Cavity flows of elastic liquids: Purely elastic instabilitiesPeyman Pakdel and Gareth H. McKinleyDivision of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138
~Received 3 September 1997; accepted 16 January 1998!
The lid-driven cavity flow is the motion of a fluid in arectangular box generated by a constant translational veloof one side while the other sides remain at rest. In the twdimensional limit, the flow consists of a planar recirculatofluid motion confined by rectangular boundaries. The ldriven cavity flow of Newtonian liquids has been the subjof extensive computational and experimental studies overpast 30 years. These studies have been motivated bygeometrical simplicity of the flow domain, the existencestress singularities at two corners, and the complex dynacal structure that arises from the onset of inertinstabilities.1
The lid-driven cavity flow poses a complex fluid mchanics problem in which regions of strong shear neartop moving plate, vortical motion in the central core, acorner flows simultaneously exist and interact in a systwith closed streamlines. Inertial effects play a dominant rin governing the kinematical structure of the fluid motioand stability of cavity flows of Newtonian liquids. Inertiaeffects are quantified by the Reynolds number, defined a
Re5LUr
m, ~1.1!
whereL is the width of the cavity,U is the magnitude of theimposed top-plate velocity, andm and r are the constanviscosity and density of the liquid, respectively. At negligbly small Reynolds numbers, the Newtonian cavity flowfore-aft symmetric. Increasing the Reynolds number brethis symmetry and eventually at a critical Reynolds numbRecrit'500, the flow becomes three dimensional via the aplification of spatial and temporal disturbances.2 The drivingmechanism of these inertial instabilities is similar to thleading to the growth of classical Taylor vortices in the Coette device3 and to the related inertial instabilities that resin the formation of Go¨rtler vortices in boundary layer flowalong curved surfaces.4
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This hydrodynamic instability is driven via the nonlinecoupling term,v–“v, in the equation of motion1 and is char-acterized by the emergence of Taylor–Go¨rtler-like ~TGL!vortices that are spatially periodic in the neutral directionthe flow and contain streamwise vorticity. Recent numerilinear stability analyses and experimental flow visualizatioprovide a consistent quantitative understanding of thenamical structure of the inertial instabilities in recirculatincavity flows of Newtonian liquids. Flow visualization experments indicate that the secondary motions are initially tiindependent (Recrit'500) and characterized by a steady sptially periodic structure in the spanwise direction. Increasthe Reynolds number beyond a value of Re'825 results in afurther flow transition that leads to evolution of a timdependent traveling-wave mode.5
Cavity flows become intrinsically more complex whethe fluid rheology is non-Newtonian. These complicatioarise as a result of the polymeric contribution to the devtoric stress field, which is strongly coupled to the fluid kinmatics and is a function of an integral history of local ratesdeformation experienced by a fluid element moving alonclosed streamline. The presence of a viscoelastic flmemory, shear thinning effects in the material functiononzero normal stress differences, and the complex exsional rheological behavior of non-Newtonian fluids ceach alter the fluid kinematics in the cavity geometry.
Leong and Ottino appear to have been the first to expmentally examine the effect of viscoelasticity in caviflows.6 They conducted a comparative experimental flowsualization study of time-periodic flows in a viscous Netonian fluid and in an ideal elastic Boger fluid.7 These idealelastic fluids are synthesized by dissolving a small amouna high molecular weight polymer in a viscous Newtonisolvent. In addition to exhibiting an almost constant shviscosity over a wide range of shear rates, they displaypreciable fluid viscoelasticity and large first normal stredifferences in steady shear flows. In their flow visualizatiexperiments with a passive dye tracer, Leong and Ottino
1059Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
served that the extent of mixing was weaker in the nNewtonian case than in the corresponding Newtonian flunder similar time-periodic boundary conditions. Analogoviscoelastic effects have also been observed in time-perimixing flows generated in the eccentric cylinder geometry8,9
Cavity flows of Generalized Newtonian fluids have bethe subject of two computational studies; Reddy and Redd10
investigated the heat transfer effects in steady thrdimensional cavity flows of power-law and Carreau fluidand Isaksson and Righdal11 examined the steady streamlinpatterns and local pressure distribution via two-dimensionumerical simulation of a power-law fluid.
In order to investigate the first effects of fluid elasticitnumerical computations of lid-driven cavity flow were peformed by Mendelsonet al.12 using the second order fluimodel. It is well known that the Newtonian velocity fielsatisfying the Stokes equations is a unique solution toequations of motion for steady planar creeping flows osecond order fluid.13 Based on these uniqueness and extence theorems, Mendelsonet al.12 show that finite elemensimulations with the second order fluid model cannot acrately resolve the steep gradients in viscoelastic stressdevelop near the upper corners of the sliding plate. Furthmore, as pointed out by these authors, such numerical slations cannot provide information on the temporal stabiof the flow and in fact, linear stability analysis has showthat the steady planar flow of the second order fluid modetemporally unstable at all finite Deborah numbers.14
Pakdel, Spiegelberg, and McKinley15 have conducted ki-nematic measurements of the steady two-dimensional moof Boger fluids in the cavity geometry at negligible Reynolnumbers using laser Doppler velocimetry~LDV ! and digitalparticle image velocimetry~DPIV!. They observe that viscoelasticity breaks the fore-aft symmetry of the flow struture observed in the Stokes flow regime. The geometric cter of the core vortex region shifts slightly in the upstreadirection ~i.e., in the opposite direction to the translationvelocity of the lid! and the magnitude of this shift increasas the imposed velocity of the driving boundary wall is icreased. The magnitude of the velocity gradients arehanced in the corner regions near the moving plate. The lmaxima in the velocity gradients are spatially locatslightly away from the corners ('0.2L) and appear in re-gions where the fluid streamlines exhibit significant curvture.
