# Elastic Turbulence of Aqueous Polymer Solution in Multi-Stream Micro-Channel Flow

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Micro-Channel

#### 2.2. Test Liquids

^{6}g/mol (POLYOX, The Dow Chemical Company, Midland, MI, USA) into an aqueous 55 wt% glycerol solvent. Similar to our previous publications [18,19,22], the center-stream liquid consists of 1 wt% PEO and is thus relatively highly elastic, while the side-stream liquid consists of 0.1 wt% PEO and has a much lower elasticity as compared to the center-stream liquid due to the reduced polymer concentration. As the size of the elastic polymer molecules were too minute for proper flow visualization and characterization, seeding particles (3 µm, Thermo Scientific, Waltham, MA, USA, 0.1 wt%) were added into both test liquids to facilitate PIV tracking for proper flow visualization. Two precision pumps (KD Scientific, Holliston, MA, USA) were used to infuse the test liquids, one for the center-stream liquid (KDS 410) and one for the two side-stream liquids (Legato

^{®}210) respectively. Table 1 shows the composition of the test liquids. The shear viscosity (η) and extensional relaxation times (λ

_{E}) of the liquids were measured using the Gemini Hr Nano Rheometer and the Capillary Breakup Extensional Rheometer (CaBER) respectively and are shown in Figure 2 and Figure 3 respectively. λ

_{E}was measured to be ≈81 ms for the center-stream liquid, and ≈26 ms for the side-stream liquid. In general, the relaxation time of the individual polymer molecules is shorter than λ

_{E}, since CaBER measures the longest relaxation time in the liquid.

#### 2.3. Optical Setup

_{c}+ d), where ρ is the liquid density, Q is the flow rate, η is the solution shear viscosity in the contraction, w

_{c}is the average contraction width occupied by each liquid stream, and d is the micro-channel depth respectively. Since the investigation was focused on the extensional effects in the contraction-expansion flow, the Deborah number is defined as De = 2λ

_{E}Q/w

_{c}d

^{2}, where λ

_{E}is the extensional relaxation time of each liquid. Table 2 shows the two flow rates investigated, the time interval (Δt) used in the image acquisition, and the corresponding average velocity ($\overline{U}$), shear rate ($\dot{\gamma}$), viscosity (η), Re, De, and El of each liquid stream in the contraction respectively. The computational methods used to obtain $\overline{U}$, $\dot{\gamma}$, and η are explained in the Supplementary Materials (Section S1). From Table 2, it can be observed that Re is slightly higher in the lower viscosity side-stream liquids, as compared to the center-stream liquids. Nevertheless, inertial forces were still considerably low (maximum Re ~ 1) in all experimental runs.

## 3. Results and Discussion

#### 3.1. Mean Flow Statistics

#### 3.2. Normalized Reynolds Stresses

#### 3.3. Single-Point Flow Statistics

#### 3.4. Velocity Power Spectra

_{1}(see Figure 13a,b). This drop-off frequency (F

_{1}) is much higher than that reported by Groisman and Steinberg [4] (≈0.1 Hz) and Bonn et al. [21] (<1 Hz) and is on the order of the lowest frequency at which the center-stream fluid relaxes (i.e., 1/λ

_{E}≈ 12 Hz). Subsequently, the spectrum follows a power law decay with a slope of ≈−5/3. This is usually observed in the inertia sub-range of high-Re turbulence, whereby energy is transferred from large to small eddies. However, throughout this experiment, inertia forces were relatively low (maximum Re ~ 1). A second drop-off frequency (F

_{2}) can be observed clearly (i.e., at ≈100–200 Hz), whereby the spectrum decays with a second, steeper power law slope of ≈−3. The second slope appears more prominent closer to the contraction exit (i.e., δ = 0.3 mm, see Figure 13a), and is slightly attenuated further downstream (i.e., δ = 1.0 mm, see Figure 13b). It is interesting to observe a double cascade here. This had also been reported earlier for elastic turbulence in a Couette-Taylor (CT) flow system [7], where the inflection frequencies were found to coincide closely to the driving frequencies of the CT flow i.e., rotation rates. In contrast, the inflection frequencies observed in our findings are nowhere near the driving frequencies of our infusion system. Instead, the presence of a double cascade could indicate the occurrence of 2D turbulence, where F

