Studies of viscoelastic flow of polymeric fluid are of practical interest since these flows are encounteredin many polymer processing applications such as injection molding, extrusion and fiber-spinning. Elas-tically driven and other flow transitions in such flows limit the processing speed and lead to a worthlessproduct. Comprehensive reviews of purely elastic flow transitions in various viscoelastic flows have beengiven in [1–3]. A brief description of various types of elastic instabilities encountered in such flows isgiven below.
Purely elastic instabilities in viscoelastic torsional flows, such as rotational shear flow between paralleldisks, between a cone and a plate, and between eccentric cylinders, have been documented by severalinvestigators [4–12]. These viscometric flows sometimes become unstable to three-dimensional distur-bances. Critical conditions for the onset of flow transitions and the spatio-temporal structure of secondaryflows generated by the flow transitions have been documented by these investigators. It has been shownthat the primary mechanism responsible for flow transitions in rotational shear flows is the occurrence ofhoop stress along the curved streamlines [4,10,13].
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Elastic flow transitions have also been observed in more complex geometries, such as contraction vis-coelastic flows and flow past an immersed object. Although the mechanism of the instability in theseflows has been shown [13] to be similar to those in simpler rotational shear flows, the spatial and thetemporal nature of the secondary flows in these geometries are very different. Flow transitions in vis-coelastic flow past a circular cylinder lead to formation of regularly spaced cells in the wake of the cylinder[14–16]. These vortices are nearly stationary and are localized into a small region downstream of thecylinder. Flow upstream of the cylinder is not influenced by thesecellular type instabilities. The cellularstructure generated by the flow transitions in this geometry is similar to Gortler vortices produced byinertially-driven instabilities in boundary layer flow over a curved surface. On the other hand,pulsatingtype instabilitieshave been observed in viscoelastic contraction flow [17–21]. Secondary flows producedby the flow transitions in these entry flows are strongly time-dependent and influence the flow far upstreamand downstream of the contraction plane [20,21].
Recently, elastic flow transitions have been reported in re-circulating viscoelastic flows [22–24]. Pakdelet al. [22] have documented cellular type instabilities that occur in re-circulating flow in a lid-driven cavity.For values of translational velocity of the top wall beyond the critical value, regularly-spaced cells areformed along the neutral direction of the cavity. Pointwise velocity measurements by the authors indicatethat the flow becomes periodic after the onset of the instability and becomes irregular in time and evenchaotic, as the translational speed of the side wall is increased further. This temporal behavior of theflow appears to be similar to the one in the pulsating type instabilities documented in contraction flow byYesilata et al. [20,21]. Hence, it is not conclusive whether the instabilities in these re-circulating flows areof the cellular type or the pulsating type. Pakdel et al. claim that these are cellular type instabilities, andthat the time-dependent nature of the flow is caused by the propagation of these cells along the neutraldirection of the cavity, but not by the pulsation or burst of the vortices. An instability of a similar type hasbeen reported by Grillet et al. [24] in re-circulating flow of viscoelastic fluid in a driven half-cavity. Thestability of viscoelastic flowpast a cavityhas not been investigated. These flows are inherently differentin nature compared to re-circulating flow in a lid-driven cavity. The flow in the re-circulation region insidethe cavity is weak, but a strong shear flow exists outside the cavity and protrudes into the cavity.
Viscoelastic flow past a hole has been employed by several researchers to measure the rheologicalproperties of polymeric fluids [25]. Cochrane et al. [26] have presented detailed flow visualization dataand numerical simulations for viscoelastic flow past a cavity. Elastic and inertial effects on the flowstructure in a square and a deep cavity have first been examined by these investigators using streaklinephotography methods with Newtonian and non-Newtonian fluids. The Newtonian fluid used in theirexperiments was a mixture of high maltose syrup and water. The non-Newtonian elastic fluid used intheir experiments was a dilute solution of polyacrylamide mixed in water and maltose syrup, with arelaxation time of 0.03 s and viscosity ranging from 0.1 to 0.3 Pa s. They have shown that the flow isnearly symmetric inside the cavity for Newtonian as well as non-Newtonian fluids at very low Reynoldsnumbers, while it becomes asymmetric as flow rate is increased. It has also been shown that the elasticityand inertia have opposing effects on the flow. The streamlines dividing the strong shear flow in the channelfrom the re-circulating region inside the cavity becomes asymmetric as inertial and elastic effects becomestronger. However, elasticity and inertia lead to asymmetry in opposite ways, as indicated in flow imagesof their deep cavity (cf. Figs. 7 and 8 in [26]). They have not examined the stability of the flow.
