2.1 Fabrication of microtubes

Tubes of diameters $390~\unicode[STIX]{x03BC}\text{m}$ and $470~\unicode[STIX]{x03BC}\text{m}$ are prepared in rectangular polydimethylsiloxane (PDMS; Sylguard 184 procured from Dow Corning) blocks. Two copper wires of different diameters (Alcoa Prime) are used as a template which is held straight using a screw mechanism and double-sided tape. A rectangular well (figure 2) is created using double-sided tape and the wire is held straight in the rectangular well using the screw mechanism. The well is prepared in a manner similar to the protocol of Neelamegam & Shankar (Reference Neelamegam and Shankar2015), except that a copper wire is used here as a template. The PDMS kit consists of an elastomer base and a curing agent. The required amount of curing agent is added to the elastomeric base, which is then mixed thoroughly. The liquid mixture is then degassed to remove air bubbles, after which it is slowly poured into the well. The well containing the PDMS mixture is now cured in a preheated oven at $80~^{\circ }\text{C}$ for 4 h. After curing, the PDMS block is kept in toluene solution for 2–3 h while swelling. The copper wire is slowly removed after swelling and thus a tubular bore is created in the PDMS block. De-swelling is carried out at low temperatures to avoid crack formation in the tube. The choice of PDMS for tube fabrication is due to its lower refractive index value ( ${\sim}1.41$ ) as compared to glass ( ${\sim}1.5$ ) or fused silica ( ${\sim}1.46$ ). Since our test fluid is water, and hence the polymer solution has a refractive index close to 1.33, using PDMS will be more effective in reducing optical distortion in our micro-PIV experiments. Furthermore, the transparent nature of the PDMS block allows for dye-stream visualization in the flow.

2.2 Preparation of polymer solutions

The polymer solution is prepared by mixing small amounts of high-molecular-weight polymers in deionized water. We use two different polymers in our study: polyacrylamide ( $\text{MW}=5\times 10^{6}~\text{g}~\text{mol}^{-1}$ ) and polyethylene oxide ( $\text{MW}=8\times 10^{6}~\text{g}~\text{mol}^{-1}$ ), both procured from Sigma-Aldrich. The required amount of polymer is added to deionized water and the resultant liquid is slowly mixed using a magnetic stirrer. The polymer solution thus prepared is used in our flow experiments within three days (for polyacrylamide) or 8 h (for polyethylene oxide) of preparation, since polyethylene oxide solutions are found to be more prone to ageing, as discussed later. The solution used is characterized before and after the experiments to check for the effect of ageing with time. The properties of the fluid do not change during the time scale of the experiments, which is approximately 2–3 h. Experiments are always performed using freshly prepared polymer solutions to avoid any ageing effects. Measurement of viscosity is done using a TA DHR-3 rheometer with the help of a concentric cylinder geometry.

2.3 Characterization of PDMS tubes

The PDMS mixture used for tube fabrication is also prepared in rectangular slabs of 4 mm thickness for characterization. Small-amplitude oscillatory experiments are performed on these slabs in order to measure the frequency-dependent shear modulus. The shear moduli of the gels prepared are more than $10^{5}~\text{Pa}$ , and hence can be considered as rigid, as has been extensively documented by the earlier studies of Verma & Kumaran (Reference Verma and Kumaran2012). In this study, we also demonstrate that the transition of a Newtonian fluid in such a tube happens at $Re\sim 2000$ , similar to rigid materials such as glass or steel. The diameter of the prepared tube is measured at different axial locations to check for uniformity of the tube dimensions using the 10 $\times$ objective of a microscope. A microscopic scale is used for calibration of the microscope. The deviation of the diameter of the fabricated tubes is less than 1 % within the tube. In order to check the extent of roughness in the inner surface of the prepared tube, we carried out imaging using scanning electron microscopy (SEM). To obtain the images, we cut open the PDMS block in which the tube is present such that the hollow cylindrical surface of the tube is exposed. The SEM images of the inner surface of the tubes are shown in figure 3. Magnifications used in the imaging are similar to Verma & Kumaran (Reference Verma and Kumaran2012). The SEM images of our tubes suggest that the roughness present in the fabricated tubes is negligible.

Figure 3.

SEM images of microtubes at different magnifications, i.e (a) 400 $\times$ , (b) 1500 $\times$ , (c) 2000 $\times$ and (d) 3000 $\times$ , to demonstrate the smoothness of the inner surface of the tubes fabricated.

2.4 Characterization of polymer solution

The viscosity of the polymer solutions is measured using a TA DHR-3 rheometer with the help of a concentric cylinder geometry. The viscosity is measured with shear rates in the range $10~\text{s}^{-1}{-}50~\text{s}^{-1}$ (figure 4). The results show that the shear-rate dependence of the viscosity is negligible in both the solutions in the observed range of shear rates. The zero-shear viscosities of the various solutions used in our experiments are summarized in table 4. However, the nominal shear rates in the flow experiments carried out later in this work are very high, but it is not possible to use the rheometer to measure the viscosity at such high shear rates encountered in the flow experiments. The effect of ageing is studied on both polyacrylamide solution and polyethylene oxide (PEO) solution by observing the change in viscosity with time. It is observed that the PEO solution is more prone to ageing. Hence viscosity is always measured before and after flow experiments to ensure that the solution had not aged sufficiently during the flow experiments. However, the viscosity of polyacrylamide solution does not change appreciably over the time scales for which such studies are performed for the PEO solution. Hence it was ensured that experiments are performed within 3 h of preparation of the PEO solution.

