# Comparison of Micro-Mixing in Time Pulsed Newtonian Fluid and Viscoelastic Fluid

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Micro-Mixer Design and Fabrication

_{A}and I

_{B}of the T-junction. The fluid on one side contains a small amount of fluorescent dye for experimental view. The two fluids are confluent at the center of the inlet channel and flow into a narrow channel before finally entering into an expanded mixing channel. The fluorescent color in the mixing channel is monitored to determine the mixing efficiency at the flow distance, S

_{0}, from the inlet channel. The length and width of the inlet channels are L

_{1}= 2 mm and W

_{1}= 100 µm, respectively. The confluence channel has dimensions of L

_{2}= 600 µm and W

_{2}= 50 µm, and the mixing channel has dimensions of L

_{3}= 2 mm and W

_{3}= 600 µm. The mixer structure has a depth of h = 100 µm, and the outlet is set at the end of the mixing channel.

#### 2.2. Materials and Underlying Physics

#### 2.3. Experimental Setup

_{i}at every pixel i. The grey value depends on the dye concentration and can be used to derive the mixing efficiency. The flow rate $Q$ is modulated by a time pulsing signal with a square wave function at I

_{A}while at I

_{B}, it is a constant. The flow rates are expressed as:

_{A}oscillates between 0 and 2${Q}_{0}$ with a period of 1/f and a duty cycle of 50%. The time pulsing signal will disturb the flow and further affect the mixing between fluid A and fluid B.

#### 2.4. Mixing Degree Characterization

_{0}from the inlet channel at a specific recorded time t

_{j}is expressed as:

_{i}is the recorded grey value at the pixel i on the cross line of the channel with the total pixel number of P. C

_{0}is the mean value of the recorded grey values within the mixing channel. It can be seen that $M{D}_{t}=1$ for the full mixing case while $M{D}_{t}=0$ for the no mixing case. $M{D}_{t}$ presents the instantaneous mixing degree. To derive the average mixing degree $\overline{MD}$, we consider the fluid flowing through the cross line S

_{0}in an adequately long recorded time range t

_{0}as a collection of the fluid flowing through S

_{0}at every recorded time t

_{i}. Therefore, the averaged mixing degree can be expressed as:

## 3. Results and Discussion

#### 3.1. Mixing in the Newtonian Fluid

_{0}from the inlet channel. Here, we derive the $\overline{MD}$ starting at distance S

_{1}= 1000 µm and ending at a distance S

_{2}= 2000 µm from the inlet channel, as shown in Figure 2b. As ${Q}_{0}$ increases from 10 µL/h (black line) to 50 µL/h (red line) and to 100 µL/h (green line), $\overline{MD}$ is greatly decreased in the observed area. The significant mixing decrease in the low flow rate regime stems from the diffusion time decrease when the flow rate increases. However, as the flow rate changes from 200 µL/h to 1000 µL/h, the decrease in $\overline{MD}$ is quite small. This indicates that the impact of the intrinsic diffusion on the mixing becomes weak in the high flow rate regime.

_{1}to S

_{2}is plotted in Figure 3a–f for different flow rates ${Q}_{0}$. When the fluid is at the low flow rate of 10 µL/h, $\overline{MD}$ is highest at f = 0 and gradually decreases as f increases to 1 Hz. A similar trend is also observed for the case when ${Q}_{0}$ = 50 µL/h. As discussed previously, the mixing largely relies on the diffusion mechanism in the low flow rate regime; we conclude that the diffusion in the microfluidic channel is affected when the disturbing frequency increases at this low flow rate.

_{2}= 2000 µm for 20 s, as shown in Figure 4a–c. For flow rate ${Q}_{0}$ = 50 µL/h, the $M{D}_{t}$ does not fluctuate for all disturbing frequencies because $\Delta Q$ is small at the low flow rate condition. $M{D}_{t}$ is highest when f = 0, which is due to the high intrinsic diffusion. At flow rate ${Q}_{0}$ = 200 µL/h, the $M{D}_{t}$ starts to oscillate temporally and the oscillation frequency is identical to the pulsing frequency. Meanwhile, the $M{D}_{t}$ for time pulsing is larger than that when f = 0. This confirms the contribution of diffusion disturbance to the mixing. At the high flow rate of 500 µL/h, the $M{D}_{t}$ oscillates significantly at f = 0.1 Hz and 0.2 Hz, which is much higher than that at f = 0.

