# Effect of Wall Proximity and Surface Tension on a Single Bubble Rising near a Vertical Wall

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{l}gd

^{2})/σ

_{l}g

^{1/2}d

^{3/2})/µ

_{l}

_{l}/ρ

_{g}

_{l}/µ

_{g}

## 2. Computational Model

#### 2.1. Governing Equations and Numerical Model

**u**represents the velocity vector shared by the two fluids throughout the flow domain. t and p denote the time and static pressure, respectively.

**F**represents the surface tension force. In a single pressure system, the normal component of the pressure gradient at a stationary non-vertical solid wall with a no-slip condition must be different for each phase due to the hydrostatic component ρ

_{σ}**g**when the phases are separated at the wall. In order to simplify the definition of boundary conditions, a modified pressure is defined as:

**x**represents the position vector of the fluid element.

**u**

_{c}is called the compression velocity, which is, in essence, the relative velocity between the two phases. Since only a single velocity for both fluids is considered in the whole domain, the compression velocity needs to be modeled [37,38] as:

^{−6}for all the governing equations. The adjustable time step setting was turned on to ensure time step convergence.

#### 2.2. Parameter Space

^{−3}, μ = 0.001 Pa s, g = 9.81 m s

^{−2}and d = 0.001 m, we obtain Ga = 99. The case of Ga = 99 and Eo = 0.14 represent a 1 mm air bubble rising near a wall in pure water. Fixing the Ga and bubble diameter and varying the Eo from 0.1 to 10 allows us to study the effect of surface tension on the near-wall bubble behavior. In practical situations, we can control the surface tension of the air–water interface by using surfactants. These Eo values range can be considered as the value of Eo for a 1 mm air bubble rising in water having different surfactant concentrations. The knowledge of the effect of surface tension on the bubble behavior in water can be leveraged in practical situations to control the flow properties by using surfactants.

## 3. Results

#### 3.1. Grid Independence and Validation

#### 3.2. Near-Wall Rising Behaviour

#### 3.3. Effect of Initial Wall Distance

#### 3.4. Effect of Surface Tension

## 4. Discussion

_{1}, t

_{2}and t

_{3}. The time between two collisions is highlighted in the figure and is almost constant, i.e., the bouncing frequency is constant. The rising motion of the bounded and unbounded bubbles have the same nature (Figure 6). Figure 6 shows the vertical distance traveled by the wall-bounded bubble between two collisions (t

_{1}, t

_{2}) and (t

_{2}, t

_{3}) and it is almost constant. Thus, the bouncing motion is very uniform with constant amplitude, frequency and rise distance in every bouncing cycle.

_{y}= 0.34 m/s. However, the y-motion of the bounded bubble cannot achieve a steady state terminal velocity. After the buoyancy and drag forces are balanced, the bubble reaches a maximum v

_{y}= 0.32 m/s. From Figure 5b and Figure 6b, we can observe that at the time of the collision, the x-velocity starts increasing, and consequently, the y-velocity drops from its maximum value. As the bubble completes its bouncing cycle, the x-velocity decreases and the y-velocity starts rising and it again reaches its maximum value and stays there for some time after which the bubble is again attracted to the wall and the rising velocity drops again.

#### 4.1. Effect of Wall Proximity

_{y}≈ 1. This implies that the terminal velocity is slightly lesser compared to the terminal velocity of the free-rising bubble. At s = 1.5, the wall effect is not strong enough to bring an effect in the wall-normal direction and induce a bouncing motion but it increases the drag, thus decreasing the y-velocity of the bubble.

#### 4.2. Effect of Eotvos Number

## 5. Conclusions

- The presence of the wall near the bubble provides a significant perturbation to the flow structures in the fluid domain, which induces an early transition of the bubble trajectory from the rectilinear to the planar zigzagging regime. At Ga = 99, the critical Eo for this transition decreases from 1.5 to 0.14.
- A 1 mm diameter air bubble rising near a vertical wall in pure water (Ga = 99, Eo = 0.14) follows a bouncing trajectory as observed in previous experimental works. The bouncing motion is characterized by constant wavelength, amplitude and frequency. There is no significant motion of the bubble along the spanwise direction. The bouncing motion is thus two-dimensional.
- Due to the presence of the wall, the drag experienced by the bubble increases. The average rise velocity of the bubble rising near the wall is less compared to the unbounded bubble.
- As the bubble–wall initial distance increases, the bubble rising trajectory changes from bouncing (planar zigzagging) to straight rising. Moreover, as the distance from the wall increases, the maximum velocity and the average rise velocity also increases, i.e., the drag decreases.
- The bubble shows bouncing motion for s < 1.5. For the bouncing regime (i.e., s < 1.5), the change in wall proximity only changes the time of onset of bouncing. It does not impact the characteristics of the bouncing trajectory. The closer the bubble is to the wall, the earlier it triggers these path instabilities.
- For Ga = 99, s = 1, the bubble shows a bouncing motion for Eo < 1.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

