# High Mixing Efficiency by Modulating Inlet Frequency of Viscoelastic Fluid in Simplified Pore Structure

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Schemes

#### 2.1. Computational Model of T-Junction Micromixer

_{1}and I

_{2}from both sides of the junction and convergence begins at the junction point X. Then, the mixing fluid flows through a narrow converging channel that feeds into an expanded mixing channel where the two fluids will mix with each other. The component concentration at a cross section S in the mixing channel is monitored to determine the mixing efficiency. The length and width of the inlet channels are L

_{0}= 20 and W

_{0}= 1, respectively, while the converging channel has dimensions of L

_{1}= 6 and W

_{1}= 0.5 and the mixing channel has L

_{2}= 20 and W

_{2}= 6. The micromixer structure has a depth of d = 1, and the outlet is set at the end of the mixing channel.

#### 2.2. Governing Equations

**U**represents the velocity vector of flow; p is the pressure; $\rho $ is the density; and $\mu $ is the dynamic viscosity of the fluid.

_{B}is the Boltzmann constant; T is the absolute temperature of the fluid; $f(r)$ is the Peterlin function; $\mathbf{C}$ is the conformation tensor of polymer molecules; and $\mathbf{I}$ is the Kronecker symbol for the unit tensor. The constitutive equation is further derived from Equation (5) as:

_{i}is the center velocity of the fluid at the inlet channel; ρ is the fluid density; and C is the component concentration in the fluid. We defined the dimensionless Reynolds number Re and Schmidt number Sc as:

_{p}is the solute kinetic viscosity; η

_{s}is the solvent kinetic viscosity. Thus, the dimensionless N–S equation for viscoelastic fluid flow and the conformation tensor transport equation are modified as:

^{−8}. In addition, in the viscoelastic fluid case, the dimensionless relaxation time λ was set at 5.0, the dimensionless solution viscosity was set at 0.4, and dimensionless solvent viscosity was set at 0.6.

#### 2.3. Numerical Methods

_{1}was 0 for the fluid entering through inlet I

_{1}, while C

_{2}= 1 for the fluid entering through inlet I

_{2}. The driving pressure is P

_{1}at inlet I

_{1}and P

_{2}at inlet I

_{2}. The no-slip condition was imposed on all channel surfaces, and a fully developed flow condition was used at the mixer outlet.

^{−3}. The convection terms in Equations (14) and (16) were discretized by the QUICK scheme, while the bounded MINMOD scheme was used to discretize the convection terms in Equation (17). Pressure–velocity coupling was handled by the PISO algorithm.

#### 2.4. The Definition of Mixing Efficiency

_{1}= 0 and C

_{2}= 1, a full mixing of the two fluids will result in a concentration of 0.5 when two averaged flow rates are the same. The degree of mixing is calculated by taking the standard deviation of concentration at all meshed points in cross-section S, defined as:

## 3. Results and Discussion

#### 3.1. Mixing in Condition of Constant Pressure

_{1}and I

_{2}are under the same constant pressure P

_{0}= 3500. Figure 2a,b compare the component concentration distribution in the Newtonian fluid and viscoelastic fluid cases after the fluid has flowed in the micromixer for a long time (t = 120). The color map indicates the normalized component concentration (red represents C = 1 and blue represents C = 0). It can be seen that the two injected fluids are well separated in the converging channel. In the mixing channel, green appears at the center area along the flow direction. The concentration gradually changed between 0 and 1 and reached 0.5 in the center, indicating a high mixing here. Compared to the Newtonian fluid case, the mixing area is larger in the viscoelastic fluid. Since the simulation conditions for two fluids were identical except that the viscoelastic fluid had a nonzero elasticity, the improvement in mixing must originate from elastic stress in the viscoelastic fluid.

_{s}= 12 below the channel expansion point. For both cases, M increases until it becomes stable starting from t = 80, but their magnitudes differ: M plateaus at 0.15 for the Newtonian fluid case and 0.25 for the viscoelastic fluid case, as shown in Figure 4. This quantitatively shows that mixing was improved in viscoelastic fluid case.