In the present study, the stability of PIB/PB/C14 fluid flow past a square cavity is studied. The criticalconditions for the onset of elastic instabilities are determined using flow visualization. The spatial andtemporal structure of the flow inside and outside the cavity is examined. Streakline photography is
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Table 1Dimensions of test sections
Dimensions of channel Dimensions of cavity
Length Gap Width Height Depth WidthL (mm) h (mm) W (mm) H (mm) D (mm) W (mm)
employed to acquire successive instantaneous images at various flow rates with different width to depthaspect ratios. Instantaneous pressure measurements inside and outside the cavity are conducted at variousflow rates to study the time-dependent nature of the flow. Such pressure measurements have been shownto be sensitive indicators of the nature of flow in complex geometries [20,21,27]. The influence of fluidelasticity on the flow structure is examined by acquiring flow images of viscoelastic fluid for a wide rangeof flow rates, and comparing them with the flow field measured and calculated for a Newtonian fluid flowpast a square cavity.
In Section 2, the test fluid rheology, geometry of the test section and experimental and numericalmethods used are described. In Section 3, the results are presented and discussed, and summarized inSection 4.
2. Experimental system and numerical procedure
2.1. Geometry
A schematic diagram of the test section is shown in Fig. 1a. Two different test channels were used inthe present experiments. The test sections are made of Plexiglas (acrylic) to facilitate flow visualization.Dimensions of these test sections are listed in Table 1.
There are three aspect ratios that are important in describing the flow geometry,
α = W
H, β = H
Dand 3 = H
h. (1)
β denotes the cavity aspect ratio, whereH is the cavity height andD is the cavity depth. Both test sectionshave square cavities,β = 1 for each geometry.3 denotes the ratio of the cavity height to the gap,h,between the channel walls.α denotes the spanwise aspect ratio, whereW is the cavity width. The firsttest channel,β = 1,3 = 1.5,α = 3.4, is referred to as C1, and the second channel,β = 1,3 = 1,α = 11,is referred to as C2.
Air from a compressor is used to force the Boger fluid from a tank into the test channel while a pressureregulator maintains constant air pressure in the tank. Pressure drops resulting from changing fluid levelin the tank are approximately 3% of the tank inlet pressure at the highest flow rates and are considerednegligible.
Flow visualization is performed by laser illuminated (streakline) photography, which has often beenused for visualization of viscoelastic flows of transparent fluids [20,26]. The technique is based on thelight scatter by tracer particles seeded within the fluid. Neutrally buoyant, silver-coated, 12mm hollowglass beads (Potter Industries) are homogeneously mixed into the fluid and are used as tracers, with aconcentration of 0.053 g/l of the test fluid. A Spectra-Physics 5 W Argon-ion laser with a cylindrical
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Fig. 1. Schematic diagram of experimental setup.
lens is used as a light source to produce a planar light sheet (Fig. 1c), which is used to illuminate thesilver-coated particles in the viscoelastic fluid and the streaks of the particles in the illuminated plane arerecorded. The light scattered from the particles is photographed using a 35 mm camera (Canon AE-1)with 400 ASA Kodak TMAX film. Long-exposure photographic images of the end view (cross sectionof cavity) and side view (neutral direction of cavity, span of the flow) of the flow field were recordedat varying flow rates. Images are digitized using a Nikon Coolscan (LS-20) scanner at a resolution of101 pixels/mm to produce streakline patterns. The resulting streakline images are saved in tagged-imagefile format (TIFF).
Schematic diagrams of the flow plane view, plane parallel to thex–z plane, and the span/neutral planeview, plane parallel to thex–y plane, inside the cavity, are also shown in Fig. 1b. Images in the flow
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plane view are acquired atY= 0 for both C1 and C2, and images in the span plane view are acquired atZ= –0.75 for C1 and Z= –0.5 for C2. Here dimensionless coordinates areZ= z/h, Y= y/handX= x/h.
Instantaneous single point pressure measurements were conducted using a Validyne DP-15 differentialpressure transducer with a 12.5 psi diaphragm (Model 3-40). The transducer is connected to pressure tapsat various locations along the side wall of the channel. The pressure taps are 2.3 mm in diameter and 40 mmlong. Since the length of the pressure tap affects the response time of the transducer, it is desirable tomaintain the shortest length possible to minimize the response time of the pressure measuring system. Thediameter of the tap must be small so that the disturbances to the flow are minimized. Pressure transducerdata was acquired using a 12-bit analog-digital (A/D) data acquisition board. The digital signals werecollected using the LABVIEW software.
