# Quantifying Uniform Droplet Formation in Microfluidics Using Variational Mode Decomposition

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Droplet Simulation

#### 2.2. Signal Decomposition

## 3. Results

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EMD | Emprirical Mode Decomposition |

VMD | Variational Mode Decomposition |

IMF | Intrinsic Mode Functions |

## Appendix A. Variational Mode Decomposition

## References

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**Figure 1.**Visualization of two-phase fluid and flow velocities in a microchannel where droplets generated from right and left drop-makers meet at the center of the channel. Color Hue represents direction of the velocity, Saturation represents the volume fraction of water, and brightness Value represents the magnitude of the velocity.

**Figure 2.**Velocity and decomposition at one position as a function of time. (

**a**) Magnitude of velocity from 12 to 200 ms (

**b**) for a 4 × 4 $\mathsf{\mu}$m${}^{2}$ bin in the center of the channel; (

**c**) First intrinsic mode function (IMF 1) of the decomposed velocity signal showing the fastest oscillations in the original signal; (

**d**) IMF 2, the second largest oscillations heavily affected by the initiation of flow; (

**e**) IMF 3 showing the bulk of the original velocity signal; (

**f**) IMF 4 slower oscillations with a transition at around 20 ms corresponding to the initiation of flow; (

**g**) IMF 5 is the slowest oscillation of the flow with an average close to velocity of the outer fluid at 40 mm/s, and a spike early in the signal at time briefly after the initiation; (

**h**) Residual of the signal.

**Figure 3.**Velocity and decomposition in the water inlet as a function of time.

**Left**: the velocity signal (

**a**) and decomposition into intrinsic mode functions (

**b**–

**f**) for the left channel of Figure 1 (Supplementary Materials) demonstrates various large fluctuations.

**Right**: velocity signal (

**g**) and decomposition (

**h**–

**l**) for the right channel with a single large fluctuation.

**Figure 4.**Velocity signal (

**a**) taken at a single position averaged for two different resolutions, 4 × 4 $\mathsf{\mu}$m${}^{2}$ (blue circles) and 20 × 20 $\mathsf{\mu}$m${}^{2}$ (orange asterisk); (

**b**–

**g**) the respective Intrinsic mode functions; (

**h**) normalized maximum cross correlation between the IMFs of the 4 × 4 $\mathsf{\mu}$m${}^{2}$ (left) and 20 × 20 $\mathsf{\mu}$m${}^{2}$ (right) with a 1 × 1 $\mathsf{\mu}$m${}^{2}$ region. Color bar and relative size represent the strength of the correlations.

**Figure 5.**Velocity signal and decomposition at one position as a function of time. (

**a**) magnitude of velocity from 12 to 200 ms (

**b**) for a 4 × 4 $\mathsf{\mu}$m${}^{2}$ (blue circles) and 20 × 20 $\mathsf{\mu}$m${}^{2}$ (orange asterisk) window size at the center of the channel; (

**c**) first intrinsic mode function (IMF 1); (

**d**) IMF 2 where the larger window decomposition swaps with IMF 3; (

**e**) IMF 3 where the larger window swaps with IMF 2; (

**f**) IMF 4; (

**g**) IMF 5, (

**h**) Residual.

**Table 1.**Maximum normalized cross correlation averaged over 12 positions with areas of 4 × 4 $\mathsf{\mu}$m${}^{2}$ region. The IMF 1 to 5 on the columns are from the 1 × 1 $\mathsf{\mu}$m${}^{2}$ and the rows are from the 4 × 4 $\mathsf{\mu}$m${}^{2}$.

IMF 1 | IMF 2 | IMF 3 | IMF 4 | IMF 5 | |
---|---|---|---|---|---|

IMF 1 | 1.0000 | 0.1003 | 0.0293 | 0.0239 | 0.0038 |

IMF 2 | 0.1003 | 0.9999 | 0.0808 | 0.0447 | 0.0052 |

IMF 3 | 0.0293 | 0.0805 | 0.9998 | 0.1500 | 0.0054 |

IMF 4 | 0.0240 | 0.0446 | 0.1487 | 0.9997 | 0.0125 |

IMF 5 | 0.0038 | 0.0052 | 0.0054 | 0.0124 | 1.0000 |

**Table 2.**Maximum normalized cross correlation averaged over 12 positions with areas of 20 × 20 $\mathsf{\mu}$m${}^{2}$. The IMF 1 to 5 on the columns are from the 1 × 1 $\mathsf{\mu}$m${}^{2}$ and the rows are from the 20 × 20 $\mathsf{\mu}$m${}^{2}$ region.

IMF 1 | IMF 2 | IMF 3 | IMF 4 | IMF 5 | |
---|---|---|---|---|---|

IMF 1 | 0.9090 | 0.1622 | 0.0326 | 0.0240 | 0.0037 |

IMF 2 | 0.1663 | 0.4976 | 0.3705 | 0.0486 | 0.0051 |

IMF 3 | 0.0332 | 0.2301 | 0.5048 | 0.3761 | 0.0053 |

IMF 4 | 0.0263 | 0.0518 | 0.2601 | 0.6468 | 0.0124 |

IMF 5 | 0.0039 | 0.0061 | 0.0061 | 0.0093 | 0.9997 |

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**MDPI and ACS Style**

Izaguirre, M.; Nearhood, L.; Parsa, S.
Quantifying Uniform Droplet Formation in Microfluidics Using Variational Mode Decomposition. *Fluids* **2022**, *7*, 174.
https://doi.org/10.3390/fluids7050174

**AMA Style**

Izaguirre M, Nearhood L, Parsa S.
Quantifying Uniform Droplet Formation in Microfluidics Using Variational Mode Decomposition. *Fluids*. 2022; 7(5):174.
https://doi.org/10.3390/fluids7050174

**Chicago/Turabian Style**

Izaguirre, Michael, Luke Nearhood, and Shima Parsa.
2022. "Quantifying Uniform Droplet Formation in Microfluidics Using Variational Mode Decomposition" *Fluids* 7, no. 5: 174.
https://doi.org/10.3390/fluids7050174