#### 3.2. Mixing in Single-Side Pressure Oscillation

_{1}at inlet I

_{1}with a sinusoidal factor while keeping the pressure P

_{2}at inlet I

_{2}constant:

_{0}was the same as that in the constant pressure case, 3500. The change in the degree of mixing in cross-section S within one period was studied when pressures in Equation (21) were applied to the viscoelastic fluid case. Under the above driving pressure, the concentration distributions at t = 120 in the Newtonian fluid and viscoelastic fluid cases are shown in Figure 5, respectively. The transition region between the blue (low concentration) and red (high concentration) fluids expanded significantly when the pressure was modulated compared to when the pressure was constant, indicating that pressure modulation greatly improves mixing efficiency. Meanwhile, the concentrations in the converging channel were no longer uniformly distributed along the channel due to the pressure changes over time. At times P

_{1}< P

_{2}, higher concentration fluid from inlet I

_{2}will enter the converging channel. The converse happens when P

_{1}> P

_{2}. In other words, the two fluids will alternatively enter the mixing channel in larger amounts. Due to the large width of the subsequent expanded channel, the small amounts of fluids introduced during one pressure cycle will start to expand and convect with each other. In this way, better mixing can be realized. In addition, we can also see that the larger discontinuous portion of fluid alternated more in the converging channel for the viscoelastic fluid case compared to the Newtonian fluid case, which indicates a higher mixing efficiency in the viscoelastic fluid. This was confirmed by investigating the degree of mixing at the cross-section S for both fluids. Figure 6 compares the mixing of a viscoelastic fluid with a different relaxation time and a Newtonian fluid. Mixing increased with the relaxation time and became stable at t = 80 for all fluids. The Newtonian fluid had the lowest degree of mixing of 0.62. Mixing gradually increased from 0.63 to 0.70 as the relaxation time λ increased from 1 to 10. This is because a higher relaxation time indicates a larger elastic stress, which leads to more convection in the fluid. Compared to the condition of constant pressure, the degree of mixing increased from 0.15 to 0.61 for the Newtonian fluid case under modulated pressure and from 0.25 to 0.68 for a viscoelastic fluid with λ of 5.

_{1}at t = 119.2 (Figure 7a,c) and mainly from inlet I

_{2}at t = 119.7 (Figure 7b,d). The streamlines were smooth, and no angular velocity was observed in the Newtonian fluid case. However, the viscoelastic fluid formed flow vortexes alternating between the right and left sides of convergence points at times t = 119.2 and t = 119.7, respectively. The fluctuation in flow vortexes indicates enhanced convection between two fluids, which in turn leads to better mixing.

#### 3.3. Mixing in Double-Sided Pressure Oscillation

_{1}and P

_{2}with sinusoidal signals of the same amplitude A and frequency f, but with independent phase delays φ

_{1}and φ

_{2}:

_{1}− φ

_{2}, the phase difference between the two pressures.

_{2}is higher than that at inlet I

_{1,}and more of the red fluid will be injected. At a lower modulating frequency, the difference in the amount of fluid flowing into the converging channel from each side is larger in each cycle; this can be observed in the converging channel and results in a higher mixing efficiency overall. The final degrees of mixing, which increase from 0.61 to 0.90 as the pressure oscillation frequency decreases from f = 2 to f = 0.5, are shown in Figure 11.

_{0}in Equation (22) is varied between 3500 to 6000 while setting f = 1 and Δφ = π. By increasing P

_{0}, the final degree of mixing in the viscoelastic fluid case also increases from 0.82 to 0.91, as shown in Figure 12.

## 4. Experimental Results

_{0}= 2 mm and W

_{0}= 100 µm, respectively. The converging channel was L

_{1}= 600 µm in length and W

_{1}= 50 µm in width. The mixing channel was L

_{2}= 2 mm in length and W

_{2}= 600 µm in width. The entire micromixer structure had a depth of d = 100 µm, and the outlet was directly at the end of the mixing channel.

