# Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometrical Design of the Micromixer

_{in}corresponding to 0.5 W for both the standard rectangular section micromixer as well as for the newly investigated design.

## 3. Numerical Model and Mixing Assessment

**u**(m∙s

^{−1}) is the velocity vector, ρ (kg∙m

^{3}) is the fluid density, η (kg∙m

^{−1}∙s

^{−1}) is the fluid viscosity, t (s) is the time, and p (Pa) is the pressure. The flow field equations are solved using a generalized minimal residual method (GMRES) iterative solver with a geometrical multigrid pre-conditioner and a Vanka algorithm for the pre- and post-smoothing. No-slip boundary conditions were set for the walls of the micromixer. A free tetrahedral mesh is used for the entire microchannel with no less than ~200,000 elements for all the geometries studied.

^{−3}) is the concentration of the species of interest, and D (m

^{2}∙s

^{−1}) is its diffusion constant, respectively. The same iterative numerical solver as for the Navier–Stokes equations is used, but the maximum element size in the mesh is constrained to less than 10 μm through the whole geometry to avoid the possible numerical errors that can be associated with this type of solution. Thus, for all the concentration simulations, the number of mesh elements in the numerical model has never been less than ~1,900,000. For all the simulations described in this work, we used the computational package COMSOL Multiphysics 5.1 (COMSOL Inc., Stockholm, Sweden) and its computational fluid dynamics/chemical engineering module. The accuracy of the numerical work employed for this type of flow has been previously validated against data obtained from other microfluidic devices [49,50]. In particular, when comparing model data with measured data on similar planar curved microchannels, as presented by Jiang et al. [50], we find mixing times to be less than 0.05 s for Dean numbers equal to or above 140, in agreement with the experimental results.

_{bins}of equal size regions. For a system with two components, the mixing index is defined as

_{1/j}and p

_{2/j}are the conditional probabilities for Components 1 and 2, respectively, to be located in bin j. They represent the fraction of Components 1 and 2, respectively, in each bin relative to the total. They are calculated as the ratio of the average bin grayscale intensity from the corresponding concentration image, normalized by the maximum intensity, i.e., 255 for grayscale image data. In this particular study, since we have two chemical species and the fluids used are incompressible the two conditional probabilities are related as p

_{2/j}= 1 − p

_{1/j}. As shown in Equation (4), the mixing index M is normalized by a factor of ln2, where 2 corresponds to the number of components. Thus, the mixing index will take the value M = 0 for completely segregated components, while it will assume the value M = 1 for the completely mixed case.

## 4. Results and Discussion

^{−3}and a viscosity η of 0.001 kg∙m

^{−1}∙s

^{−1}. The diffusion coefficient D was fixed to 1.0 × 10

^{−9}m

^{2}∙s

^{−1}corresponding to the diffusion values for most ions in aqueous solutions. For all the simulations, the working solutions were considered to be pure water introduced through one of the inlets and dyed water with a concentration c = 1 moL∙m

^{−3}introduced through the opposite inlet. The flow rates are maintained for both fluid components, with the same mean fluid velocity at both inlets. This means fluid velocity is varied from 0.0075 to 0.75 m∙s

^{−1}to span a range of Reynolds numbers corresponding to Re = 1–100.

^{−3}) and high (c = 1 moL∙m

^{−3}) concentration, respectively, left in the dyed fluid distribution map.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**3D and top views of the standard serpentine micromixer with a rectangular cross-section defined by W = 200 μm and H = 100 μm.

**Figure 2.**3D and top views of the new serpentine micromixer design employing non-rectangular cross-sections. As shown above, the orientation of the cross-section of the mixer is changed after each turn of the serpentine.

**Figure 3.**Velocity magnitude (cross-sectional maps) (

**top**) and concentration distribution (surface map) (

**bottom**) along the channel of a standard serpentine micromixer (R

_{in}= W/2 = 100 μm, and Re = 20).

**Figure 4.**Velocity magnitude (cross-sectional maps) (

**top**) and concentration distribution (surface map) (

**bottom**) along the channel of a non-rectangular cross-section serpentine micromixer (R

_{in}= W/2 = 100 μm, and Re = 20).

**Figure 5.**Transversal concentration distribution at various positions along the channel for (

**a**) a standard serpentine micromixer and (

**b**) a serpentine micromixer with a non-rectangular cross-section (Re = 20). The concentration is mapped at both the midpoint and the end of each mixing unit.

**Figure 6.**Transversal concentration distribution at various positions along the channel for (

**a**) a standard serpentine micromixer and (

**b**) a serpentine micromixer with a non-rectangular cross-section (Re = 100).

**Figure 7.**Position dependence of the mixing index along serpentine micromixers at different Reynolds numbers: (

**a**) a standard serpentine microchannel and (

**b**) a modified channel.

**Figure 8.**Arrow plots of the velocity field for the two mixers design: (

**a**) a standard micromixer and (

**b**) a non-rectangular cross-section micromixer. Both sets of data are collected for the 2nd mixing cycle at Re = 20. The left plots are for data collected at the midpoint of the first leg of the mixing cycle, while the right images are the corresponding plots for the second leg of the mixing cycle.

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**MDPI and ACS Style**

Clark, J.; Kaufman, M.; Fodor, P.S.
Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section. *Micromachines* **2018**, *9*, 107.
https://doi.org/10.3390/mi9030107

**AMA Style**

Clark J, Kaufman M, Fodor PS.
Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section. *Micromachines*. 2018; 9(3):107.
https://doi.org/10.3390/mi9030107

**Chicago/Turabian Style**

Clark, Joshua, Miron Kaufman, and Petru S. Fodor.
2018. "Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section" *Micromachines* 9, no. 3: 107.
https://doi.org/10.3390/mi9030107