Experimental study of the instability of laminar flow in a tube withdeformable wallsR. Neelamegam and V. Shankar Citation: Physics of Fluids (1994-present) 27, 024102 (2015); doi: 10.1063/1.4907246 View online: http://dx.doi.org/10.1063/1.4907246 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/27/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Numerical studies of the effects of large neutrally buoyant particles on the flow instability andtransition to turbulence in pipe flow Phys. Fluids 25, 043305 (2013); 10.1063/1.4802040 Manipulation of instabilities in core-annular flows using a deformable solid layer Phys. Fluids 25, 014104 (2013); 10.1063/1.4788712 Effect of slip on existence, uniqueness, and behavior of similarity solutions for steadyincompressible laminar flow in porous tubes and channels Phys. Fluids 18, 083601 (2006); 10.1063/1.2236302 The interdependence of friction, pressure gradient, and flow rate in unsteady laminar parallel flows Phys. Fluids 12, 518 (2000); 10.1063/1.870258 Traveling wave instability in helical coil flow Phys. Fluids 9, 407 (1997); 10.1063/1.869139

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PHYSICS OF FLUIDS 27, 024102 (2015)

Experimental study of the instability of laminar flowin a tube with deformable walls

R. Neelamegam and V. Shankara)

Department of Chemical Engineering, Indian Institute of Technology, Kanpur 208016, India

(Received 17 October 2014; accepted 13 January 2015; published online 4 February 2015)

The onset of instability of laminar flow in a tube with deformable walls is studiedexperimentally in order to characterize how the onset is affected by the elastic (shear)modulus of the deformable wall. To this end, rectangular blocks of polydimethyl-siloxane (PDMS) gels of different shear moduli are fabricated with a cylindrical hole(of diameter 1.65 mm) in which the fluid (water) flow occurs due to an imposedpressure difference. The shear moduli of the PDMS gels were in the range of 21 -608 kPa. When fluid flows through the deformable tube, we find that the tube radiuschanges slowly as a function of distance along the flow, and this change is a functionof Reynolds number (Re). The pressure drop between the two ends of the tube ismeasured, and the friction factor is calculated from this pressure drop. The frictionfactor vs. Re data shows that the expected laminar flow relation ( f = 64/Re) for flowin a rigid tube is seen in a deformable tube at lower Re, but there is a deviationfrom this relation at Re < 2000. We identify the Re at which the deviation occursas the Reynolds number at which the laminar flow in the deformable tube becomesunstable. This transition Reynolds number is as low as 500 for the 21 kPa PDMSgel, the softest gel studied in this work, and this value is much lower than the criticalReynolds number (∼ 2000) for transition in a rigid tube. The onset of the transitionis also independently corroborated using a dye-stream visualization method, and thetransition Reynolds number obtained with this method agrees well with the Reynoldsnumber at which there is a deviation in the friction-factor data from the laminar rela-tion. This transition in a deformable tube which happens at Reynolds number muchlower than 2000 could be potentially exploited in improving mixing in microscaledevices. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907246]

I. INTRODUCTION

The laminar-turbulent transition in a tube with rigid walls has been extensively studied sincethe pioneering experiments of Reynolds,1 and it is well-known that the transition happens in arigid tube at a Reynolds number, Re ∼ 2000. This transition has since attracted enormous attentionboth from the theoretical (hydrodynamic stability) and experimental standpoints.2–4 While clas-sical temporal stability analysis5,6 shows that the fully developed flow in a rigid tube is7 stable toinfinitesimal perturbations at any Reynolds number, experiments always show a robust transition atRe ∼ 2000. It is now believed that the observed transition is due to finite-amplitude disturbances,which could be triggered by linear transient (algebraic) growth of perturbations.8,9 Theoreticalstudies based on the (linear) non-modal stability analyses do exhibit transient growth of perturba-tions, but it still remains an unresolved puzzle as to how nonlinearities make the transition happenat Re ∼ 2000. While most traditional engineering applications involve rigid tubes and channels forfluid transportation, there are many instances where the walls of the fluid-conveying vessel are softand deformable. For example, most fluid-conveying vessels in biological systems10,11 are made of

a)Electronic mail: vshankar@iitk.ac.in

1070-6631/2015/27(2)/024102/18/$30.00 27, 024102-1 ©2015 AIP Publishing LLC

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024102-2 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

tissues which are softer, with shear modulus in the range of 1 MPa, in contrast to materials likesteel with shear modulus in the range of 10 GPa. Similarly, microfluidic devices12,13 are fabricatedusing soft elastomers such as polydimethylsiloxane (PDMS), and in such applications, the flowbehaviour could be different from that in rigid tubes or channels. In particular, the laminar-turbulenttransition in a tube with deformable walls could be affected by wall elasticity,14,15 and if so, thiscould be potentially of importance in both biological systems as well as microfluidic applications.While the transition in a rigid tube has been well-studied in the literature, the laminar-turbulenttransition in a tube with deformable walls is gaining attention only recently. In this study, we carryout experiments to characterize the instability of laminar flow in a tube with deformable walls. Inthe remainder of the Introduction, we briefly review relevant literature and place the present study inperspective.

To the best of our knowledge, the earliest experimental report of the effect of wall elasticityon the transition in a tube is by Krindel and Silberberg,14 who coated the inner surface of aglass tube with polyacrylamide gel and found that the transition Reynolds number for this systemcould be much lower than 2000, the value for transition in a rigid tube. Hansen and Hunston16

carried out experiments in flow over a rotating disc coated with a compliant material, and theyobserved an instability above a critical velocity. However, there have not been many experimentalstudies that have followed-up to reproduce and corroborate these observations. Nonetheless, thisobservation that wall elasticity could advance the transition in a deformable tube had spawned alarge number of theoretical studies,17–22 which we summarize below. In the interests of brevity, werestrict our discussion here only to flow in tubes, although there have been many studies23–25 thatstudy other geometries such as boundary layer flows and flow in rectangular channels. Kumaran17

first showed using linear stability analysis that the flow in a deformable tube modelled as a linearelastic solid could become unstable even in the limit of zero inertia in the fluid and solid, and thisinstability is referred to as the “viscous mode,” and the mechanism of destabilization is the shearwork done by the fluid at the fluid-solid interface. This instability in the limit of Re ≪ 1 has beennumerically continued to intermediate and high Re by Kumaran,18,26 who found that the criticalReynolds number (Re = ρUavgDavg/η) for instability is related to the nondimensional wall elasticityΣ = ρG′D2

avg/4η2 as Re ∝ Σ3/4 for Re ≫ 1. Here, G′ is the shear modulus of the wall material, Davg

is the average diameter of the deformable tube, ρ is the density of the fluid or solid, η is the viscosityof the fluid, and Uavg is the average velocity in the tube. Later, asymptotic analysis carried out byKumaran27 and Shankar and Kumaran21 showed that this instability is the equivalent of “wall mode”instability in a deformable tube. Qualitatively, however, the physical mechanism of destabilizationremains the shear work done by the fluid at the wall, with the exception that the viscous effects arenow confined to a thin “wall layer” of thickness Re−1/3(D/2).