^{TM}4400), we fixed the flow rate at 500 µL/h on one inlet, and modulated the flow rate on the other side under three different conditions: (i) Constant flow rate of 500 µL/h; (ii) alternating flow rate between 0 and 1000 µL/h at a frequency f = 0.5 Hz and duty cycle of 50%; (iii) alternating flow rate between 0 and 1000 µL/h at a frequency f = 0.1 Hz and duty cycle of 50%. The recorded flow pattern is shown in Figure 13b. The color map represents the Rhodamine B concentration, which is normalized from 0 to 1: Yellow denotes a concentration of 1 and blue, a concentration of 0. For case (i), the flow rates at both inlets were the same constant value and the mixing region was narrow, indicating a small degree of mixing. For case (ii), the mixing region was increased when the flow rate on one side alternated at f = 0.5 Hz. The mixing region was further increased when the frequency decreased to 0.1 Hz in case (iii). According to the normalized concentrations of Rhodamine B, the degree of mixing at different flow distances along the channel is calculated using Equation (20) and plotted in Figure 13c. The origin indicates the point at which the fluid enters the mixing channel. The degree of mixing gradually increases with the flow distance, and stabilizes at 0.20 for case (i), 0.25 for case (ii), and 0.37 for case (iii). This experimentally demonstrates the enhancement in mixing when the flow is modulated on one side compared to having a constant driving flow at both sides, which agrees well with the simulation results. This also confirms that mixing is increased as the alternating frequency decreases.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Stroock, A.D.; Dertinger, S.K.W.; Ajdari, A.; Mezić, I.; Stone, H.A.; Whitesides, G.M. Chaotic mixer for microchannels. Science
**2002**, 295, 647–651. [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] - 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] - Ehrfeld, W.; Hessel, V.; Löwe, H. Microreactors—New Technology for Modern Chemistry; Wiley-VCH: Weinheim, Germany, 2000; p. 288. ISBN 3-527-29590-9. [Google Scholar]
- 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, 64102. [Google Scholar] [CrossRef] [PubMed][Green Version] - 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-Newtonian Fluid Mech.
**2016**, 231, 68–78. [Google Scholar] [CrossRef] - Ye, Y.; Chiogna, G.; Cirpka, O.A.; Grathwohl, P.; Rolle, M. Experimental investigation of transverse mixing in porous media under helical flow conditions. Phys. Rev. E
**2016**, 94, 13113. [Google Scholar] [CrossRef] [PubMed] - Ouyang, Y.; Xiang, Y.; Zou, H.; Chu, G.; Chen, J. Flow characteristics and micromixing modeling in a microporous tube-in-tube microchannel reactor by CFD. Chem. Eng. J.
**2017**, 321, 533–545. [Google Scholar] [CrossRef] - Li, J.S.; Li, Q.; Cai, W.H.; Li, F.C.; Chen, C.Y. Mixing Efficiency via Alternating Injection in a Heterogeneous Porous Medium. J. Mech.
**2018**, 34, 167–176. [Google Scholar] [CrossRef] - Cai, J.; Wei, W.; Hu, X.; Liu, R.; Wang, J. Fractal characterization of dynamic fracture network extension in porous media. Fractals
**2017**, 25. [Google Scholar] [CrossRef] - Elvira, K.S.; Solvas, X.C.; Wootton, R.C.R.; Demello, A.J. The past, present and potential for microfluidic reactor technology in chemical synthesis. Nat. Chem.
**2013**, 5, 905–915. [Google Scholar] [CrossRef] [PubMed] - Floquet, C.F.A.; Sieben, V.J.; MacKay, B.A.; Mostowfi, F. Determination of boron concentration in oilfield water with a microfluidic ion exchange resin instrument. Talanta
**2016**, 154, 304–311. [Google Scholar] [CrossRef] [PubMed] - Liu, R.; Jiang, Y.; Li, B.; Wang, X. A fractal model for characterizing fluid flow in fractured rock masses based on randomly distributed rock fracture networks. Comput. Geotech.
