and at 6.5 Hz (solidline).

smooth profiles, and the condition of small defor-mations is satisfied in a wide range of frequencies.

Although the deformations are small in this fre-quency range, the whole deformation profile de-pends strongly on the frequency of driving. Thiscan be seen on the theoretical deformation pro-files of the viscoelastic fluid at 2 and 6.5 Hz, dis-played in Fig. 5. The two profiles are very differ-ent. While the deformation profile at 2 Hz is mono-tonic and very smooth, the profile at 6.5 Hz shows amore complex behaviour: instead of a single centralnode, it presents two nodes which, in the velocityprofile, correspond to a maximum at the tube cen-tre and a minimum near the tube wall. The max-imum deformation between these two nodes corre-sponds to an inflection point of the velocity profile,very close to the quiescent flow point.

In our setup the height of the liquid column is afunction of the radial coordinate r, and periodic intime. This height, h(r, t), is simply related to thelocal slope by φ(r, t) = ∂h(r, t)/∂r. Hence:

h(r, t) = h(r0, t) +

∫ r

r0

φ(r, t) dr. (5)

It is useful to choose for r0 a position where theinterface is motionless, because the reference heighth(r0, t) is then a constant that can be taken equalto 0. We have chosen r0 as the radial coordinateof the quiescent flow point (in the bulk) closest tothe cylinder axis. Implicitly, we are assuming thatthe stagnation points predicted by Eq. (3) for thebulk do not change their position as the interface isapproached. This assumption is confirmed by ourPIV and OD results (see Sec. IV).

5

Finally, the velocity profile of the interface canbe obtained from the time derivative of the localheight, v(r, t) = ∂h(r, t)/∂t.

In our experiments the grid spacings used ared = 5.0 and 5.5 mm, and the refractive indicesare n = 1.33407(01) for the CPyCl/NaSal solution(measured by Abbe refractometry), and n = 1.473for glycerol (nominal).

IV. RESULTS AND DISCUSSION

A. rms velocity at the cylinder axis

The amplitude of the velocity field given by Eq.(1) presents resonance peaks at several resonancefrequencies. As mentioned in the Introduction, thisphenomenon was demonstrated experimentally andcompared to the purely dissipative behaviour of aNewtonian fluid in Ref. [7]. These results showedthat the linear theory gives a good prediction of theresonance frequencies but overestimates the ampli-tude of the resonance peaks.

We have repeated this same kind of measure-ments in the present work, to check the depen-dence of the resonance frequencies on the rheolog-ical properties of the Maxwell fluid. To this pur-pose the driving frequency ω has been rescaled bya characteristic time τ defined as

τ = 102/5tm√

α, (6)

where α is the inverse of our Deborah numberDe = tmη/a2ρ. As it was shown in Ref. [5], thelocation of the resonance peaks becomes universal(independent of fluid parameters and system di-mensions) if ω is made dimensionless in the formωτ .

Then, we rewrite Eq. (1) in the form

V (r, ω) = ξ(r, ω)dP

dz, (7)

and plot the dimensionless response functionξ(r, ω)/ξ(r, 0) as a function of the dimensionlessdriving fequency ωτ . The results at the centre ofthe tube (r = 0), for the two different concentra-tions of the CPyCl/NaSal solution used, are shownin Fig. 6. Comparison with the linear theory isgiven by the vertical dashed lines, which give theresonance frequencies corresponding to maxima ofξ(r = 0, ω). The Figure shows that the rescalingof the frequency axis suggested by the linear the-ory leads to a satisfactory reproducibility of theresonance frequencies independently of viscoelasticfluid parameters.

0 1 2 3 4 5 6 7 8

1

10

100 CPyCl/NaSal 100/60 mM CPyCl/NaSal 40/40 mM

ξξ ξξ(ωω ωω)

/ ξξ ξξ(0

)

ωτωτωτωτ

FIG. 6: Dimensionless response function at the centreof the cylinder, as a function of dimensionless drivingfrequency, for two concentrations of the CPyCl/NaSalsolution. The vertical dashed lines give the location ofthe resonance frequencies predicted by a linear theory.

The magnitude of the response function at theresonance peaks for the surfactant solution of con-centration 60/100 is considerably lower in thepresent measurements than in the previous ones ofRef. [7]. The reason is that the present measure-ments have been carried out at 6 cm of the free air–liquid interface, while the previous ones had beentaken at 10 cm. The different measured velocitiesprovide a first evidence of the damping influence ofthe free interface on the flow.

The results presented in the next two sectionsprovide measurements of the whole velocity profile,instead of measurements at a single point in theflow [7]. The velocity profiles are obtained at twodifferent heights, in the bulk of the fluid, and at thefluid-air interface.