d | Bubble Diameter |

Eo | Eotvos Number |

g | Acceleration due to gravity |

Ga | Galilei Number |

s | Normalized bubble-wall initial distance |

t | Time |

vx | x-velocity of bubble centroid |

vy | y-velocity of bubble centroid |

vz | z-velocity of bubble centroid |

VOF | Volume of fluid |

x | x-position of bubble centroid |

y | y-position of bubble centroid |

z | z-position of bubble centroid |

α | Phase fraction |

ρ | Density |

σ | Surface Tension Coefficient |

µ | Dynamic Viscosity |

## References

- Bhaga, D.; Weber, M.E. Bubbles in viscous liquids: Shapes, wakes and velocities. J. Fluid Mech.
**1981**, 105, 61. [Google Scholar] [CrossRef] - Duineveld, P.C. The rise velocity and shape of bubbles in pure water at high Reynolds number. J. Fluid Mech.
**1995**, 292, 325–332. [Google Scholar] [CrossRef] - Vries, A.W.D.; Biesheuvel, A.; Wijngaarden, L.V. Notes on the path and wake of a gas bubble rising in pure water. Int. J. Multiph. Flow
**2003**, 28, 1823–1835. [Google Scholar] [CrossRef] - Clift, R.; Grace, J.R.; Weber, M.E. Bubbles, Drops, and Particles, 2nd ed.; Dover Publications: Mineola, NY, USA, 2005. [Google Scholar]
- Zenit, R.; Magnaudet, J. Path instability of rising spheroidal air bubbles: A shape-controlled process. Phys. Fluids
**2008**, 20, 061702. [Google Scholar] [CrossRef] - Sharaf, D.M.; Premlata, A.R.; Tripathi, M.K.; Karri, B.; Sahu, K.C. Shapes and paths of an air bubble rising in quiescent liquids. Phys. Fluids
**2017**, 29, 122104. [Google Scholar] [CrossRef] - Chen, L.; Garimella, S.V.; Reizes, J.A.; Leonardi, E. The development of a bubble rising in a viscous liquid. J. Fluid Mech.
**1999**, 387, 61–96. [Google Scholar] [CrossRef] - Mougin, G.; Magnaudet, J. Path Instability of a Rising Bubble. Phys. Rev. Lett.
**2001**, 88, 014502. [Google Scholar] [CrossRef] - Hua, J.; Lou, J. Numerical simulation of bubble rising in viscous liquid. J. Comp. Phys.
**2007**, 222, 769–795. [Google Scholar] [CrossRef] - Farhangi, M.M.; Passandideh-Fard, M.; Moin, H. Numerical study of bubble rise and interaction in a viscous liquid. Int. J. Comp. Fluid Dyn.
**2010**, 24, 13–28. [Google Scholar] [CrossRef] - Ghosh, S.; Das, A.K.; Vaidya, A.A.; Mishra, S.C.; Das, P.K. Numerical Study of Dynamics of Bubbles Using Lattice Boltzmann Method. Ind. Eng. Chem. Res.
**2012**, 51, 6364–6376. [Google Scholar] [CrossRef] - Tripathi, M.K.; Sahu, K.C.; Govindarajan, R. Dynamics of an initially spherical bubble rising in quiescent liquid. Nat. Commun.
**2015**, 6, 6268. [Google Scholar] [CrossRef] [PubMed] - Cano-Lozano, J.C.; Martínez-Bazán, C.; Magnaudet, J.; Tchoufag, J. Paths and wakes of de- formable nearly spheroidal rising bubbles close to the transition to path instability. Phys. Rev. Fluids
**2016**, 1, 053604. [Google Scholar] [CrossRef] - Gumulya, M.; Joshi, J.B.; Utikar, R.P.; Evans, G.M.; Pareek, V. Bubbles in viscous liquids: Time dependent behaviour and wake characteristics. Chem. Eng. Sci.