Figure 4.

Viscosity as a function of shear rate for (a) polyacrylamide solution ( $\text{molecular weight}=5\times 10^{6}$ ) and (b) polyethylene oxide ( $\text{molecular weight}=8\times 10^{6}$ ).

Table 4.

A comparison of solution viscosity $\unicode[STIX]{x1D702}$ of the two polymer solutions used in our study at different concentrations. PAA stands for polyacrylamide ( $\text{MW}=5\times 10^{6}$ ) and PEO for polyethylene oxide ( $\text{MW}=8\times 10^{6}$ ).

The characterization of relaxation time of a dilute polymer solution can be done in a variety of ways, and often the results from different methods do not agree accurately with each other (table 5), as has been recently noted by Srinivas & Kumaran (Reference Srinivas and Kumaran2017). In the literature, Larson et al. (Reference Larson, Shaqfeh and Muller1990) and Schiamberg et al. (Reference Schiamberg, Shereda, Hu and Larson2006) use the first normal stress coefficient to estimate relaxation time, and report relaxation times of the order of a few seconds. Larson et al. (Reference Larson, Shaqfeh and Muller1990) used low-molecular-weight polybutene and Schiamberg et al. (Reference Schiamberg, Shereda, Hu and Larson2006) used sucrose to increase viscosity of the solution. Recently, Dinic et al. (Reference Dinic, Zhang, Jimenez and Sharma2015) use the capillary breakup (‘CaBeR’) technique for calculating relaxation time. They estimate the relaxation time on the order of 1 ms for PEO solutions. Samanta et al. (Reference Samanta, Dubief, Holzner, Schäfer, Morozov, Wagner and Hof2013) also use the ‘CaBeR’ technique for measurement of relaxation time. Neelamegam, Shankar & Das (Reference Neelamegam, Shankar and Das2013) use the single-mode Maxwell model for estimation of relaxation time, where the solvent viscosity is increased by the addition of sucrose to water. However, small-amplitude oscillatory experiments do not give satisfactory data for the low-viscosity solvent (i.e. water) used in our study.

Table 5.

A comparison of relaxation times as estimated by different authors; $C_{p}$ is the polymer concentration.

To overcome this difficulty, we increased the viscosity of the solvent by adding glycerine to water, thus increasing the possibility of obtaining reliable results from small-amplitude oscillatory experiments. One consequence that must be avoided due to the addition of glycerine is that the solvent quality should not be modified significantly. This was noted in the experiments of Fang, Brown & Hvidt (Reference Fang, Brown and Hvidt1992), who measured the radius of gyration of polyacrylamide in glycerol solutions at different glycerine concentrations and showed that there is only a very modest change in the radius of gyration as the glycerine concentration is changed. In our experiments, the polyacrylamide solution was prepared by dissolving polymer in $90\,\%$ glycerine solution. Small-strain oscillatory experiments are performed on the prepared solution with increased solvent viscosity. For a truly dilute solution of polymers, the Zimm (or Rouse) relaxation time is directly proportional to solvent viscosity (Doi & Edwards Reference Doi and Edwards1988), and hence in the dilute limit it is possible to measure the relaxation time at a higher solvent viscosity and then scale it appropriately to lower solvent viscosities as desired. In the truly dilute limit, the relaxation time is independent of polymer concentration. Using the reported radius of gyration of $0.184~\unicode[STIX]{x03BC}\text{m}$ for polyacrylamide of molecular weight ${\sim}5\times 10^{6}$ (Kulicke, Kniewske & Klein Reference Kulicke, Kniewske and Klein1982), and a solvent viscosity of 0.001 Pa s, we obtain the Zimm relaxation time for our system as $1.4\times 10^{-3}~\text{s}$ . The overlap concentration for our polymer solution is obtained using $c^{\ast }=0.77/[\unicode[STIX]{x1D702}]$ , where $[\unicode[STIX]{x1D702}]$ is the intrinsic viscosity of our solution and $c^{\ast }\sim 600~\text{ppm}$ for the polyacrylamide solution used.