B. Velocity profiles in the bulk

The following set of figures presents the velocityprofiles in the bulk of the fluid column, determinedby PIV measurements, together with the theoret-ical profiles given by Eq. (3) at coincident time–phases, for comparison. The profiles have been de-termined at the driving frequencies of 2 Hz (Figs. 7and 8), 6.5 Hz (Figs. 9 and 10), and 10 Hz (Figs. 11and 12). These values coincide with the first threeresonance frequencies of our system for the 60/100viscoelastic solution, as given by Eq. (3). The time–phases have been selected by the criterion that the

6

-20 -10 0 10 20

-3

0

3

6

9

12

15

v (m

m/s

)

r (mm)

FIG. 7: Glycerol: PIV results at 2 Hz (dots), measuredat 6 cm from the upper free interface, and the corre-sponding theoretical prediction (solid line) at t = 0.125s.

-20 -10 0 10 20-10

0

10

20

30

40

v (m

m/s

)

r (mm)

FIG. 8: 60/100 CPyCl/NaSal solution: PIV results at2 Hz (dots), measured at 6 cm from the upper freeinterface, and the corresponding theoretical prediction(solid line) at t = 0.375 s.

velocity at the tube axis is a maximum.The first observation to make is that the in-

stantaneous velocity profiles of the Maxwell fluid,driven at 2 Hz, present a single defined sign of ve-locity (single direction of motion) along the wholeradius of the tube. As the driving frequency is in-creased to 6.5 and 10 Hz, however, the instanta-neous profiles display a progressively more com-plex structure, revealing the presence of annu-lar regions within the tube with alternating up-ward/downward motion. Notice that this com-plexity is inherent to the viscoelastic properties ofthe Maxwell fluid. For the Newtonian fluid (glyc-erol) the instantaneous flow in the tube goes all in

-20 -10 0 10 20-10

0

10

20

30

40

v (m

m/s

)

r (mm)

FIG. 9: Glycerol: PIV results at 6.5 Hz (dots), mea-sured at 6 cm from the upper free interface, and thecorresponding theoretical prediction (solid line) at t =0.115 s.

-20 -10 0 10 20

-100

-50

0

50

100

150

200

v (m

m/s

)

r (mm)

FIG. 10: 60/100 CPyCl/NaSal solution: PIV results at6.5 Hz (dots), measured at 6 cm from the upper freeinterface, and the corresponding theoretical prediction(solid line) at t = 0.038 s.

the same direction for the three driving frequenciestested.

For the viscoelastic fluid the frontiers betweenconsecutive annular regions with alternating signsof the velocity do not move. They correspond tothe stagnation regions of the flow which were al-ready discussed in the context of the linear theory.The PIV results show that the number of quiescentflow points along the radial direction of the tubeincreases with the driving frequency, in agreementwith the theoretical prediction (Fig. 2).

Some of the measured profiles of the viscoelasticfluid show regions near the walls with vanishinglysmall velocities and near zero velocity gradients,most noticeably the one at 2 Hz shown in Fig. 8.

7

-20 -10 0 10 20-20

0

20

40

60

v (m

m/s

)

r (mm)

FIG. 11: Glycerol: PIV results at 10 Hz (dots), mea-sured at 6 cm from the upper free interface, and thecorresponding theoretical prediction (solid line) at t =0.075 s.

-20 -10 0 10 20

-100

0

100

200

300

400

v (

mm

/s)

r (mm)

FIG. 12: 60/100 CPyCl/NaSal solution: PIV results at10 Hz (dots), measured at 6 cm from the upper freeinterface, and the corresponding theoretical prediction(solid line) at t = 0.025 s.

These profiles are reminiscent of velocity profilesobtained for systems that display shear–banding.Indeed, the CPyCl/NaSal solution is commonlyknown to show shear-banding [14], but we haveenough evidence to discard this effect in our experi-ments. We did not observe an increase in turbidityin the region of the fluid close to the walls, norchanges in the local intensity of the scattered lightthat could be attributed to inhomogeneities. Fur-thermore, we monitored possible changes on the po-larization state of the scattered light, an indicationof banding [15]. No changes in the optical proper-ties were observed for the amplitude and frequencyrange used here. In addition, Fig. 13 demonstratesthat the velocity near the walls takes values dis-

-20 -10 0 10 20-30

-20

-10

0

10

20

v (m

m/s

)

r (mm)

FIG. 13: 60/100 CPyCl/NaSal solution: PIV results at2 Hz, measured at 6 cm from the upper free interfaceand at different time phases within a driving period.

tinctly different from zero at different phases withinan oscillation period.

