# Dimensionless Analysis of the Effects of Junction Angle on the Gas-Liquid Two-Phase Flow Transition and the Scaling Law of the Microbubble Generation Characteristics in Y-Junctions

^{*}

## Abstract

**:**

_{1}< 14.38, 0.0068 < We

_{1}< 0.1723) was proposed to predict the bubble size and it showed a good agreement with the experimental results.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Material Preparation

^{−3}measured by a DIN ISO 3507-Gay-Lussac type standard density bottle. The viscosity was 1.58 mPa·s measured by Brookfield DV2TLVTJ0 Rheometer (Brookfield Engineering Laboratory Inc., Middleborough, MA, USA) and the surface tension was 34.17 mN·m measured by a Drop Shape Analysis System, Model DSA100 (Kruss GmbH, Hamburg, Germany). All the measurements above were performed at ambient temperature.

#### 2.2. Experimental Procedure and Data Analysis

^{TM}-EZ microfluidic sample pump was the part aiming to deliver fluids. The Y-junctions were the main dispersed systems to generate gas-liquid two-phase flow and the microscopy image system aimed to visualize the flow patterns and record the generation characteristics of the slug bubbles. The MFCZ

^{TM}-EZ microfluidic sample pump (Fluigent Co., Le Kremlin-Bicêtre, France) contained one flow measurement module and two fluid reservoirs. The three Y-junctions had the junction angle (φ) of 60°, 90° and 120°, respectively. The Y-junctions were manufactured by Suzhou Wen Hao Technology Company. The microchips were constructed by glass, and the microchannels were formed by hydrofluoric acid etching. After etching by hydrofluoric acid, the contact angle of the microchannels were 140° measured using the ImageJ software program. In addition, the dimensions of the microchannels were 150 μm in width and 75 μm in depth. XSZ optical microscope (Shangxin New Optical Technology Co., Shanghai, China) equipped with FASTCAM SA4 high speed camera (PHOTRON Co., Tokyo, Japan) or YM1600 high-definition digital camera (Shangxin New Optical Technology Co., Shanghai, China) was applied to capture the gas-liquid two-phase flow and collect the images of the microbubbles on the slide glass, respectively. The shooting speed of the FASTCAM SA4 high-speed camera was 125~5 × 10

^{5}fps. The frame rate was chosen to be 2 × 10

^{3}fps in this study to meet the experimental needs. The diameter of the microbubble was measured using the ImageJ software program. The multiple linear regression of dimensionless parameters was conducted by MatlabR2018a.

## 3. Results

#### 3.1. Flow Patterns and Force Analysis of the Formation of Slug Bubbles

_{c}), the pressure of the gas phase in the thread (P

_{D}) and the shear stress(τ

_{c,shear}) played a crucial role in the pinching-off process of the gas slug to resist the interfacial pressure. Based on the pressure analysis, the forces exerted on the thread are shown in Figure 3b. There were forces due to the flow of gas phase (F

_{D}), liquid phase (F

_{c}) and shear force (F

_{c,shear}). The force F

_{c}made the thread extend and causes the Rayleigh-Plateau instability in which phenomenon a liquid column could always break up into little droplets and the minimization of surface energy could be achieved. The force F

_{D}pinched off the thread and the shear force carried the slug downstream. The angle of the Y-junctions affected the distribution and magnitude of the forces when they were divided in the x-direction and y-direction. The annular flow was observed as the gas pressure continued increasing. However, in some situations, small waviness caused by the Kelvin-Helmholtz instability appeared on the interface of gas-liquid phase because of interfacial tension effect. The Kelvin-Helmholtz instability always occurs between two immiscible fluids with different velocity and leads to the unstable interfacial vortex street over their interface. Continuing increasing the gas pressure, the stratified flow was observed with obvious boundary between gas phase and liquid phase. The pressure in the two-phase flow were comparable.

#### 3.2. Flow Pattern Maps and the Transition Criteria of Dimensionless Analysis

_{g}~ 0–100 kPa, Q

_{L}~ 0–120 µL/min) changed, the variation rate of the forces in x-direction of the Y-120° system was the smallest due to the biggest value of φ/2, which led to a bigger area of the transition zone (the region of dispersed bubble flow). Different from the annular flow and dispersed bubble flow, the slug bubble flow occupied larger areas in the three flow pattern maps. The largest area of slug bubble flow appeared in the Y-90° system and the smallest area of slug bubble flow appeared in the Y-60° system. In contrast, the stratified flow occupied the largest area in the flow pattern map of the Y-60° system and the smallest area of the Y-120° system. The reason is that the forces in the Y-60° system were the smallest in y-direction and largest in x-direction, which made the two phases more likely to flow along the wall of the main channel instead of acting to form gas slug.