To quantify the non-Newtonian effects in cavity flowPakdelet al.15 define two-dimensionless groups; the Deborand the Weissenberg numbers. The Deborah number isfined as
De5lU/L, ~1.2!
and the Weissenberg number is defined as
Wi5lU/H, ~1.3!
whereU is the constant translational speed of the upperl is a characteristic relaxation time of the fluid,L is thewidth, andH is the height of the cavity. With these defintions, De quantifies a ratio of the fluid viscoelastic memoto a characteristic residence time in the system,tflow5L/U,
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and Wi provides a dimensionless measure of the magnitof the imposed shear rate,g5U/H. The non-Newtonian nor-mal stress differences in the fluid are strong nonlinear futions of the shear rate and consequently scale with Wi.
Along with these groups a purely geometrical paramenamely the aspect ratioL, can be defined as
L5H/L. ~1.4!
This parameter quantifies the relative importance of thecharacteristic length scales in the cavity geometry. Incase of deep narrow slots (H@L,L@1), the width of thecavity governs the kinematics of the main circulation regioand in case of the long shallow channels (H!L,L!1) theheight of the cavity plays the determining role in establishthe streamline patterns of the recirculating shear flow inbulk of the cavity. The collective set of dimensionless grouDe, Wi, andL, together with a knowledge of the fluid material functions, spans a parameter space that fully specthe operating condition of cavity flows of viscoelastic fluidat negligible Reynolds numbers.
Over the past eight years, viscoelastic instabilities occring at negligibly small Reynolds numbers have beensubject of intense theoretical and experimental studies. Thinstabilities, commonly referred to aspurely elastic instabili-ties, are entirely absent in the corresponding flows of Netonian fluids and are driven by mechanisms associatedelastic normal stress differences rather than inertial nonearities in the equation of motion. The nonlinear coupliamong the components of the Cauchy momentum equatis embedded in the nonlinear constitutive relationships tdescribe the evolution of the viscoelastic stress,t, in flowingpolymeric processes. This nonlinear coupling gives riseterms of the formv–¹t and¹v–t which scale independentlyof the Reynolds number in the governing equation set.
Earlier studies on the subject of elastic instabilities dback to Giesekus16 who reported onset of elastic instabilitiein Taylor–Couette flow of a shear-thinning fluid at a Renolds number of 1022. With the synthesis of ideal elastiBoger fluids,7 it became possible to isolate the effect of elaticity in the absence of additional complicating phenomesuch as shear thinning in the fluid viscosity and the assated increase in inertial effects. Purely elastic instabilitiesconstant viscosity fluids were first reported by Muller, Lason, and Shaqfeh17 who observed clear indications of a floinstability while attempting to perform rheological measurments of Boger fluids in a Taylor–Couette device. Thstudy initiated a number of computational and experimenstudies in the Taylor–Couette geometry over the past eyears~see, for example, Refs. 18–20!.
Similar studies of torsional flow in the cone-and-plageometry,21–24 and the coaxial parallel plate geometry,25,26
the axisymmetric contraction flow,27 and in the wake behinda cylinder28 indicate that elastic instabilities are not specito a flow geometry but occur in many complex flow fieldThe critical onset conditions are sensitive functions offluid rheology and of the characteristic geometric paramegoverning the flow configuration. Furthermore, these nmerical and experimental studies have all documentesimilar geometrical pattern in this class of instabilitie
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1060 Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
namely the development of a spatially periodic cellular strture in the neutral direction which may, or may not, be timdependent. The review articles of Larson29 and Shaqfeh30
provide a detailed picture of the recent research on elainstabilities, and identify specific mechanisms for the nonlear coupling between the momentum and constitutive eqtions.
In a recent study closely related to the cavity flow gometry, Grillet and Shaqfeh31 observed elastic instabilities itheir modified Taylor–Couette experiments with Boger flids. In this work a meridional block was inserted in the clindrical Taylor–Couette device which created a local recculating pressure-driven Taylor–Dean motion upon sterotation of either the inner or the outer cylindrical wall.this configuration, elastic instabilities were observed nearblock at critical speeds which were markedly lower thancritical values observed far from the block in the unmodifiTaylor–Couette flow. This indicates that the local recircution near the inserted block is responsible for initiating insbilities at lower critical speeds. In the limit of narrow gapbetween the inner and outer cylinders, the azimuthal cuture can be neglected and this geometry can be unraveledviewed as a cavity flow with a very small aspect ratL→0.
In a companion study to the present work, Pakdel aMcKinley32 conducted a series of flow visualization expements of cavity flows of Boger fluids in a more moderarange of aspect ratios 0.25<L<4.0 and reported observations of elastic instabilities for all aspect ratios. Basedthese observations and consideration of previous studieelastic instabilities occurring in various geometries, they pposed a general dimensionless stability criterion that canused to quantify the onset of elastic instabilities based onlocal kinematics of the flow and the elastic properties offluid. This criterion was further developed in a more detaistudy,33 to incorporate the effects of shear thinning in tmaterial functions, changes in the solvent viscosity, anspectrum of relaxation times. Comparisons with existing din the literature showed that the proposed stability criterionsuccessful in providing a sufficient condition that can chacterize the onset of elastic instabilities for isothermal flmotions within many geometries with curvilinear strealines. This is consistent with theoretical studies which incate that there should exist universal destabilizing mecnisms that depend on the local fluid kinematics and strdistributions, which are themselves functions of the fluid rhology and global dimensionless geometrical parameters cacterizing the flow.