**2016**, 144, 298–309. [Google Scholar] [CrossRef] - Shew, W.L.; Pinton, J.F. Dynamical Model of Bubble Path Instability. Phys. Rev. Lett.
**2006**, 97, 144508. [Google Scholar] [CrossRef] - Ern, P.; Risso, F.; Fabre, D.; Magnaudet, J. Wake-Induced Oscillatory Paths of Bodies Freely Rising or Falling in Fluids. Ann. Rev. Fluid Mech.
**2012**, 44, 97–121. [Google Scholar] [CrossRef] - Zenit, R.; Magnaudet, J. Measurements of the streamwise vorticity in the wake of an oscillating bubble. Int. J. Multiph. Flow
**2009**, 35, 195–203. [Google Scholar] [CrossRef] - Magnaudet, J.; Eames, I. The Motion of High-Reynolds-Number Bubbles in Inhomogeneous Flows. Annu. Rev. Fluid Mech.
**2000**, 32, 659–708. [Google Scholar] [CrossRef] - Gumulya, M.; Utikar, R.; Evans, G.; Joshi, J.; Pareek, V. Interaction of bubbles rising inline in quiescent liquid. Chem. Eng. Sci.
**2017**, 166, 1–10. [Google Scholar] [CrossRef] - Senapati, A.; Singh, G.; Lakkaraju, R. Numerical simulations of an inline rising unequal- sized bubble pair in a liquid column. Chem. Eng. Sci.
**2019**, 208, 115159. [Google Scholar] [CrossRef] - Tsao, H.K.; Koch, D.L. Observations of high Reynolds number bubbles interacting with a rigid wall. Phys. Fluids
**1997**, 9, 44–56. [Google Scholar] [CrossRef] - Takemura, F.; Takagi, S.; Magnaudet, J.; Matsumoto, Y. Drag and lift forces on a bubble rising near a vertical wall in a viscous liquid. J. Fluid Mech.
**2002**, 461, 277–300. [Google Scholar] [CrossRef] - Takemura, F.; Magnaudet, J. The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number. J. Fluid Mech.
**2003**, 495, 235–253. [Google Scholar] [CrossRef] - Podvin, B.; Khoja, S.; Moraga, F.; Attinger, D. Model and experimental visualizations of the interaction of a bubble with an inclined wall. Chem. Eng. Sci.
**2008**, 63, 1914–1928. [Google Scholar] [CrossRef] - Jeong, H.; Park, H. Near-wall rising behaviour of a deformable bubble at high Reynolds number. J. Fluid Mech.
**2015**, 771, 564–594. [Google Scholar] [CrossRef] - Lee, J.; Park, H. Wake structures behind an oscillating bubble rising close to a vertical wall. Int. J. Multiph. Flow
**2017**, 91, 225–242. [Google Scholar] [CrossRef] - Barbosa, C.; Legendre, D.; Zenit, R. Conditions for the sliding-bouncing transition for the interaction of a bubble with an inclined wall. Phys. Rev. Fluids
**2016**, 1, 032201. [Google Scholar] [CrossRef] - Sugioka, K.; Tsukada, T. Direct numerical simulations of drag and lift forces acting on a spherical bubble near a plane wall. Int. J. Multiph. Flow
**2015**, 71, 32–37. [Google Scholar] [CrossRef] - Zhang, Y.; Dabiri, S.; Chen, K.; You, Y. An initially spherical bubble rising near a vertical wall. Int. J. Heat Fluid Flow
**2020**, 85, 108649. [Google Scholar] [CrossRef] - Zhang, K.; Li, Y.; Chen, Q.; Lin, P. Numerical Study on the Rising Motion of Bubbles near the Wall. Appl. Sci.
