# Enhancement of Fluid Mixing with U-Shaped Channels on a Rotating Disc

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiments

_{c}= 0) located at 30 mm from the center of the disc. A capillary valve with an expansion angle of 90° was fabricated ahead of the T-junction (x

_{c}= 1.3 mm) to stop the flow in the hydrophilic channel before rotation of the disc [28,40]. The burst rotational speed for the present centrifugal micromixers was 360 rpm. It should be noted that the U-shaped channels (beginning from x

_{u}= 3.0 mm) are short and straight with an arc curve at the outer corner while keeping a sharp 90° bend at the inner corner. Such a design enables to rapidly alternate the flow direction for generating strong centrifugal force locally through the consecutive 90° bends. The U-shaped microfluidic structure was manufactured using a micro-CNC machine on a polymethylmethacrylate (PMMA) substrate of 10 cm in diameter and 2 mm in thickness. In addition to the single U-shaped structure shown in Figure 1b, micromixers with double and triple U-shaped structures were fabricated for comparison. All the microfluidic channels have the same cross-section of 300 μm in width (b) and 300 μm in depth (h). The total radial channel length is 20 mm for all the three types of U-shaped micromixers. The microchannels with double and triple U-structures also begin their first 90° bend from x

_{u}= 3.0 mm and an additional U-shaped element adds a length of 1.2 mm in the x-direction. Since the static pressure of the liquid flow in the rotating microchannels is essentially below the atmospheric pressure [33], the structured microfluidics was simply covered with transparent Scotch tape (3040-4PK, 3M, St. Paul, MN, USA) to provide good sealing as well as excellent optical access for flow visualization.

_{3}·6H

_{2}O) and ammonium thiocyanate (NH

_{4}SCN) dissolved in deionized (DI) water at the same concentration of 0.2 mol/kg. Ferric chloride solution is pale yellow and ammonium thiocyanate solution is colorless. When these two solutions come into contact, the ferric ions bind with the thiocyanate ions instead of the chloride ions to produce the blood-red color [41]. The intensity of the red color described by the RGB values for each of pixels could represent the amount of fluids that have mixed and reacted. The normalized color intensity was quantified as the index of mixing efficiency between the two fluids.

## 3. Numerical Simulation

_{c}= 1.3 mm) without a capillary valve. The fluids were assumed to have the same constant density as water ρ = 1000 kg/m

^{3}and the same viscosity μ = 0.001 kg/m

^{2}∙s. The flow field of fluids A and B in the microchannel was computed on the rotating frame. The origin of the computational coordinates is located at the center of the rotating disc. The flow field is governed by the steady, three-dimensional continuity and Navier–Stokes equations:

^{−10}m

^{2}/s, which was based on the study of Kochman et al. [22].

^{5}–6.8 × 10

^{5}were employed for the computations.

_{∞}is the completely mixed concentration, C

_{i}is the concentration at a grid i, and n is the number of grids computed on the cross-sectional plane. The mixing efficiency varies from zero (no mixing at all) to unity (complete mixing) is generally a function of the streamwise position. For the present numerical simulations, the mixing efficiency was evaluated on Section 5 as indicated in Figure 1, which was located at the immediate exit of the most downstream U-shaped element (x

_{c}= 4.2, 5.4 and 6.6 mm for single, double and triple U-shaped mixers, respectively).

## 4. Results and Discussion

#### 4.1. Simulation Results

_{ω}generated through the system rotation, the Coriolis force f

_{C}, and the Dean force f

_{D}, which is the centrifugal force locally generated through the consecutive 90° bends [36,38]. The magnitude of these three forces may be estimated as follows [42]:

_{c}= 2.7 mm. Subsequently, the 90° bends appear to stir the fluids by flipping fluids A and B in Section 2. This can be seen first in Section 2 changing from AB to BA and flipping again in Section 3. Then in Section 4 where fluid A splits to sandwich fluid B forming ABA. In addition to flipping, interface folding also becomes more significant as the flow moves to Section 4. After flowing through these bends, the two fluids are well mixed in Section 5 (x

_{c}= 4.2 mm). Similar patterns of flipping, splitting, folding and stirring were also observed in the study by La et al. [35] but with a much longer circumferential channel length of 5.2 mm (17 times the channel width) at a much higher rotational speed Ω = 2000 rpm. It should be noted that there is no Dean force generated in a long circumferential channel (R → ∞) [38], in which the secondary flow is essentially induced by the Coriolis force.