Possible systematic disturbances (i.e. hole pressure effects, dynamic response time, etc.) to pressuremeasurements has recently been discussed in detail by Yesilata et al. [27]. It has been shown by theseauthors that these disturbances can be minimized and the temporal structure of the flow can accuratelybe determined by the instantaneous pressure measurements in flows of PIB based polymer solutions.
2.2. Fluid rheology
A PIB-based polymer solution, also called PIB-Boger fluid, is used as the test fluid for the experiments.The fluid is composed of 94.86% (by weight) polybutene (PB, Amoco H-300), 4.83% tetradecane (C14)and 0.31% polyisobutylene (PIB, Scientific Polymer Products, MWv = 1.7× 106 g/gmol). Polybutenewas used for experiments studying Newtonian fluid flow behavior. It is known that the tracer particlesmixed into the Boger fluid indicate no measurable effects on the rheological properties of the test fluid[28].
Rheological testing of the fluid has been performed by Shiang et al. [28] and the results are summarizedhere. Fig. 2 shows nonlinear viscosityη (γ ), dynamic viscosityη′ (ω), first normal stress coefficient91 (γ ) and dynamic rigidity 2η′′ (ω) /ω, whereγ is the rate of deformation andω is the frequency. Theviscosity and first normal stress coefficient remain nearly constant across three decades of shear rate.Jump discontinuity in the shear viscosity and the first normal stress coefficient atγ ∼= 8 s−1 is due toelastic spiral instabilities in torsional viscoelastic flow between a cone-and-plate [6,7].
Since the fluid has little shear thinning in viscosity (η) and in the first normal stress coefficient (91)within the range of shear rates encountered in the present experiments, the material properties in the limitof zero-shear-rate are considered in the calculation of flow parameters [29]
η0 = limγ→0
η (γ ) and λ0 = limγ→0
λ1 (γ ) = limγ→0
91(γ )
2ηp(γ ). (2)
The zero-shear-rate viscosity is measured to beη0 = 43 Pa s, the viscosity of the PB/C14 solvent isηs = 28 Pa s, and the zero-shear-rate Oldroyd relaxation time of the fluid isl0 = 2.5 s. Hereηp is thepolymeric contribution to the total viscosityη (η = ηs + ηp). The viscosity of the PB is separatelydetermined to be 78 Pa s. Details of the analysis to determine the first normal stress coefficient frommeasurements of dynamic rigidity are given by Shiang et al. [28]. All experiments are performed at roomtemperature (T0 = 25◦C), and the fluid properties listed below are determined at reference temperature(TR = 25◦C).
The volumetric flow rate,Q, through the test section is used to define the bulk-average velocity in thechannel, oru ≡ Q/(Wh). The Deborah and the Reynolds numbers are defined respectively as:
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Fig. 2. Rheological properties of PIB/PB/C14 polymer solution. (a) Steady and dynamic shear viscosity, (b) first normal stressdifference coefficient and dynamic rigidity (cf. Shiang et. al., [28]).
De = λ0u
hand Re = ρuh
η0, (3)
whereρ is the density of the fluid. Since the fluid is highly viscous and elastic, highDe flows can beeasily achieved with almost no inertial effects.
2.3. Numerical procedure
Numerical simulations of Newtonian flow past a cavity were performed using the commercial programFluent v4.2. For steady, incompressible flow, the governing equations used in the present computationsare mass and momentum conservation,
∇ · uuu = 0, (4a)
ρ(uuu · ∇)uuu = −∇p + ∇ · (ηγ ). (4b)
Hereuuu = (u, v, w) denotes the velocity vector withu, v, w as thex-, z-, y-component of the velocityvector,p is pressure,η denotes fluid viscosity,γ is the rate-of-strain tensor, andρ is density. The velocityboundary conditions are no-slip and no penetration at the wall of the cavity and the channel. The inletvelocity profile is set to be planar Poiseuille flow
u = 6uz
h
(1 − z
h
). (5)
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Fluent is capable of modeling flow through complex three-dimensional geometries using a control-volumebased, finite difference scheme. Velocity distributions are discretized using a first-order, power-lawscheme. The continuity equation and the momentum equation are solved using the well-accepted SIM-PLEC algorithm [30] with an iterative line-by-line equation solver and multigrid acceleration [31]. Theresulting set of algebraic linear equations is solved by repeated applications of the solution algorithm.The number of grid points used in the computations is 28 by 117 by 52 in thez-, x- andy-direction in C1,respectively. The grid independence of the results was checked by doubling the number of mesh pointsin the direction with the largest velocity gradient (z-direction). The maximum deviation in velocity mag-nitudes at any grid point is better than 1.5%. Three dimensional velocity fields are calculated for variousvalues of Reynolds numbers for Newtonian flow past a square cavity, and the results are compared withthe experimental results for the flow of PB fluid past the cavity.