**2021**, 11, 10918. [Google Scholar] [CrossRef] - Yan, H.; Zhang, H.; Liao, Y.; Zhang, H.; Zhou, P.; Liu, L. A single bubble rising in the vicinity of a vertical wall: A numerical study based on volume of fluid method. Ocean Eng.
**2022**, 263, 112379. [Google Scholar] [CrossRef] - Hasan, S.M.M.; Hasan, A.B.M.T. Migration dynamics of an initially spherical deformable bubble in the vicinity of a corner. Phys. Fluids
**2022**, 34, 112119. [Google Scholar] [CrossRef] - Khodadadi, S.; Samkhaniani, N.; Taleghani, M.H.; Bandpy, M.G.; Ganji, D.D. Numerical simulation of single bubble motion along inclined walls: A comprehensive map of outcomes. Ocean Eng.
**2022**, 255, 111478. [Google Scholar] [CrossRef] - Tagawa, Y.; Takagi, S.; Matsumoto, Y. Surfactant effect on path instability of a rising bubble. J. Fluid Mech.
**2014**, 738, 124–142. [Google Scholar] [CrossRef] - Mukundakrishnan, K.; Quan, S.; Eckmann, D.M.; Ayyaswamy, P.S. Numerical study of wall effects on buoyant gas-bubble rise in a liquid-filled finite cylinder. Phys. Rev. E
**2007**, 76, 036308. [Google Scholar] [CrossRef] [PubMed] - Krishna, R.; Urseanu, M.; van Baten, J.; Ellenberger, J. Wall effects on the rise of single gas bubbles in liquids. Int. Comm. Heat Mass Transf.
**1999**, 26, 781–790. [Google Scholar] [CrossRef] - Klostermann, J.; Schaake, K.; Schwarze, R. Numerical simulation of a single rising bubble by VOF with surface compression. Int. J. Numer. Methods Fluids
**2013**, 71, 960–982. [Google Scholar] [CrossRef] - Deshpande, S.S.; Anumolu, L.; Trujillo, M.F. Evaluating the performance of the two-phase flow solver interFoam. Comp. Sci. Discov.
**2012**, 5, 014016. [Google Scholar] [CrossRef] - Pradeep, A.; Sharma, A.K. Numerical investigation of single bubble dynamics in liquid sodium pool. Sādhanā
**2019**, 44, 56. [Google Scholar] [CrossRef] - Senthilkumar, S.; Delauré, Y.; Murray, D.; Donnelly, B. The effect of the VOF–CSF static contact angle boundary condition on the dynamics of sliding and bouncing ellipsoidal bubbles. Int. J. Heat Fluid Flow
**2011**, 32, 964–972. [Google Scholar] [CrossRef] - Albadawi, A.; Donoghue, D.; Robinson, A.; Murray, D.; Delauré, Y. On the assessment of a VOF based compressive interface capturing scheme for the analysis of bubble impact on and bounce from a flat horizontal surface. Int. J. Multiph. Flow
**2014**, 65, 82–97. [Google Scholar] [CrossRef] - Kulkarni, A.A.; Joshi, J.B. Bubble Formation and Bubble Rise Velocity in Gas−Liquid Systems: A Review. Ind. Eng. Chem. Res.
**2005**, 44, 5873–5931. [Google Scholar] [CrossRef]