_{c}of the two forces given in Equations (6) and (7) as [42]:

_{m}and expressed as:

_{m}= 0.38 m/s and an average radius R ≈ (2

^{1/2})b = 424 µm, the ratio of the Dean force to the Coriolis force is approximated to be γ

_{c}≈ 7.1. This means that the Dean force dominates the Coriolis force in generating the secondary flow as the stream turns along the consecutive 90° bends, and the Dean–Coriolis force ratio increases with increasing rotational speed.

_{m}instead of the maximum stream velocity U

_{max}is used to scale the Coriolis force and a characteristic scaling f

_{C}∝ Ω

^{2}is invoked. As already mentioned, the mean stream velocity U

_{m}depends on Ω, and the exponent k in Equation (9) tends to decrease in the higher rotational speed range for the single-U mixer. It should be noted that in Equation (10), the Coriolis force is perpendicular to the radial direction of the system’s centrifugal force. The transverse secondary flow accompanied by the streamwise vorticity in the radial straight channel is mainly due to the Coriolis force, while the system’s centrifugal force contributes to drive the stream flow along the channel [43,44]. When examining the effect of the system’s centrifugal force on secondary flows in the circumferential channel section (orthogonal to the radial direction), great care is required because f

_{ω}is parallel to f

_{C}. This effect may be understood based on vorticity consideration by rewriting Equation (2) for a constant angular velocity Ω as [43,44]:

^{*}given by

_{c}= 20 mm) and junction position (x

_{j}= 1.3 mm) as the U-mixer. Figure 5b displays the cross-sectional concentration distributions and velocity vectors computed at x

_{c}= 4.2, which is the same x-position as Section 5 of the U-mixer, for two rotational speeds Ω = 360 ad 600 rpm (ccw). A C-shaped interface of the fluids can be clearly seen in the cross-section for both rotational speeds. In the T-mixer, the transverse secondary flow is induced essentially by the Coriolis force alone [30] since there is no Dean force in the straight channel (R → ∞). It can be seen from the transverse velocity vectors that the secondary flow drives the fluids from side walls B toward A (B→A) in the middle of the cross-section neighboring z = 0, where the maximum radial velocity occurs, and A→B near the top and bottom walls. When comparing these two cross-sectional distributions, the C-shaped interface for the higher rotational speed of 600 rpm appears to be a bit wider and lighter near the corners on side wall B than that of 360 rpm. This indicates a slightly better mixing for the higher rotational speed due to a stronger Coriolis-induced secondary flow that compensates for the shorter residence time of the mixing fluid stream.

_{c}= 4.2 mm) for the U-mixer and at the same x-position (x

_{c}= 4.2 mm) for the T-mixer. It can be seen that mixing efficiency for the U-mixer increases rapidly with rotational speed, from 20% at Ω = 120 rpm to 82% at Ω = 840 rpm. Beyond Ω = 840 rpm, the mixing efficiency appears to level off around 83–84%. The flat level efficiency in the higher speed range is due mainly to the high flow velocity that increases rapidly with rotational speed. The mean velocities at the U-mixer exit were found to be 0.67 m/s for Ω = 900 rpm and 1.0 m/s for Ω = 1200 rpm, corresponding to Reynolds numbers Re = 200 and 300. The rapidly increasing velocity reduces the fluid residence time and appears to offset the growing influence of transverse secondary flow. It is worth mentioning that the simulations with different diffusion coefficients (1.0 × 10

^{-12}and 3.0 × 10

^{-9}m

^{2}/s) also exhibit a similar trend of mixing efficiency variation with rotational speed as in Figure 6 for the same U-shaped mixer, but give a slightly higher efficiency (by an amount of 5–6%) for the larger diffusion coefficient and slightly lower efficiency (by an amount of 4–5%) for the smaller diffusion coefficient in the level-off range Ω = 900–1200 rpm. In other words, changes in mixing efficiency contributed from the diffusion coefficient variation by an order of magnitude are limited. This indicates that advection plays the key role in enhancing fluid mixing for the present centrifugal U-shaped microchannel.

_{c}for a longer channel length.

#### 4.2. Experimental Results and Comparison

_{c}≈ 6.2 and 7.4 for Ω = 600 and 900, respectively).

