# Numerical Investigation on Influence of Number of Bubbles on Laser-Induced Microjet

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## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

#### 2.1. Governing Equations

#### 2.2. Simulation Settings and Conditions

#### 2.3. Numerical Schemes

- $\rho -$THINC-MUSCL3

- MP-WENO5-JS

## 3. Result and Discussion

#### 3.1. Microjet Evolution and Time History of Impulse

#### 3.2. Pressure Field Evolution around a Single Bubble

#### 3.3. Pressure Field Evolution around Multi-Bubbles

#### 3.4. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Solver Varidation

**Figure A1.**Result of Gas–liquid Riemann problem. The exact and numerical solutions are compared at $t=0.2$.

## Appendix B. Grid Independence Study

**Figure A2.**Result of the grid independence study using $\rho $-THINC-MUSCL3 on w/o-meniscus setting.

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**Figure 1.**Illustration of the generation of microjets by focusing a pulsed laser in liquid-filled capillary. (

**a**) Laser-induced bubbles is created by a focused laser pulse. (

**b**) High-pressure laser-induced bubbles induce pressure waves toward the meniscus. (

**c**) Microjet is generated due to kinematic focusing on the meniscus.

**Figure 2.**Simulation settings for the microchannel with meniscus (w-meniscus setting). This figure shows the setting for the case of two bubbles.

**Figure 3.**Simulation settings for the microchannel without meniscus and multiple laser-induced bubbles (w/o-meniscus setting). This figure shows the setting for the case of two bubbles.

**Figure 4.**Time evolution of microjet by the single bubble. Interfaces are shown by the density gradient.

**Figure 5.**Time history of microjet velocity and microjet formations at the final state. The orange dashed line in (

**b**–

**d**) shows the position of the microjet head in the single bubble case. Interfaces are shown by the density gradient.

**Figure 6.**Time history of the impulse. This result is obtained from the simulation result of the w/o-meniscus setting.

**Figure 7.**Pressure field evolution generated by the single laser-induced bubble at (

**a**) t = 0.0, (

**b**) t = 0.9, (

**c**) t = 1.8, (

**d**) t = 3.3, (

**e**) t = 4.5, and (

**f**) t = 5.4; numerical schlieren at the upper side and pressure distribution at the lower side of each figure.

**Figure 8.**Time history of the pressure at the pressure probe in the case of the single bubble. The blue dashed line shows the arrival time of the pressure wave.

**Figure 9.**Schematics of the path of the pressure wave from the bubble to the pressure probe point. (

**a**) Path of the pressure wave through the wave reflection between the bubble and wall; (

**b**) Path of the pressure wave to the pressure probe through the wave reflection at the wall.

**Figure 10.**Time evolution of the numerical schlieren at the upper side and the pressure distribution at the lower side at (

**a**) t = 0.25, (

**b**) t = 1.0, and (

**c**) t = 2.25. The red dash line shows the inspection surface.

**Figure 12.**Path of the pressure wave from the bubble in the case of multiple bubbles. The path of the pressure wave is the same as the length shown by the dashed line. (

**a**) Path of the pressure wave between the bubbles; (

**b**) Path of the pressure wave to the inspection surface.

Number of Bubbles | 1 | 2 | 3 |
---|---|---|---|

Positions $\left(x,y\right)$ [-] | $\left(0,0\right)$ | $\left(0,-3.5\right),\left(0,3.5\right)$ | $\left(0,-4.67\right),\left(0,0\right),\left(0,4.67\right)$ |

Times [-] | $\mathit{T}$ | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ |
---|---|---|---|---|

Simulation Results (Figure 4) | 2.9 | 2.6 | 3.7 | 6.2 |

Results of Equations (26) and (27) | 2.9 | 2.5 | 3.9 | 6.4 |

Times [-] | $\mathit{T}$ | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ |
---|---|---|---|---|

Simulation Results (Figure 4) | 1.4 | 2.6 | 3.2 | 4.3 |

Results of Equation (26) and Equation (27) | 1.5 | 2.6 | 3.3 | 4.5 |

**Table 4.**Comparison between the simulation and analytical results for the case of the three bubbles.

Times [-] | $\mathit{T}$ | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | ${\mathit{t}}_{3}$ |
---|---|---|---|---|

Simulation Results (Figure 4) | 0.9 | 2.4 | 2.6 | 3.0 |

Results of Equation (26) and Equation (27) | 1.0 | 2.5 | 2.6 | 3.1 |

Number of Bubbles | 1 | $2$ | $3$ |
---|---|---|---|

Maximum Pressure [-] | 2.9 | 2.6 | 3.7 |

Time at the Maximum Pressure [-] | 2.9 | 2.5 | 3.9 |

Number of Bubbles | 1 | 2 | 3 |
---|---|---|---|

Time [-] | 2.5 | 1.0 | 0.6 |

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**MDPI and ACS Style**

Ishikawa, T.; Nishida, H.; Tagawa, Y. Numerical Investigation on Influence of Number of Bubbles on Laser-Induced Microjet. *Water* **2022**, *14*, 3707.
https://doi.org/10.3390/w14223707

**AMA Style**

Ishikawa T, Nishida H, Tagawa Y. Numerical Investigation on Influence of Number of Bubbles on Laser-Induced Microjet. *Water*. 2022; 14(22):3707.
https://doi.org/10.3390/w14223707

**Chicago/Turabian Style**

Ishikawa, Tatsumasa, Hiroyuki Nishida, and Yoshiyuki Tagawa. 2022. "Numerical Investigation on Influence of Number of Bubbles on Laser-Induced Microjet" *Water* 14, no. 22: 3707.
https://doi.org/10.3390/w14223707