The concentrations used in our transition experiments are in the range 50 ppm–800 ppm, thus placing them in the dilute regime or the cross-over to the semidilute regime. Although some of our concentrations are in the onset of cross-over to the semidilute regime, we nonetheless scale the relaxation time (from glycerol solutions to that of pure water) assuming that it is directly proportional to solvent viscosity, as one would expect for a truly dilute system. Figure 5 shows the relaxation time obtained using the glycerine–water–polymer mixture after it is scaled down according to the decrease in solvent viscosity for pure water. The results thus obtained are in fairly good agreement with the results of Samanta et al. (Reference Samanta, Dubief, Holzner, Schäfer, Morozov, Wagner and Hof2013), who used the ‘CaBeR’ technique for estimating the relaxation time. For polyacrylamide solutions, however, we were not able to reliably estimate the relaxation time for concentrations below 100 ppm. For 200 ppm, the experimentally inferred relaxation time already approaches the estimated Zimm relaxation time (valid for the truly dilute regime), and hence we use the Zimm relaxation time for $c<200~\text{ppm}$ . The weak concentration dependence of relaxation time at suboverlap concentrations was also reported in the earlier work of Clasen et al. (Reference Clasen, Plog, Kulicke, Owens, Macosko, Scriven, Verani and McKinley2006) and Samanta et al. (Reference Samanta, Dubief, Holzner, Schäfer, Morozov, Wagner and Hof2013).

Although this protocol worked for polyacrylamide solutions, it was observed that polyethylene oxide was not soluble in glycerol solutions. It was shown by Ebagninin, Benchabane & Bekkour (Reference Ebagninin, Benchabane and Bekkour2009) that low-molecular-weight polyethylene oxide (MW $1\times 10^{5}$ ) behaves as a Newtonian fluid even at concentrations up to 20 % by weight. Since polyethylene oxide (MW $8\times 10^{6}$ ) is soluble in polyethylene oxide (MW $1\times 10^{5}$ ), we were able to increase the solvent viscosity of the polymer solution prepared. We use low-molecular-weight polyethylene oxide (MW $1\times 10^{5}$ ) to obtain a mixture of PEO (MW $1\times 10^{5}$ ), PEO (MW $8\times 10^{6}$ ) and water. This gives us satisfactory oscillatory shear data that enable us to estimate the relaxation time using a single-mode Maxwell fit. Our relaxation time data obtained with this procedure are in close agreement with the relaxation time obtained using the ‘CaBeR’ technique by Dinic et al. (Reference Dinic, Zhang, Jimenez and Sharma2015). However, we were not able to obtain the relaxation time reliably for the lowest polymer concentration used in our study (50 ppm). For such low concentrations, it is reasonable to assume that the relaxation time is independent of concentration, and hence we used the relaxation time obtained using $c=100~\text{ppm}$ for $c<100~\text{ppm}$ .

Figure 5.

Relaxation time of polyacrylamide (molecular weight: $5\times 10^{6}$ ) and polyethylene oxide (molecular weight: $8\times 10^{6}$ ) solutions obtained by fitting oscillatory data into a single-mode Maxwell model. The oscillatory experiments were carried out with solvents of a much higher viscosity than water (which ensures that the data from the oscillatory experiments are reliable), and the results for the relaxation time are rescaled for water as the solvent.

2.5 Experimental protocol

The polymeric solution is prepared for micro-PIV analysis by adding fluorescent polystyrene particles (diameter $3.2~\unicode[STIX]{x03BC}\text{m}$ ) procured from Thermo Fisher Scientific, which are already dispersed in deionized water. We estimated the Stokes number to be $O(0.01)$ in our experiments, and hence the seed particles are expected to follow the fluid streamlines faithfully. Further, the Péclet numbers based on the seed particle self-diffusivity are very large, indicating that Brownian motion is negligible. While preparing the polymer solution for PIV, we first prepare a solution of seed particles in pure deionized water. This seed particle solution is sonicated for 60 s to break any agglomeration of particles. This is then added to the polymer solution to obtain the desired polymer concentration. It is to be noted that increasing seed particle concentration may increase the number of vectors obtained, but it also increases the number of out-of-plane particles. This increases noise in the experiments, and hence we use an optimum number of particles to obtain the velocity vectors. The concentration of seed particles is increased in a stepwise manner to arrive at the optimum concentration at which we get maximum number of acceptable velocity vectors. A cross-correlation method is used for processing images with an interrogation window of $64~\text{pixels}\times 64~\text{pixels}$ for inferring seed particle displacement. For micro-PIV, it is observed that volume illumination occurs because of the small diameter of the tube used. This causes visibility of low-intensity out-of-plane seed particles as well. This may result in noise during image processing. The images are run through a filter algorithm to remove the low-intensity out-of-plane seed particles. The PIV measurements are taken at a distance of 120 diameters from the entry of the flow in the microtube, which enables us to neglect any entrance effects.

Since the tubes considered in the present experiments are cast inside a rectangular PDMS block, curvature effects are minimal. However, the refractive indices of water ( ${\sim}1.3$ ) and the surrounding solid PDMS ( ${\sim}1.4$ ) are slightly different, and hence the PIV measurements are somewhat prone to optical distortion. However, optical distortion is minimal at the centre of the tube, and hence we extract all our data from the centre line of the tube in a very similar way to the experiments of Sharp & Adrian (Reference Sharp and Adrian2004) for flow of Newtonian fluids. Optical distortion is significant as we move away from the centre of the tube, as manifested in the experimentally obtained velocity profiles. The mismatch between the theoretical laminar velocity profile and the experimental velocity profile becomes more prominent as we move away from the centre of the tube. For Newtonian fluids under laminar conditions, the error between the measured and theoretical velocities is approximately $2\,\%$ near the centre and approximately $8\,\%$ near the tube wall.

