# A Symmetry Particle Method towards Implicit Non‐Newtonian Fluids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Material Model

#### 3.1. Governing Equations

#### 3.2. Carreau-Yasuda Model

#### 3.3. Smoothed Particle Hydrodynamics

_{i}) can be obtained by summation over all the sampling points x

_{j}with mass m

_{j}and density p

_{j}in the support domain:

## 4. Pressure Projection Algorithm

Algorithm 1. Pressure Projection Algorithm |

for all particles $i$ doupdate ${\rho}_{i}$ using Equation (7) update ${\alpha}_{i}$ using Equation (21) end forfor all particles $i$ doupdate ${{\mathbf{V}}_{i}}^{*}$ using Equation (12) end forwhile (${\rho}_{\mathit{avg}}-{\rho}_{0}>\eta $) or $\mathit{iter}<2$ dofor all particles $i$ doupdate ${{\rho}_{i}}^{*}$ using Equation (20) update ${k}_{i}$ using Equation (21) ${{\rho}_{i}}^{*}={{\rho}_{i}}^{*}=\frac{{k}_{i}}{{\alpha}_{i}}\Delta {t}^{2}$ end forend whilefor all particles $i$ do${{v}_{i}}^{**}={{v}_{i}}^{*}-\Delta t{\displaystyle \sum _{j}{m}_{j}(\frac{{k}_{i}}{{\rho}_{i}}+\frac{{k}_{j}}{{\rho}_{i}})\nabla {W}_{ij}}$ end for |

## 5. Implicit Viscosity Solver

Algorithm 2. Implicit non-Newtonian SPH fluid solver |

for all particles $i$ dofind neighborhoods ${N}_{i}$ compute time step size $\Delta t$ end forfor all particles $i$ docompute the external force ${F}_{i}^{ext}$ end forfor all particles $i$ doupdate velocity ${{v}_{i}}^{*}={v}_{i}(t)+\Delta t\frac{{F}_{i}^{adv}(t)}{{m}_{i}}$ end forfor all particles $i$ docompute the external force ${F}_{i}^{ext}$ end forfor all particles $i$ doupdate velocity ${{v}_{i}}^{*}={v}_{i}(t)+\Delta t\frac{{F}_{i}^{adv}(t)}{{m}_{i}}$ end forfor all particles $i$ docompute the pressure and pressure force end forfor all particles $i$ doupdate velocity ${{v}_{i}}^{**}$ // Pressure Projection end forfor all particles $i$ dosolving viscosity system end forfor all particles $i$ do${v}_{i}(t+\Delta t)={{v}_{i}}^{**}+\frac{\Delta t}{\rho}\nabla \cdot \mathsf{\sigma}$ ${x}_{i}(t+\Delta t)={x}_{i}(t)+\Delta t\xb7{v}_{i}(t+\Delta t)$ end for |

## 6. Implementation and Results

^{−4}s, fluid particles scattered in all directions with the method of Andrade et al. [25], while our algorithm still ran with the pressure projection (Figure 4d).

^{−4}s, while the method of Andrade et al. [25] could not. As a result, for a two-second animation, our method needs 32 frames and 44.5 min for computing, while the method of Andrade et al. [25] needs 67 frames and 59.5 min for computing. The simulation parameters and performance of Figure 4 are also listed in Table 3.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Illustration for smoothed particle hydrodynamics (SPHs) method. It is a symmetry particle method and the physical attributions of each particle is weighted in summation by its neighbors.

**Figure 3.**A falling water column with our method. (

**a**) is Newtonian fluid; (

**b**) is the non-Newtonian fluid.

**Figure 4.**The free fall of a water column. (

**a**) is Newtonian fluid (our method); (

**b**) is the non-Newtonian fluid (Andrade et al. [25]) with $\Delta t=2.5\times {10}^{-6}s$; (

**c**) is the non-Newtonian fluid (Andrade et al. [25]) with $\Delta t=3.0\times {10}^{-4}s$; (

**d**) is the non-Newtonian fluid (our method) with $\Delta t=3.0\times {10}^{-4}s$.

**Table 1.**The simulation parameters for Figure 3. (Both Newtonian and non-Newtonian fluid).

Simulation domain size | 6 m × 6 m × 6 m |

Fluid particles | 13,671 |

Smoothing kernel function | cubic splines |

Smoothing radius | 0.1 m |

Fluid particle width | 0.05 m |

Rendering time(per frame) | 1.14 min |

**Table 2.**The simulation parameters for Figure 5. (Both Newtonian and non-Newtonian fluid).

Simulation domain size | 8 m × 8 m × 8 m |

Fluid particles | 32,751 |

Smoothing kernel function | cubic splines |

Smoothing radius | 0.2 m |

Fluid particle width | 0.1 m |

Rendering time(per frame) | 3.85 min |

**Table 3.**The simulation parameters and performance. ${\mu}_{0}\text{}[{\text{m}}^{2}/\text{s}]$, ${\mu}_{\infty}\text{}[{\text{m}}^{2}/\text{s}]$, $K$, $\alpha $ and $n$: fluid physical parameters, $\Delta t\text{}[\text{s}]$: time step, ${t}^{total}\text{}[\text{s}]$: simulation time per frame.

Figure | ${\mathit{\mu}}_{\mathbf{0}}\text{}\mathbf{[}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{s}\mathbf{]}$ | ${\mathit{\mu}}_{\mathbf{\infty}}\text{}\mathbf{[}{\mathbf{m}}^{\mathbf{2}}\mathbf{/}\mathbf{s}\mathbf{]}$ | $\mathit{K}$ | $\mathit{\alpha}$ | $\mathit{n}$ | $\mathbf{\Delta}\mathit{t}\text{}\mathbf{[}\mathbf{s}\mathbf{]}$ | ${\mathit{t}}^{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}\text{}\mathbf{[}\mathbf{s}\mathbf{]}$ |
---|---|---|---|---|---|---|---|

Figure 4a | - | 0.2 | - | - | 1 | 3.0 × 10^{−4} | 40.8 |

Figure 4b | 2 | 0.2 | 1 | 0.5 | −0.5 | 2.5 × 10^{−6} | 53.3 |

Figure 4c | 2 | 0.2 | 1 | 0.5 | −0.5 | 3.0 × 10^{−4} | - |

Figure 4d | 2 | 0.2 | 1 | 0.5 | −0.5 | 3.0 × 10^{−4} | 83.4 |

Figure 5a | - | 0.2 | - | - | 1 | 6.0 × 10^{−4} | 178 |

Figure 5b | 0.2 | 2 | 1 | 0.5 | −0.5 | 6.0 × 10^{−4} | 216 |

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**MDPI and ACS Style**

Zhang, Y.; Ban, X.; Wang, X.; Liu, X.
A Symmetry Particle Method towards Implicit Non‐Newtonian Fluids. *Symmetry* **2017**, *9*, 26.
https://doi.org/10.3390/sym9020026

**AMA Style**

Zhang Y, Ban X, Wang X, Liu X.
A Symmetry Particle Method towards Implicit Non‐Newtonian Fluids. *Symmetry*. 2017; 9(2):26.
https://doi.org/10.3390/sym9020026

**Chicago/Turabian Style**

Zhang, Yalan, Xiaojuan Ban, Xiaokun Wang, and Xing Liu.
2017. "A Symmetry Particle Method towards Implicit Non‐Newtonian Fluids" *Symmetry* 9, no. 2: 26.
https://doi.org/10.3390/sym9020026