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Rheologica Acta ISSN 0035-4511 Rheol ActaDOI 10.1007/s00397-013-0712-8

Large Deborah number flows aroundconfined microfluidic cylinders

Stephen Kenney, Kade Poper, GaneshChapagain & Gordon F. Christopher

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Rheol ActaDOI 10.1007/s00397-013-0712-8

ORIGINAL CONTRIBUTION

Large Deborah number flows around confined microfluidiccylinders

Stephen Kenney · Kade Poper · Ganesh Chapagain ·Gordon F. Christopher

Received: 28 September 2012 / Revised: 28 February 2013 / Accepted: 19 April 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract Viscoelastic flow around a confined cylinder athigh Deborah numbers is studied using microfluidic chan-nels. By varying fluid properties and flow rates, a systematicstudy of the roles of elasticity and inertia is accomplished.Two new elastic flow instabilities that occur at high Deborahnumbers are identified. A downstream instability of disor-dered and temporally varying streamlines is observed at aDeborah number above 10. This instability is a precursorto an unsteady vortex that develops upstream of the cylin-der at higher Deborah numbers. Both instabilities occur atmoderate Reynolds numbers but are fundamentally elastic.The size and steadiness of the upstream vortex are primarilycontrolled by the Deborah and the elasticity number.

Keywords Microfluidics · Elastic instabilities ·Viscoelastic flows

Introduction

Flow of viscoelastic materials around a confined cylinderis a problem that has been studied in great detail over thelast two decades and is considered a hallmark problem offluid mechanics (Kim et al. 2005). The lack of geometricsingularity points makes this flow an ideal geometry to val-idate numerical simulation of viscoelastic fluids (Afonsoet al. 2008). Furthermore, it is an excellent model geometryfor many industrial processes such as heat exchange, extru-sion, or flow through porous media which are relevant to

Special issue devoted to novel trends in rheology.

S. Kenney · K. Poper · G. Chapagain · G. F. Christopher (�)Department of Mechanical Engineering, Texas Tech University,Lubbock, TX, USAe-mail: gordon.christopher@ttu.edu

oil recovery, energy systems, food engineering, and plasticsprocessing.

Confined cylinder flow is characterized by a number offlow regimes that include elastic and inertio-elastic instabil-ities, which can affect species diffusion, heat transfer co-efficients, and flow patterns (Alves et al. 2001; McKinleyet al. 1993; Verhelst and Nieuwstadt 2004). Although thisproblem has been studied a great deal, practical issueshave limited the range of explored phase space. In particu-lar, study of highly elastic flows has been limited,however,these flows are highly relevant to many industrial applica-tions (Baaijens et al. 1995). Experimentally, it is difficultto visualize flows of highly viscoelastic materials at highstrains, and only recently have new mathematical techniquesallowed their simulation (Afonso et al. 2009, 2011; Fattaland Kupferman 2005; Hulsen et al. 2005; Jafari et al. 2012).

However, microfluidics offers a simple and robustmethodology to examine highly elastic flows (Pakdel andMcKinley 1996; Rodd et al. 2005). This has been real-ized as increased incorporation of long-chain molecules intomicro total analysis systems has shown that flows of evenweakly viscoelastic materials display significant viscoelas-tic effects (Hohne et al. 2009; Hsieh and Doyle 2008; Kimand Doyle 2007; Lindstrom and Andersson-Svahn 2010;Sousa et al. 2010; Vyawahare et al. 2010; Christopher et al.2009; Husny and Cooper-White 2006). This occurs becauseconfinement caused by the microlength-scale of microflu-idic channels creates large extensional strains, shear rates(Pipe and McKinley 2009; Rodd et al. 2005), and stream-line curvature (Gulati et al. 2008, 2010; Pathak et al.2004). Furthermore, microfluidic channels are forced tohave rectangular cross sections due to fabrication methodol-ogy, which creates inherently 3D flows and stronger elasticeffects in comparison to macrofluidic axisymmetric chan-nels (Oliveira et al. 2008; Rodd et al. 2005). Although it is

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possible to replicate relevant dimensionless parameters andthese conditions in macroscale flows, microfluidics offersa simpler way to create the relevant high Deborah andWeissenberg numbers to cause enhanced elastic effects.

The present paper describes the examination of highlyelastic, single-phase flows around a cylinder that is bothbounded and confined for low to moderate Reynolds num-bers. Channels are created using microfluidic techniques,which to the authors’ knowledge have never been usedto study this problem previously. Fluid elasticity and flowspeed are systematically varied to examine the roles of iner-tia and elasticity on the flow patterns around the cylinder.New flow instabilities observed for large Deborah numberflows that occur both downstream and upstream of the cylin-der are described, and the critical Reynolds numbers andDeborah numbers for the onset of these flows, as well as thelength and time scales of the upstream instability, are char-acterized. Finally, the role of inertia and elasticity on theflow is evaluated

