# Experimental Study of Bubble Formation from a Micro-Tube in Non-Newtonian Fluid

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup

^{−6}–1.09 × 10

^{−6}m

^{3}/s) was measured by a bubble flowmeter, while bubble growth was captured by a high-speed video camera (HighSpec4, FASTEC, San Diego, CA, USA).

_{0}), i.e., the time required from the moment the bubble is formed on the edge of the tube until it detaches from it, can be calculated.

_{p}is the equivalent bubble diameter and H and L are the major and minor axes of the ellipsoid, respectively (Figure 3). The maximum uncertainty in measuring the length of each axis of the bubble is ± 10 μm and is attributed to the unavoidable shadows formed at the bubble interface. The minimum measured equivalent diameter in the present study was about 4.1 mm, and thus the uncertainty in measuring the size of the bubbles was around ± 8%.

_{d}is the average drag coefficient, w

_{α}is the bubble growth rate, α is the bubble growth acceleration, and d

_{α}is the μ-tube diameter.

_{g}is the air velocity, P

_{g}is the air pressure in the bubble, and P

_{L}is the average pressure exerted by the liquid.

## 3. Results and Discussion

^{−6}m

^{3}/s). In both cases, the time t

_{0}= 0 corresponds to the detachment instant.

#### 3.1. Effect of Flow Rate

^{−6}and 1.09 × 10

^{−6}m

^{3}/s, while it is estimated that a 35% increase in flow rate leads to a 10% increase in the equivalent diameter of the bubble. Similar results are obtained for Newtonian solution G1n for the same flow rates.

_{min}= 0.71 × 10

^{−6}m

^{3}/s), the gas momentum is of the order of 1 × 10

^{−4}μN, while for the maximum gas flow rate (Q

_{max}= 1.09 × 10

^{−6}m

^{3}/s), the momentum is of the order of 2 × 10

^{−4}μN. However, even for the maximum flow rate, the gas momentum remains weak comparing to all the other forces, as the buoyancy is equal to 0.53 N and the pressure is equal to 0.01N.

#### 3.2. Effect of Viscosity

^{−6}m

^{3}/s).

#### 3.3. Effect of the Type of the Liquid Phase

^{−6}m

^{3}/s.

#### 3.4. Computational Study

^{®}R1 code that employs the Volume of Fluid (VOF) method, which is an Euler–Euler approach that is suitable for similar cases [24]. The phenomenon was simulated in two dimensions using axial symmetry. Based on the grid dependence study, a computing space consisting of 306,000 cells was selected. The pressure–velocity coupling was executed using the SIMPLEC algorithm. Respectively, to avoid the effects of the pressure field on the main flow of the bubble, the PRESTO! algorithm was used as the pressure interpolation scheme.

_{∞}= 23.5 mPa∙s). The boundary conditions correspond to the experimental conditions, and thus the range of the gas velocity at the inlet of the tube equals 1.39—2.24 m/s. A non-slip condition was imposed on the wall of the cell, thus not affecting the bubble formation, while the pressure of the liquid away from the air inlet was considered equal to the atmospheric pressure.

_{∞}= 23.5 mPa∙s, while away from the bubble surface, the shear rate decreases, leading to lower viscosity values. This result interprets the experimental finding, i.e., that the bubbles formed in a non-Newtonian fluid have practically the same equivalent diameter with the bubbles formed in a Newtonian fluid with viscosity equal to the asymptotic viscosity of the non-Newtonian fluid.