In addition to the viscous mode at low Re, and the wall mode at high Re, there is anothermode of instability called the “inviscid mode,”19,20,28 for which the critical Re ∝ Σ1/2 at Re ≫ 1.This inviscid mode of instability was shown19 to be relevant to flow in the developing region orflow in a converging tube subjected to axisymmetric disturbances. For fully developed flow, it wasshown20 that nonaxisymmetric disturbances would lead to an inviscid instability. For Re ≫ 1, evenat the scaling level, the critical Re for inviscid modes grows more slowly with Σ compared to thewall modes. Detailed numerical calculations20 have further shown that for fully developed flow in adeformable tube at higher Re, the nonaxisymmetric disturbances are the most dangerous.

After the early experiments of Krindel and Silberberg, recently Verma and Kumaran15,29 carriedout experimental studies in order to characterize the instability of fluid flow in deformable tubes andchannels.30 To this end, they used an elastomer (PDMS) to fabricate the tube and used water as theworking liquid. They found by plotting the friction factor vs Re curves that the flow in a deformabletube becomes unstable at much lower Re than the rigid tube value of 2000. This transition Re wasfurther shown to reduce with decrease in the shear modulus of the gel, thereby clearly demon-strating that the flow in a deformable tube could become unstable at Re much less than 2000. Thescaling of the critical Re with Σ revealed that Re ∝ Σ5/8, and this scaling is in between the scalingof inviscid modes (Re ∝ Σ1/2) and wall modes (Re ∝ Σ3/4). However, they argued that the observedinstability is a type of wall mode that is modified by the slow deformation of the tube wall, and byconsidering the converging cross-section of the tube, they showed that the scaling Re ∝ Σ3/4 gets

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024102-3 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

modified to Re ∝ Σ5/8. Further theoretical calculations by Verma,30 which used the actual velocityprofiles in the deformed tube (computed using a computational fluid dynamics (CFD) method),showed very good agreement with experiments. A recent study by Shrivastava et al.31 showed thatmass transfer is enhanced (up to 25%) in the case of flow through a channel, with one wall made ofPDMS elastomer. This increase in mass transfer rates is attributed to a flow-induced instability onthe surface of the elastomer.

However, there has been no evidence of the inviscid mode20 in the experimental observationsof Ref. 15. The theoretical predictions show that the inviscid mode is more powerful than the wallmode, and hence ought to be observed first in the experiments. This discord between the theoreticalpredictions and observations motivates the present set of experiments, wherein we probe the flow ofwater in a deformable tube. In addition, we are not aware of experimental studies, other than that ofKrindel and Silberberg,14 and Verma and Kumaran,15 that probe the transition in a deformable tube.Thus, the motivation for the present set of experiments is to explore the possibility of an inviscidunstable mode in the flow through deformable tubes and to investigate if observations other thanwhat was reported in Verma and Kumaran15 are possible in a deformable tube.

The rest of the paper is organized as follows: Sec. II explains the fabrication of the experimentalflow setup and the rheological characterization of the gel. Section III A discusses the results ob-tained for diameter variation with Re and axial position, while Sec. III B discusses the friction factorvs Re data for flow in a deformable tube. Section III C discusses the results obtained from visualiza-tion of a dye-stream injected in the flow to characterize the transition. The salient conclusions of thisstudy are summarized in Sec. IV.

II. EXPERIMENTAL METHODS AND PROCEDURE

In our experiments to study the onset of transition, we aim to achieve low turbulence levels atthe tube inlet and also aim to make accurate measurements of the tube diameter change at differentflow rates. To achieve the low turbulence level at the tube inlet, a cuboid (rectangular) channel isfabricated (section (b) of Figure 1). We note that in the earlier study of Ref. 15, this rectangular

FIG. 1. Schematic diagram of the experimental setup: Part (a) represents the supply tank connected with a channel throughneedle value and flow sensor. The supply tank is also connected with nitrogen cylinder through pressure regulator to provideconstant flow in the tube. The needle value and a flow sensor arrangement are used to maintain the flow rate. Part (b) representsthe channel which maintains the low turbulence level at the tube inlet. It is also used to introduce a very thin dye-stream atthe tube inlet. Part (c) represents the microscope and it is used for flow visualization and to measure the tube diameter indifferent locations.

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024102-4 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 2. Soft tube preparation: (a) Preparation of the flow development section on the glass plate using capillary tube anddouble sided tapes. The PDMS gel mixture which contains 10% cross-linker is dispensed in the flow development section andis allowed to cure. (b) Adjacent to the flow development section, a test section is fabricated with PDMS gel using cross-linkerconcentration 1.9%. The PDMS block which contains capillary tube is placed in solvent (toluene) for swelling. The capillarytube is easily removed from the swollen gel, after which the swollen PDMS block is allowed to deswell at low temperature.(c) The conical tapered micro pipette which is pre-embedded in the Plexiglas is then molded with the flow developmentsection. (d) The small-sized hole for connecting pressure transducer in flow development section very nearer to the beginningof the test section. The soft tube arrangement is then fixed to the front end of the channel.

channel was not included, and no special efforts were made to reduce the turbulence level at the tubeinlet. The channel is 14.7 cm high, 14.7 cm wide, and 32.5 cm long.32 The channel is fabricatedusing 1.5 cm thick Plexiglas sheet. The corners of the channel are joined by remolding the Plexiglasby adding chloroform, and Araldite C epoxy adhesive is applied on the corners to avoid waterleakage. The back side of the channel made of (14.7 cm × 14.7 cm) Plexiglas is fixed permanentlyby adding chloroform. The front side of channel contains the tube fabricated in PDMS gel perpen-dicular to the Plexiglas sheet (shown in Figure 2(c)) and this is fixed on the channel wall using2.5 cm length screws. A 3 mm thick neoprene seal is placed between the channel wall and frontside Plexiglas sheet to avoid water leakage. At the top side of the channel (section (b) of Figure 1),two small holes are provided for dye inlet and air purge. A sponge material (11.7 cm × 11.7 cm)of 1.3 cm thickness is fixed inside the channel at the back end, where fluid enters from the supplytank. The purpose of the sponge is to damp out fluctuations at the inlet of the tube. The inlet of thechannel is connected to the supply tank using flexible PVC (polyvinyl chloride) hose (reinforcedwith high tenacity nylon yarn) through a flow sensor (FLR 1000, Omega, U.S.A) and a needle value.The supply tank (section (a) of Figure 1) is of 46 cm diameter and 70 cm high, and is made of 312grade stainless steel, and is connected to the nitrogen cylinder through a pressure regulator.