Background

Dimensionless parameters

Rheological, geometric, and flow parameters all affect flowbehavior of highly elastic flows. A review of the literatureidentifies the following dimensionless numbers as havinga particular significance on highly elastic flows: Deborahnumber (De), which is the ratio of a solution’s relaxationtime to residence time in the flow; Weissenberg number(Wi), which is a dimensionless strain rate; Reynolds num-ber (Re), which is the ratio of inertial to viscous forces; andelasticity number (El), which is the ratio of elastic to iner-tial stresses independent of kinematic processes (McKinley2005). These parameters are defined below for flows arounda confined cylinder of radius R in channels with rectangu-lar cross sections of depth D and width w, using fluids withrelaxation time λ viscosity μ, and density ρ, and in flowswith flow rate Q:

De = λUR

= λQRDw

(1)

Wi = λUw − 2R =

λQ

Dw (w − 2R) (2)

Re = ρU2Rμ

= ρQ2RμwD

(3)

El = WiRe

= λμρR(w − 2R) (4)

It can be seen that De and Wi are closely related, and theirratio is w−2R

R. For the rest of this paper, De will be exclu-

sively used with the understanding of its relationship to Wifor this geometry.

The flow around a confined and bounded cylinder ismixed with high-shear regions between the channel wallsand cylinder and high extension regions at the stagnationpoints. The ratio of the cylinder diameter to the channelwidth controls the relative strength of these two flow types(Moss and Rothstein 2010). The overall magnitude of thesestresses will also be affected by the channel depth, D. Toanalyze these flows, the following two geometric dimen-sionless parameters should also be considered: the blockageratio of the cylinder diameter to the channel width (β) andthe confinement ratio of the channel height to the channelwidth (α):

β = 2Rw

(5)

α = Dw

(6)

By using these dimensionless parameters, the work done inthis study can be compared to previous experiments.

Microfluidic rheology

The observation of strong viscoelastic effects in microflu-idics has led to increasing use of microfluidic channels asa means to characterize viscoelastic materials and flows,which is generally classified as microfluidic rheology.Microfluidic rheology is advantageous due to low samplevolumes, high shear rates, and high extension of dispersedmaterials (Pipe and McKinley 2009; Rodd et al. 2005).Using microfluidics, both shear and extensional viscosi-ties have been measured for a range of materials includ-ing dilute polymer solutions (Pipe and McKinley 2009;Kang et al. 2005), wormlike micelle solutions Pathak andHudson 2006; Soulages and McKinley 2008, and polymermelts (Moon et al. 2008).

The large extensional strains in microfluidic channelscreate elastic and inertio-elastic flow instabilities over awide range of phase space, which has led to an increas-ing use of microfluidic channels to characterize viscoelasticflow behavior. In microfluidic channels, flow instabilitiesof viscoelastic materials have been studied in a wide arrayof channel types and flow conditions (Afonso et al. 2010;Dubash et al. 2012; Galindo-Rosales et al. 2012; Gulati et al.2008, 2010; Haward et al. 2010a, b; McKinley et al. 2007;Ober et al. 2011; Oliveira et al. 2008; Pathak and Hudson2006; Pathak et al. 2004; Soulages et al. 2009; Sousa et al.2010; Steinhaus et al. 2007).

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Flow around a cylinder

Newtonian flows

For unconfined (α >> 1), unbounded (β = 0) and creep-ing flow (Re > Recrit that includereappearance of stable symmetric steady state, generationof secondary symmetric vortices, and asymmetric vortices(Sahin and Owens 2004a). Increasing β also will decreasesthe dimensionless length of the recirculation vortex (Senet al. 2009; Chakraborty et al. 2004).

When β = 0 but confinement is allowed to vary (in thiscase α = D/2R), Recrit for vortex shedding increases withdecreasing α. This begins to occur at α ∼ 40, and at α ∼ 2,Recrit grows to be as large as 135 (Lee and Budwig 1991). Ithas further been shown that recirculation length of vorticesbefore shedding decreases for α < 8 (Ribeiro et al. 2012).

Viscoelastic flows

The increased extensional viscosity of dilute polymer solu-tions has a stabilizing effect on flow around unconfinedand unbound viscoelastic flows by suppressing transversevelocity fluctuations (Richter et al. 2010). This results in theincrease of the Recrit for the onset of vortex shedding andthe decreases in St with increasing El (Pipe and Monkewtiz2006). However, if fluids are significantly shear thinning,the reverse trends are observed and Recrit is seen to decrease(Coelho and Pinho 2003; Pipe and Monkewtiz 2006).

The effect of β on viscoelastic flows around unconfined(α >> 1) cylinders has been extensively studied and mod-eled for creeping flows, especially for values of β = 0.5,which is a benchmark geometry for simulations.

Creeping flows of polymer solutions feature severalpurely elastic instabilities on the downstream cylinder sur-face that break the fore/aft symmetry of Newtonian creepingflows. These downstream instabilities are driven by stream-line curvature and high extensional stresses on dispersed

polymers. For De < 1, streamlines shift downstream behindthe cylinder and create a negative wake (Kim et al. 2005). AsDe approaches 1, 3D flow “cells” develop down the cylin-der depth. These cells are a spatially periodic variation ofthe velocity which is steady in time. When De > Decrit, thevelocity in the primary flow direction of the cells becomestime dependent. The behavior occurs in Boger solutions andalso shear-thinning materials; however, Decrit depends onthe material tested (Shiang et al. 1997, 2000; McKinley et al.1993) or the fluid model used in simulation (Oliveira andMiranda 2005; Dou and Phan-Thien 2008). However, Decritvaries from ∼1 to 10 in reported results. Decrit has beenfound to decrease with increasing β in numerical studies fora range of fluid models (Dou and Phan-Thien 2008).