**2015**, 65, 45–55. [Google Scholar] [CrossRef][Green Version] - Knight, J.B.; Vishwanath, A.; Brody, J.P.; Austin, R.H. Hydrodynamic focusing on a silicon chip: Mixing nanoliters in microseconds. Phys. Rev. Lett.
**1998**, 80, 3863–3866. [Google Scholar] [CrossRef] - Squires, T.M.; Quake, S.R. Microfluidics: Fluid physics at the nanoliter scale. Rev. Mod. Phys.
**2005**, 77, 977–1026. [Google Scholar] [CrossRef][Green Version] - 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] - Wu, J.W.; Xia, H.M.; Zhang, Y.Y.; Zhu, P. Microfluidic mixing through oscillatory transverse perturbations. Mod. Phys. Lett. B
**2018**, 32. [Google Scholar] [CrossRef] - Stone, H.A.; Stroock, A.D.; Ajdari, A. Engineering flows in small devices: Microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech.
**2004**, 36, 381–411. [Google Scholar] [CrossRef] - Schönfeld, F.; Hessel, V.; Hofmann, C. An optimised split-and-recombine micro-mixer with uniform ‘chaotic’ mixing. Lab Chip
**2004**, 4, 65–69. [Google Scholar] [CrossRef] [PubMed] - Sivashankar, S.; Agambayev, S.; Mashraei, Y.; Li, E.Q.; Thoroddsen, S.T.; Salama, K.N. A “twisted” microfluidic mixer suitable for a wide range of flow rate applications. Biomicrofluidics
**2016**, 10. [Google Scholar] [CrossRef] [PubMed] - Lee, N.Y.; Yamada, M.; Seki, M. Development of a passive micromixer based on repeated fluid twisting and flattening, and its application to DNA purification. Anal. Bioanal. Chem.
**2005**, 383, 776–782. [Google Scholar] [CrossRef] [PubMed] - Jafari, O.; Rahimi, M.; Kakavandi, F.H. Liquid-liquid extraction in twisted micromixers. Chem. Eng. Process.
**2016**, 101, 33–40. [Google Scholar] [CrossRef] - Lin, M.X.; Hyun, K.A.; Moon, H.S.; Sim, T.S.; Lee, J.G.; Park, J.C.; Lee, S.S.; Jung, H.I. Continuous labeling of circulating tumor cells with microbeads using a vortex micromixer for highly selective isolation. Biosens. Bioelectron.
**2013**, 40, 63–67. [Google Scholar] [CrossRef] [PubMed] - Bensaid, S.; Deorsola, F.A.; Marchisio, D.L.; Russo, N.; Fino, D. Flow field simulation and mixing efficiency assessment of the multi-inlet vortex mixer for molybdenum sulfide nanoparticle precipitation. Chem. Eng. J.
**2014**, 238, 66–77. [Google Scholar] [CrossRef] - Niu, X.; Lee, Y.K. Efficient spatial-temporal chaotic mixing in microchannels. J. Micromech. Microeng.
**2003**, 13, 454–462. [Google Scholar] [CrossRef][Green Version] - 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 Stat. Nonlinear Soft Matter Phys.
**2004**, 69. [Google Scholar] [CrossRef] [PubMed][Green Version] - Simonnet, C.; Groisman, A. Chaotic mixing in a steady flow in a microchannel. Phys. Rev. Lett.
**2005**, 94. [Google Scholar] [CrossRef] [PubMed] - Cai, G.; Xue, L.; Zhang, H.; Lin, J. A review on micromixers. Micromachines
**2017**, 8, 274. [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] - Gan, H.Y.; Lam, Y.C.; Nguyen, N.T. Polymer-based device for efficient mixing of viscoelastic fluids. Appl. Phys. Lett.
**2006**, 88. [Google Scholar] [CrossRef] - Xu, K.; Liang, T.; Zhu, P.; Qi, P.; Lu, J.; Huh, C.; Balhoff, M. A 2.5-D glass micromodel for investigation of multi-phase flow in porous media. Lab Chip
**2017**, 17, 640–646. [Google Scholar] [CrossRef] [PubMed] - Needham, R.B.; Doe, P.H. Polymer Flooding Review. J. Pet. Technol.
**1987**, 39, 1503–1507. [Google Scholar] [CrossRef] - Xu, K.; Zhu, P.; Tatiana, C.; Huh, C.; Balhoff, M. A microfluidic investigation of the synergistic effect of nanoparticles and surfactants in macro-emulsion based EOR. In Proceedings of the SPE—DOE Improved Oil Recovery Symposium Proceedings, Tulsa, OK, USA, 11–13 April 2016. [Google Scholar]
- Xu, K.; Zhu, P.; Colon, T.; Huh, C.; Balhoff, M. A microfluidic investigation of the synergistic effect of nanoparticles and surfactants in macro-emulsion-based enhanced oil recovery. SPE J.
