# A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. SPH Fluid Simulation

## 4. Multiple-Phase Fluids’ Simulation Using Modified Density

#### 4.1. Modified Density Model

#### 4.2. Adjusted Pressure Computation

#### 4.3. Interfacial Forces of Multiple-Phase Fluids

## 5. Surface Extraction Using Asymmetric Kernels

## 6. Asymmetric Surface Extraction for Multiple-Phase Interfaces

#### 6.1. Asymmetric Kernel for Multiple-Phase Interfaces

#### 6.2. Surface Extraction Strategy

- Initially, employ two color fields ${\varphi}^{\prime}\left(x\right)$ and $-{\varphi}^{\prime}\left(x\right)$ for the particles for one phase and then the rest of the $n-1$ phases, respectively;
- Additionally, interpolate the signed color field for one phase and the other n-1 phases, and then, select $\epsilon {\varphi}^{\prime}\left(x\right)$, $-\epsilon {\varphi}^{\prime}\left(x\right)$ separately as the surface field value;
- Furthermore, on the basis of the chosen surface field value, rebuild the surface for the one phase;
- Finally, iterate the procedures above until the surface of each phase is fully rebuilt.

## 7. Implementation and Results

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The symmetric kernel transformed to the asymmetric kernel for the multiple-phase interface.

**Figure 7.**Breaking dam experiment with two phases. (

**a**) left column, symmetric kernel method; (

**b**) middle column, asymmetric kernel method; (

**c**) right column, our own method.

**Figure 8.**Breaking dam experiment with three phases. (

**a**) first row, asymmetric kernel method; (

**b**) second row, our method.

Parameter | Value |
---|---|

Size of domain | 24 m × 24 m × 24 m |

Smoothing kernel | Cubic splines |

Number of blue particles | 126 k |

Number of yellow particles | 126 k |

Density of blue phase | 200 $\mathrm{kg}/\phantom{\mathrm{kg}{\mathrm{m}}^{3}}\phantom{\rule{0.0pt}{0ex}}{\mathrm{m}}^{3}$ |

Density of yellow phase | 1000 $\mathrm{kg}/\phantom{\mathrm{kg}{\mathrm{m}}^{3}}\phantom{\rule{0.0pt}{0ex}}{\mathrm{m}}^{3}$ |

Support radius | 0.2 m |

Diameter of fluid particle | 0.1 m |

Parameter | Value |
---|---|

Size of domain | 24 m × 24 m × m |

Smoothing kernel | Cubic splines |

Number of blue particles | 13,325 |

Number of yellow particles | 13,325 |

Number of red particles | 13,325 |

Density of red phase | 300 $\mathrm{kg}/\phantom{\mathrm{kg}{\mathrm{m}}^{3}}\phantom{\rule{0.0pt}{0ex}}{\mathrm{m}}^{3}$ |

Density of blue phase | 900 $\mathrm{kg}/\phantom{\mathrm{kg}{\mathrm{m}}^{3}}\phantom{\rule{0.0pt}{0ex}}{\mathrm{m}}^{3}$ |

Density of yellow phase | 100 $\mathrm{kg}/\phantom{\mathrm{kg}{\mathrm{m}}^{3}}\phantom{\rule{0.0pt}{0ex}}{\mathrm{m}}^{3}$ |

Support radius | 0.2 m |

Diameter of fluid particle | 0.1 m |

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**MDPI and ACS Style**

Wang, X.; Xu, Y.; Ban, X.; Liu, S.; Xu, Y.
A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model. *Symmetry* **2019**, *11*, 745.
https://doi.org/10.3390/sym11060745

**AMA Style**

Wang X, Xu Y, Ban X, Liu S, Xu Y.
A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model. *Symmetry*. 2019; 11(6):745.
https://doi.org/10.3390/sym11060745

**Chicago/Turabian Style**

Wang, Xiaokun, Yanrui Xu, Xiaojuan Ban, Sinuo Liu, and Yuting Xu.
2019. "A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model" *Symmetry* 11, no. 6: 745.
https://doi.org/10.3390/sym11060745