_{2}, more red fluid oscillates across the center and mixes well with the blue fluid at the peak value A; for peak value C, more blue fluid oscillates across the center and mixes well with the red fluid. For the dip value B, the blue fluid occupies most of the channel and the mixing is low. For the dip value D, both the two fluids occupy half of the channel and little mixing occurs at the center. Therefore, we conclude the enhancement of the mixing for the Newtonian fluid can be induced by the time pulsing flow with a relatively low oscillation frequency.

#### 3.2. Mixing in Viscoelastic Fluid

_{2}= 2000 µm.

_{2}= 2000 µm is 0.82. As ${Q}_{0}$ increases to 50 µL/h, 100 µL/h and 200 µL/h, the mixing has a similar mechanism to that when ${Q}_{0}$ = 10 µL/h, except that the diffusion time is decreased, thus $\overline{MD}$ decreases gradually. When ${Q}_{0}$ reaches 500 µL/h, the oscillation amplitude is sufficiently high so that the two fluids alternatively enter into the confluence channel; therefore, the mixing in the confluence channel is limited compared to that at a lower flow rate. This explains the large mixing decrease from 0.60 to 0.46 when ${Q}_{0}$ increases from 200 µL/h to 500 µL/h, respectively. As ${Q}_{0}$ increases from 1000 µL/h, the oscillation is so large that the fluids can reach the channel wall on the other side. As explained previously, the fluid will rebound at the wall, which increases the mixing degree.

_{1}= 600 µm to flow distance S

_{2}= 2000 µm in the channel is measured for different ${Q}_{0}$, and each sub-figure compares the $\overline{MD}$ modulated at frequency f = 0, 0.1 Hz, 0.2 Hz, and 1 Hz. At a fixed ${Q}_{0}$, lower than 200 µL/h, the $\overline{MD}$ is almost the same for different pulsing frequencies. This means the disturbance from the pulsing does not affect the mixing of the viscoelastic fluid. With ${Q}_{0}$ = 500 µL/h, $\overline{MD}$ is increased from 0.46 to 0.55 at S

_{2}when the pulsing frequency increases from 0 to 1 Hz, as shown in Figure 7d. To understand the mixing improvement at the non-zero frequency condition, we investigate the instantaneous mixing degree $M{D}_{t}$ for a sufficient time in the mixing channel area between S

_{1}and S

_{2}, as shown in the contour of Figure 8. The x-axis of the contour map is the flow distance and the y-axis is the flow time. It is noticed that some high mixing instances, for example, B, C and D in the contour map of Figure 8b–d, are induced during the flow and contribute to the averaged mixing. Comparably, there is no high mixing region in the constant flow condition and the flow remains almost the same at a relatively low mixing degree, like at point A in Figure 8a. The concentration profiles at points B, C and D are plotted correspondingly and compared with that at point A in the insertion in Figure 8. At point A, the red and blue fluid flow on two sides of the mixing channel and oscillate as a crescent shape due to the elastic stress. While at B, C and D, the flows have similar profiles and the red and the blue fluid flow alternatingly along the channel. This indicates that the oscillation in the constant flow condition is amplified by the time pulsing and results in a higher $\overline{MD}$. As previously discussed in Section 3.1, $\overline{MD}$ is larger at a lower frequency f for the mix in the Newtonian fluid. Here, in contrast, the mixing in the viscoelastic fluid, $\overline{MD}$ is larger at a higher pulsing frequency. The difference of the mixing enhancement between the Newtonian fluid and the viscoelastic fluid stems from the elastic stress in the viscoelastic fluid.