In this study, we expand the results of Pakdel aMcKinley32 and Pakdelet al.15 and provide detailed experimental results on the spatial and temporal dynamical stture of the purely elastic instabilities that develop in cavflows over a range of aspect ratios (0.25<L<4.0). We useLDV and DPIV to probe the kinematics of the secondamotions in the cavity flows of two ideal elastic Boger fluidIn the next section we describe our experimental apparand operating conditions, and characterize the rheologthe viscoelastic fluids used in our experiments. We thenport our experimental results including global flow visualiz
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tion of the instabilities, plus quantitative measurementsthe spatial and temporal frequency of the unstable modesconclude with an analysis of the recently proposed dimsionless criterion for elastic instabilities and its applicationviscoelastic cavity flows.
II. EXPERIMENTAL CONDITIONS
A. Geometry
A schematic diagram of the apparatus is shown in FigThe geometric specifications of the test cell have beenscribed previously,15 and here we briefly describe the pernent information related to the present experimental stuThe cavity cell is constructed with12-in. Plexiglas and thecavity dimensions areL52.54 cm,W510.16 cm, whereWis the length in the neutral direction. The depth of the cavH, can be varied in the range 1/4L<H<4L, by using Plexi-glass inserts to provide aspect ratios in the range of 0<L<4, respectively. The aspect ratio in the spanwise dirtion is W/L54 and can be altered toW/L58 by using suit-able inserts. The fluid motion is generated by translatinsmooth continuous polyester belt over the top of the cavThe maximum linear belt speed is approximatelyU'5.0 cm/s or 2L/s.
The cavity cell provides visual access from all thrprincipal planes in the Cartesian coordinate system showFig. 1. Thex-y plane at the midpoint of the channel widththe cross section of the flow in which our previous twdimensional cavity flow measurements were performed. Flowing the onset of hydrodynamic instabilities, the orthognal views of thex-z and y-z planes provide additionainformation about the kinematic structure of the disturbanin the neutralz-direction. The test fluids are seeded wiminute amounts of small Mica flakes which reflect the indent light with varying intensity depending on the directioand uniformity of the local velocity field. In the steady baflow regime, these particles reflect a uniform backgroulight intensity in thex-z andy-z planes. However, following
FIG. 1. Schematic diagram of the experimental apparatus:~a! cavity cell;~b!translating belt;~c! pressure plate.
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1061Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
the onset of an hydrodynamic instability, they develop nouniform effective light patterns in the neutral direction dpending on the relative fluid velocity and particle orientatiwith respect to the incident light. A similar visualizatiotechnique has been used by Baumert and Muller34 to docu-ment the complex spatio-temporal dynamics following tonset of purely elastic instabilities in Taylor–Couette flow
In addition to visualization of the global dynamicquantitative kinematic measurements are performed usinser Doppler velocimetry and digital particle image velocietry, details of which are given elsewhere.15,35 The onset ofinstabilities are detected both by visual inspection andLDV measurements. Spatial wave numbers are determfrom photographs and verified by rapidly scanning the cwith the laser probe using a translating stage while simuneously collecting LDV data. The temporal frequenciesobtained from time-series analysis of velocity measurememade using the LDV system at a fixed point in space. Eexperimental run is carried out with a fresh batch of eachfluid in order to reduce possible effects of polymer degration.
B. Fluid rheology
A viscous polybutene~PB! oil ~Amoco Indopol H300!with a mass-averaged molecular weight of approximat1000 g/mole is used as the Newtonian base fluid. The elaBoger fluids are prepared by dissolving 0.20 wt. % and 0wt. % of high molecular weight polyisobutylene~PIB!~Exxon L-120,Mw;1.23106 g/mole! respectively in the PBoil.
The rheological properties of these fluids show simicharacteristics to those reported previously inliterature.36,15 The viscometric properties of both solutionare summarized in Table I.
As the rheological characterization of Quinzaniet al.36
demonstrates, the first normal stress coefficient of semidiBoger fluids generally show two plateaulike regions; a zeshear-rate plateau at very small deformation rates, and aond region at an intermediate shear rate range of 0.1<g<10 s21. At higher shear rates ofg.10 s21, the first normalstress coefficient monotonically shear thins. For these fluthe zero-shear-rate limitC10 is difficult to measure directlywith reasonable accuracy since the normal force becoindetectably small, although its magnitude can be inferfrom linear viscoelastic measurements of the quantity 2h9/vat low frequencies.36 However, in Table I, we choose to report the average values ofC1 measured experimentally i
TABLE I. Viscometric properties of the two PIB Boger fluids.
the second plateau region since this range is consistentthe operating range of shear rates in our cavity flow geoetry.
The shear viscosity of both fluids remains relatively costant over a wide range of shear ratesg<100 s21 beyondwhich it slowly shear thins. The values of the dimensionleparameterb5hs /h0 , characterizing the relative contribution of the Newtonian solvent to the total viscosity in vicoelastic constitutive equations such as the Oldroydmodel, are in the range of 0.6<b<0.8.