Figure 6.

Ratio of the time-averaged peak velocity to cross-section average velocity in Newtonian fluid through a rigid microtube. Transition is detected by deviation of the ratio of peak velocity to average velocity from the laminar value of 2. Tubes of two different diameters are used in our experiments. Transition occurs at approximately the same $Re$ for both the tubes, which is consistent with the established result that transition occurs at approximately $Re\sim 2000$ for a Newtonian fluid.

Two types of image processing methods are used. In the first method, multiple images are averaged over time to obtain the time-averaged velocity profile. The centre-line time-averaged velocity is divided by the (cross-sectional) average velocity to facilitate comparison with theoretically expected laminar flow values. The cross-section average velocity in the flow is calculated from the flow rate of the syringe pump, and this is further verified by collecting the liquid at the outlet of the flow. The ratio of the centre-line to cross-section average velocity should be ideally equal to 2 for laminar flow of a Newtonian fluid through a rigid tube. The value tends to reach 1.2 for fully turbulent flows and is between 1.2 and 2 in the transition regime. In the second method, images are processed separately at every time instant. The centre-line velocity at each time instant is calculated by taking the average of velocities at the centre line of the tube at different axial locations. The centre-line velocity thus obtained at each time instant is averaged over time. The r.m.s. velocity fluctuations are normalized by the time-averaged centre-line velocity.

We performed three sets of experiments for each flow rate used in our experiments. The data is presented with error bars which represent deviation in our experiments from the average over different experimental runs.

3.1 Onset of transition in the flow of polyacrylamide solutions

Figure 9 shows the variation of the ratio $V_{peak}/V_{avg}$ with $Re$ for flow in tubes of diameter $390~\unicode[STIX]{x03BC}\text{m}$ and $470~\unicode[STIX]{x03BC}\text{m}$ . At low polymer concentrations, as the concentration is increased in the range 50 ppm–100 ppm, it is observed that transition Reynolds number increases systematically from 2000 (for a Newtonian fluid) to approximately 2500 for the polymer solution. Furthermore, the nature of the variation of $V_{peak}/V_{avg}$ with $Re$ after transition is very different from that for a Newtonian fluid. There is no marked difference in the onset of the transition as we change the tube diameter (figure 9 b) except that the variation of $V_{peak}/V_{avg}$ with $Re$ seems to be gentler for $470~\unicode[STIX]{x03BC}\text{m}$ tubes. This could be attributed to the increased shear rates in tubes of smaller diameter at a given Reynolds number.

Figure 9.

Comparison of time-averaged centre-line velocity for the flow of 50 ppm and 100 ppm polyacrylamide solution in tubes of diameters (a) $390~\unicode[STIX]{x03BC}\text{m}$ and (b) $470~\unicode[STIX]{x03BC}\text{m}$ . The horizontal line is the ratio of $V_{peak}/V_{avg}$ in the laminar regime.

Figure 10.

Comparison of magnitude of centre-line velocity fluctuations for 50 ppm and 100 ppm polyacrylamide solutions for flow in tubes of diameters (a) $390~\unicode[STIX]{x03BC}\text{m}$ and (b) $470~\unicode[STIX]{x03BC}\text{m}$ . The horizontal line is an indication of the magnitude of velocity fluctuations in the laminar regime. The vertical dotted line is an indicator of the transition $Re$ .

The earlier study of Srinivas & Kumaran (Reference Srinivas and Kumaran2017) observed that for channel flows, upon addition of small amounts of polymer, the velocity fluctuations are decreased even in the laminar regime. They also noted a decrease in velocity fluctuations in the post-transition regime. Figure 10 shows the normalized r.m.s. velocity fluctuations of the velocity (at the centre line) in the flow direction as a function of $Re$ . Beyond a certain $Re$ , there is a sharp increase in the magnitude of the r.m.s. velocity fluctuations, and this $Re$ is taken as the point of transition from laminar to turbulent flow. This Reynolds number value agrees reasonably with the Reynolds number at which there is a deviation of $V_{peak}/V_{avg}$ from its expected laminar value, thus corroborating the two different ways of marking the onset of transition. There is also a prominent decrease in the magnitude of the velocity fluctuations (compared to the Newtonian case) in the post-transition regime for low-concentration polymer solutions. There is no distinct change in the post-transition flow behaviour, with regard to velocity fluctuations, when we change the diameter of the tube from $390~\unicode[STIX]{x03BC}\text{m}$ to $470~\unicode[STIX]{x03BC}\text{m}$ (figure 10 b). The velocity profile for a 100 ppm solution also becomes flatter in the post-transition regime (figure 11), but the profile is not as flat as in the case of a Newtonian fluid flow under a similar $Re$ range. The velocity profiles in the transition regime reveal a very small but consistent asymmetry in the radial direction, and similar observations were made for more concentrated solutions in the literature (Escudier et al. Reference Escudier, Poole, Presti, Dales, Nouar, Desaubry, Graham and Pullum2005; Wen et al. Reference Wen, Poole, Willis and Dennis2017).