High De creeping flows with β = 0.5 have recentlybeen simulated using new numerical methods. However,the range of De is limited by fluid model (Afonso et al.2009, 2011; Hulsen et al. 2005), the simulations cannotconverge at all points (Hulsen et al. 2005; Jafari et al.2012), and results are not always consistent with experi-mental work (Afonso et al. 2011). Simulations of high Deflow around a confined cylinder have been conducted forOldroyd-B, Giesekus, and PTT fluids (Afonso et al. 2009;Hulsen et al. 2005). The results of Oldroyd-B fluids are ofparticular interest, since this model mimics the rheology ofBoger fluids which are used in this work. Simulations ofOldroyd-B fluids around a confined cylinder have been ableto reach De ∼ 3 and found that flows were unsteady withoutwell-defined periodicity (Afonso et al. 2009).

As Re exceeds 1 for β = 0.5, inertial instabilitiesobserved in Newtonian bounded flows are also observedfor viscoelastic flows, but affected by elasticity similar tothe effects elasticity has on unbounded flows. A wake withtwo recirculating vortices forms behind the cylinder, butelasticity decreases the size of the wake in comparisonto Newtonian fluids at similar Re (Kim et al. 2004). Theresult of this suppression of the wake’s growth is a delayin the onset of vortex shedding to larger Recrit. In fact,because Recrit is increased and vortex shedding is delayed,the wake eventually grows larger than is possible for Newto-nian flows. St decreases for elastic solutions in comparisonto Newtonian systems. The fluid rheology or rheologicalmodel parameters impact the relative magnitude of theseeffects (Sahin and Owens 2004b).

Experimental methods

Channel fabrication

A microfluidic confined cylinder geometry was fabricatedusing standard soft lithography methods (Duffy et al. 1998).A high-resolution transparency mask, with a minimum

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feature size of 10 μm (CAD/ART Services, Inc.) was usedto fabricate SU-8 (Su-8 2035, Microchem) molds on sili-con wafers. The mask was printed emulsion side down andplaced in direct contract on the Su-8 and clamped in place toensure vertical side walls and high fidelity of pattern trans-fer. The Su-8 was exposed through a 380-nm filter usinga UV flood exposure system (DYMAX, 2000-EC series).Once cured and solidified, molds were spin-coated with 3-mercaptopropyl trimethoxysilane to ensure easy removal ofmicrochannels and high fidelity of transfer from the mold tothe channels.

Channels were fabricated by pouring polydimethylsilox-ane (PDMS) (SYLGARD 184, Dow Corning) over themolds, which was allowed to crosslink overnight afterdegassing. The PDMS channels were removed from themold, and access ports were plugged using a 0.75-mm holepunch. The channel bottom was made by spinning PDMSonto a glass slide. The PDMS channels and bottoms arebonded together using air plasma (Plasma Cleaner, HarrickPlasma). Bonding was done using the medium setting for35 s. The channels were left overnight in an 80 ◦C oven andfinally stored at room temperature until use.

Microfluidic protocol

The channel geometry and relevant dimensions used in allexperimental work are shown in Fig. 1; an inset of an actualmicrofluidic confined cylinder is included. Based on maskresolution and cylinder size, the resulting cylinder is rela-tively smooth. However, roughness on the cylinder surfaceof ∼ 1 μm is expected. Channels are 100 μm wide andheights were 35 μm. The cylinder spans the entire heightof the channel, is 65 μm wide, and is placed directly at thechannel center. This sets α = 0.35 and β = 0.65. Thechannels are 5 cm long, and the cylinder is located directly

Fig. 1 Channel schematic and dimensions of microfluidic channelsused to study confined and blocked cylinder flows. The inset is apicture of the actual microfluidic channel

in the middle, far away from any entry effects. Channelsare used for a single experimental session. A single experi-mental session tests one fluid but may examine several flowrates.

Flow is controlled by a syringe pump (BS-8000, Brain-tree). Syringes (Gastight, Hamilton) are used in sizes rang-ing from 500 μL to 5 mL. Flow rates are varied from1 to 20,000 μL/h. Syringe sizes are selected such thatthe pump operates at a frequency of at least ten timesgreater than its minimum operating parameters. Flows arestarted and allowed to develop for 20 min before data isrecorded.

Fluid rheology

Four aqueous solutions containing polyethylene oxide(PEO) (Sigma-Aldrich, used as received) are used to cre-ate low-viscosity Boger solutions, which are elastic withoutshear thinning (Boger 1977; James 2009). To formulate,PEO of varying molecular weights is dissolved in a glyc-erol/water base solution at compositions shown in Table 1.Solutions were mixed for 48 h and then allowed to restfor 48 h before characterization in the rheometer. Solu-tions were tested a second time several days later toensure consistency. No observable changes in rheology werenoted.