The time constantls computed from the viscometricdata measured in steady simple shear flow is defined as
ls5C1,plateau
2~h02hs!,
where hs is the solvent viscosity. As it has been noteelsewhere,30 different rheological tests provide differentlweighted moments of the relaxation spectrum present inmacromolecular material. For completeness we also reanother characteristic time constantl r in Table I obtainedfrom measurements of the normal force relaxation followithe cessation of steady shear flow as shown in Fig. 2. Maing the entire nonlinear relaxation observed in the decayfirst normal stress differenceN1
2(t) requires consideration oa multimodal constitutive model; however, the data beyothe first few seconds can be accurately represented vsingle relaxation time,l r . The ratio of relaxation timesl r
A/l rB for the two fluidsA andB, appears to scale well with
the corresponding ratio oflsA/ls
B . Relevant values of thedimensionless numbers characterizing viscoelastic effectthe cavity flow can be based on either of these charactertime scales and for clarity are denoted Des or Der as appro-priate.
In the context of elastic instabilities, it is appropriateask which relaxation time~or Deborah number! is appropri-ate for correlating experimental results. This issue has bbriefly discussed in Ref. 30 and has been considered in dby Larsonet al.37 In the latter work, experimental measur
FIG. 2. Relaxation of the first normal stress differenceN12(t) with time
following cessation of steady shear flow for the 0.20 and 0.35 wt. % Psolutions. The initial steady shear flow was at a shear rate ofg0510 s21.
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1062 Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
ments of the steady and transient shear rheology were fiwith a K-BKZ integral constitutive model with a continuouspectrum of relaxation times characterized by a power-exponent,p. For this model the ratio of average to the lonest relaxation time~denoted byl/l1 in Ref. 37! is given byls /l r5(12p). For solutions of 0.1 wt. % PIB in PB anpolystyrene in oligomeric styrene, Larsonet al. report thatthe values ofp'0.75 andp'0.82, respectively, provide thbest correlations to the rheological data. For the fluid prerties given in Table I, we findpA'0.81 andpB'0.83, ingood agreement with the previous study. Larsonet al.37 alsocompared theoretical predictions of elastic instabilities inTaylor–Couette flow~using the K-BKZ model! with experi-mental stability observations and suggested that a Debnumber based on the geometric mean of the averagelongest relaxation times may provide the best measure oelasticity in the flow. In the present work we report our sbility observations in terms of both Der and Des , and a geo-metric mean can easily be computed from these valuedesired.
III. RESULTS
A. Global flow visualization
To detect the evolution in the secondary flow structufollowing onset of an elastic instability, the flow cell is illuminated with a diffuse background white light source. In tabsence of any flow structure in the neutral direction,Mica flakes reflect the incident light uniformly. When thflow becomes nonuniform, bands of darker and brighter liintensities develop in the neutralz-direction.
The belt speed is increased in small discrete steps wample observation time~5–10 min! between increments igiven for the possible development of elastic instabilitiWhen a discernable flow structure appears in the neutrarection, the critical linear belt speedUcrit is recorded. In Fig.3 an end viewof the y-z plane of the cavity cell is shownalong thex-axis and the spatial scale of the cell is showna ruler with centimeter gradations. The direction of motiof the belt is normal to the viewing plane and outward frothe page. In the stable region, the reflected light intenappears uniform@Fig. 3~a!#. However, at the critical onsecondition, Des,crit'0.35, a new spatial structure developsthe flow as shown in Fig. 3~b!. The aspect ratio of the cavitin Fig. 3 isL51, and the aspect ratio in the neutral directiis W/L54. The bright regions in Fig. 3~b! divide the flowdomain into three cellular regions.
In Fig. 4, the flow structure at the onset conditionsshown for a cavity with an identical aspect ratio ofL51 butwith a neutral direction aspect ratio ofW/L58. The criticalDeborah number is found to be the same for both casesthe number of cellular regions in theW/L58 case is doubledto six. This observation indicates that the elastic instabilitare driven primarily from consideration of the stable twdimensional planar base flow in thex-y plane and the threedimensional flow regions near the end walls do not driveflow into the three-dimensional regime, at least for cavitof sizeW/L>4.
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As we show below, these cellular structures are in ftraveling waves which propagate along the neutral (z) direc-tion of the cavity. The LDV time series presented in SeIII B show that in a time-averaged sense the flow structurperiodic at small supercritical Deborah numbers. Howevat any instant of time, analysis of the video recording shothat cells are continuously created and destroyed near eend wall, and this accounts for the uneven spanwise disbution observed in Figs. 3~b! and 4. In analogous Taylor–Couette studies, the neutral direction aspect ratio,W/L, istypically very large due to the small gap width in the flocell. However, in cavity geometries appearing in industrapplications, this aspect ratio is much smaller and nonsmetric structures arising from the influence of end effeand geometric imperfections are more likely to be observ
In Fig. 5 we show similar end views~in they-z plane! ofthe cellular structures following onset of elastic instabilfor deeper cavities with aspect ratios ofL52 andL53. Thecritical Deborah numbers are approximately the samethose recorded for the square cavityL51. Furthermore, thenumber of primary cells remains unchanged at three, asserved for the case ofL51. The separation between thprimary recirculating vortex structure near the translatplate and the weaker secondary vortex near the statio
FIG. 3. Flow visualization of elastic instability in a square cavity~L51;plan view!. The end view of the cavity flow is presented in which thimposed velocity is normal to the plane of view and outward from the pa~a! The stable flow at Des5Wis50.25; ~b! the unstable flow following theonset of elastic instability at critical flow conditions of Des5Wis50.35.
FIG. 4. Flow visualization of the elastic instability in a square cavity~L51, end view!. The aspect ratio in the neutral direction isW/L58. Theimposed velocity is normal to the plane of view in the outward directi(Des5Wis50.35).
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1063Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
base of these deep cavities can be clearly discerned.uniformity of the reflected light in these lower regions indcates that, at least within the time scale of an experimerun ('1 h), there is little dynamical evolution in this wearecirculation region for either aspect ratio.
