# On the Simulations of Thermal Liquid Foams Using Lattice Boltzmann Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Lattice Boltzmann Method

#### 2.2. Shan-Chen Multiphase Multicomponent Model

#### 2.3. Forcing Scheme

#### 2.4. Two Relaxation Time

#### 2.5. Mid-Range Interaction Model

#### 2.6. Thermal Field

#### 2.7. Units and Non-Dimensional Groups

- Reynolds Number:

- 2.
- Atwood Number:

- 3.
- Bonds Number:

- 4.
- Prandtl Number:

## 3. Model Verification and Validation

#### 3.1. Young–Laplace Test

#### 3.2. Rayleigh–Taylor Instability

#### 3.3. Bubble Rise Diagram

- Bo = 100, Re = 500
- Bo = 100, Re = 100
- Bo = 10, Re = 100
- Bo = 10, Re = 10

## 4. Application

#### 4.1. Thermal Field in Liquid Foam

#### 4.2. Locally Variable Surface Tension Due to Temperature

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**D2Q9 Lattice for short range nodes (red only). D2Q25 Lattice for short- and mid-range nodes (red and blue).

**Figure 2.**Young–Laplace test for the two limiting surface tension cases for density ratio ≈ 1 (single phase two components). Surface tension and density contours are presented in LU.

**Figure 3.**Young–Laplace test for the two limiting surface tension cases and density ratio ≈ 10 (two phase two component). Surface tensions and density contours are presented in LU.

**Figure 4.**RTI with time plotted on the same density contour levels (Re = 256, At ≈ 0.5, Bo ≈ 100). (

**a**) Resolution: 256 × 1024. (

**b**) Resolution: 1024 × 4096.

**Figure 5.**(

**a**) Shape regimes for bubble rise after Clift et al. [29]. (

**b**) Four simulation samples from different points on the diagram.

**Figure 6.**Different time steps starting from the rising of random bubbles to foam generation. Temperature contours are shown, while the bubbles and liquid interfaces are clearly visible.

**Figure 7.**(

**a**) Bubble with lamella showing the drainage direction and Marangoni convection, after Poulain et al. [6]. (

**b**) Representation of surface tension variation due to temperature gradient.

**Figure 8.**(

**a**) Bubble rise simulation case with locally variable surface tension. (

**b**) Bubble rise simulation with constant surface tension.

$\mathbf{Direction}{\widehat{\mathit{e}}}_{\mathit{i}}$ | ${\mathit{w}}_{\mathit{i}}$ |
---|---|

0 | 4/9 |

1, 2, 3, 4 | 1/9 |

5, 6, 7, 8 | 1/36 |

$\mathbf{Direction}{\widehat{\mathit{e}}}_{\tilde{\mathit{i}}}$ | ${\tilde{\mathit{w}}}_{\tilde{\mathit{i}}}$ |
---|---|

0 | 247/420 |

1, 2, 3, 4 | 4/63 |

5, 6, 7, 8 | 4/135 |

9, 10, 11, 12 | 1/180 |

13, 14, …, 20 | 2/945 |

21, 22, 23, 24 | 1/15,120 |

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

Resolution | 1000 × 1000 |

Re | 10 |

Bo | 0.1 |

${\mathrm{Pr}}_{\mathit{component}1}$ | 0.72 |

${\mathrm{Pr}}_{\mathit{component}2}$ | 6.0 |

G_{12} = G_{21} | 5.45 |

G_{11} = G_{22} | −2.0 |

${\tilde{G}}_{11}={\tilde{G}}_{22}$ | 5.0 |

θ * | 20˚ |

${T}_{\mathit{source}}/{T}_{\mathit{ref}}$ | 1.0 |

${T}_{\mathit{walls}}/{T}_{\mathit{ref}}$ | 0.25 |

${R}_{initial}$ | 20 LU |

${\mathit{v}}_{\mathit{component}1}={\mathit{v}}_{\mathit{component}2}$ | - |

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**MDPI and ACS Style**

Mobarak, M.; Gatternig, B.; Delgado, A.
On the Simulations of Thermal Liquid Foams Using Lattice Boltzmann Method. *Energies* **2023**, *16*, 195.
https://doi.org/10.3390/en16010195

**AMA Style**

Mobarak M, Gatternig B, Delgado A.
On the Simulations of Thermal Liquid Foams Using Lattice Boltzmann Method. *Energies*. 2023; 16(1):195.
https://doi.org/10.3390/en16010195

**Chicago/Turabian Style**

Mobarak, Mohammad, Bernhard Gatternig, and Antonio Delgado.
2023. "On the Simulations of Thermal Liquid Foams Using Lattice Boltzmann Method" *Energies* 16, no. 1: 195.
https://doi.org/10.3390/en16010195