Similar formulations to these have been used in a num-ber of microfluidic studies (Rodd et al. 2005; Tirtaatmadjaet al. 2006; Christopher and Anna 2009; Husny and Cooper-White 2006). The formulations used in this work are iden-tical to the solutions used by Christanti and Walker; thepolymer concentration is below overlap concentration, andthe solutions are considered dilute (Christanti and Walker2002). A Newtonian solution of glycerol and water witha viscosity of 5.00 ± 0.05 mPa s is used as a controlNewtonian fluid.

Table 1 Composition of four dilute polymer solutions used in thisstudy

Fluid systems

Mv C Glycerol η λZimm λa

[g/mol] [wt%] [wt%] [mPa s] [ms] [ms]

1 × 105 0.3 40.7 5.09 ± 0.05 0.013 × 105 0.1 41.9 5.37 ± 0.05 0.05 0.491 × 106 0.05 40 4.64 ± 0.05 0.38 2.45 × 106 0.043 34 5.11 ± 0.05 4.64 42

PEO of four molecular weights was dissolved in glycerol/water solu-tions. The viscosities listed are averaged values from rheometermeasurements shown in Fig. 1aλ values are from the work of Christanti and Walker

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The shear rheology of these solutions is shown in Fig. 2and was measured using a double-wall Couette cell in aTA AR-G2 rheometer at shear rates ranging from 0.01 to500 s−1. The fluids display minimal shear thinning, lessthan 5 % change over the range of shear rates tested. Allsolutions have an average viscosity over the shear rangeof ∼5 mPa s and are within 10 % of the values reportedby Christanti and Walker. Average values of viscosity areshown in Table 1.

The dashed lines in Fig. 2 represent the limits of viscositymeasurement. The lower limit is set by the minimum torqueresolution, 10−5 mN m (Rodd et al. 2005). Using the knowngeometry parameters and the minimum torque resolution,the minimum measured viscosity as a function of shearrate can be calculated. To ensure accurate measurement, thelimit of machine resolution is set to ten times the minimumtorque resolution. When the samples approach this limit, theviscosity increases significantly and the measurement is nolonger reliable (Fig. 2).

The upper theoretical limit is taken from the assumptionof the Taylor–Couette inertial instability for Newtonian flu-ids (Rodd et al. 2005). However, as can be seen from Fig. 2,the viscosity measurements are ended before they reach thislimit in all cases. This is due to low-viscosity solutions ris-ing out of the geometry due to inertia at high shear rates. Alltests are stopped once this inertial rise is observed, which issignificantly below the maximum upper limit as defined bythe Taylor–Couette instability.

For the purposes of comparison to experiments, the esti-mated shear rates between the cylinder and the wall for themicrofluidic channels are also shown in Fig. 2. The rangeof shear rates tested in the rheometer cover a large section

Fig. 2 Results of steady flow tests on low-viscosity Boger solutions:1×105 (plus signs), 3×105 (minus signs), 1×106 (asterisks), and 5×106 (multiplication signs). Dashed line represents minimum viscositymeasurement limit. Dot dashed line represents maximum limit

of shear rates in the microfluidic experiments. Based on thenegligible shear thinning and the appearance that viscositieshave reached their high-shear plateau, it is concluded thatthe fluids are not shear thinning in the experimental rangeand the solutions are considered to be low-viscosity Bogerfluids. The average viscosity over the measured shear ratesis used for all analysis.

Due to their low viscosity, the relaxation times of thefluid could not be measured using small-amplitude oscilla-tory shear (Rodd et al. 2005). Molecular parameters are usedto estimate the Zimm longest relaxation time for the fluidusing the relationship

λZimm = [η]Mwηsξ (3υ) NAkBT

(7)

where [η] is the intrinsic viscosity, Mw is the weight-averaged molecular weight, ηs is the solvent viscosity, NAis Avogadro’s number, kB is the Boltzmann constant, andT is the temperature (Ottinger 1996). Using typical solventparameters for PEO in water/glycerol mixtures and estimat-ing intrinsic viscosity with the Mark–Houwink–Sakuradaequation (Christopher and Anna 2009; Larson 1988;Tirtaatmadja et al. 2006), Zimm relaxation times at 20 ◦Care listed Table 1.

A number of recent studies have demonstrated a siz-able discrepancy between experimentally measured andZimm relaxation times for dilute concentrations of PEO inwater/glycerol. This is especially true when these solutionsare highly confined or used in microfluidics (Christopherand Anna 2009; Amarouchene et al. 2001; Christanti andWalker 2002; Tirtaatmadja et al. 2006; Wagner et al. 2005;Steinhaus et al. 2007; Eggers and Villermaux 2008).

Due to the use of microfluidic channels and the observeddiscrepancies between Zimm and experimental relaxationtimes, we choose to use these solutions experimentally mea-sured relaxation times, as measured through filament self-thinning by Christanti and Walker (2002). For the lowestmolecular weight PEO solution, no observable relaxationwas measured in the work of Christanti and Walker. Foranalysis purposes, a value 0.001 ms is used as this solution’srelaxation time.