_{2}corresponds to the energy injection point into the turbulence [37,38,39]. Note that here we have invoked the Taylor’s ‘frozen eddy’ hypothesis, which was assumed to be valid (refer to Supplementary Materials). The significance of F

_{2}here is yet to be fully understood; however, it is clear that the turbulence observed is driven by elasticity (i.e., molecular relaxation) at frequencies > 12 Hz (i.e., 1/λ

_{E}), since λ

_{E}is a measure of the longest molecular extensional relaxation time in the center-stream liquid.

#### 3.5. Flow Field Visualization

## 4. Conclusions

_{1}and F

_{2}). This characteristic is similar to that of 2D turbulence, where F

_{2}is the energy injection point into the turbulence. The turbulence observed here could be predominantly 2D in nature, due to the 3-stream flow configuration presented which promotes the manifestation of the center-stream liquid in two dimensions. This is also supported by evidence from the Galilean decomposition technique, where an up-down ‘sweeping’ flow was observed. Since elastic turbulence is driven by molecular stretching and relaxation, it would be interesting to investigate in the future, on the possibility of any correlation between F

_{2}(i.e., possibly the energy injection point into the turbulence) and λ

_{E}(extensional relaxation time). Furthermore, the contraction length has been reported to affect the upstream flow dynamics in a contraction-expansion flow [20]. This could be further explored, and a correlation study between the downstream and upstream flow dynamics could also be examined.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Tai, J.; Lim, C.P.; Lam, Y.C. Visualization of polymer relaxation in viscoelastic turbulent micro-channel flow. Sci. Rep.
**2016**, 5, 16633. [Google Scholar] [CrossRef] [PubMed] - Vinogradov, G.V.; Ivanova, L.I. Wall slippage and elastic turbulence of polymers in the rubbery state. Rheol. Acta
**1968**, 7, 243–254. [Google Scholar] [CrossRef] - Giesekus, H. On instabilities in Poiseuille and Couette flows of viscoelastic fluids. Prog. Heat Mass Transf.
**1972**, 5, 187–193. [Google Scholar] - Groisman, A.; Steinberg, V. Elastic turbulence in a polymer solution flow. Nature
**2000**, 405, 53–55. [Google Scholar] [CrossRef] [PubMed] - Groisman, A.; Steinberg, V. Stretching of polymers in a random three-dimensional flow. Phys. Rev. Lett.
**2001**, 86, 934. [Google Scholar] [CrossRef] [PubMed] - Liu, Y.; Steinberg, V. Molecular sensor of elastic stress in a random flow. EPL
**2010**, 90, 44002. [Google Scholar] [CrossRef] - Groisman, A.; Steinberg, V. Elastic turbulence in curvilinear flows of polymer solutions. New J. Phys.
**2004**, 6, 29. [Google Scholar] [CrossRef] - Latrache, N.; Crumeyrolle, O.; Abcha, N.; Mutabazi, I. Destabilization of inertio-elastic mode via spatiotemporal intermittency in a Couette-Taylor viscoelastic flow. JPCS
**2008**, 137, 012022. [Google Scholar] [CrossRef] - Dutcher, C.S.; Muller, S.J. Effects of moderate elasticity on the stability of co-and counter-rotating Taylor–Couette flows. J. Rheol.
**2013**, 57, 791–812. [Google Scholar] [CrossRef] - Arratia, P.E.; Thomas, C.C.; Diorio, J.; Gollub, J.P. Elastic instabilities of polymer solutions in cross-channel flow. Phys. Rev. Lett.