_{c}= 4.2, 5.4 and 6.6 mm for the single-, double- and triple-U mixer, respectively. The mixing efficiency computed from the pixel intensity of averaged RGB values in the imaged area is given by [16]:

_{max}is observed in a fully mixed image and the minimum intensity I

_{min}is observed in an image of DI water. The technique of quantifying mixing efficiency from concentration measurements was employed previously by Chen et al. [48]. For all the three U-mixers, the mixing efficiency measured from the visualization experiments increases sharply with increasing rotational speed in the lower range, Ω = 360 to 600 rpm for the single- and double-U mixers and an even lower range of Ω = 360 to 480 rpm for the triple-U mixer. As the speed further increases to 720 and 900 rpm, the efficiency gradually levels off for the single-U mixer. Nearly complete mixing is attained at Ω = 720 rpm for the double-U mixer and at an even lower speed of Ω = 600 rpm for the triple-U mixer. This increasing trend is observed in both counterclockwise and clockwise cases and is well predicted by the simulations with a constant diffusion coefficient of 3 × 10

^{−10}m

^{2}/s.

## 5. Conclusions

_{c}, similar to the one used by Zhang et al. [42], is introduced to examine effects of the Dean force as compared to the Coriolis force. Stretching, twisting and folding of mixing interface caused by the transverse secondary flow were observed in flow visualization and closely resembled in numerical simulation as well. It is also found that the secondary flow becomes stronger with increasing rotational speed and with more U-shaped structures, resulting in a larger area of mixed fluids and hence higher mixing performance. The mixing efficiency measured for the three types of U-shaped mixers shows a sharp increase with increasing rotational speed in the lower range Ω ≤ 600 rpm. As the speed further increases, nearly complete mixing can be achieved at Ω = 600 for the triple-U mixer and at Ω = 720 rpm for the double-U mixer, while a maximum level between 83 and 86% is reached for the single-U mixer at Ω = 720 and 900 rpm. The variation of simulated mixing efficiency with rotational speed in a wider range (120–1200 rpm) agrees well with the measurements. Moreover, both the simulation and measurement results show no discernible difference in mixing efficiency between clockwise and counter-clockwise cases. The centrifugal U-shaped micromixers presented in this study enabling to assist fluid mixing effectively within a short channel length are especially suitable for use in CD-based microfluidic systems.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Whitesides, G. The lab finally comes to the chip! Lab Chip
**2014**, 14, 3125–3126. [Google Scholar] [CrossRef] [PubMed] - Rapp, B.; Gruhl, F.J.; Länge, K. Biosensors with label-free detection designed for diagnostic applications. Anal. Bioanal. Chem.
**2000**, 398, 2403–2412. [Google Scholar] [CrossRef] [PubMed] - Rajendran, S.T.; Scarano, E.; Bergkamp, M.H.; Capria, A.M.; Cheng, C.H.; Sanger, K.; Ferrari, G.; Nielsen, L.H.; Hwu, E.T.; Zór, K.; et al. Modular, lightweight, wireless potentiostat-on-a-disc for electrochemical detection in centrifugal microfluidics. Anal. Chem.
**2019**, 91, 11620–11628. [Google Scholar] [CrossRef] - Jeong, G.S.; Chung, S.; Kim, C.B.; Lee, S.H. Applications of micromixing technology. Analyst
**2010**, 135, 460–473. [Google Scholar] [CrossRef] [PubMed] - Yoshida, J.; Nagaki, A.; Iwasaki, T. Enhancement of chemical selectivity by microreactors. Chem. Eng. Technol.
**2005**, 28, 259–266. [Google Scholar] [CrossRef] - Doku, G.; Verboom, W.; Reinhoudt, D.; van den Berg, A. On-microchip multiphase chemistry—A review of microreactor design principles and reagent contacting modes. Tetrahedron
**2005**, 61, 2733–2742. [Google Scholar] [CrossRef] - Su, Y.; Chen, G.; Yuan, Q. Ideal micromixing performance in packed microchannels. Chem. Eng. Sci.