_{g}, liquid flow rate Q

_{L}, liquid viscosity μ

_{l}, liquid density ρ

_{l}and surface tension σ. In this study, the ∏ theorem was used to analyze the relationship between each physical quantity. The physical quantities v

_{l}, ρ

_{l}and σ with different basic dimensions were selected as basic quantities, and the remaining physical quantities were taken as derived quantities. Then, the expressions of the basic quantities were established,

_{i}, b

_{i}and c

_{i}were obtained, and the expressions of dimensionless parameters were,

_{c}), shear force (F

_{c,shear}), pressure force (F

_{D}) and surface tension force were involved. The surface tension force prevented the thread from pinching-off while the inertia force, shear force and pressure force helped the thread break up into gas slug. The Weber number represented for the ratio of F

_{c}and surface tension force, the Euler number represented for the ratio of F

_{D}and F

_{c}, while the Capillary number represented for the ratio of F

_{c,shear}and surface tension force. Therefore, from the point of view of mechanical analysis, the correlation of We, Eu and Ca should be used to describe the dynamics of gas-liquid interaction in the microfluidic Y-junctions. Normally, a function f was proposed to describe the transitions of the flow patterns,

^{3}< a(We × Eu)

^{b}(a = 6.5130 − 10.176(πφ) + 3.8645(πφ)

^{2}, b = −1.5698 + 4.2417(πφ) − 1.5850(πφ)

^{2}) with 0.002 < Ca < 0.02. Figure 5c,d illustrate the proposed transition line of dispersed bubble flow to slug bubble flow could be expressed as Ca × 10

^{3}= a(We × Eu)

^{b}(a = 0.00596 − 0.00222(πφ) + 0.00951(πφ)

^{2}, b = 2.20619 − 0.07656(πφ) + 0.14824(πφ)

^{2}) with 0.002 < Ca < 0.015. Finally, the transition line of slug bubble flow to annular flow could be described by Ca × 10

^{3}= a(We × Eu)

^{b}(a = 0.3634 + 0.1319(πφ) − 0.0049(πφ)

^{2}, b = 0.8997 − 0.3468(πφ) − 0.1766(πφ)

^{2}) with 0.002 < Ca < 0.010. Generally speaking, the generation region of gas-liquid two-phase flow in Y-junction with different angle could be obtained by employing the transition criteria given above in this study.

#### 3.3. Effect of the Angle Factor on the Generation Characteristics of Microbubbles

_{c}and F

_{D}which acted more powerfully to shorten the pinching-off process. The decreasing stage occurred when the component of F

_{c}in y-direction was so large that it acted not only to squeeze the gas thread but also prevent the gas phase from flowing into the main channel. In Figure 6, the generation frequency f obeyed the law that f

_{Y-60°}> f

_{Y-90°}> f

_{Y-120}

_{°}at constant air pressures and liquid flow rates. It was deduced that the component of F

_{c}caused by liquid inertia had the order of F

_{c-x-Y-60°}> F

_{c-x-Y-90°}> F

_{c-x-Y-120°}, and the larger F

_{c-x}caused the shorter generation time of gas slug.

_{Y-60°}> d

_{Y-90°}> d

_{Y-120°}until the liquid flow rate Q

_{L}reached 60 μL/min, where the influence of the angle factor was weakened as the liquid flow rate was large enough to make the shear force Fc,shear play a crucial role in the pinching-off process. Lei et al. [34] experimentally and numerically studied the size of microdroplets generated in Y-junctions with different angles (30°–120°). The study also found that the junction angle has a significant effect on the droplet size and the variation of droplet length with inlet angle obeyed a regular law under the experimental conditions. Kucuk et al. [35] prepared microbubbles in Y-junctions with different angles (0°–60°) and reported a similar law that the microbubble size decreased with increasing inlet angle. However, in this work, the law of d

_{Y-60°}> d

_{Y-90°}> d

_{Y-120°}was found to be broken up at large liquid flow rates when the shear force Fc,shear in x-direction dominated the generation process of the microbubble, as shown in Figure 7a–c. In contrast of air pressure, the microbubble size decreased with increasing liquid flow rate as shown in Figure 8. The microbubble size still obeyed the law of d

_{Y-60°}> d

_{Y-90°}> d

_{Y-120°}until the air pressure P

_{g}reached 40 kPa, where the influence of the angle factor was weakened as the air pressure was large enough to make the shear force F

_{D}play a crucial role in the pinching-off process. Generally speaking, the influence of the angle factor on microbubble generation characteristics only existed in a certain range of injection conditions, once the air pressure/liquid flow rate was large enough, the influence of the angle was weakened and vanished then. The applicable conditions of d

_{Y-60°}> d

_{Y-90°}> d

_{Y-120°}needed to be determined.