## 4. Conclusions

^{−6}–1.09 × 10

^{−6}m

^{3}/s) the bubble reaches the detachment point in less time, while its equivalent diameter is larger. As the viscosity increases, the release time and the equivalent diameter also increase. Thus, the bubble size increases with increasing asymptotic viscosity, while around the bubble, the shear rate is high, and as a result, the viscosity around the bubble in a non-Newtonian solution has a value equal to that of asymptotic viscosity. The bubbles formed in Newtonian fluid with viscosity equal to the asymptotic viscosity of the non-Newtonian have the same equivalent diameter with the bubbles formed in the non-Newtonian fluid. The bubble behavior is strongly affected by the gas flow rate and the viscosity. Future study on the effect of the other parameters would lead to an even better interpretation of the phenomenon.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- 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] - Skurtys, O.; Bouchon, P.; Aguilera, J. Formation of bubbles and foams in gelatine solutions within a vertical glass tube. Food Hydrocoll.
**2008**, 22, 706–714. [Google Scholar] [CrossRef] - Zhou, F.; Wang, L.; Xu, Z.; Liu, Q.; Chi, R. Reactive oily bubble technology for flotation of apatite, dolomite and quartz. Int. J. Miner. Process.
**2015**, 134, 74–81. [Google Scholar] [CrossRef] - Dijkmans, P.; Juffermans, L.; Musters, R.; Van Wamel, A.; Cate, F.T.; Van Gilst, W.; Visser, C.; De Jong, N.; Kamp, O. Microbubbles and ultrasound: From diagnosis to therapy. Eur. J. Echocardiogr.
**2004**, 5, 245–256. [Google Scholar] [CrossRef] - Felgner, P.L. Nonviral Strategies for Gene Therapy. Sci. Am.
**1997**, 276, 102–106. [Google Scholar] [CrossRef] [PubMed] - Fischer, M.; Zinovik, I.; Poulikakos, D. Diffusion and reaction controlled dissolution of oxygen microbubbles in blood. Int. J. Heat Mass Transf.
**2009**, 52, 5013–5019. [Google Scholar] [CrossRef] - Lewandowski, K. Extracorporeal membrane oxygenation for severe acute respiratory failure. Crit. Care
**2000**, 4, 156–168. [Google Scholar] [CrossRef] [Green Version] - Makdisi, G.; Wang, I.-W. Extra Corporeal Membrane Oxygenation (ECMO) review of a lifesaving technology. J. Thorac. Dis.
**2015**, 7, E166–E176. [Google Scholar] - Hernot, S.; Klibanov, A.L. Microbubbles in ultrasound-triggered drug and gene delivery. Adv. Drug Deliv. Rev.
**2008**, 60, 1153–1166. [Google Scholar] [CrossRef] [Green Version] - Sirsi, S.R.; Borden, M.A. Microbubble Compositions, Properties and Biomedical Applications. Bubble Sci. Eng. Technol.
**2009**, 1, 3–17. [Google Scholar] [CrossRef] - De Menech, M.; Garstecki, P.; Jousse, F.; Stone, H.A. Transition from squeezing to dripping in a microfluidic T-shaped junction. J. Fluid Mech.
**2008**, 595, 141–161. [Google Scholar] [CrossRef] - Fu, T.; Ma, Y.; Funfschilling, D.; Li, H.Z. Gas–liquid flow stability and bubble formation in non-Newtonian fluids in microfluidic flow-focusing devices. Microfluid. Nanofluidics
**2010**, 10, 1135–1140. [Google Scholar] [CrossRef] - Kazakis, N.A.; Mouza, A.A.; Paras, S.V. Coalescence during Bubble Formation at Two Neighbouring Pores: An Ex-perimental Study in Microscopic Scale. Chem. Eng. Sci.
**2008**, 63, 5160–5178. [Google Scholar] [CrossRef] - Tsamopoulos, J.; Dimakopoulos, Y.; Chatzidai, N.; Karapetsas, G.; Pavlidis, M. Steady bubble rise and deformation in Newtonian and viscoplastic fluids and conditions for bubble entrapment. J. Fluid Mech.
**2008**, 601, 123–164. [Google Scholar] [CrossRef] - Acharya, A.; Mashelkar, R.A.; Ulbrecht, J.J. Bubble Formation in Non-Newtonian Liquids. Ind. Eng. Chem. Fundam.
**1978**, 17, 230–232. [Google Scholar] [CrossRef] - Fan, W.; Jiang, S.; Zhu, C.; Ma, Y.; Li, H. Study on Bubble Formation in Non-Newtonian Fluids by Laser Image Technique. Opt. Laser Technol.
**2008**, 40, 389–393. [Google Scholar] [CrossRef] - Fu, T.; Ma, Y.; Funfschilling, D.; Li, H.Z. Bubble formation in non-Newtonian fluids in a microfluidic T-junction. Chem. Eng. Process. Process. Intensif.
**2011**, 50, 438–442. [Google Scholar] [CrossRef] - Martinez, C.J. Bubble generation in microfluidic devices. Bubble Sci. Eng. Technol.
**2009**, 1, 40–52. [Google Scholar] [CrossRef] - Li, H.Z.; Frank, X.; Funfschilling, D.; Mouline, Y. Towards the understanding of bubble interactions and coalescence in non-Newtonian fluids: A cognitive approach. Chem. Eng. Sci.
**2001**, 56, 6419–6425. [Google Scholar] [CrossRef] - Lin, T.-J.; Lin, G.-M. Mechanisms of in-line coalescence of two-unequal bubbles in a non-Newtonian fluid. Chem. Eng. J.
**2009**, 155, 750–756. [Google Scholar] [CrossRef] - Snabre, P.; Magnifotcham, F.I. Formation and rise of a bubble stream in a viscous liquid. Eur. Phys. J. B
**1998**, 4, 369–377. [Google Scholar] [CrossRef] - Gaddis, E.; Vogelpohl, A. Bubble formation in quiescent liquids under constant flow conditions. Chem. Eng. Sci.
**1986**, 41, 97–105. [Google Scholar] [CrossRef] - Khurana, A.K.; Kumar, R. Studies in Bubble Formation—III. Chem. Eng. Sci.
**1969**, 24, 1711–1723. [Google Scholar] [CrossRef] - Sokolichin, A.; Eigenberger, G.; Lapin, A.; Lübert, A. Dynamic numerical simulation of gas-liquid two-phase flows Euler/Euler versus Euler/Lagrange. Chem. Eng. Sci.
**1997**, 52, 611–626. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setup. (1) Gas flowmeter, (2) Valve, (3) Three-way valve, (4), Bubble flowmeter, (5) Plexiglass cell, (6) High-speed digital video camera, and (7) PC.