**2006**, 96, 144502. [Google Scholar] [CrossRef] - Haward, S.J.; Ober, T.J.; Oliveira, M.S.; Alves, M.A.; McKinley, G.H. Extensional rheology and elastic instabilities of a wormlike micellar solution in a microfluidic cross-slot device. Soft Matter
**2012**, 8, 536–555. [Google Scholar] [CrossRef] - Groisman, A.; Steinberg, V. Efficient mixing at low Reynolds numbers using polymer additives. Nature
**2001**, 410, 905–908. [Google Scholar] [CrossRef] [PubMed] - Burghelea, T.; Segre, E.; Bar-Joseph, I.; Groisman, A.; Steinberg, V. Chaotic flow and efficient mixing in a microchannel with a polymer solution. Phys. Rev. E
**2004**, 69, 066305. [Google Scholar] [CrossRef] [PubMed] - Jun, Y.; Steinberg, V. Elastic turbulence in a curvilinear channel flow. Phys. Rev. E
**2011**, 84, 056325. [Google Scholar] [CrossRef] [PubMed] - Tatsumi, K.; Takeda, Y.; Suga, K.; Nakabe, K. Turbulence characteristics and mixing performances of viscoelastic fluid flow in a serpentine microchannel. JPCS
**2011**, 318, 092020. [Google Scholar] [CrossRef] - Rodd, L.E.; Scott, T.P.; Boger, D.V.; Cooper-White, J.J.; McKinley, G.H. The inertio-elastic planar entry flow of low-viscosity elastic fluids in micro-fabricated geometries. J. Nonnewton. Fluid Mech.
**2005**, 129, 1–22. [Google Scholar] [CrossRef] - Rodd, L.E.; Cooper-White, J.J.; Boger, D.V.; McKinley, G.H. Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries. J. Nonnewton. Fluid Mech.
**2007**, 143, 170–191. [Google Scholar] [CrossRef] - Gan, H.Y.; Lam, Y.C.; Nguyen, N.T.; Tam, K.C.; Yang, C. Efficient mixing of viscoelastic fluids in a microchannel at low Reynolds number. Microfluid. Nanofluid.
**2007**, 3, 101–108. [Google Scholar] [CrossRef] - Lam, Y.C.; Gan, H.Y.; Nguyen, N.T.; Lie, H. Micromixer based on viscoelastic flow instability at low Reynolds number. Biomicrofluidics
**2009**, 3, 014106. [Google Scholar] [CrossRef] - Rodd, L.E.; Lee, D.; Ahn, K.H.; Cooper-White, J.J. The importance of downstream events in microfluidic viscoelastic entry flows: Consequences of increasing the constriction length. J. Nonnewton. Fluid Mech.
**2010**, 165, 1189–1203. [Google Scholar] [CrossRef] - Bonn, D.; Ingremeau, F.; Amarouchene, Y.; Kellay, H. Large velocity fluctuations in small-Reynolds-number pipe flow of polymer solutions. Phys. Rev. E
**2011**, 84, 045301. [Google Scholar] [CrossRef] [PubMed] - Gan, H.Y.; Lam, Y.C. Experimental observations of flow instabilities and rapid mixing of two dissimilar viscoelastic liquids. AIP Adv.
**2012**, 2, 042146. [Google Scholar] [CrossRef] - Li, F.C.; Kinoshita, H.; Li, X.B.; Oishi, M.; Fujii, T.; Oshima, M. Creation of very-low-Reynolds-number chaotic fluid motions in microchannels using viscoelastic surfactant solution. Exp. Therm. Fluid Sci.
**2010**, 34, 20–27. [Google Scholar] [CrossRef] - Grilli, M.; Vázquez-Quesada, A.; Ellero, M. Transition to turbulence and mixing in a viscoelastic fluid flowing inside a channel with a periodic array of cylindrical obstacles. Phys. Rev. Lett.