The flow rate in the deformable tube is measured by using the following procedure. Water ispumped from the collecting tank to the supply tank and the latter is filled till three-fourths of itscapacity. A constant pressure of 2 kg/cm2 is applied in the supply tank using the nitrogen cylinderand a pressure regulator. The needle value connected to the supply tank outlet is used for controllingthe water flow rate. The flow sensor (FLR 1000, Omega, U.S.A.) placed between the channel andthe needle value is connected to a computer. From the measured flow rate, the position of the needlevalve is accordingly varied to maintain a constant flow rate. The flow rate is also measured by col-lecting (for 30 s) and weighing the water at outlet of the soft tube. In order to measure the pressuredrop between the ends of a test section, a pressure transducer (PCB Piezotronics, Depew, NY) isfixed in the development section very close to the test section. The pressure transducer is connectedto a computer through NI (USB-6009, National Instruments) card. The sample rate for the NI cardis 1 kHz. The data (in volts) are averaged over time, and this average voltage is correlated withpressure using the calibration chart. The pressure transducer measures absolute pressure, and from

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024102-5 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

this pressure drop, the friction factor is determined. The ambient temperature of the fluid and solidis maintained (at 25 ◦C) to investigate the instability characteristics. The water bath (HAAKE K10)is connected in the annulus of supply tank which maintains the temperature at 25 ◦C. The roomtemperature was also maintained at nearly 25 ◦C using air conditioners, and changes in the shearmoduli of the solid and the fluid viscosity (with temperature) are, therefore, negligible.

The deformable tube (section c of Figure 2) fabricated using the PDMS elastomer (as describedbelow) is connected to the channel. A tapering micro-tip arrangement is present in the soft tubeinlet which helps in maintaining the smooth flow (from the channel to the tube) and to avoidentry effects (as reported in Refs. 1 and 33). The water from the deformable tube is collected in acollecting tank. For accurate measurement of the tube diameter, an inverted microscope is used. Themicroscope (CKX41, Olympus, Japan) with the light source (high-intensity halogen bulb) is placedin the illuminating column. The 4X objective (PlanCN 4X) is used for bright-field observation andmeasurement of the tube diameter. The TV adapter (U-TV1X-2) and TV camera mount adapter(U-CMAD3) are attached to the microscope using a triangular tube. The CCD camera (LU165C,Lumenra) is fixed on the camera mount adopter and connected to the personal computer to visualizeand capture the images. The light intensity is kept constant for all the experiments.

The measurements of the tube diameter variation at various downstream locations are carriedout using the following procedure. The arrangement that contains the PDMS gel (with the tubularbore) and the pressure transducer fixed in the development section is placed on the microscopestage. The markings along the downstream direction were made at every 3 cm distance in the testsection beginning from the end of the development section. A constant flow rate is maintained inthe tube. The objective lens in the microscope is adjusted to observe the diameter variation througha CCD camera, which is connected to a computer. The snapshots at different downstream locationswere taken by moving the PDMS gel block horizontally. This process is repeated for all the flowrates studied. The snapshots of the galvanized iron (GI) capillary tube (template) were also taken,and from this known diameter, the actual tubular bore diameter is determined for all the downstreamlocations at different flow rates.

Dye-stream experiments are carried out to visualize the flow patterns that ensue during thetransition from laminar to turbulent flow inside the soft tubes. Because the PDMS block (in whichthe tubular bore is present) is transparent, it is possible to visualize the flow events inside the softtube by monitoring the motion of a dye-stream. A smooth GI capillary tube of 1 mm diameter (bentin the form of “L-shape”) is used to deliver the dye at the soft tube inlet as follows (Figure 1). Oneend of this GI tube is fixed at the top part of the channel (as shown in Figure 1, section (b)) andis connected with a syringe pump (PHD 2000, Harvard Apparatus, MA) using a silicone tubing. A100 µm size syringe needle is attached with the other end of the GI tube using Araldite C adhesive.This syringe needle delivers the dye-stream of diameter approximately 100 µm, and this has beenplaced at the center of the tapering micro-tip arrangement for entry of dye into the soft tube. Ablue-coloured water-soluble dye is filled in the syringe, which is placed in the syringe pump. Thedye flow rate is programmed to be maintained in the range from 50 to 250 µl/min, and the specificvalue depends on the Reynolds number of the flow in soft tube. Care was taken to ensure that thedye-stream is injected at the center of the tapering micro-tip. A high-intensity halogen bulb presentin the microscope is used as a light source for illuminating the dye-stream inside the soft tube.A high speed camera (Integrated Design tools, SN:13-0907-0353) is fixed on the camera mountadapter in the place of CCD camera. Using the Red-Lake Motion Pro software, 50–1000 images areacquired in the 1 kHz sample (images) rate. The images were acquired at 1.5 cm distance from thebeginning of the test section. The contrast and brightness of the images are adjusted, and based onthe image sequence, a movie is constructed using the software TMPGEnc (version: 2.525.64.184).

A. Fabrication of the deformable tube and the PDMS gel characterization

Deformable tubes of varying shear moduli are fabricated using a template in the PDMS gelmixture in this work, but the tube could also prepared in a polyacrylamide gel.14 The tube comprisesof two sections: the first is the relatively rigid flow development section, and the second is the moresofter test section where transition is to be studied. The length of the development section is chosen

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024102-6 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

to be 15 cm, and this ensures that the flow that enters the (softer) test section is always fully devel-oped even at the largest Reynolds numbers studied in our experiments. The test section is fabricatedwith different percentages of cross-linker in order to obtain tubes of varying deformability, whichwill aid in understanding the dynamic coupling between the fluid and the gel, and its role in the flowinstability. The PDMS gel is prepared by mixing the elastomer base (Sylgard 184 elastomer, DowCorning C ) and cross-linker (curing agent) in required amounts to get the specific shear modulus.