Figure 11.

Velocity profile comparison for (a) laminar ( $Re=1500$ ) and (b) transition regime ( $Re=3000$ ) of a 100 ppm polyacrylamide solution through a rigid microtube of diameter $470~\unicode[STIX]{x03BC}\text{m}$ . Here, $r$ is the coordinate in the direction perpendicular to the flow and $r=0$ denotes the tube centre. The continuous line represents theoretical velocity profile for a Newtonian fluid in the laminar regime and symbols represent the experimentally obtained velocity profile.

Figure 12.

The variation of the velocity profile for a 100 ppm polyacrylamide solution through a $390~\unicode[STIX]{x03BC}\text{m}$ tube with increasing $Re$ . It is seen that the profile matches with the Newtonian velocity profile upto a certain $Re$ and then starts to become more flat. The continuous line represents theoretical velocity profile for a Newtonian fluid in the laminar regime and symbols represent the experimentally obtained velocity profile.

Figure 12 shows the evolution of velocity profiles with increasing $Re$ for the 100 ppm polyacrylamide solution. It is observed that the velocity profile matches with the theoretical velocity profile reasonably well up to $Re=2288$ . The velocity profile becomes flatter as we go to $Re=2542$ and higher. Theoretically, the area under these normalized velocity profiles should be $1/2$ . However, we find that the areas under the curves for different Reynolds number are slightly different from $1/2$ . This is a consequence of the use of $V_{avg}$ from volumetric flow-rate measurement in this study. If we had calculated $V_{avg}$ from the PIV data then the area under the normalized velocity profile would be very close to $1/2$ and the small errors encountered are due to fitting the data points. Thus, the discrepancy in the area in these curves is due to the difference in flow rates obtained from the PIV measurements and from direct volumetric flow-rate measurements. In the PIV measurements, it becomes increasingly difficult to measure velocity near the walls at high $Re$ . However, the difference in the area under the curve between the lowest $Re$ used and the highest transition $Re$ used is less than $3\,\%$ .

As the concentration of the added polymer is increased to 300 ppm (figure 13), the transition $Re$ decreases again, and is close to the Newtonian value of $2000$ . Beyond this critical concentration of polymer, the transition is initiated at a lower $Re$ as compared to the Newtonian case. However, the post-transition characteristics are completely different for polymeric solutions and Newtonian fluids. For example, the velocity fluctuations are much lower for a polymeric solution (figure 13) as compared to Newtonian fluids. For the same concentration of polymer (i.e. 300 ppm), the transition happens at a lower $Re$ (figure 13) for the tube of diameter $390~\unicode[STIX]{x03BC}\text{m}$ compared to $470~\unicode[STIX]{x03BC}\text{m}$ , presumably because the elasticity numbers in the two cases are different due to the different diameters of the two tubes. This is markedly different from the Newtonian transition (figure 7), where the transition $Re$ is independent of tube diameter. The elasticity number is higher for the $390~\unicode[STIX]{x03BC}\text{m}$ tube, and transition occurs at a lower $Re$ for this tube compared to the $470~\unicode[STIX]{x03BC}\text{m}$ tube, suggesting that the transition characteristics are very different for a polymeric solution as compared to a Newtonian fluid.

Figure 13.

Comparison of magnitude of centre-line velocity fluctuations in the flow of 300 ppm polyacrylamide solution with that in deionized water. The horizontal line is an indication of the magnitude of velocity fluctuations in the laminar regime. The vertical dotted line is an indicator of the transition $Re$ .

Figure 14.

Comparison of time-averaged centre-line velocity of polyacrylamide solution (of concentrations 800 ppm, 600 ppm, 500 ppm) for flow in tubes of diameter (a) $390~\unicode[STIX]{x03BC}\text{m}$ and (b) $470~\unicode[STIX]{x03BC}\text{m}$ . The horizontal line is an indication of the ratio of $V_{peak}/V_{avg}$ in the laminar regime.

The time-averaged centre-line velocity for the flow of relatively higher concentration (in the range 500 ppm–800 ppm) polyacrylamide solution (figure 14 a) shows that, for such higher concentrations of the added polymer, the transition occurs at a lower $Re$ compared to pure deionized water. It is further observed that there is a shift of the non-dimensional parameter $V_{peak}/V_{avg}$ from its expected laminar value of 2 to approximately 1.95, which can be attributed to shear-thinning behaviour of the polymeric solution at the high shear rates (the nominal shear rate at $Re=500$ is approximately $5\times 10^{4}~\text{s}^{-1}$ ) prevailing in the microtubes. Pinho & Whitelaw (Reference Pinho and Whitelaw1990) carried out experiments of highly shear-thinning sodium carboxymethyl cellulose solutions (with water as solvent), and they too observed that the ratio of centre-line to average velocity was lower than 2 for polymer solutions under laminar flow conditions. Changing the diameter of the tube enables us to operate at different shear rates in a comparable Reynolds number range. There appears to be a change in the variation of the ratio $V_{peak}/V_{avg}$ in the post-transition regime when we change the diameter of the tube (figure 14 b).