3. Results and discussions
The structure of viscoelastic creeping flow of a polymeric solution past a square cavity is examined.Results of streakline photography and instantaneous pressure measurements are presented for variousvalues of Deborah number to examine the spatio-temporal nature of the flow. The critical conditions forthe onset of the elastic instabilities and the secondary flow structure produced by the flow transitions inthe cavity are described.
3.1. Flow structure in the cavity
Flow images in thex–z plane across the cavity in C1 (α = 3.4, β = 1, 3 = 1.5) are shown in Fig. 3for various values ofDe numbers. The direction of flow is downward for all images. At low flow rates,the streamline dividing the high shear flow region from the re-circulating region inside the cavity issymmetric, as shown in Fig. 3a forDe= 0.22. This penetration of dividing streamline into the cavity isa result of a larger area available for the flow and deceleration of the fluid. The elasticity of the fluidplays little or no role in the above phenomenon. The dividing streamline becomes slightly asymmetricas flow rate is increased, and it shifts deeper into the cavity (Fig. 3b). Asymmetry of the re-circulatingregion becomes pronounced as flow rate is increased toDe= 2.4, as shown in Fig. 3d. The center ofthe re-circulating vortex moves closer to the top wall inside the cavity. The nature of flow near thecavity at highDe resembles the die-swell of a viscoelastic jet discharging from a long capillary or anorifice [29,32]. Like the relaxation of normal stress (or release of normal stress) observed in the die-swellphenomenon, here in the present flow the fluid expands into a larger area (cavity) near the top corner. Thatintroduces asymmetry in the dividing streamline. The critical value of the Deborah number for the onsetof asymmetry is 0.38< Deasy< 0.83 for C1 and 0.53< Deasy< 0.97 for C2. It is remarkable to note thatthe value ofDeasy is very close to the criticalDe for the onset of die-swell (Deswell = 0.55) of PIB/PB/C14extrudate from a capillary [32]. We also note that the extent of penetration of the dividing streamline intothe cavity appears not to be monotonic withDe. This is particularly obvious from a comparison of flowimages atDe= 2.4, 3.6 and 7.5. This is due to the fact that flow becomes time-dependent at higher flowrates and the dividing streamline oscillates in and out of the cavity with time. This will be discussed indetail in Section 3.3. Cochrane and colleagues [26] have reported flow visualization in viscoelastic flow
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Fig. 3. Images in the flow plane view atY= 0 in C1 (α = 3.4,β = 1,3 = 1.5) for various indicatedDenumbers. Exposure timesused in the streakline photography are (a) 6 s, (b) 8 s, (c) 8 s, (d) 8 s, (e) 8 s, (f) 4 s.
past a cavity forDe up to only 0.75. The flow structure in viscoelastic flow past a cavity at highDe (forDeup to 12.9) has not been documented previously.
Flow images in the plane parallel to thex–z plane were also acquired for creeping flow of essentiallyNewtonian fluid, PB, for various values of Reynolds number. Fig. 4a shows that the flow structure of thePB fluid for Re= 4.8× 10−3 is similar to the structure of the PIB-based polymer solution at low flowrates (De= 0.22). AtRe= 4.8× 10−3, the flow rate of PB fluid corresponds to the flow rate atDe= 4.5for PIB-polymer solution. At this flow rate, the flow structure in PB fluid is very different from the flowstructure of Boger fluid. This shows that the inertial effects are practically eliminated in the viscoelasticflow of PIB Boger fluid, and that the observed flow structure is solely caused by the elastic nature of thefluid.
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Fig. 4. (a) Streakline image in flow plane view atY= 0 of PB fluid in C1 (α = 3.4,β = 1,3 = 1.5, exposure time is 4 s). Velocityvectors calculated for values ofRe(b) 4.8× 10−3, (c) 1, (d) 5, (e) 10. Also shown in (e) is magnified velocity vectors forRe= 10.