The tube is fabricated as per the following procedure. The initial flow development section isfabricated by pasting double sided tapes (2.25 cm width) on the top of a glass plate that has dimen-sions 6.5 cm × 30 cm width and length, respectively (Figure 2), and after leaving 2 cm wide gap,another set of double sided tapes is pasted on the glass plate. The height of both the double-sidedtapes is 1 cm. The tube template, a galvanised iron capillary tube with 1.65 mm outer diameterand 35 cm length, is used to make tubular bore in the gel. One end of the template (i.e., the GIcapillary tube) is fixed using small strips (of height 0.5 cm) of double sided tape in the gap, and thetemplate is held similarly at a distance of 15 cm from the starting point. This arrangement formsa rectangular well with 15 cm × 2 cm × 1 cm length, width, and height, respectively. The PDMSmixture consists of Sylgard 184 base and cross-linker in 10% concentration. The PDMS mixture isdispensed in the rectangular well without any air entrapment and cured at 70 ◦C at 12 h. Adjacentto this harder PDMS gel, another rectangular well is prepared with the above-mentioned dimensionsby pasting small strips of double sided tape. The PDMS mixture consisting 1.9 % cross-linker andSylgard base is dispensed in the well, and is cured to prepare the softer, test section. The first (15 cmlength) section is the development section for the fluid flow, and the adjacent deformable section(of length 15 cm) is the test section to study the transition. The PDMS gel containing the capillarytube is placed in toluene for few (3 to 4) h to allow for swelling of the gel. The capillary tube wasgently removed from the gel, and the gel was allowed to deswell at low temperature (by placing ina refrigerator, in order to reduce solvent evaporation rate). The development section (beginning ofsection (c) in Figure 1) is carefully attached to the channel (end of section (b) in Figure 1), withthe help of a conical microtip which was already embedded with the Plexiglass sheet. The conicalmicrotip ensures smooth entry of the fluid, similar to the “bell-mouth” entry.32

The PDMS gel mixture containing 10% cross-linker is added to the tiny gaps (if present) andcured to avoid leakage of fluid. The Plexiglas sheet (that forms the front end of the section (b) ofFigure 1) is attached to the low-turbulence channel. Using a surgical needle, a hole is made forconnecting the pressure transducer in the tube (Figure 2). The hole is made in the developmentsection very near to the test section to avoid leakages. The same procedure is followed to fabricatethe soft tube (test section) in PDMS gel with cross-linker concentrations 2 %, 2.1 %, and 2.25 %.

The PDMS gels were cast in the shape of rectangular slabs (4 mm thickness) in order tomeasure their linear viscoelastic properties using the parallel plate geometry (40 mm diameter) ofthe rheometer (Discovery HR-3, T.A instruments, U.S.A). The gel slab is placed on the bottom plateof the rheometer, and the top plate is brought in contact with the gel surface till the normal force ismeasured 0.2 N. The top plate position is taken as an indicator of zero gap between the plate andthe gel surface. The rheometer is operated in the oscillatory mode at an amplitude of 50 Pa. Thefrequency spectrum of the shear modulus is shown in the Figure 3 in the range of 0.1-100 Hz. Theshear and loss moduli increase with frequency. In order to relate the transition Reynolds number(Re = ρUavgDavg/η) for instability with the nondimensional wall elasticity Σ = ρG′D2

avg/4η2, we

average the shear and loss moduli in two different frequency ranges. Here, G′ is the shear modulusof the wall material, Davg is the average diameter of the deformable tube, ρ is the density of the fluidor solid, η is the viscosity of the fluid, and Uavg is the average velocity in the tube. First, we haveaveraged the shear and loss moduli over the frequency range from 0.1 to 10 Hz, and the results aresummarized in Table I. Second, we averaged the shear and loss moduli over the frequency rangeof 10 to 100 Hz, and the results are summarized in Table II. The results of 2 % cross-linker PDMSgel shear modulus (G′) from our experiments agree well with those reported in Ref. 15 preparedunder identical conditions. In this article, we use the shear and loss moduli averaged over the highfrequency (6 to 100 Hz) range, because the shear modulus shows the good agreement with reportedresults of Ref. 15. The standard deviation is large for shear and loss moduli calculated in thisfrequency range.

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024102-7 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 3. Characterization of the PDMS gels: Storage modulus G′(ω) (panel (a)) and loss modulus G′′(ω) (panel (b)) forthe 1.9% [open square], 2% [open triangle up], 2.1% [open diamond], 2.25% [open triangle right], and 10% [open circle]cross-linker PDMS gels. The measurements are performed at 25 ◦C using the parallel-plate geometry of DHR3 rheometer.The PDMS gels are prepared in 4 mm thickness. Error bars are within size of the legends for the PDMS gels prepared with1.9%, 2.1%, and 2.25% cross-linker, and the error bars are not clearly visible in the graph.

B. Characterization of varying diameter in the deformable tube

In order to observe the critical Re, the flow transition, from laminar flow to turbulence, weconstructed the friction factor versus Re chart. The friction factor follows the relation f = 64/Rewhen the flow is laminar. For flow in the deformable tube, the point at which there is a departureof friction factor from its laminar value is considered as the transition Re. The expression for the(Darcy) friction factor for flow through a rigid tube is written as

f = 2τw/ρU2avg. (1)

The wall shear stress is given by τw = 0.25D(∆P/L) and the average velocity of flow Uavg = (Q/A),where D is the diameter of the deformable tube at the down stream location, (∆P/L) is the pressuredrop across length of the soft tube, Q is the flow rate, ρ is the density of the fluid, and A is the crosssectional area of the tube. Simplifying the above equation results in following equation:

f = (π2∆P/32LQ2ρ)D5. (2)

When fluid flows through the soft tube at high flow rates, the soft tube diameter varies along theflow direction, and this variation depends on the flow rate. Due to this reason, the friction factor

TABLE I. Rheological properties of the PDMS gels used for tube prepa-ration. The shear and loss moduli are shown for various % cross-linkerconcentrations. The data are averaged over the frequency range of 0.1 to10 Hz.

Cross-linkerconcentration wt/wt%

G′

(kPa)G′′

(kPa)

1.9 8.84 ± 2.31 3.49 ± 2.402 11.24 ± 2.56 3.85 ± 2.592.1 13.47 ± 2.91 4.27 ± 2.822.25 15.79 ± 3.32 4.82 ± 3.1710 511.82 ± 30.13 38.91 ± 12.29

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024102-8 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

TABLE II. The shear and loss moduli of PDMS gel fabricated with different% of cross-linker. The data are averaged over the frequency range from 10to 100 Hz.