Decreasing the aspect ratio of the cavity, however, hadifferent effect. In Fig. 6, the cellular structures observedthe critical onset conditions are shown in aplan view~i.e., inthe x-z plane! for shallow cavities with aspect ratios ofL50.5 andL50.25. The critical values of the Deborah number and Weissenberg number are significantly different frthose recorded for the square cavity (L51). The criticalDeborah number decreases as the aspect ratio decreasedicating that the instabilities are initiated at smaller bspeeds than for the square cavity. On the other hand,critical Weissenberg number increases with decreasingpect ratios, indicating the increasing magnitude of the shrate near the moving belt and the resulting enhancemenelastic normal stress differences in the cell. The relationtween these two dimensionless groups at the critical oconditions is a delicate balance between these two len
FIG. 5. Flow visualization of elastic instabilities in cavity flows~end view!with ~a! L53, Des50.37, Wis50.12,~b! L52, Des50.35, Wis50.18. Theimposed velocity is normal to the plane of view and oriented outward frpaper.
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scales in the system, namely,L and H, which in turn giverise to two time scales in the cavity geometry.
Increasing the belt speed to values significantly greathanUcrit excites higher order wave number disturbancesthis region the cavity operates in a time-dependent mixflow regime with the cellular structures rapidly fluctuatingextent and configuration. This transition to the mixing floregime can be better characterized by the spatio-tempfrequency measurements presented in the next section.
B. Local frequency measurements
To probe the dynamical structure of the instabilities oserved via flow visualization, temporal and spatial frequecies are measured as the imposed lid velocity is increaseFig. 7, time-series measurements of they-component of thevelocity are shown as a function of increasing Deborah nuber. These measurements are performed close to themetrical center of the primary recirculation in the cavityx/L50.5, y/H50.8, z/W50.0 for an aspect ratio ofL51.
In the stable region, Des50.2, the vertical component othe velocity at this position remains almost zero as a functof time. At the critical onset condition, Des,crit'0.35, theinstability initially develops as a slowly traveling sinusoidwave in the cavity with a period of approximately 600Increasing the Deborah number beyond the critical limitDes50.72 decreases the period of the traveling wave to vues on the order of 100 s. At higher Deborah numbers,primary mode is combined with higher frequency distubances which ultimately result in rapid aperiodic fluctuatioof the local velocity field in the cavity.
In Fig. 8, the power spectral density~PSD! is shown fora set of time-series measurements obtained at the sametion in the cavity. At weakly supercritical Deborah numbe(Des /Des,crit'1.3), the instability is characterized bysingle mode with a frequencyf 157.931023 Hz. Increasing
FIG. 6. Flow visualization of elastic instabilities in cavity flows~plan view!with ~a! L50.5, Des50.29, Wis50.58, ~b! L50.25, Des50.25, Wis51.0. The imposed velocity is parallel to the plane of view and orientedthe upward direction.
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1064 Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
the Deborah number further, excites the second harmand other higher frequency modes ultimately creatingmixing-type flow regime.
In Fig. 9, the primary temporal frequency,f 1 , is plottedagainst the Deborah number for the three dynamicallytinct aspect ratios ofL51,0.5,0.25. The spectral resolutioof the slowly varying time-periodic flow that develops byond a critical Deborah number is governed by the toduration of the velocity time series,T. For most of the datapresented here, the time-series span a period ofT'200– 300 s leading to a spectral resolution of appromately 61/T'0.003 Hz indicated by the error bar on thfirst data point in Fig. 9.
The temporal frequencies measured in deeper cavwith aspect ratios greater than one are similar to theL51case. As is shown in Fig. 9~a!, the magnitude of the primaryfrequency initially increases with Deborah number agradually approaches a plateau value as the flow evolvesthe mixing regime. The frequencies are higher in theL50.25 cavity than in the larger aspect ratios. In Fig. 9~b! thesame frequencies are scaled with the characteristic residtime of the systemL/U and are plotted against the Debornumber. The primary dimensionless frequency appears tocline dramatically forL50.25 while remaining relativelyflat, particularly forL51. When the cavity is operated abelt speeds beyond the limits shown in Fig. 9, the spec
FIG. 7. Time-series measurements of the verticaly-component of the ve-locity field at x/L50.5, y/H50.8, z50.0 in a square cavity (L51); theevolution of the elastic instability as a function of the imposed lid veloc
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content of the secondary flow can no longer be simply chacterized by a single dominant frequency as the mixing flregime develops.
In Fig. 10, the primary dimensionless spatial wave nubers (aL), measured via LDV and photographic analysesplotted against the Deborah number and the Weissennumber respectively~based on the steady shear relaxatitime!. The spatial wave numbera is extracted from Fourieranalysis of the rapid LDV scans ofvy(z) and vx(z) in theneutral z-direction of the cell and they monotonically increase with increasing Des and Wis , consistent with our flowvisualization observations. The wave numbers are contently higher in smaller aspect ratios and, as observed in5, the spatial frequency for deep cavities with higher aspratios L52,3 remain similar to theL51 case. Increasingthe Deborah number beyond the limits shown in Fig.creates a mixing flow in which many spatial modes arecited, making accurate determination of the primary spawave number difficult.
IV. OPERATING STABILITY DIAGRAMS
Cavity flows are encountered in many important indutrial processes and the knowledge of the boundaries ooperating stability for a given fluid formulation has impotant design and processing implications.2 In Fig. 11, the criti-cal Deborah number is plotted versus the aspect ratioboth PIB fluids. For completeness, the Deborah numbersfined with both time constantsls and l r are shown on theleft and right ordinate axes, respectively. Within the accracy of the experimental measurements, the data for bfluids show an excellent superposition. The critical Debonumbers remain almost the same for aspect ratios grethanL51, and decrease progressively for shallower cavitwith small aspect ratios.