_{Y-60°}> d

_{Y-90°}> d

_{Y-120°}and measure the relation of the forces involved in the law, a dimensionless analysis of Capillary number, Weber number and Euler number was conducted. Figure 9 shows the relationship between the microbubble size and dimensionless parameters Ca and (We·Eu). The boundary line was obtained by projecting the boundary points to the surface of x(We·Eu)/y(Ca). Then, it could be seen obviously that the law was suitable for Ca < (148.52 × 10

^{−3})·(We·Eu)

^{−0.69}when Ca ranging from 0.002 to 0.02. In other words, the effect of the angle factor on the microbubble size became complicated out of this range.

_{g}, liquid flow rate Q

_{L}, liquid viscosity μ

_{l}, liquid density ρ

_{l}and surface tension σ. The microbubble diameter could be expressed as,

_{L}and liquid phase density ρ

_{l}were chosen to be the independent basic quantities with different basic dimensions of length, time and mass. Took the other physical quantities as derived quantities, and the following expressions were obtained,

_{i}, b

_{i}, c

_{i}were obtained, and the expressions of ∏

_{i}were,

_{1}, Re

_{1}

^{−1}and We

_{1}

^{−1}as,

_{1}< 14.38 and 0.0068 < We

_{1}< 0.1723. Figure 10a–c shows the comparison of the experimental data d with dimensionless expressions. The results show a good agreement with a maximum deviation within ±10%.

_{1}

^{b}Re

_{1}

^{c}We

_{1}

^{D}, where a = 62,843.7837−69,889.1621(πφ/180) + 19,066.6994(πφ/180)

^{2}, b = −0.2199 + 0.7317(πφ/180) − 0.2309(πφ/180), c = −8.4217 + 7.2621(πφ/180) − 1.9181(πφ/180)

^{2}, D = 0.9144 + 0.2550(πφ/180) − 0.1072(πφ/180)

^{2}. Figure 10 compares the microbubble diameter d between experimental values and predicted values. It shows a good agreement and the deviation is within ±12%.

## 4. Conclusions

_{Y-60°}> f

_{Y-90°}> f

_{Y-120}

_{°}at constant air pressures and liquid flow rates.

_{Y-60°}> d

_{Y-90°}> d

_{Y-120°}could be broken in inapplicable conditions. In other words, the effect of the junction angle on the microbubble size became complicated as the liquid flow rate or air pressure rose up to a constant, which has never been reported by the previous studies. The applicable region of the law could be expressed using the equation of Capillary number, Weber number and Euler number as Ca < (148.52 × 10

^{−3})·(We·Eu)

^{−0.69}with Ca ranging from 0.002 to 0.02.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