**2017**, 22, 459–469. [Google Scholar] [CrossRef] - Olajire, A.A. Review of ASP EOR (alkaline surfactant polymer enhanced oil recovery) technology in the petroleum industry: Prospects and challenges. Energy
**2014**, 77, 963–982. [Google Scholar] [CrossRef] - Jha, B.; Cueto-Felgueroso, L.; Juanes, R. Fluid mixing from viscous fingering. Phys. Rev. Lett.
**2011**, 106. [Google Scholar] [CrossRef] [PubMed] - Jha, B.; Cueto-Felgueroso, L.; Juanes, R. Synergetic fluid mixing from viscous fingering and alternating injection. Phys. Rev. Lett.
**2013**, 111. [Google Scholar] [CrossRef] [PubMed] - James, D.F.; McLaren, D.R. The laminar flow of dilute polymer solutions through porous media. J. Fluid Mech.
**1975**, 70, 733–752. [Google Scholar] [CrossRef] - Peters, E.C.; Petro, M.; Svec, F.; Fréchet, J.M.J. Molded Rigid Polymer Monoliths as Separation Media for Capillary Electrochromatography. 1. Fine Control of Porous Properties and Surface Chemistry. Anal. Chem.
**1998**, 70, 2288–2295. [Google Scholar] [CrossRef] [PubMed] - Rodriguez, S.; Romero, C.; Sargenti, M.L.; Müller, A.J.; Sáez, A.E.; Odell, J.A. Flow of polymer solutions through porous media. J. Non-Newtonian Fluid Mech.
**1993**, 49, 63–85. [Google Scholar] [CrossRef] - Stavland, A.; Jonsbråten, H.C.; Lohne, A.; Moen, A.; Giske, N.H. Polymer flooding—Flow properties in porous media versus rheological parameters. In Proceedings of the 72nd European Association of Geoscientists and Engineers Conference and Exhibition 2010: A New Spring for Geoscience, Barcelona, Spain, 14–17 June 2010; pp. 3292–3306. [Google Scholar]
- Liu, R.; Li, B.; Jiang, Y. Critical hydraulic gradient for nonlinear flow through rock fracture networks: The roles of aperture, surface roughness, and number of intersections. Adv. Water Resour.
**2016**, 88, 53–65. [Google Scholar] [CrossRef] - Liu, R.; Li, B.; Jiang, Y. A fractal model based on a new governing equation of fluid flow in fractures for characterizing hydraulic properties of rock fracture networks. Comput. Geotech.
**2016**, 75, 57–68. [Google Scholar] [CrossRef] - Li, Z.; Kim, S.J. Pulsatile micromixing using water-head-driven microfluidic oscillators. Chem. Eng. J.
**2017**, 313, 1364–1369. [Google Scholar] [CrossRef] - Glasgow, I.; Aubry, N. Enhancement of microfluidic mixing using time pulsing. Lab Chip
**2003**, 3, 114–120. [Google Scholar] [CrossRef] [PubMed] - Krupa, K.; Nunes, M.I.; Santos, R.J.; Bourne, J.R. Characterization of micromixing in T-jet mixers. Chem. Eng. Sci.
**2014**, 111, 48–55. [Google Scholar] [CrossRef] - Gao, Z.; Han, J.; Bao, Y.; Li, Z. Micromixing efficiency in a T-shaped confined impinging jet reactor. Chin. J. Chem. Eng.
**2015**, 23, 350–355. [Google Scholar] [CrossRef] - Liu, Z.; Guo, L.; Huang, T.; Wen, L.; Chen, J. Experimental and CFD studies on the intensified micromixing performance of micro-impinging stream reactors built from commercial T-junctions. Chem. Eng. Sci.
**2014**, 119, 124–133. [Google Scholar] [CrossRef] - Oualha, K.; Ben Amar, M.; Michau, A.; Kanaev, A. Observation of cavitation in exocentric T-mixer. Chem. Eng. J.
**2017**, 321, 146–150. [Google Scholar] [CrossRef] - Li, B.; Liu, R.; Jiang, Y. Influences of hydraulic gradient, surface roughness, intersecting angle, and scale effect on nonlinear flow behavior at single fracture intersections. J. Hydrol.
**2016**, 538, 440–453. [Google Scholar] [CrossRef] - Giesekus, H. A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility. J. Non-Newtonian Fluid Mech.
**1982**, 11, 69–109. [Google Scholar] [CrossRef]