Recognizing the importance of both length scalesL andH in the cavity geometry, an operating stability diagram cbe developed by plotting the critical Weissenberg numagainst the critical Deborah number for all aspect ratiosshown in Fig. 12. For each aspect ratio, the operatingbegins at the origin where the imposed velocity is zeroU50) and the fluid is at rest. Increasing the imposed drivvelocity U describes a unique set of operating conditiolying along a straight line which passes through the oriwith a slope ofL21. The stability boundary defines the regions of stable versus unstable operation of the lid-drivcavity for the range of aspect ratios 0.25<L<4.0 consideredin the present study. More data are clearly required for shlower cavity flows with smaller aspect ratiosL,0.25 to ex-pand the process stability diagram. However, the presentbility diagram in Fig. 12 spans most industrially importacases in coating and extrusion processes.
V. KINEMATIC CHARACTERISTICS OF THEUNSTABLE FLOW
The flow visualization photographs provide qualitatiinformation on the global structure of the three-dimensioflow that develops in the neutral direction following onsetthe elastic instability. However, further details of the tran
.
1065Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
FIG. 8. Power spectral density~PSD! for the time-series measurements of they-component of the fluid velocity atx/L50.5, y/H50.8, z50.0 in a squarecavity (L51).
rieoecn
orsearedting
two
tion from the stable to unstable region and of the symmetand kinematic characteristics of the resulting unsteady flare needed to make a more clear assessment of the mnisms that drive the growth of the observed temporal aspatial disturbances.
In Fig. 13, streak images of the flow field in thex-y
swha-d
plane following onset of the cellular instability are shown faspect ratios ofL50.5, 1 and 2. The exposure times of theimages are less than 1 second and are very small compto the period of the traveling wave disturbances propagain the neutralz-direction ~orthogonal to the imaging plane!.Hence the streaklines in these images appear smooth and
linnhoys.neher-en
akesa
ity
n
inof
laneeedthee
-on.
e
%F
vitynd
1066 Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
dimensional. In all cases the fore-aft symmetry about thex50.5L, observed for the creeping motion of Newtonialiquids in the corresponding cavity geometry, is broken. Tupstream shift of the geometric center of the main core vtex flow is enhanced compared to that reported previouslsteady two-dimensional flows at lower Deborah number15
Laser Doppler measurements of the out-of-plane compoof the velocity field at the critical conditions indicate that tinstability is initiated in the vicinity of the downstream coner where the curvature of the fluid streamlines is mosthanced.
This region is highlighted by the black box in the strephotographs. The similarity in each image is readily apparand, furthermore, it is clear that the representative radiucurvature of the streamlines in this region is not solely chacterized by either the heightH or width L of the test cell.
In Fig. 14 an approximate DPIV analysis of the veloc
FIG. 9. ~a! Dimensional and~b! dimensionless temporal frequency of thprimary mode of elastic instability in cavity flows with aspect ratiosL50.25,L50.5, andL51.0 as a function of Deborah number for 0.35 wt.PIB solution. The error bars indicate the spectral resolution limit of the Fanalysis.
e
er-in
nt
-
ntofr-
field in they-z plane atx/L50.5 is attempted. The directioof the imposed velocityU is outward from the illuminatedplane; however the out-of-plane displacement occurringthe time interval of 1/30th s is smaller than the thicknessthe laser light sheet, hence we are able to resolve the in-pcomponents of the three-dimensional trajectories of the sparticles in each subimage. In the stable flow regime,dominant component of the vectorial fluid velocity at thmidplanex/L50.5 is projected in thex-direction and there-fore DPIV observations in they-z plane generate null displacement cross-correlations indicating no in-plane motiFollowing the onset of instability, they- andz-componentsT
FIG. 10. Dimensionless spatial wave numbers of secondary flow in caflows with aspect ratiosL50.25, 0.5, and 1.0 as a function of Deborah aWeissenberg numbers. The data include LDV measurements~hollow sym-bols! and photographic image analysis~solid symbols! for 0.35 wt. % PIBsolution.
FIG. 11. The critical Deborah number Des,crit and Der ,crit ~based on therelaxation timesls andl r , respectively! as a function of aspect ratioL forboth PIB fluids.
rersruoue
r
ta-ctr-
i--
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ity
IB
a
thea
ofthe
1067Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
of the secondary flow are apparent in the DPIV measuments as shown in Fig. 14. This measurement is, of couonly an instantaneous snap shot of the evolving spatial stture in the cavity geometry but it shows the complexitythe traveling waves in the flow domain. A similar techniqhas been used very recently by Baumertet al.34 to resolvesecondary cellular structures in the viscoelastic TayloCouette instability.
FIG. 12. An operating stability diagram for isothermal viscoelastic cavflows. The solid line divides the stable and unstable operating regions.
FIG. 13. Streak images of the fluid streamlines for cavity flows of the Psolution with aspect ratios ofL50.5, L51.0, andL52.0. The Deborahnumber is Des51.2 for all cases. The highlighted areas are the downstrecorner regions where the instability initiates.
-e,c-f
–
VI. DISCUSSION
We have documented the onset of purely elastic insbilities in cavity flows of Boger fluids in moderate asperatios (0.25<L<4.0). Complementary experimental obsevations have also been reported by Grillet and Shaqfeh31 inrecirculating flows of ideal elastic Boger fluids in the semcavity problem (L→0). The structural patterns of these instabilities are similar to the purely elastic instabilities oserved in the Taylor–Couette device, planar contractflows, and the stagnation flow in the wake of a stationacylinder. These instabilities are characterized by evolutiona three-dimensional motion in the neutral direction whicthrough flow visualization, appear as cellular structures silar to those presented in Figs. 3, 4, and 5.