**2011**, 13, 2912–2919. [Google Scholar] [CrossRef] - Benz, K.; Jackel, K.P.; Regenauer, K.J.; Schiewe, J.; Drese, K.; Ehrfeld, W.; Hessel, V.; Lowe, H. Utilization of Micromixers for Extraction Processes. Chem. Eng. Technol.
**2001**, 24, 11–17. [Google Scholar] [CrossRef] - Okubo, Y.; Toma, M.; Ueda, H.; Maki, T.; KazuhiroMae, K. Microchannel devices for the coalescence of dispersed droplets produced for use in rapid extraction processes. Chem. Eng. J.
**2004**, 101, 39–48. [Google Scholar] [CrossRef] - Zhang, Y.; Ozdemir, P. Microfluidic DNA amplification—A review. Anal. Chim. Acta
**2009**, 638, 115–125. [Google Scholar] [CrossRef] [Green Version] - Nguyen, N.T.; Wu, Z. Micromixers—A review. J. Micromech. Microeng.
**2005**, 15, R1–R16. [Google Scholar] [CrossRef] - Wang, L.; Li, P.C.H. Microfluidic DNA microarray analysis: A review. Anal. Chim. Acta
**2011**, 687, 12–27. [Google Scholar] [CrossRef] [PubMed] - Chang, C.C.; Yang, R.J. Electrokinetic mixing in microfluidic systems. Microfluid. Nanofluidics
**2007**, 3, 501–525. [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] - Raza, W.; Hossain, S.; Kim, K.Y. A review of passive micromixers with a comparative analysis. Micromachines
**2020**, 11, 455. [Google Scholar] [CrossRef] [PubMed] - Liu, R.H.; Stremler, M.A.; Sharp, K.V.; Olsen, M.G.; Santiago, J.G.; Adrian, R.J.; Aref, H.; Beeb, D.J. Passive mixing in a three-dimensional serpentine microchannel. J. Microelectromech. Syst.
**2000**, 9, 190–197. [Google Scholar] [CrossRef] - Stroock, A.D.; Dertinger, S.K.W.; Ajdari, A.; Mezic, I.; Stone, H.A.; Whitesides, G.M. Chaotic mixer for microchannels. Science
**2005**, 295, 647–651. [Google Scholar] [CrossRef] [Green Version] - Xia, H.M.; Wan, S.Y.M.; Shu, C.; Chew, Y.T. Chaotic micromixers using two-layer crossing channels to exhibit fast mixing at low Reynolds numbers. Lab Chip
**2005**, 5, 748–755. [Google Scholar] [CrossRef] - Park, J.M.; Kim, D.S.; Kang, T.G.; Kwon, T.H. Improved serpentine laminating micromixer with enhanced local advection. Microfluid. Nanofluidics
**2008**, 4, 513–523. [Google Scholar] [CrossRef] - Ansari, M.A.; Kim, K.Y. Parametric study on mixing of two fluids in a three-dimensional serpentine microchannel. Chem. Eng. J.
**2009**, 146, 439–448. [Google Scholar] [CrossRef] - Hossain, S.; Lee, I.; Kim, S.M.; Kim, K.Y. A micromixer with two-layer serpentine crossing channels having excellent mixing performance at low Reynolds numbers. Chem. Eng. J.
**2017**, 327, 268–277. [Google Scholar] [CrossRef] - Kockmann, N.; Kiefer, T.; Engler, M. Silicon microstructures for high throughput mixing devices. Microfluid. Nanofluidics
**2006**, 2, 327–335. [Google Scholar] [CrossRef] - Hong, C.C.; Choi, J.W.; Ahn, C.H. A novel in-plane passive microfluidic mixer with modified tesla structures. Lab Chip
**2004**, 4, 109–113. [Google Scholar] [CrossRef] [PubMed] - Hossain, S.; Ansari, M.A.; Husain, A.; Kim, K.Y. Analysis and optimization of a micromixer with a modified Tesla structure. Chem. Eng. J.
**2010**, 158, 305–314. [Google Scholar] [CrossRef] - Madou, M.; Zoval, J.; Jia, G.; Kido, H.; Kim, J.; Kim, N. Lab on a CD. Annu. Rev. Biomed. Eng.
**2006**, 8, 601–628. [Google Scholar] [CrossRef] [Green Version] - Ducrée, J.; Haeberle, S.; Lutz, S.; Pausch, S.; von Stetten, F.; Zengerle, R. The centrifugal microfluidic Bio-Disk platform. J. Micromech. Microeng.
**2007**, 17, S103–S115. [Google Scholar] [CrossRef] - Gorkin, R.; Park, J.; Siegrist, J.; Amasia, M.; Lee, B.S.; Park, J.M.; Kim, J.; Kim, H.; Madou, M.; Cho, Y.K. Centrifugal microfluidics for biomedical applications. Lab Chip
**2010**, 10, 1758–1773. [Google Scholar] [CrossRef] [Green Version] - Duffy, D.C.; Gills, H.L.; Lin, J.; Sheppard, N.F.; Kellogg, G.J. Microfabricated centrifugal microfluidic systems: Characterization and multiple enzymatic assays. Analy. Chem.
