# The Relative Contribution of Solutal Marangoni Convection to Thermal Marangoni Flow Instabilities in a Liquid Bridge of Smaller Aspect Ratios under Zero Gravity

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## Abstract

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## 1. Introduction

## 2. Numerical Method

#### 2.1. Numerical Simulation

#### 2.2. Dynamic Mode Decomposition (DMD)

## 3. Results and Discussion

#### 3.1. Critical $M{a}_{\mathrm{C}}$ at Small Thermal Marangoni Numbers, $M{a}_{\mathrm{T}}\le 1800{A}_{s}^{-1}$

#### 3.2. Critical $M{a}_{\mathrm{T}}$ under a Weak Solutal Marangoni Convection

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

LSA | Linear stability analysis |

POD | Proper orthogonal decomposition |

DMD | Dynamic mode decomposition |

## References

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**Figure 1.**Geometry of a half zone liquid bridge system. The dashed arrows represent the direction of Marangoni force driven by temperature and concentration gradients along the free surface. Note that the top/bottom temperature and concentration values are dimensionless defined with respect to the differences of $\Delta T$ and $\Delta C$.

**Figure 2.**Flow map, ($M{a}_{\mathrm{T}},\phantom{\rule{3.33333pt}{0ex}}M{a}_{\mathrm{C}}$) for (

**a**) ${A}_{s}=0.25$, (

**b**) ${A}_{s}=0.5$ (Minakuchi et al. [22]), and (

**c**) ${A}_{s}=1$. The different symbols represent the wavenumbers of oscillatory modes. The dashed lines represent $M{a}_{\mathrm{T},\mathrm{Cri}}=3150{A}_{s}^{-1}$ ($400<M{a}_{\mathrm{C}}{A}_{s}<800$), roughly indicating the critical value.

**Figure 3.**The non-dimensional concentration distributions in the middle plane ($z=0.5{A}_{s}$) for ${A}_{s}=0.25$ of DNS: ($M{a}_{\mathrm{T}}$, $M{a}_{\mathrm{C}}$) = (

**a**) (2100, 3572), (

**b**) (1400, 3572), (

**c**) (9100, 3572), (

**d**) (14000, 1786), and (

**e**) (13300, 1786).

**Figure 4.**The azimuthal wavenumbers of the bifurcated oscillatory modes (m) as a function of the aspect ratio of the liquid bridge.

**Figure 5.**(

**a**) Dependency of critical solutal Marangoni numbers, $M{a}_{\mathrm{C},\mathrm{Cri}}$, on the aspect ratio of the liquid bridge. The lines are the stability curves: $M{a}_{\mathrm{T}}=0$ and $M{a}_{\mathrm{T}}=1400$. (

**b**) Critical thermal Marangoni number $M{a}_{\mathrm{T},\mathrm{Cri}}$ as a function of ${A}_{s}$: ——, Present (at $M{a}_{\mathrm{C}}=714$ and $Pr=0.006$); — · —, Present (at $M{a}_{\mathrm{C}}=0$ and $Pr=0.006$); $\xb7\xb7\xb7\xb7\xb7\xb7$, Imaishi et al. [19] ($M{a}_{\mathrm{C}}=0$, $Pr=0$); – – – –, Yasuhiro et al. [3] ($M{a}_{\mathrm{C}}=0$, $Pr=1$). The different symbols represent the wavenumbers of oscillatory modes.

**Figure 6.**The time evolution of the non-dimensional concentration for ${A}_{s}=0.25$ at the sampling point of $(r,\theta ,z)=(0.75,0,0.5{A}_{s})$ when ($M{a}_{\mathrm{T}},M{a}_{\mathrm{C}})=(9100,3572)$ (

**a**) and the frequency spectrum (

**b**).

**Figure 7.**Representative dynamic modes corresponding to the primary dominant eigenvalues, visualized by contours of the concentration field in the middle plane ($z=0.5{A}_{s}$) for a flat liquid bridge, ${A}_{s}=0.25$: ($M{a}_{\mathrm{T}}$, $M{a}_{\mathrm{C}}$) = (

**a**) (2100, 3572), (

**b**) (1400, 3572), (

**c**) (9100, 3572), (

**d**) (14000, 1786), and (

**e**) (13300, 1786).

**Figure 8.**The time evolution of the non-dimensional concentration for ${A}_{s}=0.25$ at the sampling point of $(r,\theta ,z)=(0.75,0,0.5{A}_{s})$.