Fig. 4(b-e) show velocity vectors in the flow plane atY= 0 calculated for C1 at various flow rates. Flowimage in C1 acquired by streakline photography shown in Fig. 4a and the velocity vectors calculated forRe= 4.8× 10−3 shown in Fig. 4b are similar, and both indicate that the dividing streamline is symmetric.As the flow rate increases toRe= 5, the velocity vectors indicate that the flow inside the cavity becomesslightly asymmetric, as shown in Fig. 4d. ForRe= 10, the re-circulating vortex becomes definitively
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Fig. 5. Velocity vectors shown in the flow plane for PB flow in C1. Numerical simulations are performed atRe= 4.8× 10−3 ata location (a) near a side wall (Y= 2.45) and (b) near the midplane (Y= 0).
asymmetric (see Fig. 4e). The asymmetry of the dividing streamline caused by the inertial effects isvery different compared to that caused by the elastic effects. The inertial effects pull the re-circulatingvortex in the flow direction, and as a result the dividing streamline penetrates deeper into the cavitynear the bottom wall. This introduces an asymmetry in the dividing streamline that is opposite to theasymmetry caused by the elastic effects. For all flow rates considered in this study, the re-circulatingflow in the cavity is rather weak. These results are similar to numerical simulation results reported byCochrane et al. (cf. Fig. 19 in [26]). In order to examine the influence of the side wall on the flow structure,velocity vectors on a plane close to the side wall (Fig. 5a) are compared to the results at the midplane(Fig. 5b) of the cavity in the spanwise direction. At both planes, the flow structure is very similar,indicating that the effect of the side wall on the flow is weak for flow rates considered in the presentexperiments.
Flow images in the plane parallel to thex–zplane in C2 (α = 11,β = 1, 3 = 1) acquired by streaklinephotography for various values ofDeare shown in Fig. 6. At low flow rates, the flow structure is similar tothose for Newtonian fluid flow with a symmetric dividing streamline. Flow structures for high flow rates,especially atDe= 11, are remarkable. The re-circulating region becomes very small, nearly nonexistent
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Fig. 6. Images in the flow plane view atY= 0 in C2 (α = 11,β = 1, 3 = 1) for various indicatedDe numbers. Exposure timesare (a) 8 s, (b) 8 s, (c) 8 s, (d) 2 s, (e) 1 s, (f) 1 s.
at the top wall and the asymmetry of the dividing streamline is very clear. The re-circulating vortex in thecavity is also very asymmetric.
The results presented here for lowDeare similar to those observed by Cochrane et al. [26] for viscoelas-tic flow past a square cavity. They performed visualization using both Newtonian (syrup–water mixture)and elastic non-Newtonian fluid (polyacrylamide solution, PAC). They documented flow images in asimilar plane for various values ofDe (0 ≤ De ≤ 0.75) andRe(1 ≤ Re ≤ 20), and compared the resultsof non-Newtonian (PAC) flow to those of Newtonian flow (syrup–water mixture). They have shown thatfor PAC fluid the re-circulating vortex inside the cavity is symmetric only when bothReandDeare verysmall. It has been shown that elastic and inertial effects lead to an asymmetric re-circulating vortex, andthat they have opposing influences on the flow structure. In the present experiments, however, the elasticeffects on the flow structure are essentially isolated. The maximum value ofRenumber is around 10−3,and thus inertial effects in the flow of PIB-based polymer solution are negligible.
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Fig. 7. Flow images along the spanwise/neutral direction atZ= −0.75 in C1 (α = 3.4,β = 1, 3 = 1.5) for PIB Boger fluid atvarious indicatedDenumbers, for exposure times of (a) 30 s, (b) 30 s, (c) 30 s, (d) 30 s, (e) 4 s, (f) 6 s, (g) 4 s, (h) 2 s. Flow imagealong the neutral direction atZ= −0.75 in C1 shown in (i) for PB fluid atRe= 4.26× 10−3 for exposure time of 2 s.
3.2. Elastic flow transitions
In order to examine the stability of the viscoelastic flow past a cavity, we acquired flow images in theplane atZ= –0.75, along the span of the flow. Fig. 7 shows the flow inside the cavity in the neutral planeview for α = 3.4 (in C1). At lowDenumbers, the flow is nearly uniform along the neutral direction awayfrom the side wall. The end-wall vortices are present near the side wall, a large re-circulating vortexexists in the middle of the cavity, as clearly shown in Fig. 7b forDe= 0.62. AsDe is increased to 0.64,there are three cells formed inside the cavity beside the end-wall vortices. This is a clear indication ofacellular flow transitionfrom a single re-circulating vortex to multiple counter-rotating vortices inside the
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Fig. 7. (Continued).
cavity (excluding the end-wall vortices). The critical value of the Deborah number for the onset of theelastic flow transition lies between 0.62< Decrit < 0.64. We note that the values ofDeasy andDecrit aresimilar (0.38< Deasy< 0.83 and 0.62< Decrit < 0.64). There appears to be a close relationship betweenthe asymmetry of the dividing streamline and the elastic cellular instability. This conclusion is drawnfrom the fact that similar magnitudes of elastic effects are required for transition to 3-D cellular flow andasymmetric flow into the cavity.
