# Fluid-Solid Boundary Handling Using Pairwise Interaction Model for Non-Newtonian Fluid

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## Abstract

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## 1. Introduction

## 2. Related Works

## 3. Non-Newtonian Fluid Simulation

#### 3.1. Comparison of Newtonian and Non-Newtonian Fluid

#### 3.2. Control Equation with Cross Model

#### 3.3. Non-Newtonian Fluid Simulation with SPH

#### 3.4. Fluid-Solid Boundary Conditions

## 4. Fluid-Solid Boundary Handling of Non-Newtonian Fluids

#### 4.1. Pairwise Interaction Model

#### 4.1.1. Density Estimation of Solid Particle

#### 4.1.2. Fluid-solid Interaction Forces

#### 4.2. Friction Boundary Condition

#### 4.3. Optimization with Predictive-Corrective Method

## 5. Implementation and Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 7.**Friction boundary conditions of non-Newtonian fluid.

**Upper row**: No friction boundary conditions.

**Lower row**: Add friction boundary conditions.

**Figure 8.**Fluid volume dropping on a planar solid object.

**Upper row**: Newtonian fluid.

**Middle row**: non-Newtonian fluid.

**Lower row**: non-Newtonian fluid with our friction boundary condition.

Item | Experiment 1 | Experiment 2 | Experiment 3 |
---|---|---|---|

Simulation domain size | 12 m × 12 m × 12 m | 20 m × 20 m × 20 m | 32 m × 30 m × 32 m |

Smooth and kernel function | B- Spline function | B- Spline function | B- Spline function |

Smooth radius | 0.2 m | 0.2 m | 0.3 m |

Width of the fluid particles | 0.1 m | 0.1 m | 0.15 m |

Number of fluid particles | 3.8 k | 13.67 k | 5.2 k |

viscosity parameter I | 1 | 1 | 1 |

viscosity parameter n | 0.5 | 0.5 | 0.5 |

Upper limit of viscosity ${\upsilon}_{0}$ | 15 m${}^{2}$/s | 2 m${}^{2}$/s | 2 m${}^{2}$/s |

Lower limit of viscosity ${\upsilon}_{\infty}$ | 2 m${}^{2}$/s | 0.2 m${}^{2}$/s | 0.2 m${}^{2}$/s |

Total simulation time | 10 s | 20 s | 1.5 s |

Item | Experiment 4 | Experiment 5 | Experiment 6 |
---|---|---|---|

Simulation domain size | 12 m × 12 m × 12 m | 32 m × 30 m × 32 m | 40 m × 40 m × 40 m |

Smooth and kernel function | B- Spline function | B- Spline function | B- Spline function |

Smooth radius | 0.36 m | 0.3 m | 0.26 m |

Width of the fluid particles | 0.18 m | 0.15 m | 0.13 m |

Number of fluid particles | 32.7 k | 5.2 k | 57 k |

viscosity parameter I | 1 | 1 | 1 |

viscosity parameter n | 0.5 | 0.5 | 0.5 |

Upper limit of viscosity ${\upsilon}_{0}$ | 2 m${}^{2}$/s | 2 m${}^{2}$/s | 2 m${}^{2}$/s |

Lower limit of viscosity ${\upsilon}_{\infty}$ | 0.2 m${}^{2}$/s | 0.2 m${}^{2}$/s | 0.2 m${}^{2}$/s |

Total simulation time | 15 s | 1.5 s | 20 s |

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**MDPI and ACS Style**

Wang, X.; Ban, X.; He, R.; Wu, D.; Liu, X.; Xu, Y.
Fluid-Solid Boundary Handling Using Pairwise Interaction Model for Non-Newtonian Fluid. *Symmetry* **2018**, *10*, 94.
https://doi.org/10.3390/sym10040094

**AMA Style**

Wang X, Ban X, He R, Wu D, Liu X, Xu Y.
Fluid-Solid Boundary Handling Using Pairwise Interaction Model for Non-Newtonian Fluid. *Symmetry*. 2018; 10(4):94.
https://doi.org/10.3390/sym10040094

**Chicago/Turabian Style**

Wang, Xiaokun, Xiaojuan Ban, Runzi He, Di Wu, Xing Liu, and Yuting Xu.
2018. "Fluid-Solid Boundary Handling Using Pairwise Interaction Model for Non-Newtonian Fluid" *Symmetry* 10, no. 4: 94.
https://doi.org/10.3390/sym10040094