# Controlling Shapes in a Coaxial Flow Focusing Microfluidic Device: Experiments and Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{2}plasma corona discharge (Corona SB, Black Hole Lab) before contact bonding.

## 3. Results

#### 3.1. Mathematical Model

_{int}is the applied flow rate for the inner flow, x

_{i}is the minor diameter, and y

_{i}the major diameter.

_{t}is the total flow rate applied to the whole system (Q

_{t}= Q

_{int}+ Q

_{ext}), and X is the side of the square section of our microchannel.

_{i}and y

_{i}that depends on the flow rate ratio (Q

_{int}/Q

_{t}):

_{i},y

_{i}) as a function of the flow rate ratio Q

_{int}/Q

_{t}with a predicted slope of m

_{th}= 3.64 × 10

^{8}μm

^{4}. This value should be compared with the one obtained from experimental results.

#### 3.2. Experimental Results

_{i}) and major axis (y

_{i}) versus the flow rate ratio. From the data obtained, the difference in changing inner flow cross-section shape could be quantitatively observed. While the minor-diameter (x

_{i}) of the initial elliptical cross-section shape increases rapidly, the major-diameter (y

_{i}) showed a slower increase; however, this was significant and could not be considered to be constant. Moreover, both diameters were mathematically related to each other and the flow rate ratio. Therefore, the mathematical model developed in the previous section allowed us to better characterize the geometrical shape of the coaxial inner flow (Equation (3)).

_{i},y

_{i}) versus the flow rate ratio has been also plotted (Figure 6). This experimental data shows a linear tendency, as expected from the theoretical model, with a slope of m

_{exp}= 1.41 × 10

^{8}μm

^{4}(Figure 6). This value is slightly smaller from the one obtained from the theory (m

_{th}= 3.64 × 10

^{8}μm

^{4}directly derived from Equation (3)) The difference between the theoretical and experimental slope comes from the behavior of the system in the lower regime (Q

_{int}<< Q

_{t}), where the height y

_{i}of the focused fluid is not accurately defined (see the highly dispersed data presented in Figure 5 for this lower regime). In this regime where the sample focused flow rate is small compared to the focusing fluids, the stability of the flow is compromised and then, the condition of elliptical cross-section is disturbed. That makes the correct measurement of the ellipse dimensions impractical and thus gives rise to a considerably big error, which explains the discrepancy between experimental and theoretical data.

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Design of the coaxial flow focusing microfluidic device: (

**a**) Representation of a general (x,z)-plane view of the design; (

**b**) longitudinal section of the device (y,z). The coaxial microchannel is circled in green, the observation zone, where the inner flow is formed, is marked in red; (

**c**) A 3D amplification of the junction where the external flow and inner flow are combined (marked in black in Figure 1a). The coaxial inner flow is represented with a green cylinder. The diagrams are not to scale.

**Figure 2.**(

**left**) Image of the polydimethylsiloxane (PDMS) microfluidic device structure once aligned and bounded. There are a total of three coaxial systems in a single piece of PDMS, which is similar to the size of a one cent coin. (

**right**) AutoCAD design of the microfluidic system with representative dimensions shown in mm. The region where the coaxial inner flow is formed has 2 mm of length.

**Figure 3.**Representation of the Poiseuille velocity profile of our microfluidic system in the (x,z)-plane. The inner flow is represented in blue and the outer flow in white. A single parabolic profile is the result of the miscibility of the fluids. As consequence, the pressure drop from point P2 to P1 of the total flow (inner and outer flow) and the pressure drop in the inner flow from P2 to P1 are the same.

**Figure 4.**Top: (

**a**) an example of 3D reconstruction with ImageJ software from the confocal microscope images. The blue feature corresponds to the walls of the microchannel where the coaxial inner flow (Q

_{int}) is formed, which is represented in green. Bottom: (

**b**–

**d**) There are three different examples of reconstructed cross sections (x,y-plane) of the inner flow at different flow ratios (b: 5 mL/min, c: 30 mL/min, d: 75 mL/min). Scale bar: 50 μm.

**Figure 5.**Plots of the behavior of the minor axis of the coaxial ellipse jet (x

_{i}) and major axis (y

_{i}) versus the flow rate ratio (Q

_{int}/Q

_{t}where Q

_{t}= Q

_{int}+ Q

_{ext}), shown as blue triangles and red diamonds respectively. The growth of both axes is not the same as the flow rate ratio is increased. The aspect ratio x

_{i}/y

_{i}of the inner flow varies by controlling Q

_{int}for a fixed Q

_{ext}. It is possible to change from an ellipse cross-section to a circle by increasing Q

_{int}.

**Figure 6.**Plot of function f(x

_{i}, y

_{i}) defined in Equation (3) for different values of Q

_{i}/Q

_{t}. The values of x

_{i}and y

_{i}are experimental measurements. The continuous line is the regression line of the data with a slope m

_{exp}= 1.41 × 10

^{8}μm

^{4}with an R

^{2}value of 9.55. The theoretical model predicted a slope of m

_{th}= 3.64 × 10

^{8}μm

^{4}.

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**MDPI and ACS Style**

Rodriguez-Trujillo, R.; Kim-Im, Y.-H.; Hernandez-Machado, A.
Controlling Shapes in a Coaxial Flow Focusing Microfluidic Device: Experiments and Theory. *Micromachines* **2020**, *11*, 85.
https://doi.org/10.3390/mi11010085

**AMA Style**

Rodriguez-Trujillo R, Kim-Im Y-H, Hernandez-Machado A.
Controlling Shapes in a Coaxial Flow Focusing Microfluidic Device: Experiments and Theory. *Micromachines*. 2020; 11(1):85.
https://doi.org/10.3390/mi11010085

**Chicago/Turabian Style**

Rodriguez-Trujillo, Romen, Yu-Han Kim-Im, and Aurora Hernandez-Machado.
2020. "Controlling Shapes in a Coaxial Flow Focusing Microfluidic Device: Experiments and Theory" *Micromachines* 11, no. 1: 85.
https://doi.org/10.3390/mi11010085