For De= 2.4, there are three or possibly more cells present in the cavity. The cells are closer to eachother and the center of cell is shifted towards the top wall. At even higher values ofDe the number of cellsincreases, as seen in Fig. 7f and 7h. The development of cells withDe at flow rates beyond the criticalvalue resembles the well-known nonlinear phenomena called cell-splitting, as seen in natural convectionand other hydrodynamic instabilities [33].
The flow structure becomes rather remarkable at higher flow rates, as shown in Fig. 7e and 7g. In Fig.7f for De= 5.8, a pair of eddies is clearly visible with the centers closer to the top wall, and the flowstructure becomes very complicated. Several pairs of eddies are observed at even higher flow rates (Fig.7g for De= 8.0), along with smaller scale eddies. In Fig. 7h, several flower-like structures are present,
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Fig. 8. Flow images along the spanwise/neutral direction atZ= −0.5 in C2 (α = 11, β = 1, 3 = 1) for various indicatedDenumbers, for exposure times of (a) 10 s, (b) 10 s, (c) 4 s, (d) 0.5 s.
indicating that these cellular structures are strongly three-dimensional. We also note that at this highflow rate cells become so strong that they even invade the region very close to the side wall. In orderto display the influence of elastic effects in these flows, PB flow in the neutral plane atZ= −0.75 forRe= 4.26× 10−3 is shown in Fig. 7i. At thisRe, the average velocity in the flow channel would matchthat atDe= 4.2. There are no cellular structures in this flow; the structure of the flow is very similar tothat for PIB-Boger fluid flow at a lowDenumber (De= 0.21). These results lead to a conclusion that theflow transitions observed here are driven by the elastic nature of the fluid.
Similar flower-like structures have been documented by Chiba et al. in entry flow through a forward-facing step channel [34] and in entry flow through a planar contraction [35]. Dilute aqueous solutionof polyacrylamide has been used in both of these experiments. These authors have reported cellularinstabilities producing bundle-like flow structure in plane view streak photographs behind the contractionplane or the planar step. It has to be pointed out that the instabilities in these flows are very similar to thecentrifugal instabilities, Taylor–Gortler type, and that the inertial effects undoubtedly play a major rolein such flow transitions. Elastic effects in these flows [34,35] modify the critical conditions for the onsetof the instabilities and the resulting secondary flow structure, but are not the cause of these instabilities.Additionally, the present flow and the flow in [34,35] are very different. Nevertheless, there is a remarkablesimilarity in the spatial structure of the secondary flow between the present experiments and experimentsreported in [34,35].
Flow images in C2 (α = 11) along the span of the flow inside the cavity are shown in Fig. 8 for variousflow rates. ForDe= 0.27, no cellular structures are visible and there is a single re-circulating vortex
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occupying the region inside the cavity away from the side-wall where two end-wall vortices are present.As De is increased to 0.58, multiple cells appear along the neutral direction, indicating a cellular flowtransition in the cavity. The critical Deborah number for the onset of flow transition, 0.58< Decrit < 0.6,is very similar to that determined for the instabilities in C1 (α = 3.4). This implies that the influence ofthe side wall on the onset of flow transition is not strong. This result is very similar to that observedby Pakdel and McKinley [22] in viscoelastic lid-driven cavity flow. They have reported that the criticalconditions for the onset of elastic instabilities in these closed cavity flows are not altered significantlyeven when the length of the cavity in the neutral direction is doubled.
At De= 7.8 flower-like cellular structures are observed in C2, as shown in Fig. 8c. The number ofcells increases and the cells become well organized as the Deborah number is increased to 11.3. At thesehigh flow rates, the cells are regularly spaced along the span of the flow (neutral direction), as shown inFig. 8d.
3.3. Temporal structure
The temporal nature of the flow is examined by observing a time series of flow images along withsingle point pressure measurements inside the cavity and in the channel upstream and downstream of thecavity. Fig. 9 shows streaklines in the flow plane view atY= 0 (midplane of the cavity) in C2 (α = 11)for De= 1.1 andDe= 8.5 at successive times. ForDe= 1.1, the flow images are nearly identical, andthe flow inside the cavity is nearly steady with time. The vortices inside the cavity are stationary at thisflow rate. However, forDe= 8.5, the flow structure changes with time. The dividing streamlines movesin and out of the cavity and the size and shape of the re-circulating region varies with time. When theimage plane intersects a three dimensional flow cell, the re-circulating region appears larger and the highshear flow region does not penetrate deep into the cavity. When the re-circulating flow cell in the cavityis out of the image plane, the re-circulating region appears to be smaller and the high shear flow regionpenetrates deeper into the cavity. This phenomenon is repeating with time, but it is difficult to determinethe periodicity by examining the series of flow images.