Achievable phase space

Based on the measured flow properties and the channelgeometries, the range of dimensionless parameters are cal-culated and shown in Table 2. Changing the Boger fluidtested and keeping the flow rate constant allows inde-pendent control of the De while working at constant Re.Adjusting both flow rate and Boger solutions allows testingat near constant De with changing Re. This allows indi-vidual effects of elasticity and inertia on the flow to bemethodically evaluated.

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Table 2 Calculated minimum and maximum values of De, Wi, Re,and El based on the known geometry of microfluidic channels, fluidproperties from Table 1, and range of tested flow rates

Experimental range

Min Max

De 0.01 1,500

Wi 0.01 1,000

Re 0.001 20

El 0.1 8

Flow visualization

All fluids, including the Newtonian control, are seededwith 1-μm fluorescent polystyrene particles (580/605 Emis-sion/Excitation) at a concentration of 0.02 wt%. A smallconcentration of sodium dodecyl sulfate, 0.1 wt%, is addedto all solutions to avoid particles adhering to the PDMSchannels. The particles and surfactant result in negligiblechanges to solution viscosities, which was confirmed bymeasuring the viscosity of solutions with and without theseadditional components.

An inverted transmission microscope (IX-71, Olympus)is used in epifluorescence to visualize the particles. Fluo-rescent streakline images are captured using digital camerasattached to the microscope. Several movies are taken of eachexperiment at a range of camera settings and magnifica-tions. Depending on flow speeds, different frame rates areneeded to accurately capture the flow patterns. For lowerspeeds, a PixeLINK PL-E531MU (55 fps at 1,280 × 1,024,6-μm pixels) is used, and for high speeds, a NAC Memre-cam GX-3 (2,000 fps at 1,024 × 1,024, 20,000 fps at 512 ×512, 20 μm pixels) is used. The large pixels of the Memre-cam allow fluorescent visualization of the particles at framerates up to 2,000 fps. Streakline images are taken at a singledepth in the channels; the depth of focus is estimated to be36, 18, and 11 μm when using a ×10, ×20, and ×40 objec-tive, respectively (Rodd et al. 2005). Streakline images areassumed to encompass the entire channel depth, limiting ourability to comment on the variation of the velocity throughthe depth of the channel.

Data analysis

In order to characterize observed flow patterns, image anal-ysis of streakline images is done using ImageJ (NIH). Alldata from experimental work are in the form of monochro-matic movie files. These files are imported into ImageJand analyzed frame by frame using built-in measurementfunctions. Based on camera specifications and magnifica-tion, spatial resolution is approximately 1 pixel which isequivalent to 1.1 μm at ×20.

Temporal resolution is typically high; measurements ofthe faster of the observed instabilities are recorded at 1,000frames per second, resulting in a temporal resolution of0.001 s. Based on this resolution and the number of analyzedframes (300), the range of frequencies that can be observedare ∼1 to 100 Hz.

Time-averaged values are typically reported throughoutthe results. Parameters that are within temporal and spa-tial resolution and are not observed to vary with time,as determined by qualitative observation, are considerednonoscillatory. To evaluate error on these measurements, thestandard deviation of the average value is used and typicallyhas a relative error of ±15 %. These values are larger thanthe spatial resolution and represent the systematic error dueto analysis techniques and/or experimenter bias.

For values which were oscillating, determined by qualita-tive observation, the time-averaged values are reported withthe amplitude of oscillation rather than an error. This valueindicates the degree of variation of these measurements.

Results

Flow patterns

In Fig. 3, flow visualization for both the Newtonian controland the highest molecular weight solution tested (El = 88)over a range of Re are shown. For Re < 1, flow for bothNewtonian and Boger solution looks nominally identicalwith the symmetric flow patterns both upstream and down-stream of the cylinder. However, as flow rate is increased,increasing both Re and De, flow patterns of the viscoelasticsolution diverge in comparison to the Newtonian flow.

At Re ∼ 1, viscoelastic flows downstream of the cylin-der display divergent behavior (Fig. 4). The streamlinesbehind the cylinder diverge from the steady symmetricshapes of Newtonian flows and begin to vary both spatiallyand temporally. Directly behind the cylinder, the streamlinesbecome disordered and appear to cross one another withinthe focus depth. These disordered and crossing streamlinesindicate that the flow is no longer planar; it is varying spa-tially down the cylinder depth and is 3D. Furthermore, thestreamlines are no longer steady, and their position becomestime dependent, indicating that the flow is also temporallyvarying. Observation of the disorderly pattern does not showany periodicity within measurement resolution; it appears tobe unsteady with no obvious periodicity.