Performing PIV measurements at two differentheights of the liquid column allows studying the in-fluence of the upper free interface on the velocityprofiles. Thus, it is interesting to compare the re-sults presented above, which have been performedat 6 cm from the upper interface, to the resultsshown in Figs. 14 and 15, which have been per-formed at 10 cm from the upper interface. Thefirst conclusion to draw from corresponding mea-surements at different heights is that the locationof the quiescent flow points is not affected by thepresence of the upper free interface. The secondconclusion is that the magnitude of the velocityprofile is smaller when the measurement is carriedout closer to the free interface. The damping ef-fect of the free interface, disregarded in the theory,originates from the air–liquid surface tension. Thisobservation applies equally to the two fluids inves-tigated (Newtonian and Maxwellian).

The experimental velocity profiles are in reason-able agreement with the theoretical ones. The num-ber of quiescent flow points is the same, and theirlocation very similar. As a general trend, however,the theory overestimates the measured velocity.

A first explanation would be that the theory dis-regards nonlinearities, and these tend to limit thelargest velocity values. Nonlinearities could arisefrom either the hydrodynamic equations or the con-stitutive relation of the fluid. In our case, how-ever, since the small amplitude of the piston oscilla-tions ensures that Re< 10−4, the linearized momen-tum equation is a very good approximation. Onthe other hand, taking into account the cylindricalsymmetry of the problem and assuming that the

8

-20 -10 0 10 20

r (mm)

160

v (m

m/s

)120

80

40

0

-40

-80

FIG. 14: 60/100 CPyCl/NaSal solution: PIV results at6.5 Hz (dots), obtained at 10 cm from the upper freeinterface, and the corresponding theoretical prediction(solid line) at t = 0.057 s.

-20 -10 0 10 20

r (mm)

400

300

200

100

0

-100

v (m

m/s

)

FIG. 15: 60/100 CPyCl/NaSal solution: PIV results at10 Hz (dots), obtained at 10 cm from the upper freeinterface, and the corresponding theoretical prediction(solid line) at t = 0.022 s.

velocity depends only on the radial coordinate (asour results confirm to a very good approximation),it turns out that the first nonlinear correction tothe constitutive equation of the fluid cancels outexactly.

The disagreement between theory and experi-ment is probably due to shear–thinning of the vis-coelastic fluid. As shown in Ref. [11], our fluid isproperly described by a Maxwell model up to shearrates γ ≃ 0.6 s−1, and experiences shear–thinningbeyond that value. A close inspection of Figs. 8,

10, and 12 reveals that the shear rate actually ex-perienced by the fluid in some phase intervals of theoscillation is larger [7]. In these conditions the vis-cosity of the fluid decreases with shear. The theorypredicts that the dynamic response of the systemat the resonant frequencies becomes smaller as theviscosity is reduced [5], and thus would support theview that the measured velocity profiles are system-atically smaller than the theoretical ones for theviscoelastic fluid (and not for the newtonian fluid)because of shear–thinning.

C. Deflection of the air–liquid interface

The results obtained by Optical Deflectometryare presented on top of Figs. 16 and 17 (2 Hz), andFigs. 18 and 19 (6.5 Hz). There are no results at 10Hz because, at this driving frequency, the deforma-tions of the free interface of the Maxwellian fluidare so large that the technique is not applicable.Each figure displays measurements at two differenttime–phases.

The experimental deformation profiles of the in-terface at 2 and 6.5 Hz for the viscoelastic fluid canbe compared to the theoretical ones (computed forthe bulk) displayed in Fig. 5. We observe that theirshapes are fully coincident for each of the two driv-ing frequencies. This shows that the quiescent flowpoints of the velocity profiles do not change theirposition along the vertical direction, and thus theirlocation in the bulk can be used as reference tocompute the velocity profiles at the interface fromthe deformation profiles.

The velocity profiles are presented on the bottomof Figs. 16 and 17 (2 Hz), and Figs. 18 and 19 (6.5Hz). The magnitude of the interfacial velocities isin all cases much lower than the one measured inthe bulk of the fluid. This is mostly due to the sta-bilizing role of surface tension, as discussed above.

It is interesting to note that the velocity profilesof the interface depend also on the direction of mo-tion of the driving piston. The magnitude of the ve-locity is systematically lower for positive displace-ments of the piston (liquid displacing air) than fornegative ones (air displacing liquid). We attributethis asymmetry to the fact that the large viscositycontrast across the interface stabilizes the displace-ment of a nearly inviscid fluid (air) by a viscousliquid, and destabilizes the opposite displacement.