As our LDV time-series measurements indicate~Fig. 7!,the elastic instability is time dependent, initially having velong periods on the order of several hundreds of secondscontrast, Grillet and Shaqfeh31 report that for the semi-cavityflow, the secondary flow that develops at small supercritiDeborah numbers appears to be stationary. However,important to note that the aspect ratiosW/H in the neutraldirection for these two sets of experiments are substantidifferent. Recent experimental observations in the placontraction geometry have also shown that the mode ofstability ~i.e., traveling or standing waves! is modified by theaspect ratio of the flow cell in the neutral direction.38 Previ-ous linear stability analyses in the Taylor–Couegeometry30 have shown that for a Couette cell of infinitaxial length both traveling and standing waves are admsible modes of elastic instability for an Oldroyd-B fluid, anrecent numerical studies on the effects of eccentricity in tgeometry have shown that even very small geometric impfections can have a pronounced effect on the most unstmode of the resulting elastic instability.39 Since the period ofour measured velocity fluctuations close to the onset cotions is much longer than any fluid relaxation time scale oexperimental observations suggest that for a twdimensional cavity withW/H→`, where the effect of endwalls are entirely absent, the elastic instability may initia
m
FIG. 14. DPIV measurements of the time-dependent secondary flow onplane x/L50.5 that develops following onset of elastic instabilities insquare cavity at Des51.5. The imposed velocity is normal to the planeview and in the outward direction. The dashed lines are used to illustratefluid trajectories and are not DPIV measurements.
xtio
nlitilew
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vef td
b
obecma
o-ot
gu
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crea
mtnd
ow
chavity
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nd
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n
pli-
ra-
tes
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1068 Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
develop in the form of standing waves. Clearly, further eperimental investigations in cavities with fixed aspect raL5L/H, but with varying aspect ratioin the neutral direc-tion, W/H, are required to resolve this question.
Within the resolution of our experimental apparatus,detectable hysteresis effect was observed and all instabiwere supercritical. The hysteresis is, of course, a compphenomenon and it could occur within a very small windoof operating conditions which is experimentally difficultconclusively detect. Certainly numerical stability analysisneeded to clarify the behavior of the three-dimensional timdependent solution of the equations of motion, near, orthe critical onset conditions.
The spatial frequencies of the secondary flow that deops at the onset conditions appear to be independent oneutral direction aspect ratioW/H as observed in Figs. 3 an4. The spatial frequencies are, however, strongly affectedthe variation in the aspect ratioL. From Figs. 3 to 6, it canbe seen that the number of distinct cellular structuresserved forL53,2,1,0.5,0.25 are 2, 2, 2, 4–5, 11–12, resptively. The functional dependency of the spatial wave nuber onL is complex because the aspect ratio is a relevdynamic parameter only when both length scalesH and Lequally affect the local configuration of the planar twdimensional base flow. In the limiting cases of deep sl(L!1) or shallow cavities (L>1) only one length scaleaffects the fluid kinematics, and therefore any scaling arment should be able to explain these limiting cases.
In general, the spatial frequencies increase with increing Deborah and Weissenberg numbers as shown in FigUpon transition to the mixing regime (De>3Decrit), it is ex-perimentally difficult to decompose the spatial frequeninto a single predominant mode, since many higher fquency disturbances are also excited and mask the primmode. The kinematical structure of these instabilities is silar to the Taylor–Go¨rtler vortices observed following onseof inertially driven hydrodynamic instabilities in Newtoniacavity flows.2 The DPIV snapshot of the flow field, presentein Fig. 14, is also consistent with the laser light-sheet flvisualization of Grillet and Shaqfeh.31 Both observations in-dicate the existence of a ‘‘mushroomlike’’ structure whipossibly arises from the pairwise interactions of weak treling vortices near the floor of the cavity. Nonlinear stabilcalculations have shown that similar structures evolveform Gortler vortices in Newtonian fluids and these strutures lead to the rapid mixing of fluid regions with high alow momentum.4
As we have argued previously,15 the kinematics of thetwo-dimensional flow in the downstream corner play an iportant role in characterizing the critical onset conditioThe two-dimensional view of the unstable flow shownFig. 13 illustrates the key corner region where these elainstabilities initiate. At the critical onset conditions it is clethat the spatial structure of the primary flow in this regionvery similar regardless of the global aspect ratio of the city. At high Deborah numbers, the fore-aft asymmetry of tbase two-dimensional flow becomes increasingly apparThe center of the core vortex flow shifts in the upstredirection and the streamline radius of curvature in the dow
-,
oesx
s-t,
l-he
y
---nt
s
-
s-0.
y-ry
i-
-
o
-.
ic
-
t:
-
stream corner concomitantly decreases which, in turn, amfies the magnitude of the streamwise hoop stress (tuu), andits gradient (tuu /r ) in this region. Here,r is a local coordi-nate normal to the fluid streamline that characterizes thedius of curvature of the streamline andtuu is the uu-component of polymeric stress. In curvilinear coordinaterms such astuu andtuu /r provide the coupling mechanismbetween the individual components of the momentum eqtion and the convected derivatives of stress appearing in qsilinear constitutive equations. In their study of the pureelastic instability arising in Taylor–Couette flow, Larsoet al.17 point out that the systematic increase in the magtude of such terms with increasing shear rate is the primphysical mechanism for driving the elastic instability. Simlar couplings between the kinematics and radial gradientthe hoop stresses lead to instabilities in Taylor–Dean floof elastic fluids.40
In more complex two-dimensional flows such as the city flow considered here, both the curvature of the flustreamlines and the magnitude of the polymeric hoop stvary throughout the flow domain. Recently, a dimensionlcriterion for unifying the critical onset conditions of elastinstabilities in various flow geometries has been propose32
This criterion for onset of elastic instability can be writtenthe following form,
H l
RWisJ >M crit
2 , ~6.1!