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Yu, C.; Kim, G.B.; Clark, P.M.; Zubkov, L.; Papazoglou, E.S.; Noh, M. A microfabricated quantum dot-linked immuno-diagnostic assay (μQLIDA) with an electrohydrodynamic mixing element. Sens. Actuators B Chem.
**2015**, 209, 722–728. [Google Scholar] [CrossRef] - Kefala, I.N.; Papadopoulos, V.E.; Karpou, G.; Kokkoris, G.; Papadakis, G.; Tserepi, A. A labyrinth split and merge micromixer for bioanalytical applications. Microfluid. Nanofluid.
**2015**, 19, 1047–1059. [Google Scholar] [CrossRef] - Lang, Q.; Ren, Y.; Hobson, D.; Tao, Y.; Hou, L.; Jia, Y.; Hu, Q.; Liu, J.; Zhao, X.; Jiang, H. In-plane microvortices micromixer-based AC electrothermal for testing drug induced death of tumor cells. Biomicrofluidics
**2016**, 10, 064102. [Google Scholar] [CrossRef] - Yeh, S.-I.; Sheen, H.-J.; Yang, J.-T. Chemical reaction and mixing inside a coalesced droplet after a head-on collision. Microfluid. Nanofluid.
**2015**, 18, 1355–1363. [Google Scholar] [CrossRef] - Roberge, D.M.; Ducry, L.; Bieler, N.; Cretton, P.; Zimmermann, B. Microreactor technology: A revolution for the fine chemical and pharmaceutical industries? Chem. Eng. Technol.
**2005**, 28, 318–323. [Google Scholar] [CrossRef] - Microreactors—New Technology for Modern Chemistry Wolfgang Ehrfeld Volker Hessel Holger Löwe Wiley-VCH: Weinheim. 2000. 288 pp. Price £80. ISBN3-527-29590-9. Org. Proc. Res. Dev.
**2001**, 5, 89. [CrossRef] - Hessel, V.; Löwe, H.; Schönfeld, F. Micromixers—A review on passive and active mixing principles. Chem. Eng. Sci.
**2005**, 60, 2479–2501. [Google Scholar] [CrossRef] - Abed, W.M.; Whalley, R.D.; Dennis, D.J.C.; Poole, R.J. Experimental investigation of the impact of elastic turbulence on heat transfer in a serpentine channel. J. Non-Newt. Fluid Mechan.
**2016**, 231, 68–78. [Google Scholar] [CrossRef] - Xia, H.M.; Wang, Z.P.; Koh, Y.X.; May, K.T. A microfluidic mixer with self-excited ‘turbulent’ fluid motion for wide viscosity ratio applications. Lab Chip
**2010**, 10, 1712–1716. [Google Scholar] [CrossRef] [PubMed] - Lemenand, T.; Della Valle, D.; Habchi, C.; Peerhossaini, H. Micro-mixing measurement by chemical probe in homogeneous and isotropic turbulence. Chem. Eng. J.
**2017**, 314, 453–465. [Google Scholar] [CrossRef] - Ward, K.; Fan, Z.H. Mixing in microfluidic devices and enhancement methods. J. Micromech. Microeng.
**2015**, 25, 094001. [Google Scholar] [CrossRef] [PubMed] - Lee, C.-Y.; Chang, C.-L.; Wang, Y.-N.; Fu, L.-M. Microfluidic mixing: A review. Int. J. Mol. Sci.
**2011**, 12, 3263–3287. [Google Scholar] [CrossRef] [PubMed] - Lee, C.Y.; Fu, L.M. Recent advances and applications of micromixers. Sens. Actuators B Chem.
**2018**, 259, 677–702. [Google Scholar] [CrossRef] - Cai, G.; Xue, L.; Zhang, H.; Lin, J. A review on micromixers. Micromachines
**2017**, 8, 274. [Google Scholar] [CrossRef] [PubMed] - Suh, Y.K.; Kang, S. A review on mixing in microfluidics. Micromachines
**2010**, 1, 82–111. [Google Scholar] [CrossRef] - Lee, C.-Y.; Wang, W.-T.; Liu, C.-C.; Fu, L.-M. Passive mixers in microfluidic systems: A review. Chem. Eng. J.
**2016**, 288, 146–160. [Google Scholar] [CrossRef] - Buchegger, W.; Wagner, C.; Lendl, B.; Kraft, M.; Vellekoop, M.J. A highly uniform lamination micromixer with wedge shaped inlet channels for time resolved infrared spectroscopy. Microfluid. Nanofluid.
**2011**, 10, 889–897. [Google Scholar] [CrossRef] - Tofteberg, T.; Skolimowski, M.; Andreassen, E.; Geschke, O. A novel passive micromixer: Lamination in a planar channel system. Microfluid. Nanofluid.
**2010**, 8, 209–215. [Google Scholar] [CrossRef] - Zhang, Y.; Hu, Y.; Wu, H. Design and simulation of passive micromixers based on capillary. Microfluid. Nanofluid.
**2012**, 13, 809–818. [Google Scholar] [CrossRef] - Li, L.; Chen, Q.D.; Tsai, C. Three dimensional triangle chaotic micromixer. Adv. Mater. Res.