As flow rate is increased, a second flow pattern emergeson the upstream face of the cylinder at Re = 3 (Fig. 3).The streamlines “pinch” directly upstream of the stagnationpoint on the upstream cylinder surface, forming a wineglass-shaped flow pattern directly in front of the cylinder. Asflow rate is increased, this “pinch” becomes more severe,

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Fig. 3 Flow patterns for Newtonian control and El = 88 Boger fluid.Re on the left applies to both the Newtonian and El = 88 flows; Deon the right only applies to El = 88. Values of Re and De are typicalof experimental work conducted. Streakline images represent typicalresults from a given flow condition. All images here are shown at ×20magnification, but each image was taken at a frame rate necessary toaccurately capture flow phenomenon. These frame rates varied from10 fps to as high as 2,000 fps. Arrow indicates flow direction

incorporating more streamlines and narrowing in width(Re = 7, Fig. 3). The location of the pinch moves upstreamfrom the cylinder, forcing the upstream stagnation pointon the cylinder to detach (Re > 7, Fig. 3); this creates an

Fig. 4 Two consecutive streakline images of downstream instabilityobserved at moderate De and Re. Images are taken at 10 fps at a mag-nification of ×40. Close inspection of images reveals that streamlineis disordered and varying with time

upstream vortex of slow-moving fluid between the stagna-tion point and the cylinder face which is encapsulated byfast-moving flow on its edge. The fast-moving outer flowreattaches to the cylinder’s upstream face at a fixed angle.The streamlines then stay in contact with the cylinder anddetach off the downstream face of the cylinder.

The observed upstream vortex has a constant length ini-tially, but increasing flow rate results in the length becomingunsteady in time as shown in Fig. 5a. The vortex growsto a maximum size, which it maintains for some period oftime, and then it rapidly decays and collapses back ontothe cylinder face. This growth and collapse cycle is cyclic(Fig. 5b).

Although results are only shown for single Bogersolution in Fig. 2, identical changes in flow behaviorfor the three highest molecular weight Boger solutionsare observed. Transitions between symmetric downstreaminstability and upstream instability occur at different criticalRe and De for each fluid. However, the basic progressionand nature of the observed flow patterns are identical.

Examining the Newtonian flow, the basic fore/aft sym-metry is observed up to Re = 10. For Re > 10, a downstreamwake directly behind the cylinder begins to develop, but thewake is still small and below measurement resolution. Novortex formation or shedding is observed. El = 0.1 behavesidentically to the Newtonian sample. It shows no observ-able transitions from symmetric flow, and no instabilities areobserved.

Phase space of instabilities

A series of tests are conducted to identify the Recrit andDecrit for the onset of the observed instabilities. The resultsare presented in Fig. 6a. The downstream instability isobserved within 1 < De < 100 and 0.1 < Re 10 and De > 10.

Both the Recrit and Decrit for onset of both the upstreamand downstream instabilities depend on El (Fig. 6b). Forboth instabilities, Recrit decreases with increasing El; theopposite trend is observed for Decrit. Although the func-tional form of these trends appears to be monotonic, thereare not enough data points to confidently fit a functionalform to this data.

Geometry of upstream instability

In order to better characterize the observed flow, the rel-evant geometric parameters of the upstream instabilityare measured. Figure 7a displays the relevant propertiesof the upstream instability that have been identified formeasurement.

Figure 5b displays the distance from the upstream sur-face of the cylinder to the pinch point, Fp, as a function of

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Fig. 5 a Streakline imagesshowing one cycle of collapseand growth of upstreaminstability observed at high Deflows. Images were taken at1,000 fps at a magnification of×20. b Measurement of vortexlength versus time for same flowin a. Slow growth of the vortexleads to rapid collapse back tothe surface of the cylinder. Thisevent is clearly periodic

time. As mentioned earlier, length of the upstream recircula-tion zone is clearly periodic. Fp grows, reaches a maximumvalue, maintains that position, rapidly collapses, and thenrepeats this process. No other geometric parameters displaysimilar periodic time-dependent behavior within measure-ment resolution. This oscillatory behavior begins whenFp / D > 1. For smaller vortices, Fp is constant.

The time-averaged Fp of the instability is a clear functionof both De and El (Fig. 7b). For the larger elasticity fluid,F̄p increases by approximately an order of magnitude withan order of magnitude change in De; however, F̄p is con-stant with De for the lower elasticity fluid. Both fluids’ F̄pshow significant oscillation, which increases with increas-ing De as observed by size of the amplitude oscillation barsin Fig. 7b. The fact that the lower elasticity fluid has a largervortex length at lower De indicates to some degree the roleof inertia on vortex length since upstream instabilities forthe lower elasticity fluid are occurring at larger Re than thehigher elasticity fluids.

To explore this further, the change in vortex length forRe = 10 and Re = 18 as a function of De is examined inFig. 7c. To create Fig. 7c, individual data points at the spec-ified Re from three sample fluids (El = 1, 5, 88) are used to

create each line of constant Re. There seems to be a differenttrend for the Re = 10 and Re = 18 lines. However, withinmeasurement resolution, no significant difference betweenthe two lines in Fig. 7c is observed.

Figure 7d shows time-averaged location of the pinch, cp.The instability is centrally located on the cylinder diame-ter. Although the upstream instability’s length oscillates, cpappears to be constant for all El within the resolution andbias error of measurement techniques with average relativeerror below 15 %.

Figure 7e shows time-averaged angle of attachment, θF,and angle of detachment, θB, as a function of El. Theseappear to be constant over El with values of ∼75◦ and∼125◦, respectively. The relative error on these measure-ments was 20–30 %. It is possible that these values oscillatewith the upstream instability; however, current measure-ments techniques do not allow us to definitively say whetherthese are steady or unsteady.