Figure 15.

Comparison of centre-line velocity fluctuations in polyacrylamide solutions (800 ppm, 600 ppm, 500 ppm) for tubes of diameter (a) $390~\unicode[STIX]{x03BC}\text{m}$ and (b) $470~\unicode[STIX]{x03BC}\text{m}$ . The horizontal line is an indication of the laminar velocity fluctuation values. The vertical dotted line is an indicator of the transition $Re$ .

The magnitude of the centre-line velocity fluctuations (figure 15 a) reaffirms the trend that the transition $Re$ decreases from the Newtonian value of 2000 in agreement with the results from the time-averaged velocity profile. However, we are able to see variation in the $Re$ value for the onset of transition with the tube diameter, for the same polymer solution (figure 15 b). This can be ascribed to the increased shear rate in the smaller tube, leading to an increase in the elasticity number. Further, the magnitude of the velocity fluctuations in the transition regime is lower as compared to a Newtonian fluid.

Figure 16.

Velocity profile comparison for laminar and transition regime in the flow of a 500 ppm polyacrylamide solution through a rigid microtube of diameter $470~\unicode[STIX]{x03BC}\text{m}$ : (a) flow at $Re=600$ and (b) at $Re=2400$ . Here, $r$ is the coordinate in the direction perpendicular to the flow and $r=0$ denotes the tube centre. The velocity profile is flatter after transition in flow of the polymeric solution. The continuous line represents the theoretical velocity profile for a Newtonian fluid in the laminar regime and symbols represent the experimentally obtained velocity profile.

In the case of concentrated polymer solution (500 ppm) there is a modification in the velocity profile and it starts to become flatter near the centre line with increasing $Re$ (figure 16), compared to the case of the 100 ppm polymer concentration (figure 11), suggesting that the 500 ppm case is more turbulent at $Re=2400$ compared to the 100 ppm case at $Re=3000$ . In order to establish that the observed instability in the flow of polymer solutions is not because of the lower shear modulus of PDMS (compared to rigid materials like steel or glass) used to fabricate the tubes, we carried out experiments in tubes fabricated with PDMS of two different shear moduli. It is important to ascertain this, because it is known that when the shear moduli of the tubes are in the range of ${\sim}10~\text{kPa}$ , the deformable nature of the tube wall itself can induce new instabilities (Verma & Kumaran Reference Verma and Kumaran2012). We compared $V_{peak}/V_{avg}$ and the centre-line velocity fluctuations in tubes fabricated in $10\,\%$ PDMS tube (shear modulus ${\sim}0.35~\text{MPa}$ ) and $15\,\%$ PDMS tube (shear modulus ${\sim}0.5~\text{MPa}$ ). The data (not shown, in the interests of brevity) for the onset of transition for these two different tubes agreed reasonably well. Hence, we conclude that the shear modulus of the tube prepared plays no role in the onset of transition observed in our experiments.

3.2 Onset of transition in the flow of polyethylene oxide solutions

To establish the robustness of the observation that transition happens at $Re$ much lower than 2000 due to the added polymer, in this section, we discuss the results obtained using a different polymer, viz., polyethylene oxide with a $\text{molecular weight}=8\times 10^{6}$ . At low concentration of the added polymer, addition of the polymer delays transition in PEO solutions similar to the case of polyacrylamide solutions. However, the post-transition flow seems to be rather similar to the Newtonian fluid flow. There is no marked reduction in the magnitude of the fluctuations with PEO solution as seen from velocity fluctuations plot (figure 17). For higher concentrations, PEO solutions also exhibit an early transition at $Re<2000$ , similar to PAA solutions (figure 18). Hence PEO solutions exhibit a different post-transition flow behaviour (as compared to polyacrylamide), but the onset of transition is strikingly similar to polyacrylamide for solutions of similar viscosities. There is a shift in the value of $V_{peak}/V_{avg}$ from $2$ to approximately 1.95 in the average velocity data. This may perhaps be attributed to shear-thinning behaviour at high shear rates.

Figure 17.

Comparison of centre-line velocity fluctuations in 50 ppm and 100 ppm polyethylene oxide solutions. The horizontal line is an indication of magnitude of velocity fluctuations in the laminar regime. The vertical dotted line is an indicator of the transition $Re$ .

Figure 18.

Comparison of the velocity fluctuations at the centre line in high-concentration (300 ppm and 200 ppm) polyethylene oxide solutions. The horizontal line is an indication of the magnitude of velocity fluctuations in the laminar regime. The vertical dotted line is an indicator of the transition $Re$ .

Figure 19.

Velocity profile comparison for laminar and transition regime of a 300 ppm PEO polymeric solution through a $390~\unicode[STIX]{x03BC}\text{m}$ rigid microtube: (a) flow at $Re=300$ and (b) $Re=1720$ . Here, $r$ is the coordinate in the direction perpendicular to the flow and $r=0$ denotes the tube centre. The velocity profile is flatter for post-transition flow of the polyethylene oxide solution. The continuous line represents the theoretical velocity profile for a Newtonian fluid in the laminar regime and symbols represent the experimentally obtained velocity profile.