**2014**, 875–877, 1189–1193. [Google Scholar] [CrossRef] - Westerhausen, C.; Schnitzler, L.G.; Wendel, D.; Krzysztoń, R.; Lächelt, U.; Wagner, E.; Rädler, J.O.; Wixforth, A. Controllable acoustic mixing of fluids in microchannels for the fabrication of therapeutic nanoparticles. Micromachines
**2016**, 7, 150. [Google Scholar] [CrossRef] [PubMed] - Rife, J.; Bell, M.I.; Horwitz, J.S.; Kabler, M.N.; Auyeung, R.C.Y.; Kim, W.J. Miniature valveless ultrasonic pumps and mixers. Sens. Actuators A Phys.
**2000**, 86, 135–140. [Google Scholar] [CrossRef] - Sounart, T.; Baygents, J. Electrically-driven fluid motion in channels with streamwise gradients of the electrical conductivity. Colloid. Surf. A Physicochem. Eng. Aspect.
**2001**, 195, 59–75. [Google Scholar] [CrossRef] - El Moctar, A.O.; Aubry, N.; Batton, J. Electro-hydrodynamic micro-fluidic mixer. Lab Chip
**2003**, 3, 273–280. [Google Scholar] [CrossRef] [PubMed] - Bau, H.H.; Zhong, J.; Yi, M. A minute magneto hydro dynamic (MHD) mixer. Sens. Actuators B Chem.
**2001**, 79, 207–215. [Google Scholar] [CrossRef] - Glasgow, I.; Aubry, N. Enhancement of microfluidic mixing using time pulsing. Lab Chip
**2003**, 3, 114–120. [Google Scholar] [CrossRef] [PubMed] - Goullet, A.; Glasgow, I.; Aubry, N. Effects of microchannel geometry on pulsed flow mixing. Mechan. Res. Commun.
**2006**, 33, 739–746. [Google Scholar] [CrossRef] - Goullet, A.; Glasgow, I.; Aubry, N. Dynamics of Microfluidic Mixing Using Time Pulsing. Available online: https://www.researchgate.net/profile/Ian_Glasgow/publication/228678991_Dynamics_of_microfluidic_mixing_using_time_pulsing/links/570931c908aed09e916f931c.pdf (accessed on 17 April 2019).
- Zhang, M.; Cui, Y.; Cai, W.; Wu, Z.; Li, Y.; Li, F.; Zhang, W. High Mixing Efficiency by Modulating Inlet Frequency of Viscoelastic Fluid in Simplified Pore Structure. Processes
**2018**, 6, 210. [Google Scholar] [CrossRef] - Glasgow, I.K.; Aubry, N. Mixing Enhancement in Simple Geometry Microchannels. In Proceedings of the ASME 2003 International Mechanical Engineering Congress and Exposition, Washington, DC, USA, 15–21 November 2003; pp. 565–572. [Google Scholar]
- Cho, C.-C.; Chen, C.-L.; Tsai, R.-T. A novel microfluidic mixer using aperiodic perturbation flows. Chem. Eng. Sci.
**2011**, 66, 6159–6167. [Google Scholar] [CrossRef] - 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] - Pathak, J.A.; Ross, D.; Migler, K.B. Elastic flow instability, curved streamlines, and mixing in microfluidic flows. Phys. Fluid.
**2004**, 16, 4028–4034. [Google Scholar] [CrossRef] - Hong, S.O.; Cooper-White, J.J.; Kim, J.M. Inertio-elastic mixing in a straight microchannel with side wells. Appl. Phys. Lett.
**2016**, 108, 014103. [Google Scholar] [CrossRef] - Julius, L.A.N.; Jagannadh, V.K.; Michael, I.J.; Srinivasan, R.; Gorthi, S.S. Design and validation of on-chip planar mixer based on advection and viscoelastic effects. BioChip J.
**2016**, 10, 16–24. [Google Scholar] [CrossRef] - Grigoriev, R.; Schuster, H.G. Transport and Mixing in Laminar Flows; John Wiley & Sons: Hoboken, NJ, USA, 2012. [Google Scholar]
- Krishnan, J.M.; Deshpande, A.P.; Sunil Kumar, P.B. Rheology of Complex Fluids; Springer: New York, NY, USA, 2010; pp. 3–34. [Google Scholar]
- Pakdel, P.; McKinley, G.H. Elastic instability and curved streamlines. Phys. Rev. Lett.
**1996**, 77, 2459. [Google Scholar] [CrossRef] - Larson, R.G.; Shaqfeh, E.S.; Muller, S.J. A purely elastic instability in Taylor–Couette flow. J. Fluid Mechan.
**1990**, 218, 573–600. [Google Scholar] [CrossRef] - McKinley, G.H.; Pakdel, P.; Öztekin, A. Rheological and geometric scaling of purely elastic flow instabilities. J. Non-Newtonian Fluid Mechan.
**1996**, 67, 19–47. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Concentration profile of Newtonian fluid at different constant flow rates; (