Frequency of upstream instability

The frequency of oscillation of the instability is found bytaking a fast Fourier transform of Fp. The dominant peak of

Fig. 6 a Phase space ofobserved instabilities:symmetric fore/aft flow aroundcylinder (open triangles),downstream instability ofunsteady and disorderlystreamlines (filled circles), andupstream instability withdetached upstream stagnationpoint and upstream vortex (filledsquares). Symbols on De =0.001 are Newtonian controlfluid. b Recrit and Decrit for bothdownstream (filled circles) andupstream instabilities taken froma (filled squares)

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Fig. 7 a Imageoutlining-relevant geometricparameters measured tocharacterize upstreaminstability. b Average width ofpinch (filled triangles) andaverage center of pinch (filleddiamonds) as a function of El.For all experiments conducted,these values are essentiallyconstant. c Average downstreamangle of separation (filledsquares) and average upstreamangle of attachment (minussigns) as a function of El. For allexperiments conducted, thesevalues are essentially constant.d Dimensionless vortex lengthfor El = 2.5 (filled circles) andEl = 88 (filled squares) as afunction of De. The higherelasticity flow shows muchgreater dependence onincreasing De in comparison toEl = 5, which shows novariation. e Dimensionlessvortex length for Re = 10 (filledcircles) and Re = 18 (filledsquares) as a function of De. Nodependence of vortex length onRe for the range of De tested isseen

the transform is used as the frequency of the periodic oscil-lations. Results are shown in Fig. 8. El significantly impacts

Fig. 8 Frequency of upstream instability oscillation for El = 2.5(filled circles) and El = 88 (filled squares) as a function of De. Thehigher elasticity flow shows much greater dependence on increasingDe in comparison to El = 5, which shows no variation in frequency

the frequency of oscillation. For El = 5, the frequency ofoscillation is independent of De at a value of approximately40 Hz. For El = 88, frequency appears to monotonicallyincrease with De. In this figure, a frequency of 0 denotes alack of oscillation of vortex size.

Discussion

Phase space

For the Newtonian control and El = 0.1, symmetric andsmooth flow for all the Re tested is observed, except at thehighest Re, where a small wake in the Newtonian flowsis observed. No recirculation wakes or vortex shedding inthese flows is observed, which typically can occur at Reon the order of magnitude seen in this study; however, dueto high confinement and blockage, recirculation and vortexshedding are known to be suppressed to larger Recrit (Leeand Budwig 1991; Sahin and Owens 2004a).

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The elastic instabilities are observed first at moderate Reand De. There is a clear relationship between Recrit andDecrit on El. For both instabilities, Recrit is decreasing func-tion of El and Decrit is an increasing function of El. Forinertial instabilities affected by elasticity, increasing El hasbeen observed to delay Recrit for Boger solutions (Pipe andMonkewtiz 2006; Sahin and Owens 2004b). The oppositetrend is observed here, which leads to the conclusion thatthe mechanism of the observed instabilities is not inertial.

If trends in Fig. 6b are extrapolated, it is reasonableto conclude that at sufficiently large El, Recrit would gobelow 0.01. This indicates that the instability would occurfor creeping flow for sufficiently high El. However, unlikethe creeping flow, purely elastic instabilities around a con-fined cylinder discovered by McKinley et al. (1993), inertiadoes not appear to prevent the growth of the instabilitiesobserved in this work. In fact, increased inertial energyallows the development of these instabilities for less elasticsolutions. This leads to the conclusion that the instabilitiesobserved in the presented results are purely elastic in originbut affected by inertia. The relative impact of inertia on theelastic instability is likely affected by the high blockage andconfinement of these channels, and changes in both α and βwould affect the relationship of Recrit and Decrit with El.

Downstream instability

The downstream instability of disordered and temporallyvarying streamlines is observed at moderate De and Re.Looking in literature, it is well established that confinedcylinder flows have a number of possible downstream insta-bilities that could possibly be the cause of this observedflow.

It is possible that the observed downstream instabilitiesare related to the purely elastic instabilities first reportedby McKinley et al. (1993). Similar to these instabilities, thephenomenon observed in the present work is both spatiallyand temporally varying. Due to the depth of focus limit tothe resolution of the streakline images, the 3D nature of thevelocity profiles cannot truly be characterized. This makesit impossible to quantitatively characterize this flow andcompare it to previous work. However, the McKinley et al.elastic instabilities occur for De ∼ 1 and Re

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do not seem to vary with time within measurement reso-lution, but the flow inside appears to be unsteady, similarto the precursor downstream instability observed at mod-erate De. The downstream wake is much longer than theupstream vortex and could not be measured due to its sizeextending out of the image frame. The angle of detach-ment of this wake is ∼120◦, which is close to the knownseparation angle for turbulent wakes of Newtonian fluids(Williamson 1996).

Role of inertia and elasticity on upstream instability

The role of inertia on the development and growth of theupstream instability is unclear. The lower elasticity flu-ids appear to have larger upstream vortex lengths thanlarger elasticity fluids in Fig. 7b. We suspect that the largervortex length for the lower elasticity fluid is due to theenhanced contribution of inertial energy. For El = 5 fluid,the upstream instability occurred for Re > 10; however,for the El = 88 fluid, Re > 10 only occurred for the lasttwo points in Fig. 7b. However, Fig. 7c demonstrates thatupstream vortex length appears to be independent of Re.Further examination of the role of inertia will require morefluids to allow study of lines of constant Re.