**1999**, 71, 4669–4678. [Google Scholar] [CrossRef] - Madou, M.J.; Lee, L.J.; Daunert, S.; Lai, S.; Shih, C.-H. Design and fabrication of CD-like microfluidic platforms for diagnostics: Microfluidic functions. Biomed. Microdevices
**2001**, 3, 245–254. [Google Scholar] [CrossRef] - Ducree, J.; Haeberle, S.; Brenner, T.; Glatzel, T.; Zengerle, R. Patterning of flow and mixing in rotating radial microchannels. Microfluid. Nanofluidics
**2006**, 2, 97–105. [Google Scholar] [CrossRef] - Gilmore, J.; Islam, M.; Martinez-Duarte, R. Challenges in the use of compact disc-based centrifugal microfluidics for healthcare diagnostics at the extreme point of care. Micromachines
**2016**, 7, 52. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Tang, M.; Wang, G.; Siu-Kai Kong, S.K.; Ho, H.P. A review of biomedical centrifugal microfluidic platforms. Micromachines
**2016**, 7, 26. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kim, D.S.; Kwon, T.H. Modeling, analysis and design of centrifugal force-driven transient filling flow into a circular microchannel. Microfluid. Nanofluidics
**2006**, 2, 125–140. [Google Scholar] [CrossRef] - Chiu, H.C.; Chen, J.M. Numerical study fluid mixing in a microchannel with a circular chamber on a rotating disk. Adv. Mat. Res.
**2012**, 542, 1113–1119. [Google Scholar] [CrossRef] - La, M.; Park, S.J.; Kim, H.W.; Park, J.J.; Ahn, K.T.; Ryew, S.M.; Kim, D.S. A centrifugal force-based serpentine micromixer (CSM) on a plastic lab-on-a-disk for biochemical assays. Microfluid. Nanofluidics
**2013**, 15, 87–98. [Google Scholar] [CrossRef] - Kuo, J.N.; Jiang, L.R. Design optimization of micromixer with square-wave microchannel on compact disk microfluidic platform. Microsyst. Technol.
**2014**, 20, 91–99. [Google Scholar] [CrossRef] - Kuo, J.N.; Li, Y.S. Centrifugal-based micromixer with three-dimensional square-wave microchannel for blood plasma mixing. Microsyst. Technol.
**2017**, 23, 2343–2354. [Google Scholar] [CrossRef] - Shamloo, A.; Vatankhah, P.; Akbari, A. Analyzing mixing quality in a curved centrifugal micromixer through numerical simulation. Chem. Eng. Process.
**2017**, 116, 9–16. [Google Scholar] [CrossRef] - Shamloo, A.; Madadelahi, M.; Akbari, A. Numerical simulation of centrifugal serpentine micromixers and analyzing mixing quality parameters. Chem. Eng. Process.
**2016**, 104, 243–252. [Google Scholar] [CrossRef] [Green Version] - Chen, J.M.; Huang, P.C.; Lin, M.G. Analysis and experiment of capillary valves for microfluidics on a rotating disk. Microfluid. Nanofluidics
**2008**, 4, 427–437. [Google Scholar] [CrossRef] - Shakhashiri, B.Z. Chemical Demonstrations: A Handbook for Teachers of Chemistry; University of Wisconsin Press: Madison, WI, USA, 1983; Volume 1. [Google Scholar]
- Zhang, J.; Guo, Q.; Liu, M.; Yang, J. A lab-on-CD prototype for high-speed blood separation. J. Micromech. Microeng.
**2008**, 18, 125025. [Google Scholar] [CrossRef] - Lee, G.H.; Baek, J.H. A numerical study of the similarity of fully developed laminar flows in orthogonally rotating rectangular ducts and stationary curved rectangular ducts of arbitrary aspect ratio. Comput. Mech.
**2002**, 29, 183–190. [Google Scholar] [CrossRef] - Dai, Y.J.; Huang, W.X.; Xu, C.X.; Cui, G.X. Direct numerical simulation of turbulent flow in a rotating square duct. Phys. Fluids
**2015**, 27, 065104. [Google Scholar] [CrossRef] - Currie, I.G. Fundamental Mechanics of Fluids; McGraw-Hill: New York, NY, USA, 1974. [Google Scholar]
- Batchelor, G.K. An Introduction to Fluid Dynamics; Cambridge University Press: Cambridge, UK, 1967. [Google Scholar]
- Panton, R.L. Incompressible Flow; John Wiley & Sons: New York, NY, USA, 1984. [Google Scholar]
- Chen, J.M.; Horng, T.L.; Tan, W.Y. Analysis and measurements of mixing in pressure-driven microchannel flow. Microfluid. Nanofluidics
**2006**, 2, 455–469. [Google Scholar] [CrossRef]