**b**) the averaged mixing degree $\overline{MD}$ at different constant flow rate mixing.

**Figure 3.**Averaged mixing degree $\overline{MD}$ of time pulsing Newtonian flow with different pulsing frequency at flow rates of (

**a**) ${Q}_{0}$ = 10 µL/h; (

**b**) ${Q}_{0}$ = 50 µL/h; (

**c**) ${Q}_{0}$ = 100 µL/h; (

**d**) ${Q}_{0}$ = 200 µL/h; (

**e**) ${Q}_{0}$ = 500 µL/h and (

**f**) ${Q}_{0}$ = 1000 µL/h.

**Figure 4.**The instantaneous mixing degree $M{D}_{t}$ of time pulsing flow with different frequency at flow rates of (

**a**) ${Q}_{0}$ = 50 µL/h; (

**b**) ${Q}_{0}$ = 200 µL/h; (

**c**) ${Q}_{0}$ = 500 µL/h at flow distance 2000 µm; (

**d**) the concentration profiles at the peak-value and dip-value time instances when ${Q}_{0}$ = 500 µL/h.

**Figure 5.**Concentration profile of viscoelastic fluid at different constant flow rates of (

**a**) ${Q}_{0}$ = 10 µL/h; (

**b**) ${Q}_{0}$ = 50 µL/h; (

**c**) ${Q}_{0}$ = 100 µL/h; (

**d**) ${Q}_{0}$ = 200 µL/h; (

**e**) ${Q}_{0}$ = 500 µL/h and (

**f**) ${Q}_{0}$ = 1000 µL/h; (

**g**) the averaged mixing degree $\overline{MD}$ of viscoelastic fluid at different constant flow rates mixing.

**Figure 6.**Mixing degree $\overline{MD}$ at S

_{2}= 2000 µm at different flow rates for both glycerol and polyacrylamide (PAM).

**Figure 7.**Averaged mixing degree $\overline{MD}$ of time pulsing viscoelastic flow with different frequency at flow rates of (

**a**) ${Q}_{0}$ = 50 µL/h; (

**b**) ${Q}_{0}$ = 100 µL/h; (

**c**) ${Q}_{0}$ = 200 µL/h and (

**d**) ${Q}_{0}$ = 500 µL/h.

**Figure 8.**Top: The instantaneous mixing degree along the flow distance at different flow times with a flow rate of 500 µL/h at a pulsing frequency of (

**a**) 0 ; (

**b**) 0.1 Hz; (

**c**) 0.2 Hz and (

**d**) 1 Hz. Bottom: The concentration profiles at the instance of A, B, C, and D, correspondingly.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Zhang, W.; Wu, Z.; Shen, Y.; Chen, Y.; Lan, C.; Li, F.; Cai, W.
Comparison of Micro-Mixing in Time Pulsed Newtonian Fluid and Viscoelastic Fluid. *Micromachines* **2019**, *10*, 262.
https://doi.org/10.3390/mi10040262

**AMA Style**

Zhang M, Zhang W, Wu Z, Shen Y, Chen Y, Lan C, Li F, Cai W.
Comparison of Micro-Mixing in Time Pulsed Newtonian Fluid and Viscoelastic Fluid. *Micromachines*. 2019; 10(4):262.
https://doi.org/10.3390/mi10040262

**Chicago/Turabian Style**

Zhang, Meng, Wu Zhang, Zhengwei Wu, Yinan Shen, Yicheng Chen, Chaofeng Lan, Fengchen Li, and Weihua Cai.
2019. "Comparison of Micro-Mixing in Time Pulsed Newtonian Fluid and Viscoelastic Fluid" *Micromachines* 10, no. 4: 262.
https://doi.org/10.3390/mi10040262