# On the Three Dimensional Interaction between Flexible Fibers and Fluid Flow

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## Abstract

**:**

## 1. Introduction

## 2. Three Dimensional Numerical Simulations

#### 2.1. Governing Equations

#### 2.2. Numerical Setup

#### 2.2.1. Flow Around a Sphere

#### 2.2.2. Convergence Study for Ball and Fiber System

#### 2.2.3. Parametric Study

## 3. Results

## 4. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**These plots show the change of drag force as a function of velocity and are computed for angles of ${45}^{\circ}$ (◯) and ${90}^{\circ}$ (□) for fibers of lenghts (

**a**) 2 cm and (

**b**) 4 cm.

**Figure 4.**The φ Deformation is considered to be measured in the $ZX$ plane (up and down). These angles are measured for (

**a**) ${45}^{\circ}$ and (

**b**) ${90}^{\circ}$ configuration angles for fiber lengths of 2 cm (◇) and 4 cm (□).

**Table 1.**Numerical values of the relative error for drag coefficients with respect to Mikhailov [27].

Mesh Number of Elements | ${\mathit{C}}_{\mathit{D}}$ | Relative Error |
---|---|---|

5847 | 0.473984641 | 1.97127 |

11,771 | 0.477832654 | 2.79912 |

26,030 | 0.496393657 | 6.79226 |

50,843 | 0.48586821 | 4.52786 |

155,780 | 0.469231214 | 0.94864 |

309,889 | 0.465496378 | 0.14514 |

Element Size Parameters | Value |
---|---|

Maximum element size | 0.00824 [m] |

Minimum element size | 0.001 [m] |

Maximum element growth rate | 1.13 |

Resolution of curvature | 0.5 |

Resolution of narrow regions | 0.8 |

Mesh Elements | ${\mathit{F}}_{\mathit{D}}$ | Relative Error |
---|---|---|

21,617 | 0.02097 | 0.0605 |

31,004 | 0.0221 | 0.0134 |

48,493 | 0.02251 | 0.0068 |

73,857 | 0.02282 | 0.00071022 |

Fiber Length | 2 cm | 2 cm | 4 cm | 4 cm |
---|---|---|---|---|

Angle | 45 deg | 90 deg | 45 deg | 90 deg |

${C}_{D}$ | 0.1092 | 0.0838 | 0.1231 | 0.08825 |

α | 1.984 | 1.98 | 1.977 | 1.966 |

**Table 5.**Percentage increase in 2 cm fiber length bending at $u=0.4510$ m/s relative to $u=0.009$ m/s for initial configurations of ${90}^{\circ}$ and ${45}^{\circ}$ for both 2-D and 3-D simulations.

Angle | 2D % | 3D % |
---|---|---|

${90}^{\circ}$ | $0.50908$ | $0.147211349$ |

${45}^{\circ}$ | $0.19281$ | $0.064313345$ |

**Table 6.**Percentage increase in 4 cm fiber length bending at $u=0.4510$ m/s relative to $u=0.009$ m/s for initial configurations of ${90}^{\circ}$ and ${45}^{\circ}$ for both 2-D and 3-D simulations.

Angle | 2D % | 3D % |
---|---|---|

${90}^{\circ}$ | $4.66557$ | $0.946241708$ |

${45}^{\circ}$ | $1.61049$ | $0.492157133$ |

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Nita, B.G.; Allaire, R.
On the Three Dimensional Interaction between Flexible Fibers and Fluid Flow. *Fluids* **2017**, *2*, 4.
https://doi.org/10.3390/fluids2010004

**AMA Style**

Nita BG, Allaire R.
On the Three Dimensional Interaction between Flexible Fibers and Fluid Flow. *Fluids*. 2017; 2(1):4.
https://doi.org/10.3390/fluids2010004

**Chicago/Turabian Style**

Nita, Bogdan G., and Ryan Allaire.
2017. "On the Three Dimensional Interaction between Flexible Fibers and Fluid Flow" *Fluids* 2, no. 1: 4.
https://doi.org/10.3390/fluids2010004