9

-0.008

-0.004

0.000

0.004

0.008

φv

(mm

/s)

h (m

m)

-0.0100

-0.0075

-0.0050

-0.0025

0.0000

0.0025

r(mm)-20 -10 0 10 20

-0.4

-0.2

0.0

0.2

0.4

FIG. 16: Glycerol: deflectometry results at 2 Hz, mea-sured at two different time–phases within an oscillation.

V. CONCLUSIONS

We have used three different optical techniquesto characterize the oscillating flow of a viscoelas-tic fluid (a solution of CPyCl/NaSal in water) con-tained in a vertical cylinder and subjected to anoscillating pressure gradient. A Newtonian fluid(glycerol) has also been studied for the sake of com-parison.

LDA measurements of the fluid velocity at thesymmetry axis of the cylinder as a function of driv-ing frequency, for two different compositions of the

-0.030

-0.015

0.000

0.015

0.030

v (m

m/s

)h

(mm

)

φ-0.2

-0.1

0.0

0.1

0.2

r(mm)-20 -10 0 10 20

-6

-4

-2

0

2

4

6

FIG. 17: 60/100 CPyCl/NaSal solution: deflectometryresults at 2 Hz, measured at two different time–phaseswithin an oscillation.

surfactant solution, have enabled us to show thatthe frequencies of the resonance peaks can be pre-dicted accurately in terms of the fluid rheologicalproperties by a simple linear theory. This theoryneglects inertial effects, and makes use of a linearMaxwell model as constitutive relation for the vis-coelastic fluid.

Systematic PIV measurements of the radial ve-locity field in the bulk of the fluid have been per-formed at three different driving frequencies, closeto the first three resonance frequencies of the vis-

10

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

v (m

m/s

)h

(mm

)

φ

-0.09

0.00

0.09

0.18

0.27

0.36

r(mm)-20 -10 0 10 20

-6

-4

-2

0

2

4

6

FIG. 18: Glycerol: deflectometry results at 6.5 Hz,measured at two different time–phases within an os-cillation.

coelastic system. While the velocity profile of theNewtonian fluid along the radial direction does notchange sign, this is not the case for the Maxwellianfluid. The profiles measured at 6.5 and 10 Hzpresent regions with alternating signs of the veloc-ity, separated by quiescent flow points. The numberof quiescent flow points increases with the drivingfrequency, revealing the increasing complexity ofthe flow. Measurements within the fluid columnat two different heights show that these quiescentflow points do not shift as one moves along the

-0.04

-0.02

0.00

0.02

0.04

0.06

v (m

m/s

)h

(mm

)

φ-0.30

-0.15

0.00

0.15

0.30

r(mm)-20 -10 0 10 20

-20

-15

-10

-5

0

5

10

15

20

FIG. 19: 60/100 CPyCl/NaSal solution: deflectometryresults at 6.5 Hz, measured at two different time–phaseswithin an oscillation.

vertical direction, and that their radial location isaccurately reproduced by the linear theory.

The presence of a free interface at the top of theliquid column has a damping effect on the velocityamplitude. This observation is visible, both withLDA and PIV, when the results of measurementscarried out at two different heights within the liquidcolumn are compared.

Optical Deflectometry measurements of the freeinterface confirm that the velocity field is severelydamped by the surface tension of the air–liquid in-

11

terface, compared to the velocity field within thebulk. Interestingly, the deflectometry results showalso that the oscillations of the velocity field at theinterface are asymmetric, the profiles correspond-ing to positive displacements of the piston having aslightly but systematically smaller amplitude thanthose corresponding to negative displacements. Weattribute this asymmetry to the fact that the up-ward motion of the interface (liquid displacing air)is stabilized by the viscous pressure gradient in theliquid. Recently it has been suggested that the vis-cous fingering interfacial instability of a viscoelasticfluid in a Hele–Shaw cell will be enhanced by mak-ing the flow oscillate at the resonant frequencies[16]. Our deflectometry results show that the sta-bilizing role of the surface tension will reduce thiseffect to some extent.

These results, as part of the study of resonancefrequencies in viscoelastics, are potentially relevantin biorheology to interpret the frequencies at which

biofluids are optimally pumped in living organisms[17].

VI. ACKNOWLEDGEMENTS

We acknowledge A. Morozov (Universiteit Lei-den) for illuminating discussions, and J. Soriano(Universitat Bayreuth) and G. Hernandez (UNAM)for their technical assistance. This researchhas received financial support through projectsBQU2003-05042-C02-02 and BFM2003-07749-C05-04 (MEC, Spain), SGR-2000-00433 (DURSI, Gen-eralitat de Catalunya), and CONACyT 38538(Mexico). JO acknowledges additional supportfrom the DURSI. JRCP acknowledges the sup-port by CONACyT, SEP, and the ORS AwardScheme. AACP acknowledges the support givenby the Dorothy Hodgkin Award.

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