Elasticity clearly affects vortex length and dynamics.Looking at Fig. 7b, c, increasing De causes increasing vor-tex length and amplitude of oscillation. Furthermore, as Deincreases, frequency also increases (Fig. 8). This temporallyunstable length is consistent with the upstream instabilitiesof contraction flows (Afonso et al. 2011). These results indi-cate the strong role of elasticity on the shape and stability ofthe upstream vortex.

Although the evolution of flow behavior is consistentbetween the three highest elasticity solutions tested, insta-bility size and frequency depend on El. For the El = 5 fluid,F̄p is essentially constant with De and is larger than El = 88fluid, for which F̄p increases with increasing De (Fig. 7a, b).Frequency of the upstream instability shows similar trends(Fig. 8). The decrease in frequency with increasing El isconsistent with the effects of El on St for inertial instabilities(Pipe and Monkewtiz 2006; Sahin and Owens 2004b); how-ever, since the observed instabilities are elastic but affectedby inertia, this result is puzzling.

Given the range of tested phase space, we cannot con-clude if these trends would continue or if the lower El fluidwould eventually converge to the higher El results. Further-more, we suspect that the high confinement and blockageimpact the relative effects of inertia and elasticity on theseflows and warrant further study.

The other measured geometric parameters includinglocation of the pinch as well as angle of attachment anddetachment show no dependence on El, De, or Re andless temporal instability in comparison to the vortex length.

These parameters are not observed to change significantly,but we expect them to be a function of confinement andblockage.

Mechanism of instability onset

Based on the analysis of the critical De and Re for theonset of the observed instabilities, both the upstream anddownstream instabilities are purely elastic instabilities thatare affected by inertia. Unlike purely elastic instabilities forcreeping flow around unconfined and bounded cylinders,inertial effects do not prevent the instabilities observed inthis work from developing but in fact aid in their onset asobserved by the ability of lower El fluids to still manifestthese instabilities if Re is increased sufficiently.

The instabilities appear to be primarily driven byenhanced extensional stresses indicated by large Decrit,which increases as Recrit decreases. However, the very highblockage, β = 0.65, and confinement, α = 0.35 of thestudied channels also play a significant role by in the abilityof these instabilities to form. High confinement and block-age enhance the three dimensionality of the flow, resultingin greater extensional stresses that instigate the elastic insta-bilities. To better understand the mechanisms of these flowsrequires a study of the effects of β and α on the instabilityas well as more precise flow visualization methods. Finally,the roughness of the microfluidic cylinder may also increasethe three dimensionality of the flow and hence aid in theonset of instabilities.

Conclusions

Using microfluidic channels, we have discovered a newclass of elastic instabilities for viscoelastic flows aroundconfined cylinders. First, a disorderly streamline on thedownstream face of the cylinder develops at moderate De.This is a precursor to a much larger and unstable upstreamvortex that occurs when the upstream stagnation pointdetaches from the cylinder face. These instabilities occurat large De and moderate Re. However, based on analysisof the critical values for their onset, the instabilities are infact elastic in nature and would occur for creeping flows ofsufficiently high De.

Based on analysis of the geometry of these instabilities,increased elastic stresses at high confinement and blockageprimarily shape the upstream vortex. Increased De increasessize, frequency, and degree of oscillation of the observedupstream instability. Inertia clearly does affect the observedflows, but its exact role remains unclear.

Furthermore, we believe that high confinement andblockage of the channels is also necessary for the onsetof these instabilities, and there are likely critical α and β

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that channel geometry must be below in order for theseinstabilities to occur. The current study has only examinedsingle channel geometry. Due to the importance of geomet-ric parameters on the other dimensionless parameters and onthe relative strength of shear and extensional stresses aroundthe cylinder, the onset and flow patterns of both instabili-ties will be significantly affected by changes in geometricparameters.

In summary, we present experimental results outliningnew flow regimes for viscoelastic flows around confinedcylinders. The primary dimensionless numbers of impor-tance to the observed phenomenon are El and De, but a morecomplete analysis of the phase space, including geometricdimensionless groups, is necessary to better characterize theobserved instabilities. The results presented should enablethe recreation and analysis of these instabilities in differ-ent systems and hopefully assist modeling efforts of highlyelastic flows.

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Large Deborah number flows around confined microfluidic cylindersAbstractIntroduction[-18pt]Kindly check if the section headings are assigned to appropriate levels.BackgroundDimensionless parametersMicrofluidic rheologyFlow around a cylinderNewtonian flowsViscoelastic flows

Experimental methodsChannel fabricationMicrofluidic protocolFluid rheologyAchievable phase spaceFlow visualizationData analysis

ResultsFlow patternsPhase space of instabilitiesGeometry of upstream instabilityFrequency of upstream instability

DiscussionPhase spaceDownstream instabilityUpstream instabilityRole of inertia and elasticity on upstream instabilityMechanism of instability onset

ConclusionsReferences