**Figure 1.**Schematic of the centrifugal U-shaped micromixer: (

**a**) experimental arrangement for flow visualization; (

**b**) the microchannel geometry and coordinates for single U-shaped structure.

**Figure 2.**Top view of the U-shaped channel and concentration distribution of fluids A and B for the single-U mixer rotating at Ω = 600 rpm (counter-clockwise; ccw): (

**a**) top view with detailed dimensions and cross-sections selected for examination as well as the directions of Coriolis and Dean forces acting on the fluids; (

**b**) distribution on mid-plane along the channel; (

**c**–

**g**) on cross-sections 1–5 normal to the fluid stream as indicated in (

**b**).

**Figure 3.**Two-dimensional streamlines on cross-sections 2–5 as indicated in Figure 2a for the single U-shaped mixer: (

**a**) rotating at Ω = 360 rpm (ccw); (

**b**) rotating at Ω = 600 rpm (ccw).

**Figure 4.**Velocity vector field with the directions of Coriolis and Dean forces on cross sections 2–5 as indicated in Figure 2a for the single U-shaped mixer: (

**a**) rotating at Ω = 360 rpm (ccw); (

**b**) rotating at Ω = 600 rpm (ccw).

**Figure 5.**Centrifugal T-type micromixer and cross-sectional concentration distribution: (

**a**) schematic of the T-type micromixer showing the channel geometry and coordinates; (

**b**) concentration distributions and velocity vectors at x

_{c}= 4.2 mm for Ω = 360 and 600 rpm (ccw).

**Figure 6.**Comparison of mixing efficiency for the single U-shaped mixer and T-type mixer rotating at different speeds from 120 to 1200 rpm (ccw).

**Figure 7.**(

**a**–

**d**) Top view visualization of the mixed fluids for the single U-shaped mixer rotating from Ω = 360 to 900 rpm (ccw), where the flow enters from the left.

**Figure 8.**(

**a**–

**d**) Top view visualization of the mixed fluids for the single U-shaped mixer rotating from Ω = 360 to 900 rpm (clockwise; cw), where the flow enters from the left.

**Figure 9.**(

**a**–

**d**) Top view images of fluid mixing obtained from numerical simulations for the single U-shaped mixer at Ω = 360 and 600 rpm undergoing both counterclockwise and clockwise rotations, where the flow enters from the left.

**Figure 10.**(

**a**–

**d**) Top view visualization of the mixed fluids for the double-U mixer undergoing counterclockwise (ccw) and clockwise (cw) rotations at 360 and 600 rpm, where the flow enters from the left.

**Figure 11.**Variations of simulation mixing efficiency in downstream channel distance for the double-U mixer undergoing counterclockwise rotations at 360, 600 and 900 rpm with indication of downstream positions for the cross-sections of the first and second U-structures.

**Figure 12.**Top view images of fluid mixing from visualization experiments (

**left**, where only the second and third U-structures are displayed) and numerical simulations (

**right**) for the triple-U mixer at Ω = 360 rpm: (

**a**) counterclockwise rotation; (

**b**) clockwise rotation. The flow enters from the left.

**Figure 13.**Comparison of mixing efficiency between experiments and simulations for the single-, double- and triple-U mixers undergoing counterclockwise rotation in the range 120–1200 rpm (360–900 rpm for experiments).

**Figure 14.**Comparison of mixing efficiency between experiments and simulations for the single-, double- and triple-U mixers undergoing clockwise rotation in the range 120–1200 rpm (360–900 rpm for experiments).

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**MDPI and ACS Style**

Hsu, C.-W.; Shih, P.-T.; Chen, J.M.
Enhancement of Fluid Mixing with U-Shaped Channels on a Rotating Disc. *Micromachines* **2020**, *11*, 1110.
https://doi.org/10.3390/mi11121110

**AMA Style**

Hsu C-W, Shih P-T, Chen JM.
Enhancement of Fluid Mixing with U-Shaped Channels on a Rotating Disc. *Micromachines*. 2020; 11(12):1110.
https://doi.org/10.3390/mi11121110

**Chicago/Turabian Style**

Hsu, Chi-Wei, Po-Tin Shih, and Jerry M. Chen.
2020. "Enhancement of Fluid Mixing with U-Shaped Channels on a Rotating Disc" *Micromachines* 11, no. 12: 1110.
https://doi.org/10.3390/mi11121110