**2013**, 110, 174501. [Google Scholar] [CrossRef] [PubMed] - White, J.L.; Kondo, A. Flow patterns in polyethylene and polystyrene melts during extrusion through a die entry region: Measurement and interpretation. J. Nonnewton. Fluid Mech.
**1977**, 3, 41–64. [Google Scholar] [CrossRef] - Nguyen, H.; Boger, D.V. The kinematics and stability of die entry flows. J. Nonnewton. Fluid Mech.
**1979**, 5, 353–368. [Google Scholar] [CrossRef] - Lawler, J.V.; Muller, S.J.; Brown, R.A.; Armstrong, R.C. Laser Doppler velocimetry measurements of velocity fields and transitions in viscoelastic fluids. J. Nonnewton. Fluid Mech.
**1986**, 20, 51–92. [Google Scholar] [CrossRef] - Yesilata, B.; Öztekin, A.; Neti, S. Instabilities in viscoelastic flow through an axisymmetric sudden contraction. J. Nonnewton. Fluid Mech.
**1999**, 85, 35–62. [Google Scholar] [CrossRef] - Zhang, H.N.; Li, F.C.; Cao, Y.; Kunugi, T.; Yu, B. Direct numerical simulation of elastic turbulence and its mixing-enhancement effect in a straight channel flow. Chin. Phys. B
**2013**, 22, 024703. [Google Scholar] [CrossRef] - Whalley, R.D.; Abed, W.M.; Dennis, D.J.C.; Poole, R.J. Enhancing heat transfer at the micro-scale using elastic turbulence. TAML
**2015**, 5, 103–106. [Google Scholar] [CrossRef] - Abed, W.M.; Whalley, R.D.; Dennis, D.J.; Poole, R.J. Experimental investigation of the impact of elastic turbulence on heat transfer in a serpentine channel. J. Nonnewton. Fluid Mech.
**2016**, 231, 68–78. [Google Scholar] [CrossRef] - Ligrani, P.; Copeland, D.; Ren, C.; Su, M.; Suzuki, M. Heat transfer enhancements from elastic turbulence using sucrose-based polymer solutions. J. Thermophys. Heat Transf.
**2018**, 32, 51–60. [Google Scholar] [CrossRef] - Taylor, Z.J.; Gurka, R.; Kopp, G.; Liberzon, A. Long-duration time-resolved PIV to study unsteady aerodynamics. IEEE Trans. Instrum. Meas.
**2010**, 59, 3262–3269. [Google Scholar] [CrossRef] - Barnes, H.A.; Hutton, J.F.; Walters, K. An Introduction to Rheology; Elsevier: Amsterdam, The Netherlands, 1989; Volume 3, p. 62. [Google Scholar]
- Tamano, S.; Itoh, M.; Inoue, T.; Kato, K.; Yokota, K. Turbulence statistics and structures of drag-reducing turbulent boundary layer in homogeneous aqueous surfactant solutions. Phys. Fluids
**2009**, 21, 045101. [Google Scholar] [CrossRef] - Japper-Jaafar, A.; Escudier, M.P.; Poole, R.J. Laminar, transitional and turbulent annular flow of drag-reducing polymer solutions. J. Nonnewton. Fluid Mech.
**2010**, 165, 1357–1372. [Google Scholar] [CrossRef] - Kraichnan, R.H. Inertial ranges in two-dimensional turbulence. Phys. Fluids
**1967**, 10, 1417. [Google Scholar] [CrossRef] - Leith, C.E. Diffusion approximation for two-dimensional turbulence. Phys. Fluids
**1968**, 11, 671–672. [Google Scholar] [CrossRef] - Batchelor, G.K. Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids
**1969**, 12, II–233. [Google Scholar] [CrossRef] - Raffel, M.; Willert, C.; Wereley, S.; Kompenhans, J. Particle Image Velocimetry—A Practical Guide, 2nd ed.; Springer: Berlin/Heidelberg, Germany, 2007; p. 193. [Google Scholar]
- Adrian, R.J.; Christensen, K.T.; Liu, Z.C. Analysis and interpretation of instantaneous turbulent velocity fields. Exp. Fluids
**2000**, 29, 275–290. [Google Scholar] [CrossRef]