Cross-linkerconcentration wt/wt%

G′

(kPa)G′′

(kPa)

1.9 20.47 ± 6.83 16.10 ± 7.412 23.84 ± 7.27 17.04 ± 7.612.1 27.47 ± 7.89 18.46 ± 8.092.25 31.84 ± 9.15 20.47 ± 8.8310 607.73 ± 59.93 81.24 ± 26.87

equation is written15 as

f = (π2∆P/32LQ2ρ) 1

L

L

0D(x)5 dx. (3)

The friction factor is determined from the measured pressure drop and is plotted as a function ofRe. The Re at which there is a departure of f from its laminar value of 64/Re is considered asthe transition Reynolds number. The Reynolds number for flow through a rigid tube is defined asRe = ρDavgUavg/η, where Davg is average diameter of the tube, Uavg is the average velocity of theflow, and η is viscosity of the fluid. The average Reynolds number for flow through the soft tube iscalculated14 as

Re = (2Qρ/πη) 1L

L

0(D(x)/2)−1 dx. (4)

III. RESULTS

In this section, we first discuss results for the variation of diameter of the deformable tube withdownstream location as well as with Reynolds number and then proceed to discuss the friction factorvs Reynolds number results and dye-stream visualization results for flow in a deformable tube.

A. Soft tube diameter change at downstream locations and for the different Reynoldsnumbers

Figure 4 shows the soft tube diameter in test section as function of distance from the beginning ofthe test section. The change in the tube diameter is observed at Re = 500 for the PDMS gel fabricatedwith cross-linker concentration 1.9%. When there is no flow, the tube diameter (unexpanded shape)is nearly 1.58 mm and is nearly 1.63 mm at Re = 500 at the downstream location 3 cm away from thebeginning of the test section. The cross-section of the test section diverges initially and then exhibits aconverging behavior, as also observed in Ref. 15. The maximum change of the tube diameter is nearly3% in the test section, and the change in diameter is considerably small compared to the originaldiameter. The change in tube diameter is more in the beginning of the test section, because the pressureis more upstream, and the diameter change becomes small in the downstream direction. The lowestpossible shear modulus of PDMS tube fabricated is the one with 1.9% cross-linker concentration,and with increase in percentage of cross-linker, there is a decrease of change in tube diameter. Asimilar behavior is observed for soft tubes fabricated with cross-linker concentrations 2%, 2.1%, and2.25%. When we prepared the PDMS gel with 10% cross-linker (shear modulus 608 kPa), we observenegligible changes in tube diameter. Due to the above reason, the development section is preparedwith PDMS gel that has shear modulus 608 kPa, and this can be considered to be nearly rigid. Wealso demonstrate below (Figure 6) that the transition of laminar flow in a tube fabricated with this(relatively harder) gel is very similar to the rigid tube transition. It must also be noted here that thevariation of tube diameter with downstream distance observed in this study is somewhat different

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024102-9 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 4. Normalized soft tube diameter (actual diameter normalized with undeformed uniform tube diameter (D0)) as functionof the downstream locations in test section. The open triangles and the dashed line represent data for Re= 500. The shearmodulus of the tube is 21 and 608 kPa for test and development section, respectively. Note that the x and y-axes scales arevastly different. The change of diameter at a specific location is negligible for low flow rates, however the error bar is notincluded.

from that reported in Ref. 15: the slope of diameter variation (in both converging and diverging parts)is gentler in the current work. This could presumably be attributed to the following differences inthe PDMS gel block in which the tubular bore was fabricated. The current study uses a block withwidth 2 cm and height 1 cm, but this information is not available from the experiments of Ref. 15.Further, the length of the test section used in the current experiments is 15 cm, while that used inRef. 15 is 10 cm. These differences could have lead to the differences in the variation of diameterwith downstream distance between the present work and Ref. 15.

In addition to the above observations of the tube diameter change for different downstreamlocations in test section and for different Re (at 3 cm from the end of development section), theaverage diameter (averaged over the downstream length) was also determined for the test section atvarious flow rates. Figure 5 shows the average diameter change for the different tubes prepared with

FIG. 5. Average diameter of soft tube as a function of Reynolds number: The diameter is averaged over the length of thetest section. The diameter of the undeformed tube is 1.6 mm. The soft tubes are fabricated in PDMS gels with cross-linkerconcentrations 1.9% [open square], 2% [open circle], 2.1% [triangle up], and 2.25% [triangle right], respectively.

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024102-10 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 6. Data showing the flow transition similar to that in rigid tubes: Friction factor as a function of Reynolds number forflow the harder PDMS gel fabricated. The departure of friction factor [open triangle] at transition Reynolds number for thetube prepared using PDMS gel with shear modulus 608 kPa. The diameter of the tubular bore is 1.65 mm and length of thetube 30 cm. Since the tube is relatively harder, the diameter of the soft tubular bore does not change for all the flow rates.

shear moduli 21, 24, 28, and 32 kPa. The average diameter increases with increase in Re, howeverthe variation is large in the high Re regime. The average diameter (as function of Re) is slightlyhigher for PDMS gels fabricated in cross-linker concentrations 1.9%, and 2%.

B. Friction factor vs. Re

In order to determine the transition of flow through the soft tube, the friction factor ( f ) versusRe chart is constructed for different PDMS gel systems. The soft tubes are prepared in the PDMSgel with shear moduli of 21, 24, 28, and 32 kPa for 1.9%, 2%, 2.1%, and 2.25% cross-linker, respec-tively. The departure f from the 64/Re line is considered to be the Reynolds number for transition inthe deformable tube. Before determining flow transition in soft tube, the relatively harder PDMS gel(shear modulus 608 kPa) is prepared and experiments are carried out (results shown in Figure 6) tobenchmark the experimental setup with the results from a truly rigid tube. At low Reynolds number,the observed friction factor follows the 64/Re line until Re ∼ 2000, when f departs from its laminarvalue. The observed results show that the transition is very similar to that in a rigid tube.