**Figure 2.**Concentration distribution in the Newtonian fluid case (

**a**) and viscoelastic fluid case (

**b**) under constant driving pressure at both inlets.

**Figure 3.**Streamlines and angular velocity in the Newtonian fluid case (

**a**) and viscoelastic fluid case (

**b**) under constant driving pressure at both inlets.

**Figure 4.**Degree of mixing in the Newtonian fluid case (

**black line**) and viscoelastic fluid case (

**red line**) under constant driving pressure at both inlets.

**Figure 5.**Concentration distribution in the Newtonian fluid case (

**a**) and viscoelastic fluid case (

**b**) under modulated pressure at the inlet I

_{1}and constant pressure at the inlet I

_{2}.

**Figure 6.**Degree of mixing in the Newtonian fluid case and viscoelastic fluid case with different relaxation time under modulating driving pressure at the inlet I

_{1}and constant pressure at the inlet I

_{2.}

**Figure 7.**Streamlines and angular velocity in the Newtonian fluid case (

**a**,

**c**) and viscoelastic fluid case (

**b**,

**d**) under modulated pressure at the inlet I

_{1}and constant pressure at the inlet I

_{2}.

**Figure 8.**Concentration distribution in viscoelastic fluid case under modulated pressure at both inlets with phase difference (

**a**) 0, (

**b**) π/6, (

**c**) π/3, (

**d**) π/2, (

**e**) 2π/3, (

**f**) 5π/6, and (

**g**) π.

**Figure 9.**Degree of mixing in viscoelastic fluid case under modulated pressure at both inlets with phase difference from 0 to π.

**Figure 10.**Concentration distribution in viscoelastic fluid case under modulated pressure at both inlets with modulating frequency of f = 1 (

**a**) and f = 1.5 (

**b**).

**Figure 11.**Degree of mixing in viscoelastic fluid case under modulated pressure at both inlets with different modulating frequencies.

**Figure 12.**Degree of mixing in viscoelastic fluid case under modulated pressure at both inlets with different amplitudes.

**Figure 13.**(

**a**) The fabricated microfluidic chip for the T-mixer; (

**b**) Recorded flow pattern of the glycerol solution with/without Rhodamine B on left/right side and the flow rate for right side (

**i**) remains constant at 500 µL/h; (

**ii**) alternates with frequency f = 0.5 Hz; (

**iii**) alternates with frequency f = 0.1 Hz. (

**c**) The measured degree of mixing for case (

**i**–

**iii**).

© 2018 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.; 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.
https://doi.org/10.3390/pr6110210

**AMA Style**

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(11):210.
https://doi.org/10.3390/pr6110210

**Chicago/Turabian Style**

Zhang, Meng, Yunfeng Cui, Weihua Cai, Zhengwei Wu, Yongyao Li, Fengchen Li, and Wu Zhang. 2018. "High Mixing Efficiency by Modulating Inlet Frequency of Viscoelastic Fluid in Simplified Pore Structure" *Processes* 6, no. 11: 210.
https://doi.org/10.3390/pr6110210