**Figure 5.**Typical sequence of bubble detachment in non-Newtonian fluid G1 (Q = 1.09 × 10

^{−6}m

^{3}/s).

**Figure 6.**Typical sequence of bubble detachment in non-Newtonian fluid G2 (Q = 1.09 × 10

^{−6}m

^{3}/s).

**Figure 8.**Bubble size comparison during release time t

_{0}from the tube for (

**a**) G1, (

**b**) G2 (Q = 0.81 × 10

^{−6}m

^{3}/s).

**Figure 10.**Frame of bubble 1 ms after the detachment: (

**a**) Computational Fluid Dynamics (CFD), (

**b**) Experiment G2 (Q = 1.09 × 10

^{−6}m

^{3}/s).

**Figure 11.**Distribution of (

**a**) shear rate and (

**b**) viscosity around a formed bubble (G2, Q = 1.09 × 10

^{−6}m

^{3}/s).

Liquid Phase | Content | ρ (kg/m^{3}) | μ (mPa∙s) | μ_{∞}(mPa∙s) | σ (mN/m) | ||
---|---|---|---|---|---|---|---|

Water (%v/v) | Glycerin (%v/v) | Xanthan (g/100 mL) | |||||

W | 100 | - | - | 997 | 1 | - | 72 |

G1n | 45 | 55 | - | 1140 | 8.5 | - | 68 |

G1 | 50 | 50 | 0.025 | 1126 | 0.0332 + 0.0421γ^{0.75} | 8.5 | 68 |

G2n | 28 | 72 | - | 1186 | 23.5 | - | 67 |

G2 | 30 | 70 | 0.025 | 1181 | 0.0098 + 0.0271γ^{0.98} | 23.5 | 67 |

Flow Rate Q (m ^{3}/s) | Equivalent Diameter d _{p} (mm) | Release Time t (s) | Bubble Volume V (10 ^{−8} m^{3}) |
---|---|---|---|

0.71 | 3.9 | 0.027 | 1.92 |

0.81 | 4.1 | 0.017 | 1.38 |

0.94 | 4.4 | 0.015 | 1.41 |

1.09 | 4.5 | 0.013 | 1.42 |

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**MDPI and ACS Style**

Kontaxi, G.; Stergiou, Y.G.; Mouza, A.A.
Experimental Study of Bubble Formation from a Micro-Tube in Non-Newtonian Fluid. *Micromachines* **2021**, *12*, 71.
https://doi.org/10.3390/mi12010071

**AMA Style**

Kontaxi G, Stergiou YG, Mouza AA.
Experimental Study of Bubble Formation from a Micro-Tube in Non-Newtonian Fluid. *Micromachines*. 2021; 12(1):71.
https://doi.org/10.3390/mi12010071

**Chicago/Turabian Style**

Kontaxi, Georgia, Yorgos G. Stergiou, and Aikaterini A. Mouza.
2021. "Experimental Study of Bubble Formation from a Micro-Tube in Non-Newtonian Fluid" *Micromachines* 12, no. 1: 71.
https://doi.org/10.3390/mi12010071