# Fluid Flow Dynamics in Partially Saturated Paper

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials and Experiment Flow

#### 2.2. Modeling—Flow in Saturated Paper

#### 2.3. Numerical Method

## 3. Results

#### 3.1. Empirical Fluid Flow

#### 3.2. Numerical Fluid Flow

#### 3.3. Parametric Study

#### 3.4. Limiting Case

## 4. Discussion

#### 4.1. Insights into Fluid Behavior across Saturation Levels in µPADs

#### 4.2. Predictability and Applicability of the Numerical Model

#### 4.3. Impact of Pore Size and Fiber Orientation on Fluid Dynamics

#### 4.4. Validating the Numerical Model through Empirical Data

#### 4.5. Advancing µPAD Design: Implications from Fluid Dynamics Studies

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental model (on the

**left**) and laboratory setup (on the

**right**) for the flow dynamics in the paper.

**Figure 10.**Comparison for empirical data and numerical prediction for fluid flow in saturated paper.

Variable | Value | Description/Reference |
---|---|---|

${\psi}^{\prime}$ | 2.8 | Diffusion Coefficient of W41 ^{1} Filter Paper [20] |

${\varphi}_{dry}$ | 0.75 | Porosity of Dry W41 ^{1} Filter Paper |

$C/D$ | 9 | Parameter for LW Flow in W41 ^{1} Filter Paper [21] |

$v$ | $1\times {10}^{-6}$ ${\mathrm{m}}^{2}/\mathrm{s}$ | Kinematic Viscosity of Water ^{2} |

$\rho $ | $1000\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$ | Density of Water ^{2} |

^{1}Whatman Grade 41 Filter Paper.

^{2}Water Servers as Both Primary (Saturating) and Secondary (Loading) Fluid.

**Table 2.**Empirical Fluid Flow Rate in Whatman Grade 41 and Whatman Grade 4 Filter Papers [18].

Wetted Length (l) [mm] | 0.0 | 4.0 | 8.0 | 12.0 | 16.0 | 20.0 |

Time for Whatman 41 (t) [s] | 0.0 | 1.1 | 4.8 | 11.9 | 22.9 | 39.0 |

Time for Whatman 4 (t) [s] | 0.0 | 1.2 | 4.4 | 8.9 | 15.7 | 23.0 |

**Table 3.**Empirical Fluid Flow Rate in Whatman Grade 41 Filter Paper for Fiber Orientation in Cross-Machine Direction (CMD) and Machine Direction (MD) [18].

Wetted Length (l) [mm] | 0.0 | 4.0 | 8.0 | 12.0 | 16.0 | 20.0 |

Time for CMD (t) [s] | 0.0 | 1.1 | 4.8 | 11.9 | 22.9 | 39.0 |

Time for MD (t) [s] | 0.0 | 0.8 | 4.0 | 9.9 | 19.3 | 32.4 |

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**MDPI and ACS Style**

Kumar, A.; Hatayama, J.; Soucy, A.; Carpio, E.; Rahmani, N.; Anagnostopoulos, C.; Faghri, M.
Fluid Flow Dynamics in Partially Saturated Paper. *Micromachines* **2024**, *15*, 212.
https://doi.org/10.3390/mi15020212

**AMA Style**

Kumar A, Hatayama J, Soucy A, Carpio E, Rahmani N, Anagnostopoulos C, Faghri M.
Fluid Flow Dynamics in Partially Saturated Paper. *Micromachines*. 2024; 15(2):212.
https://doi.org/10.3390/mi15020212

**Chicago/Turabian Style**

Kumar, Ashutosh, Jun Hatayama, Alex Soucy, Ethan Carpio, Nassim Rahmani, Constantine Anagnostopoulos, and Mohammad Faghri.
2024. "Fluid Flow Dynamics in Partially Saturated Paper" *Micromachines* 15, no. 2: 212.
https://doi.org/10.3390/mi15020212