Fig. 10 shows a time series of flow images along the spanwise/neutral direction atZ= −0.5 in C1(α = 3.4) for De= 2.4 and 8.0. AtDe= 2.4, the cells are stationary, as evidenced by the time-invariantflow structure in the cavity. However, forDe= 8.0, the structures are time-dependent. It is not obvious ifthe cells are moving in any direction hence it is not possible to conclude whether the time-dependencyis caused by the propagation of the cells or from a weak pulsation of the flow. The temporal structureobtained from this view is consistent with that observed in the flow plane view.
The temporal structure of the flow inside and outside of the cavity is determined quantitatively byconducting a series of instantaneous pressure measurements at differentDe. Local pressure measure-ments (single-ended) were conducted at three different locations in theα = 3.4 channel. Time series ofthese pressure measurements are shown in Fig. 11a for various values of the Deborah number,De. Thepressure tap for these measurements are located in the cavity (X= −0.75,Y= 0, Z= −1.5), where thedimensionless coordinates are defined in terms of (X= x/h, Y= y/h, Z= z/h). ForDe< 2.3, the pressure atthis point is nearly uniform in time. This implies that forDe< 2.3 the flow inside the cavity is steady andthe re-circulating cells are stationary. The flow becomes weakly time-dependent as the Deborah number isincreased beyond 4.2. The intensity of the pressure fluctuations increases slightly as the Deborah numberis increased to 9.5, but a well-defined frequency at these flow rates is not easily discernible. AsDe isincreased further toDe= 11.3, the fluctuations have more definite temporal structure and become more
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Fig. 9. Time series of flow images in the flow plane atY= 0 in C2 (α = 11,β = 1, 3 = 1) for (a)De= 1.1 at times, t= 0 s, 8 s,16 s, 24 s, and for (b)De= 8.5 at times, t= 0 s, 4 s, 8 s, 12 s.
periodic in nature. Power spectral density of pressure data obtained from FFT are shown in Fig. 11(b-d)for De= 2.3, 6.9 and 11.3. There is no discernible peak in the power spectral density forDe= 2.3. ForDe= 6.9, there is a peak atf ∼= 0.065 Hz, just slightly above the noise level. However, forDe= 11.3, thereare two definite peaks atf ∼= 0.067 Hz and 0.09 Hz, indicating that the secondary flow becomes moreregular and periodic at these high flow rates.
Time dependent pressure measurements (a) upstream of the cavity, (b) inside the cavity, and (c) down-stream of the cavity are shown in Fig. 12 for variousDe numbers. The upstream pressure tap is locatedat (X= −4.15,Y= 0, Z= 0) and the downstream tap is located at (X= 9.85,Y= 0, Z= 0). Each pressure
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Fig. 10. Time series of flow images along the spanwise/neutral direction atZ= −0.5 in C1 (α = 3.4,β = 1, 3 = 1.5), for (a)De= 2.4 at time, t= 0 s, 4 s, 16 s, 20 s, and for (b)De= 8 at times,t = 0 s, 8 s, 16 s, 24 s.
traceP(t) at a givenDe is normalized by the corresponding time-averaged nominal value of pressure,(<P(t)>). Each trace is drawn about a local value ofP(t)/ < P (t) > equal to unity. The distance betweensuccessive divisions is equal to 0.1. The temporal structure of the flow at each location is very similarand the critical Deborah number for the onset of the time-dependent flow at each locations is nearly thesame (3.2< De< 3.8). The intensity of fluctuations downstream of the cavity appears to be slightly largerthan that upstream of the cavity. This is due to the fact that nominal value of pressure downstream ofthe cavity is smaller since the pressure tap at this location is closer to the exit. These results suggest thattime dependent flow exists in the shear flow in the channel at high flow rates and is not caused by theflow transitions occurring in the cavity. In other words, temporal structure of the channel flow prior toreaching the cavity is imposed on the time dependent nature of the re-circulating flow in the cavity. Theelastic flow transition produces secondary vortices that are nearly stationary. The time dependent natureof these re-circulating cells in the cavity is consistent with the temporal structure of cells generated byGortler instabilities in Newtonian fluid flow when the streamlines are curved [36]. A stationary patternof Gortler vortices is observed in those flows.
The flow structure produced by the flow transition for the present experiments is very similar to thatobserved in viscoelastic flow past a cylinder [14–16]. In both flows, standing cells are formed following
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Fig. 11. (a) Time series of single point pressure measurements inside the cavity in C1 (α = 3.4,β = 1, 3 = 1.5) for variousDe.The pressure tap is located atX= − 0.75,Y= 0, Z= −1.5. Power spectral density of the pressure data forDeof (b) 2.3, (c) 6.9and (d) 11.3.
the onset of the cellular type of flow transition. These localized secondary flows do not influence the globalflow strongly. This is in contrast with the type of flow transition observed in viscoelastic contraction flow.In the contraction flow, secondary flows produced by apulsatingtype of flow transition modifies thetemporal structure of the global flow [20,21].