The deformable tubes prepared in the PDMS gel with lower shear moduli exhibit diverging andconverging cross-sections under the applied pressure gradient. To understand the flow transition inthe soft tube with fully developed flow entry in the test section, the development section is preparedwith PDMS gel of shear modulus 608 kPa. The test section is prepared in PDMS gel with muchlower shear modulus material 21 kPa. Figure 7(a) shows the friction factor as function of Reynoldsnumber for flow of water in tube made of PDMS with shear modulus 21 kPa. The departure ofthe friction factor is observed at Re ∼ 500 and this is considered as the transition Re. The dashedline is theoretical friction factor f = 64/Re line and triangles are the explicit data transition in rigidtube. The diameter of the tubular bore is 1.65 mm before the experiment was started. The length ofboth the development and test sections is 15 cm. Since the tubes are relatively softer, the diameterof the soft tubular bore changes along the length. The earlier study of Verma and Kumaran15 alsoreported that the diameter of the tube varies along the flow direction, and they further demonstratedthat the transition is not due to the slow variation in the tube diameter. This was accomplished byfabricating a glass tube with the same cross-section as the deformable PDMS tube and by showingthat the transition in that glass tube was similar to transition in a rigid tube. Thus, the deviationof the friction factor vs Re curve from its behavior in the laminar-regime can be attributed to thedeformable nature of the wall. Figures 7(b), 7(c), and 7(d) show f as a function of Re for the soft

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024102-11 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 7. Data showing flow transition in soft tubes: Friction factor as function of Reynolds number for flow in tubes preparedwith PDMS gel. Panel (a) indicates the departure of friction factor [open square] at critical Reynolds number for the tubeprepared with cross-linker concentration 1.9%. The panels (b), (c), and (d) indicate the departure of friction factor for thetubes prepared in PDMS gel with cross-linker concentrations 2% [open diamond], 2.1% [open triangle right], and 2.25%[open circle]. The dashed line is the theoretical friction factor f = 64/Re relation and open triangles [blue] show data for rigidflow transition (608 kPa tube). The diameter of the tubular bore is 1.65 mm before and after the experiments. The lengths ofthe development and test sections are kept at 15 cm.

tube test section prepared in PDMS gel with shear moduli 24, 28, and 32 kPa, respectively, and theobserved transition Re is 700, 900, and 1260. Thus, as the shear modulus of the deformable tubeincreases, there is an increase in the transition Re.

C. Results from dye-stream visualization

The dye-stream patterns are shown in Figure 8 for flow through soft tubes fabricated in PDMSgels with shear moduli of 21, 24, and 32 kPa. The patterns were recorded for a wide range of Reynoldsnumbers from 300 to 3000, in steps of 20. Thus, we can only provide a range of Re (of width 20) inwhich the dye stream motion becomes irregular. However, here we present three salient images (foreach PDMS gel) that demonstrate laminar flow, the point of transition, and the apparently turbulentstate. As noted in Sec. III A, the images were acquired at 1.5 cm distance from the beginning ofthe test section. At this location, the soft tube diameter is diverging in the flow direction, and thediameter starts to decrease at around 3 cm from the beginning of the test section (see Figure 4).

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024102-12 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 8. Representative images of the dye-stream pattern: The change of dye-stream pattern inside the soft tubes wasfabricated using PDMS gel. The images (a), (b), and (c) are the dye-stream patterns correspond to flow Reynolds numbers520, 540, and 662 for tube fabricated with cross-linker concentration 1.9%. The dye-stream patterns representing the laminarflow, the point of transition, and apparently turbulent state. The images (d), (e), and (f) are the dye-stream patterns of flow Re503, 629, and 915 (cross-linker concentration 2%), and (g), (h), and (i) for the flow Re 1432, 1460, and 1977 (cross-linkerconcentration 2.25%) for the tube fabricated with different shear moduli. (Multimedia view) [URL: http://dx.doi.org/10.1063/1.4907246.1] [URL: http://dx.doi.org/10.1063/1.4907246.2]

This location for observation of the dye stream is different from the earlier experiments of Ref. 15,where the converging section of the tube was chosen for recording the dye motion. The motion of theobserved dye-stream is straight at the center of the tube for the Reynolds number 520 (Figure 8(a)),which indicates that the flow is laminar for the tube fabricated with shear modulus 21 kPa. At higherReynolds number (around 540), we observe that the shape of the dye-stream resembles a sinusoidalwave-like pattern (these sinusoidal oscillations are better viewed in the videos corresponding to Fig-ures 8(b) and 8(e) (Multimedia view)). This suggests that the flow undergoes rather complicatedmotion, and the corresponding Reynolds number (540) is considered as the transition Re, where thereis an onset of the instability. Upon further increase in Reynolds number, the amplitude of the wavepattern increases, and the dye mixes throughout the tube at Re = 662 (Figure 8(c)). Figures 8(d) to8(i) show the dye-stream patterns (at different Reynolds numbers) for deformable tubes with shearmoduli 24 and 32 kPa. The dye-stream patterns are similar in the laminar regime, and the amplitudeand frequency of wave pattern show variation at transition Reynolds number (Figures 8(e) and 8(h)).At high Re, the mixing pattern of the dye-stream in the soft tubes is different (Figures 8(f) and 8(i)).It appears that for the softest tube in which experiments were carried out, there is complete mixing atRe ∼ 662, while for harder tubes, the mixing of the dye-stream is incomplete. The transition Reynoldsnumber as obtained from the dye-stream visualization is summarized in Table III for the soft tubes

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024102-13 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

TABLE III. Transition Reynolds number Ret determined from the f -Rechart and dye-stream experiments for tubes with different shear moduli. Forthe dye-stream experiments, we provide the range of Reynolds numbers inwhich the transition would happen.

Shear modulus(kPa) Ret from f –Re Ret from dye-stream

21 487 ± 17 520−54024 706 ± 21 610−62928 841 ± 21 840−86732 1260 ± 34 1440−1460

fabricated with different shear moduli. The increase in shear modulus of the tube results in increase intransition Reynolds number for flow instability. The transition Reynolds number observed from thefriction factor data and the dye-stream experiments shows reasonably good agreement. The transitionReynolds number observed for flow through soft tubes is much lower compared to rigid tube transitionRe ∼ 2100. Because the Re at which there is an onset of instability decreases with increase in the shearmodulus, this strongly suggests that the flow transition that occurs in the soft tubes is due a dynamicalinstability of fluid-solid interface (absent in truly rigid tubes), and this brings down the transition Resubstantially. The dye-stream experiments are also carried out in the (harder) development sectionbefore the beginning of the (softer) test section, where we observe that laminar flow persists until theRe is 2000.