**Figure 4.**Comparison of the rising trajectory of a bubble released near a wall with that of an unbounded bubble: (

**a**) 3D trajectory of the bubble released close to the wall; (

**b**) 3D trajectory of the bubble released away from the wall. d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 5.**Comparison of x-motion of the bounded and unbounded bubble: (

**a**) Time variation of x position of bubble centroid; (

**b**) Time variation of x velocity of bubble centroid d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 6.**Comparison of y-motion of the bounded and unbounded bubble: (

**a**) Time variation of y-position of bubble cen troid; (

**b**) Time variation of y-velocity of bubble cen troid. d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 7.**Comparison of z-motion of the bounded and unbounded bubble: (

**a**) Time variation of z-position of bubble centroid; (

**b**) Time variation of z-velocity of bubble centroid. d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 12.**Comparison of the bubble trajectory when released at different distances from the wall (Eo = 0.14, Ga = 99).

**Figure 13.**Comparison of x-motion of the bubble released at different distances from the wall: (

**a**) Time variation of x-position of bubble centroid; (

**b**) Time variation of x-velocity of bubble centroid. d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 14.**Comparison of y-motion of the bubble released at different distances from the wall: (

**a**) Time variation of y-position of bubble centroid; (

**b**) Time variation of y-velocity of bubble centroid. d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 16.**Comparison of z-motion of the bubble released at different distances from the wall: (

**a**) Time variation of z-position of bubble centroid; (

**b**) Time variation of z-velocity of bubble centroid. d = 1 mm, Eo = 0.14, Ga = 99.

**Figure 18.**Comparison of x-motion of the bubble for different Eo: (

**a**) Time variation of x-position of bubble centroid; (

**b**) Time variation of x-velocity of bubble centroid. d = 1 mm, Ga = 99, s = 1.

**Figure 19.**Comparison of y-motion of the bubble for different Eo: (

**a**) Time variation of y-position of bubble centroid; (

**b**) Time variation of y-velocity of bubble centroid. d = 1 mm, Ga = 99, s = 1.

**Figure 21.**Comparison of z-motion of the bubble for different Eo: (

**a**) Time variation of z-position of bubble centroid; (

**b**) Time variation of z-velocity of bubble centroid. d = 1 mm, Ga = 99, s = 1.

**Figure 22.**The bubble shape as viewed from the XY (

**left**) and YZ (

**right**) planes for different values of surface tension.

Term | Discretization Scheme | OpenFOAM Terminology |
---|---|---|

$\frac{\partial}{\partial t}(\rho \mathit{u})$ | Euler implicit time scheme | Euler |

$\nabla \xb7(\rho \mathit{u}\mathit{u})$ | Total Variation Diminishing | Limited linearV 1 |

$\nabla \xb7(\alpha \mathit{u})$ | Total Variation Diminishing | Gauss vanLeer |

$\nabla \xb7\left[\alpha (1-\alpha ){\mathit{u}}_{C}\right]$ | Bounded limited scheme | InterfaceCompression |

$\nabla \xb7\mathit{u}$ | Central Differencing Scheme | Linear |

$\nabla \alpha $ | Central Differencing Scheme | Linear |

$\nabla \cdot \left[\mu \left(\nabla \mathit{u}+\nabla {\mathit{u}}^{T}\right)\right]$ | Central Differencing Scheme | Linear Corrected |

Researchers | Eo | Ga | s |
---|---|---|---|

Zhang et al. [29] | 16 | 0.57, 63.36, 90.51 | 0.75, 1, 2 |

Yan et al. [31] | 2 | 8.8, 51, 95, 133 | 0.75, 1, 2 |

Our work | 0.1, 0.14, 1, 2, 10 | 99 | 0.75, 1, 1.2, 1.5 |

Refinement Level | Number of Cells per Bubble Diameter | Terminal Velocity (m/s) |
---|---|---|

1 | 10 | 0.3 |

2 | 15 | 0.357 |

3 | 20 | 0.34 |

Numerical (Mesh 3) | Experimental | % Error | |
---|---|---|---|

Terminal Velocity | 0.34 | 0.344 | 1.16 |

Drag Coefficient | 0.113 | 0.123 | 8.13 |

s | Amplitude (A) (mm) | Wavelength (λ) (mm) | Time Period (T) (s) |
---|---|---|---|

0.75 | 0.56 | 18.5 | 0.06 |

1 | 0.6 | 18 | 0.06 |

σ | 0.1 | 0.07 | 0.01 | 0.005 | 0.001 |
---|---|---|---|---|---|

Eo | 0.0981 | 0.14 | 0.981 | 1.96 | 9.81 |

Eo | Amplitude (A) (mm) | Wavelength (λ) (mm) | Time Period (T) (s) |
---|---|---|---|

0.1 | 0.73 | 25.1 | 0.07 |

0.14 | 0.6 | 18 | 0.06 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mundhra, R.; Lakkaraju, R.; Das, P.K.; Pakhomov, M.A.; Lobanov, P.D.
Effect of Wall Proximity and Surface Tension on a Single Bubble Rising near a Vertical Wall. *Water* **2023**, *15*, 1567.
https://doi.org/10.3390/w15081567

**AMA Style**

Mundhra R, Lakkaraju R, Das PK, Pakhomov MA, Lobanov PD.
Effect of Wall Proximity and Surface Tension on a Single Bubble Rising near a Vertical Wall. *Water*. 2023; 15(8):1567.
https://doi.org/10.3390/w15081567

**Chicago/Turabian Style**

Mundhra, Raghav, Rajaram Lakkaraju, Prasanta Kumar Das, Maksim A. Pakhomov, and Pavel D. Lobanov.
2023. "Effect of Wall Proximity and Surface Tension on a Single Bubble Rising near a Vertical Wall" *Water* 15, no. 8: 1567.
https://doi.org/10.3390/w15081567