**Figure 9.**The non-dimensional concentration distributions in the middle plane (at $z=0.5{A}_{s}$ and ${A}_{s}=1$) of DNS: ($M{a}_{\mathrm{T}}$, $M{a}_{\mathrm{C}}$) = (

**a**) (1050, 1786), (

**b**) (3500, 1786), and (

**c**) (3150,1786).

**Figure 10.**Representative dynamic modes corresponding to the second leading eigenvalues, visualized by contours of the concentration field in the middle plane (at $z=0.5{A}_{s}$ and ${A}_{s}=0.25$): ($M{a}_{\mathrm{T}}$, $M{a}_{\mathrm{C}}$, ${\lambda}_{2}$) = (

**a**) (13,300, 1784, 0.00005$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.2764\mathrm{i}$), (

**b**) (13,300, 2144, −0.0029$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.3518\mathrm{i}$), and (

**c**) (13,300, 2500, −0.0006$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.3351\mathrm{i}$).

**Figure 11.**The non-dimensional concentration distributions in the middle plane at $z=0.5{A}_{s}$ obtained DMD analysis for ${A}_{s}=1$: ($M{a}_{\mathrm{T}}$, $M{a}_{\mathrm{C}}$) = (

**a**) (3150, 1490) and (

**b**) (3150, 1786).

**Figure 12.**The time variation of the non-dimensional concentration for ${A}_{s}=1$ at the sampling point of $(r,\theta ,z)=(0.5,0,0.5{A}_{s})$.

**Table 1.**The dominant eigenvalues obtained from DMD and non-dimensional frequencies ((i) at ${A}_{s}=0.25$ and (ii) at ${A}_{s}=1$).

Case | ${\mathit{A}}_{\mathit{s}}$ | ${\mathit{Ma}}_{\mathit{T}}$ | ${\mathit{Ma}}_{\mathit{C}}$ | Eigenvalue (${\mathit{\lambda}}_{1}$) | Frequency | m |
---|---|---|---|---|---|---|

(i) | 0.25 | 14,000 | 1786 | 0.0002$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.1719\mathrm{i}$ | 19 | 6 |

13,300 | 1786 | 0.0002$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.2936\mathrm{i}$ | 33 | 7 | ||

2100 | 3572 | 0.0052$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.4877\mathrm{i}$ | 55 | 11 | ||

1400 | 3572 | 0.0219$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.4468\mathrm{i}$ | 50 | 12 | ||

9100 | 3572 | 0.00006$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}1.0629\mathrm{i}$ | 120 | 13 | ||

(ii) | 1 | 3150 | 1490 | 0.006$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.0545\mathrm{i}$ | 6.19 | 2 |

3150 | 1786 | 0.0032$\phantom{\rule{3.33333pt}{0ex}}\pm \phantom{\rule{3.33333pt}{0ex}}0.0556\mathrm{i}$ | 6.32 | 2 |

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**MDPI and ACS Style**

Agampodi Mendis, R.L.; Sekimoto, A.; Okano, Y.; Minakuchi, H.; Dost, S. The Relative Contribution of Solutal Marangoni Convection to Thermal Marangoni Flow Instabilities in a Liquid Bridge of Smaller Aspect Ratios under Zero Gravity. *Crystals* **2021**, *11*, 116.
https://doi.org/10.3390/cryst11020116

**AMA Style**

Agampodi Mendis RL, Sekimoto A, Okano Y, Minakuchi H, Dost S. The Relative Contribution of Solutal Marangoni Convection to Thermal Marangoni Flow Instabilities in a Liquid Bridge of Smaller Aspect Ratios under Zero Gravity. *Crystals*. 2021; 11(2):116.
https://doi.org/10.3390/cryst11020116

**Chicago/Turabian Style**

Agampodi Mendis, Radeesha Laknath, Atsushi Sekimoto, Yasunori Okano, Hisashi Minakuchi, and Sadik Dost. 2021. "The Relative Contribution of Solutal Marangoni Convection to Thermal Marangoni Flow Instabilities in a Liquid Bridge of Smaller Aspect Ratios under Zero Gravity" *Crystals* 11, no. 2: 116.
https://doi.org/10.3390/cryst11020116