The wave properties of the dye-stream motion such as amplitude, frequency of oscillation,wavelength, and propagation velocity are determined using an image analysis of the snapshots,and the results obtained are shown in Figure 9. The trajectory of the dye-stream is reconstructedby tracking the dye-stream locations in the image, and the same procedure is repeated for all theimages. The wave amplitude is determined as the distance between wave crests and troughs, andthe amplitude is normalized with the tube diameter. The normalized wave amplitude as a functionof Reynolds number is shown in Figure 9(a). Our results suggest that the variation of amplitudewith Re is continuous at the point of transition, thus suggesting that the nature of the bifurcation issupercritical. This agrees qualitatively with the results of the weakly non-linear analysis of Ref. 34.This is in stark contrast with results obtained for flow in rigid tubes, wherein the dye-stream breaksdown rather abruptly at the point of transition. This trend of continuous increase in the dye-streamamplitude is observed for other soft tubes prepared with different shear moduli. The spread of thedata points shown in Figure 9(a) is the variation of amplitude (obtained from the images) with timeand not the standard deviation over different experiments. The maximum wave amplitude observedis nearly 0.15 times the tube radius, and upon further increase in Re, the dye-stream eventuallybreaks down and mixes throughout the tube cross section.

The frequency of dye-stream oscillation as a function of non-dimensional elasticity Σ is shownin Figure 9(b). The frequency is determined from the images acquired at the 1 ms interval. Thetime taken for the dye-stream oscillation (radial movement) was calculated at a fixed location in theimages. The observed dominant frequencies of oscillations are 244 ± 29, 323 ± 09, 422 ± 41, and94 Hz for tubes prepared using PDMS gels shear moduli 21, 24, 28, and 32 kPa, respectively. Theobserved frequency shows good agreement with the reported theoretical frequency value rangingfrom 100 to 500 Hz.21 The frequencies were scaled with (G′/ρD2)1/2, and the scaled frequencyshows very little increase with Σ (Figure 9(b)). The average value of the frequency was calculatedfrom nearly 40 images, and the spread of the data about the average is shown in Figures 9(b), 9(c),and 9(d).

The normalized wavelength (λ/D) of the dye-stream is shown as a function of Σ in Figure 9(c).The dye-stream is reconstructed as a graph from all the images. The distance between crest to crest(or trough to trough) of the wave is determined and the result is divided by the diameter of thetube to obtain the normalized wavelength. The nondimensional wavelength is nearly 1, and wave-length is calculated as λ = 2π/k. This suggests that the most unstable mode is a finite wavelength

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024102-14 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

FIG. 9. Panel (a) shows the dye-stream amplitude obtained from the images as a function of Reynolds number. Thenon-dimensional frequency of oscillation (b), normalized wavelength (c), and normalized wave speed (d) of dye-streamas a function Σ determined for soft tubes fabricated with shear moduli 21, 24, 28, and 32 kPa.

mode. The propagation velocity of the dye-stream is shown as a function of Σ in Figure 9(d). Thewaves are superposed upon one another by horizontal shifting. The wave speed was calculatedfrom the time difference between two images and the horizontal shift value. The normalized wavespeed is calculated by dividing the dimensional wave speed by (G′/ρ)1/2, where the G′ and ρare shear modulus of soft gel and density of the water, respectively. The non-dimensional wavespeed c∗/(G′/ρ)1/2 should be independent of Σ, as predicted by theory,19,21 and this is the case inthe experiments as well. Previous theoretical studies show that at higher Re, the unstable modesin a tube will have wave speeds which are of the same order as the wave speed of shear waves(∼ (G′/ρ)1/2) in an elastic solid. Thus, the dimensional wave speeds of dye-stream propagationobtained from the image processing are nondimensionalised by (G′/ρ)1/2; the resulting quantitymust be independent of the nondimensional wall elasticity Σ. Figure 9(d) indeed shows that thenondimensional wave speed is approximately a constant and does not show any systematic variationwith the nondimensional wall elasticity Σ.

D. Scaling of Re with non-dimensional elasticity Σ

As mentioned in the Introduction, previous theoretical studies have broadly identified threeclasses of instabilities in fluid flow through deformable tubes, which are distinguished by the

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024102-15 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

manner in which the Reynolds number for transition scales with the non-dimensional elasticityΣ = ρG′D2

avg/4η2 of the deformable wall. At higher Re (of relevance to this study), two types of

modes were predicted, viz., the inviscid mode with Re ∝ Σ1/2 and the wall mode with Re ∝ Σ3/4.It must be mentioned here that in the earlier theoretical studies,21,22 the fluid is assumed to besurrounded by an annular gel layer, while in our experiments and that of Ref. 15, this symmetry isnot present as the height and width of the gel block are different. This could have some effect on thecomparisons between theory and experiments, since theory shows that the critical Reynolds numberis indeed a function of the annular thickness. Another difference between theory and experimentsis that in theoretical analyses, the solid has been assumed to be incompressible, while in reality,the gel is slightly compressible. The previous experimental study of Verma and Kumaran15 foundthat Re ∝ Σ5/8, which is in between 1/2 and 3/4 (since 4/8 < 5/8 < 6/8). They argued that theobserved scaling is consistent with the wall mode result of Re ∝ Σ3/4, if one accounts for the slowconvergence of the tube, and the fact that the laminar velocity profile in the converging section issignificantly different from the fully developed velocity profile when Re ≫ 1. This modifies theshear rate at the interface, and by accounting for this increased shear rate, they found that, at thescaling level, Re ∝ Σ5/8. However, in the experiments, instability was observed at a much lower Re(of about 1000) less than the theoretical prediction. Figure 10 shows data from our experiments fortransition Re as function of the nondimensional wall elasticity Σ = ρG′D2

avg/4η2 for the soft tubes

prepared in PDMS gels with shear moduli in the range of 21-32 kPa. The data for shear modulusof the gels (shown in Figure 3(a)) used to indicate that the shear modulus increases with increase infrequency and does not show a well-defined plateau region. In this situation, it is not immediatelyobvious as to which value of shear modulus one must use in the calculation of the non-dimensionalparameter Σ. Hence, we averaged the shear modulus in the range of 0.1-10 Hz, as well as in therange of 10 - 100 Hz, and the Re vs Σ plots are shown for both these averages in Figure 10. Thetheoretical predictions show that the frequency of oscillation of the wall mode instability21 is in therange of 100 Hz. The frequency of oscillation of the dye stream (shown in Figure 9(b)) inferredfrom our experiments is also in the same range. Hence, if the high-frequency average (in the range10-100 Hz) is used, one obtains Figure 10(a), while if the low-frequency (in the range of 0.1-10 Hz)average is used, then one obtains Figure 10(b). Since the average shear modulus changes by afactor of 2 from the low-frequency range to the high-frequency range, the Re vs. Σ curve shiftshorizontally, without any shift in the vertical direction. However, the slope of the Re-Σ curve (inlog-log scale) is much higher in our data, viz., ≃ 3/2, in marked contrast to 5/8 seen in the earlier

FIG. 10. Critical Reynolds number as function of solid elasticity parameter Σ. Open squares represent the transition Reynoldsnumber obtained from f vs. Re chart, and open diamonds denote the transition Reynolds number obtained from dye-streamexperiments for flow through soft tubes prepared in PDMS gel with shear moduli range of 21-32 kPa. Open circles are datafrom the earlier studies of Verma and Kumaran15 for flow trough a deformable tube. (a) High-frequency range shear modulusused in calculation of Σ. (b) Low-frequency range shear modulus used in calculation of Σ.