**Figure 1.**Schematic 2-D diagram of the micro-channel. Arrows depict the flow of the center-stream (blue) and side-stream (red) liquids respectively.

**Figure 3.**Plot of filament diameter over time as measured by CaBER for the test liquids. The fitted black line was used to obtain the extensional relaxation time for each liquid.

**Figure 6.**Schematic diagram of the contraction-expansion micro-channel, showing the plot of a mean axial velocity profile at a distance δ downstream of the contraction. Data in the shaded region has been removed due to high noise level.

**Figure 7.**Mean flow statistics at δ = 0.3 mm. (

**a**) Normalized axial velocity and (

**b**) Turbulent intensities.

**Figure 8.**Mean flow statistics at δ = 1.0 mm. (

**a**) Normalized axial velocity and (

**b**) Turbulent intensities.

**Figure 11.**Normalized low frequency axial velocity measurements at (

**a**) interrogation window A (δ = 0.3 mm) and (

**b**) interrogation window B (δ = 1.0 mm) obtained using recording mode 1.

**Figure 12.**Normalized high frequency axial velocity measurements at (

**a**) interrogation window A (δ = 0.3 mm) and (

**b**) interrogation window B (δ = 1.0 mm) obtained using recording mode 2.

**Figure 13.**Power spectrum of the axial velocity fluctuations for the two flow rates investigated at (

**a**) interrogation window A (δ = 0.3 mm) and (

**b**) interrogation window B (δ = 1.0 mm).

**Figure 14.**An instantaneous plot of velocity and vorticity (color plot) downstream of the contraction, from δ = 0.1–1.4 mm. Insets a–c show structures (drawn to scale) at various time instances located in different regions, obtained using the Galilean decomposition method with a fraction of the convective velocity removed. (

**a**) U − ${\overline{U}}_{bulk}$, (

**b**,

**c**) U − 0.6${\overline{U}}_{bulk}$ (at two different time instants).

Test Liquid | PEO (g) | Glycerol (g) | Water (g) | Microsphere (g) |
---|---|---|---|---|

Center-stream | 1.00 | 55 | 44 | 0.03 |

Side-stream | 0.10 | 55 | 44 | 0.03 |

Properties | Extensional Relaxation Time (ms) | Flow Rate (mL/h) | Δt (s) | Mean Velocity ($\overline{\mathit{U}}$, m/s) | Shear Rate ($\dot{\mathit{\gamma}}$, s ^{−1}) | Viscosity ($\mathit{\eta}$, Pa·s) | Re | De | |
---|---|---|---|---|---|---|---|---|---|

Fluid | |||||||||

Center-stream Liquid | 81 ± 4 | 2 | 0.004 | 0.03 | 364 | 0.126 | 0.022 | 27 | |

20 | 0.00025 | 0.31 | 4262 | 0.052 | 0.528 | 278 | |||

Side-stream Liquid | 26 ± 1 | 2 | 0.004 | 0.02 | 1107 | 0.01 | 0.145 | 6 | |

20 | 0.00025 | 0.21 | 10514 | 0.01 | 1.45 | 59 |

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**MDPI and ACS Style**

Tai, J.; Lam, Y.C.
Elastic Turbulence of Aqueous Polymer Solution in Multi-Stream Micro-Channel Flow. *Micromachines* **2019**, *10*, 110.
https://doi.org/10.3390/mi10020110

**AMA Style**

Tai J, Lam YC.
Elastic Turbulence of Aqueous Polymer Solution in Multi-Stream Micro-Channel Flow. *Micromachines*. 2019; 10(2):110.
https://doi.org/10.3390/mi10020110

**Chicago/Turabian Style**

Tai, Jiayan, and Yee Cheong Lam.
2019. "Elastic Turbulence of Aqueous Polymer Solution in Multi-Stream Micro-Channel Flow" *Micromachines* 10, no. 2: 110.
https://doi.org/10.3390/mi10020110