# Fluid Viscosity Measurement by Means of Secondary Flow in a Curved Channel

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of the Approach

#### 2.2. Microfluidic Device Fabrication

#### 2.3. The Tested Liquids

#### 2.4. Experimental Setup and Procedure

#### 2.5. Computational Fluid Dynamics (CFD)

#### 2.6. Secondary Flow Quantification

## 3. Results and Discussion

#### 3.1. Flow Patterns Analysis

#### 3.2. Determining Fluid Viscosity

#### 3.3. Methodology for Design of Microfluidic Viscometer

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A microfluidic chip from PMMA with U-shaped microchannel with a width w = 920 $\mathsf{\mu}$m, a height of 500 $\mathsf{\mu}$m, and a radius of curvature r = 10 mm was used for experimental investigation. The liquid with two different dyes was pumped into the medial and lateral inlets with the same flow rate. The angle $\theta $ was measured from the start of curvature.

**Figure 2.**Analysis of experimental image of flow pattern for water at Q = 5 mL/min by means of developed application. (

**a**) Automatic detection of curvature center and demonstration of selecting pixels set by the beam at fixed angle; (

**b**) hue distribution in the channel in 5 various cross-sections; (

**c**) dependence of switching index ($SI$, normalized standard deviation of selected pixels hue) on the angle $\theta $ with labeled extremes corresponding to various flow recirculation angles.

**Figure 3.**Results of computer simulation performed in application AnsysFluent illustrates appearance of secondary vortex in curved channel with water flow rate at 5 mL/min. Isosurface of the volumetric fraction of the liquid phase. The arrows show the velocity vector field. The flows at angle 50.5${}^{\circ}$ corresponding to the $S{I}_{90}$ are used for measuring viscosity.

**Figure 4.**Experimental (left pictures) and calculated (CFD) (center pictures) flow patterns for water at different $De$ numbers. On the right pictures are the distributions of $SI$ depending on the angle $\theta $ (the red line is the experiment, the blue line is the CFD). The inserts in the calculated flow patterns are the phase distribution contours in sections 0${}^{\circ}$, 90${}^{\circ}$, and 180${}^{\circ}$, which clearly display the secondary Dean flow in the U-microchannel. (

**a**) flow patterns for water at $De$ = 1.93; (

**b**) flow patterns for water at $De$ = 5.79; (

**c**) flow patterns for water at $De$ = 10.6; (

**d**) flow patterns for water at $De$ = 20.3.

**Figure 5.**Experimental flow patterns of the water and aqueous solutions of glycerol at Q = 6 mL/min.

**Figure 6.**The results of experimental study. (

**a**) Dependence of flow recirculation angle $S{I}_{90}$ on fluid flow rate; (

**b**) Dean number vs. flow recirculation angle; (

**c**) inverse Dean number vs. flow recirculation angle; (

**d**) viscosity vs. flow recirculation angle; (

**e**) relative deviation between the dynamic viscosity coefficient calculated according to (3) and the reference data [42] in the investigated range of the Dean number.

**Table 1.**Density and dynamic viscosity coefficient of the tested liquids at a temperature of 25 ${}^{\circ}$C.

Property | Water | 30 wt% | 50 wt% |
---|---|---|---|

Viscosity [mPa·s] | 0.94 ± 0.03 | 2.09 ± 0.03 | 5.11 ± 0.03 |

Density [kg·m${}^{-3}$] | 997 | 1068 | 1137 |

**Table 2.**Fitting coefficients (7), analyzed ranges of the $De$ numbers, and the minimum angle ${\theta}_{min}$.

$\mathit{SI}$ | $\mathit{a}\xb7{10}^{2}$ | $\mathit{b}\xb7{10}^{2}$ | ${\mathit{R}}^{2}$ | AAD | MAD | $\mathit{De}$ | ${\mathit{\theta}}_{\mathit{min}}$ |
---|---|---|---|---|---|---|---|

$S{I}_{90}$ | 6.787 | −1.604 | 0.999 | 0.56% | 1.25% | 5 ÷ 20 | 45${}^{\circ}$ |

$S{I}_{180}$ | 3.414 | −1.261 | 0.997 | 1.01% | 2.18% | 10 ÷ 30 | 80${}^{\circ}$ |

$S{I}_{270}$ | 2.843 | −2.283 | 0.992 | 1.35% | 3.18% | 15 ÷ 35 | 110${}^{\circ}$ |

$S{I}_{360}$ | 2.874 | −3.942 | 0.990 | 2.08% | 4.06% | 20 ÷ 45 | 125${}^{\circ}$ |

Mass | ${\mathit{\mu}}_{\mathit{exp}}$ | ${\mathit{\mu}}_{\mathit{rot}}$ | ${\mathit{\mu}}_{\mathit{ref}}$ [42] | ${\mathit{\epsilon}}_{\mathit{exp}}$ | ${\mathit{\epsilon}}_{\mathit{rot}}$ |
---|---|---|---|---|---|

Fraction | [mPa·s] | [mPa·s] | [mPa·s] | [%] | [%] |

0 | 0.8897 | 0.94 | 0.8927 | −0.34 | 5.3 |

0.3 | 2.081 | 2.09 | 2.124 | −2.0 | −1.1 |

0.5 | 4.864 | 5.11 | 5.004 | −2.8 | 1.9 |

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**MDPI and ACS Style**

Pryazhnikov, M.I.; Yakimov, A.S.; Denisov, I.A.; Pryazhnikov, A.I.; Minakov, A.V.; Belobrov, P.I.
Fluid Viscosity Measurement by Means of Secondary Flow in a Curved Channel. *Micromachines* **2022**, *13*, 1452.
https://doi.org/10.3390/mi13091452

**AMA Style**

Pryazhnikov MI, Yakimov AS, Denisov IA, Pryazhnikov AI, Minakov AV, Belobrov PI.
Fluid Viscosity Measurement by Means of Secondary Flow in a Curved Channel. *Micromachines*. 2022; 13(9):1452.
https://doi.org/10.3390/mi13091452

**Chicago/Turabian Style**

Pryazhnikov, Maxim I., Anton S. Yakimov, Ivan A. Denisov, Andrey I. Pryazhnikov, Andrey V. Minakov, and Peter I. Belobrov.
2022. "Fluid Viscosity Measurement by Means of Secondary Flow in a Curved Channel" *Micromachines* 13, no. 9: 1452.
https://doi.org/10.3390/mi13091452