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experiments of Ref. 15. It must be mentioned that while inferring the exponent in the power law,the range of Σ in our data is rather limited, and one needs a larger range to unambiguously infer thepower law exponent. In either case, the critical Re obtained in our experiments is somewhat lowerthan that reported in Ref. 15.

The question arises whether the scaling suggested by our experiments, viz., Re ∝ Σ3/2 corre-sponds to the scaling of the “inviscid mode,”20 albeit being altered by the change in tube diameteralong the flow direction. There is a major difference in the location of observation of the dye streamin our work and the earlier study of Verma and Kumaran.15 While we recorded the dye streamin the initial diverging section, Verma and Kumaran15 recorded the dye stream in the convergingsection of the tube. The velocity profile at Re ∼ 500 in the slightly converging section would notbe parabolic, but will have a plug-like region in the center and with sharp velocity gradients nearthe wall. For a taper of slope α ≪ 1, the velocity profile in the converging region of the deformabletube would be very different from the fully developed profile when αRe ≫ 1. The thickness ofthis boundary layer is proportional to (αRe)−1/2, implying that the actual shear rate at the wallscales as (αRe)1/2(2Umax/D), where D is the undeformed diameter of the tube, and Umax is themaximum velocity at the center of the tube. Because of the higher velocity gradient at the wall,the transition Reynolds number would decrease by a factor (αRe)−1/2 compared to the case of fullydeveloped flow. If Re ∝ Σa (with the exponent a = 3/4 for wall modes, and a = 1/2 for inviscidmodes) for the fully developed flow, for flow in the converging section, Re ∝ (αRe)−1/2Σa by thepreceding physical argument. Now, the slope α ∝ (2Umaxη)/(G′D), and hence α ∝ Re/Σ, implying(αRe)−1/2 ∼ (Re2/Σ)−1/2. Thus, the transition Reynolds number in the converging section wouldscale as Re ∼ (Re2/Σ)−1/2Σa, and this yields Re2 ∼ Σa+1/2. For wall modes, a = 3/4, and henceRe ∼ Σ5/8, while for inviscid modes a = 1/2, and thus Re ∼ Σ1/2. Thus, while the scaling Re ∝ Σ3/4

changes for the wall mode to Re ∝ Σ5/8 in the converging section, the scaling Re ∝ Σ1/2 remainsthe same in the converging section for inviscid modes. Thus, the modified velocity profile in theconverging section is not enough to explain the difference in the scaling.

However, in the diverging section (relevant to the present dye-stream observations), the velocitygradient at the wall is smaller compared to the case of fully developed flow.35 However, as theReynolds number increases (for a given taper), the velocity gradient at the wall decreases further,unlike in the converging tube where the velocity gradient increases with Re for a given taper. Hence,a boundary-layer like scaling argument (described above for the converging case) cannot be madeto estimate the re-scaled velocity gradient at the wall as a function of the Reynolds number. Inthe converging case, the exponent for wall modes Re ∝ Σ3/4 decreases from 3/4 to 5/8 owing tothe convergence. However, since the behaviour of the velocity gradient is opposite to that of theconverging case, we can anticipate that the exponent for the slightly diverging case would increasecompared to 3/4. Indeed, our experimental data indicate an exponent of 3/2. But it is not possibleto quantitatively estimate the exponent using a boundary layer-type analysis, due to the lack of athin region where the velocity changes rapidly in the diverging tube. This further means that it isnot possible to clearly identify the mode of instability, since the increase in exponent (to 3/2) couldbe either from inviscid mode or wall mode. Thus, more detailed experimental observations andtheoretical calculations are needed in order to understand this further.

IV. CONCLUSION

We have carried out experimental studies to analyze the instability of pressure-driven flow ina tube with deformable walls. To this end, we measured the pressure-drop across the two ends ofthe deformable tube, and from these data, we plotted the f vs. Re curves. The deviation of thef vs. Re curve from its expected laminar behavior is taken to be the point at which laminar flowbecomes unstable. In addition, dye-stream visualization was also employed to locate the transitionRe. In general, we find good agreement between both the procedures for locating the transition. Ourexperiments show that the Reynolds number for transition increases with increase in the solid shearmodulus, and for the lowest shear modulus studied in this work, we find that the transition Re could

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024102-17 R. Neelamegam and V. Shankar Phys. Fluids 27, 024102 (2015)

be as low as 500, which is much lower compared to the transition Re for a rigid tube (∼ 2000).Our results suggest that the transition Re scales with the non-dimensional wall elasticity parameterΣ as Re ∼ Σ3/2. This observed scaling is quite different from the results of Verma and Kumaran15

which suggested the scaling Re ∼ Σ5/8. Thus, while the present study independently corroboratesthe general conclusion of Ref. 15 that fluid flow in a deformable tube could be destabilized by walldeformability, our study differs in the conclusions regarding details of the transition. In order toprecisely isolate the mechanism of instability, more accurate experimental characterization of theinstability (such as PIV (particle image velocimetry) measurements of flow velocities) is required,which is beyond the scope of the present study. This should form the basis for future studies in thisarea. If the instability observed in this study is accurately characterized and understood, it could bepotentially exploited in promoting mixing in small-scale flows.

ACKNOWLEDGMENTS

We thank Professor Debopam Das, Professor V. Kumaran, and Dr. M. K. S. Verma for manyhelpful discussions. The Board of Research in Nuclear Sciences (BRNS), Department of AtomicEnergy (DAE), Government of India (Grant No. 2007/36/48-BRNS) is acknowledged for financialsupport of this work.

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