Figure 19 shows that the velocity profile for PEO solution deviates from the Newtonian flow profile even at low $Re$ . This can attributed to the shear-thinning behaviour at such high shear rates at which the experiments are being performed. Studies performed on PEO solutions with viscosities close to polyacrylamide solutions show similar transition characteristics. Velocity fluctuations in the flow of PEO solutions also show early transition as compared to deionized water. These observations show that the phenomenon of early transition is quite robust, as the observations are similar for two different polymer solutions.

Table 6.

A comparison of different dimensional and non-dimensional parameters used for characterizing polyacrylamide solutions in the experiments. Here, $Wi_{t}$ is the Weissenberg number required for onset of transition in our experiments.

Figure 20.

Reynolds number for transition for the two polymer solutions, i.e polyethylene oxide (MW $8\times 10^{6}$ ), and polyacrylamide (MW $5\times 10^{6}$ ), for different concentrations.

Table 7.

A comparison of different dimensional and non-dimensional parameters used for characterizing polyethylene oxide solutions in the experiments. Here, $Wi_{t}$ is the Weissenberg number required for onset of transition in our experiments.

3.3 Data collapse of experimental observations

It is instructive to examine whether the experimental observations obtained using two different polymer solutions and tubes of different diameters can be reduced into a single plot using suitable dimensionless groups.

Tables 6 and 7 summarize the various dimensional and non-dimensional parameters for polyacrylamide and polyethylene oxide solutions used in our experiments. In addition, we also mention the Weissenberg number required for transition in our experiments. Figure 20 shows the variation of the transition Reynolds number with the concentration of the added polymer. There seems to be a critical concentration up to which the addition of polymer delays the transition, and beyond which addition of polymer tends to initiate transition at $Re$ lower than $2000$ , the transition $Re$ for a Newtonian fluid. From the standpoint of dimensional analysis, the transition $Re$ for the present system is a function of the elasticity number $E$ and the ratio of solvent to total viscosity $\unicode[STIX]{x1D6FD}$ . Here, it is assumed that the polymer solution is characterized only by its longest relaxation time. It is useful to plot the transition $Re$ in terms of the non-dimensional parameters $E$ and $\unicode[STIX]{x1D6FD}$ . The limits of $E\rightarrow 0$ and $\unicode[STIX]{x1D6FD}\rightarrow 1$ correspond to those of a Newtonian fluid. Instead of plotting $Re$ as a function of the two dimensionless variables $E$ and $(1-\unicode[STIX]{x1D6FD})$ separately, we propose to combine the two as $E(1-\unicode[STIX]{x1D6FD})$ , and $Re_{t}$ is plotted as a function of this product in figure 21(a). This combination can also be understood by the following physical argument (Poole Reference Poole2012). The usual Weissenberg number can be thought of as a ratio of elastic to viscous forces in a viscoelastic fluid. The elastic forces are proportional to the polymer viscosity while the viscous forces are proportional to the total viscosity. Thus, for polymer solutions, the relevant parameter is $(1-\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D706}V/R$ . Thus, the dual limit of both $E\ll 1$ and $\unicode[STIX]{x1D6FD}\rightarrow 1$ represents the approach to a Newtonian fluid, and deviations from the Newtonian limit can be represented by the product $E(1-\unicode[STIX]{x1D6FD})$ . Here, the (longest) relaxation time of both the polymer solutions at different concentrations are taken from our small-amplitude oscillatory experiments. Plotted in this way, the transition $Re$ shows a reasonable data collapse with $E(1-\unicode[STIX]{x1D6FD})$ for tubes with different diameters, as well as for two different polymers at different concentrations of added polymer. The limit $E(1-\unicode[STIX]{x1D6FD})=0$ is the case of a Newtonian fluid, where the transition happens at $Re\sim 2000$ . Upon increase of $E(1-\unicode[STIX]{x1D6FD})$ , there is an initial increase in transition $Re$ , and after a further increase in $E(1-\unicode[STIX]{x1D6FD})$ , there is a decrease of transition $Re$ to as low as $800$ for the highest values of $E(1-\unicode[STIX]{x1D6FD})$ probed in our experiments. This plot thus shows that the ‘early’ transition observed for the two different polymer solutions can indeed be collapsed on a single curve, when plotted in terms of the suitable dimensionless combination $E(1-\unicode[STIX]{x1D6FD})$ . Based on this plot, it can be concluded that, for very small values of $E(1-\unicode[STIX]{x1D6FD})$ , the addition of very small amounts of polymer interferes with the finite-amplitude (subcritical) instability of a Newtonian fluid, but has a stabilizing effect on that transition. For sufficiently large values of $E(1-\unicode[STIX]{x1D6FD})$ , the role of added polymer is destabilizing in the regime of ‘early transition’ or elasto-inertial turbulence. When plotted in a $\log$ – $\log$ plot (figure 21 b), the data for $Re_{t}$ seem to scale as $1/\sqrt{E(1-\unicode[STIX]{x1D6FD})}$ in the regime where the polymer is destabilizing. It remains an open question whether this transition is due to an instability of the flow to infinitesimal disturbances, unlike in the case of a Newtonian fluid, where infinitesimal disturbances are always stable in the linear stability analysis. However, since the transition Reynolds number decreases with $E(1-\unicode[STIX]{x1D6FD})$ , it can be unambiguously inferred that fluid elasticity has a destabilizing effect on the transition.