Slowly moving re-circulating flow cells in viscoelastic re-circulating flow in a lid-driven cavity [22,23]and in a driven half-cavity [24] have been documented. These instabilities appear to be of thecellulartype, similar to those observed in the present experiments. However, the spatio-temporal structure of thecells observed in the re-circulating flow in the lid-driven cavity is very different compared to that of thepresent re-circulating flow.
We are intrigued by the time-dependent nature of the shear flow in the channel. It is possible thatthe channel flow becomes unstable at higher flow rates (De> 3.8). It is also possible that geometric
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Fig. 12. Normalized time series of single point pressure measurements (a) upstream of the cavity (X= −4.15,Y= 0, Z= 0), (b)inside the cavity (X= −0.75,Y= 0, Z= −1.5), and (c) downstream of the cavity (X= 9.85,Y= 0, Z= 0) for variousDe. EachtraceP(t) at a givenDe is normalized by the corresponding time-averaged nominal value of pressure, (〈P(t)〉), and fluctuationsaround unity are shown. The distance between successive major tick division is equal to 0.1.
irregularities upstream of the channel introduce systematic disturbances to the flow at these high flow rates.There are various flow disruptions upstream of the channel due to fittings (i.e. valves, contractions andexpansions associated with the connection to the fluid tank). These types of disruptions are usually presentin most experimental setups as described here. Instabilities in flows through such configurations have beendocumented [20,21]. These studies have shown that such flow transitions can lead to time-dependencein the global flow field. It has recently been reported by Yesilata et al. [20,21] that pulsating type ofinstabilities in contraction flow influences the flow behavior both upstream and downstream far awayfrom the contraction plane. In order to minimize the effects of these systematic disturbances, the lengthof the present experimental flow channel upstream of the cavity was designed to be 17 diameters long(170 mm), which corresponds to approximately 3 s (1.7λ0) for transit time in the flow channel. Even atthe highestDe number under consideration, this should provide enough distance and time, such that the
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travel time of the fluid is longer than the relaxation time of the fluid. The transport of any deformationhistory of the fluid into the test section due to such upstream irregularities should have been minimized.Further investigations are needed to understand and resolve some of the issues raised above.
4. Conclusions
• Nonlinear dynamics of viscoelastic flow past a square cavity have been examined for Deborah numbersreaching 12.9. Flow structure of the creeping viscoelastic flow past a cavity is governed by the elasticnature of the fluid.
• Elastic flow transitions are observed for flow rates beyond a critical value. The critical value of theDenumber for the onset of cellular flow transition in a square cavity is 0.62< Decrit < 0.64 in C1 (α = 3.4,β = 1, 3 = 1.5) and 0.58< Decrit < 0.6 in C2 (α = 11,β = 1, 3 = 1). Comparison of flow images ofviscoelastic PIB-Boger fluid with images of Newtonian PB fluid proves that these instabilities aredriven by the elastic nature of the fluid.
• Re-circulating flow cells are formed in the cavity along the neutral direction for flow rates beyond thecritical value for the onset of elastic flow transition. Three-dimensional nature of the flow due to thepresence of the side walls influences the spatial structure of the cells but not the critical value ofDenumber,Decrit.
• At low Deflow, the inertialess viscoelastic flow inside the cavity is symmetric and the direction of theflow cannot be determined from flow images. At higherDe, flow becomes asymmetric, and the criticalvalue for theDe for the onset of asymmetry has been determined to be 0.38< Deasy< 0.83 in C1 and0.53< Deasy< 0.97 in C2. We believe that the mechanism responsible for the asymmetric flow intothe cavity is similar to that for well-known die-swell phenomena. It is probably not a coincidence thatthe critical value of the Deborah number for the onset of asymmetric flow into the cavity and for theonset of die-swell (Deswell = 0.55) of the same test fluid extrudate from a capillary are very similar.
• The flow inside and outside of the cavity is steady at low flow rates. The critical Deborah number forthe onset of time dependent flow is 3.2< De< 3.8. Measurements of instantaneous pressure inside,upstream and downstream of the cavity suggest that time-dependency on the flow may not be causedby the presence of the cavity. It is possibly introduced either by irregularities upstream of the channelor by the instabilities in the channel flow. In that case, the re-circulating flow cells in the cavity areessentially standing vortices.
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