wherel[lsU is interpreted as the characteristic length scover which the perturbations to the viscoelastic base flrelax, andR as the characteristic streamline radius of curvture in the system. At the critical conditions, this dimensioless group attains a critical magnitudeM crit
2 , beyond whichthe flow is unstable.
In many unidirectional flow geometries, such as flowgenerated in a cone-plate rheometer, between two conceparallel disks, and in the Taylor–Couette geometry, the chacteristic streamline radius of curvature can be readily idtified. More importantly, the streamline radius of curvaturemains constant when the magnitude of the imposed drivvelocity is increased. However, in more complex flows suas the cavity flow, the streamline radius of curvature varthroughout the flow geometry and is furthermore a functof the imposed velocity or throughput. Under such circustances, a simple scaling expression can be constructequantify the streamline radius of curvature by combiningprincipal radii of curvature of any two-dimensional flow fiein the form
1
R5
a
L1
b
H, ~6.2!
wherea andb are two dimensionless weighting parametethat identify the relative importance of the two primalength scalesL and H in modulating the geometrical structure of the flow.
A more detailed study33 provides further insight into thisdimensionless criterion and applies the formula successf
as
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-
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-
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1069Phys. Fluids, Vol. 10, No. 5, May 1998 P. Pakdel and G. H. McKinley
to a broad selection of experimental and theoretical cstudies. In this work the proposed dimensionless groupexpressed in a more general form as
S lsU
R
t11
t12D>M crit
2 , ~6.3!
wheret11 is the tensile stress in the local streamwise dirtion, andt12;h0g is the characteristic shear stress acrthe stream lines. This definition, which provides an expldefinition for the Weissenberg number in terms of a strratio, can be obtained directly from dimensional considation of the convected derivative terms encountered in qsilinear and nonlinear constitutive equations. The dimensless criterion in Eq.~6.3! can also be extended to includespectrum of relaxation times and, more importantly, thefect of shear thinning in the material properties.33
If we follow a fluid element along a closed streamlinthen the streamwise tensiont11 will decrease, and can evechange sign, as fluid particles pass through regions of lo~Lagrangian! deceleration. In these regions, Eq.~6.3! sug-gests that the flow is stable in agreement with our expmental observation. By contrast, regions characterizedcurved streamlines coupled with strong streamwise accetion are expected to be prone to elastic instability. The spavariation of the scalar magnitude ofM2 in a stagnation flowhas recently been investigated by O¨ ztekin et al.41
In the cavity geometry, the instability initiates in thdownstream corner where the streamlines of the baseexhibit significant curvature, and fluid elements acceleraway from the corner. In this region, the deformation rscales withg5U/R, which defines a local WeissenbenumberWis,R5lsU/R. Substituting the local Weissenbernumber and Eq.~6.2! into Eq. ~6.1!, we arrive at the follow-ing equation at the critical onset conditions,
lsUcritS a
L1
b
H D5M crit . ~6.4!
Rearranging terms, we obtain
aL1b51
Wis,crit, ~6.5!
with a5a/M crit and b5b/M crit . Although a and b are un-known a priori, the reciprocal of the critical Weissenbenumber at the onset of instability is thus expected to blinear function of the aspect ratioL.
In Fig. 15, the critical Weissenberg number is plottagainst the aspect ratio and there is an excellent agreewith the proposed dimensionless scaling analysis.32,33 Theexperimental measurement of Grillet and Shaqfeh31 for asemi-infinite cavity geometry (L→0) can also be represented in this form, and is shown in Fig. 15 by the asteclose to the abscissa. Clearly this experimental observawhich was performed with a very similar Boger fluid, is alconsistent with the proposed form of the stability criterioSince Eq.~6.4! involves three dimensionless parameters aonly two values may be determined from linear regressionthe experimental data in Fig. 15, it is not possible to unabiguously determine the critical magnitude of the propos
eis
-sts-a-n-
f-
,
al
i-ya-al
wtee
a
ent
xn,
.df-d
stability criterion,M crit , without consideration of a numerical linear stability analysis. AsL→0 the critical onset con-dition reduces tob50.14 or Wis,crit57.2.
As we have documented in the present study, thererich dynamical structure in the cavity flow of non-Newtoniafluids. The scaling arguments proposed recently32,33,42can beused to systematically describe the effects of fluid rheoloand geometry on the operational stability of isothermal ldriven cavity flows. Ultimately the critical conditions represented by a criterion such as Eq.~6.3! may even be utilizedin a predictive or design capacity to expand the operatiocapabilities of processing operations involving complflows of viscoelastic liquids. Screw extruders embody maof the kinematic elements of cavity flows, however such pcesses are commonly nonisothermal, and it would be ofterest to extend these arguments to such processes.
ACKNOWLEDGMENTS
This research was supported by National Science Fodation Grant No. CTS9553216 to G. H. McKinley. The athors would like to thank Dr. Stephen Spiegelberg for asstance in constructing the flow cell and the anonymoreviewers for their insightful comments.
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