Figure 21.

Comparison of transition Reynolds number ( $Re_{t}$ ) with $E(1-\unicode[STIX]{x1D6FD})$ in a (a) linear plot and (b) log–log plot. Here $\unicode[STIX]{x1D6FD}$ is the ratio of solvent viscosity to the solution viscosity.

3.4 Dye-stream visualization

When injected in a laminar flow a dye stream remains straight but breaks up as soon as transition sets in the flow. The phenomenon is well documented for Newtonian flows (Verma & Kumaran Reference Verma and Kumaran2012; Neelamegam & Shankar Reference Neelamegam and Shankar2015) but surprisingly is not frequently used to visualize transition in the flow of polymer solutions. We first carried out dye-stream experiments for deionized water through a $470~\unicode[STIX]{x03BC}\text{m}$ microtube. A hollow (GI) tube is bent and placed at the centre of the tube just before the tube inlet. A needle of inner diameter ${\sim}100~\unicode[STIX]{x03BC}\text{m}$ is joined at the outlet of the GI tube using an adhesive (Araldite). The dye-stream experiments are performed by injecting a black-coloured water-soluble (Parker) dye stream of approximately $100~\unicode[STIX]{x03BC}\text{m}$ in diameter at the centre of the $470~\unicode[STIX]{x03BC}\text{m}$ microtube inlet. The dye is injected using a separate syringe pump (Fusion 200, Chemyx, Inc) and the flow rate of the dye stream is adjusted according to the $Re$ of the flow. Images are acquired at 1000 frames per second using Redlake MotionPro software by connecting a high-speed camera (Integrated Design Tools, SN:13-0907-0353, 1 megapixel) to a microscope (CKX41, Olympus, Japan). A 4 $\times$ objective (PlanCN 4X) is used to magnify the view of the $470~\unicode[STIX]{x03BC}\text{m}$ microtube. A halogen bulb of high intensity is used to illuminate the $470~\unicode[STIX]{x03BC}\text{m}$ PDMS microtube. The phenomenon was first validated for a Newtonian fluid and then we performed a similar analysis for the flow of polymer solutions. We observe (figure 22) a breakup of the dye stream for a Newtonian fluid when we increase the $Re$ above 2000, which is in close agreement with our micro-PIV results for the Newtonian fluid. We carried out a similar experiment for a 800 ppm polyacrylamide solution through $470~\unicode[STIX]{x03BC}\text{m}$ tube and observed a breakup of the dye stream at $Re\sim 1040$ (figure 22). The velocity fluctuations as measured from the micro-PIV analysis showed a jump for the 800 ppm polyacrylamide solution through a $470~\unicode[STIX]{x03BC}\text{m}$ microtube at $Re\sim 926$ . Hence the results from our dye-stream experiments are consistent with the observations of our micro-PIV experiments. Since polymer solutions are prone to ageing, we performed all the dye-stream visualization experiments within 48 h of the preparation of the polyacrylamide solution. We were not able to perform dye-stream visualization experiments for tubes of diameter less than $470~\unicode[STIX]{x03BC}\text{m}$ .

Figure 23 shows the images obtained for the flow of 100 ppm PEO solutions. As shown in figure 17, the onset of transition happens at $Re\sim 2750$ . In figure 23, the dye-stream image for $Re=2600$ shows that the flow is laminar, while the image for $Re=2800$ shows that the dye stream has broken down, indicating the onset of transition. Thus, the dye-stream experiments corroborate both early transition for higher-concentration polymer solutions and the transition delay in polymer solutions of relatively lower concentration.

Figure 22.

Snapshots of visualization using dye-stream injection in a $470~\unicode[STIX]{x03BC}\text{m}$ tube. (a) Movement of the dye stream for water in the laminar regime for $Re\sim 1750$ . (b) Motion of dye in water at $Re\sim 2250$ . (c) Motion of dye in a 800 ppm polyacrylamide solution at $Re\sim 810$ . (d) Motion of dye in a 800 ppm polyacrylamide solution at $Re\sim 1040$ . These snapshots show that for polymer solutions the transition happens at $Re$ much lower than $2000$ .

Figure 23.

Snapshots of visualization using dye-stream injection in a $470~\unicode[STIX]{x03BC}\text{m}$ tube. (a) The image for 100 ppm PEO solution at $Re\sim 2600$ and (b) the image for 100 ppm PEO solution at $Re\sim 2800$ . These snapshots demonstrate that laminar flow is maintained at $Re\sim 2600$ and the transition happens only for $Re>2600$ .