Symbols | |

a_{i} | the exponent of the Πi expression |

b_{i} | the exponent of the Πi expression |

c_{i} | the exponent of the Πi expression |

Ca | Capillary number |

d | microbubble diameter [μm] |

Eu | Euler number |

f | microbubble generation frequency [Hz] |

F_{D} | the force exerted by dispersed phase [N] |

F_{c} | the force exerted by continuous phase [N] |

F_{c,shear} | shear force [N] |

l | microchannel equivalent diameter [μm] |

P_{g} | air pressure [kPa] |

Q_{L} | liquid phase flow rate [μL/min] |

Re | Reynolds number |

v_{l} | liquid phase flow velocity [m/s] |

We | Weber number |

Greek | |

µ_{l} | liquid phase viscosity [Pa·s] |

ρ_{l} | liquid phase density [kg/m^{3}] |

σ | surface tension [N/m] |

## References

- Vladisavljević, G.T.; Ekanem, E.E.; Zhang, Z.; Khalid, N.; Kobayashi, I.; Nakajima, M. Long-term stability of droplet production by microchannel (step) emulsification in microfluidic silicon chips with large number of terraced microchannels. Chem. Eng. J.
**2018**, 333, 380–391. [Google Scholar] [CrossRef] [Green Version] - Charmet, J.; Arosio, P.; Knowles, T. Microfluidics for Protein Biophysics. J. Mol. Biol.
**2018**, 430, 565–580. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rong, N.; Zhou, H.; Liu, R.; Wang, Y.; Fan, Z. Ultrasound and microbubble mediated plasmid DNA uptake: A fast, global and multi-mechanisms involved process. J. Control. Release
**2018**, 273, 40–50. [Google Scholar] [CrossRef] [PubMed] - Liu, P.; Zhong, Y.; Luo, Y. Preparation of monodisperse biodegradable magnetic microspheres using a T-shaped microchannel reactor. Mater. Lett.
**2014**, 117, 37–40. [Google Scholar] [CrossRef] - Ansari, M.; Bokhari, H.H.; Turney, D.E. Energy efficiency and performance of bubble generating systems. Chem. Eng. Process.
**2018**, 125, 44–55. [Google Scholar] [CrossRef] - Walunj, A.; Sathyabhama, A. Comparative study of pool boiling heat transfer from various microchannel geometries. Appl. Therm. Eng.
**2018**, 128, 672–683. [Google Scholar] [CrossRef] - Wang, S.; Chen, H.; Chen, C. Enhanced flow boiling in silicon nanowire-coated manifold microchannels. Appl. Therm. Eng.
**2019**, 148, 1043–1057. [Google Scholar] [CrossRef] - Guo, R.; Fu, T.; Zhu, C.; Yin, Y.; Ma, Y. The effect of flow distribution on mass transfer of gas-liquid two-phase flow in two parallelized microchannels in a microfluidic loop. Int. J. Heat Mass Trans.
**2019**, 130, 266–273. [Google Scholar] [CrossRef] - Marques, M.; Boyd, A.S.; Polizzi, K.; Szita, N. Microfluidic devices towards personalized health and wellbeing. J. Chem. Technol. Biotechnol.
**2019**, 94, 2412–2415. [Google Scholar] [CrossRef] - Zhu, L.-L.; Zhu, C.-T.; Xiong, M.; Jin, C.-Q.; Sheng, S.; Wu, F.-A.; Wang, J. Enzyme immobilization on photopatterned temperature-response poly (N-isopropylacrylamide) for microfluidic biocatalysis. J. Chem. Technol. Biotechnol.
**2019**, 94, 1670–1678. [Google Scholar] [CrossRef] - Zhu, Y.; Bai, Z.; Luo, W.; Wang, B.; Zhai, L. A facile ion imprinted synthesis of selective biosorbent for Cu
^{2+}via microfluidic technology. J. Chem. Technol. Biotechnol.**2017**, 92, 2009–2022. [Google Scholar] [CrossRef] - Chiu, F.W.Y.; Bagci, H.; Fisher, A.G.; Demello, A.J.; Elvira, K.S. A microfluidic toolbox for cell fusion. J. Chem. Technol. Biotechnol.
**2016**, 91, 16–24. [Google Scholar] [CrossRef] [Green Version] - Yu, W.; Liu, X.; Li, B.; Chen, Y. Experiment and prediction of droplet formation in microfluidic cross-junctions with different bifurcation angles. Int. J. Multiph. Flow
**2022**, 149, 103973. [Google Scholar] [CrossRef] - Yu, W.; Liu, X.; Zhao, Y.; Chen, Y. Droplet generation hydrodynamics in the microfluidic cross-junction with different junction angles. Chem. Eng. Sci.
**2019**, 203, 259–284. [Google Scholar] [CrossRef] - Yin, Y.; Chen, K.; Qiao, X.; Lin, M.; Lin, Z.; Wang, Q. Mean pressure distributions on the vanes and flow loss in the branch in a T pipe junction with different angles. Energy Procedia
**2017**, 105, 3239–3244. [Google Scholar] [CrossRef] - Yue, J.; Luo, L.; Gonthier, Y.; Chen, G.; Yuan, Q. An experimental investigation of gas-liquid two-phase flow in single microchannel contactors. Chem. Eng. Sci.
**2008**, 63, 4189–4202. [Google Scholar] [CrossRef] - Zhao, C.; Middelberg, A.P.J. Two-phase microfluidic flows. Chem. Eng. Sci.
**2011**, 66, 1394–1411. [Google Scholar] [CrossRef] - Liu, Y.; Hansen, A.; Block, E.; Morrow, N.R.; Squier, J.; Oakey, J. Two-phase displacements in microchannels of triangular cross-section. J. Colloid Interface Sci.
**2017**, 507, 234–241. [Google Scholar] [CrossRef] - Fourar, M.; Bories, S. Experimental study of air-water two-phase flow through a fracture (narrow channel). Int. J. Multiph. Flow
**1995**, 21, 621–637. [Google Scholar] [CrossRef] - Mandhane, J.M.; Gregory, G.A.; Aziz, K. A flow pattern map for gas-liquid flow in horizontal pipes. Int. J. Multiph. Flow
**1974**, 1, 537–553. [Google Scholar] [CrossRef] - Triplett, K.; Ghiaasiaan, S.; Abdel-Khalik, S.; Sadowski, D. Gas-liquid two-phase flow in microchannels Part I: Two-phase flow patterns. Int. J. Multiph. Flow
**1999**, 25, 377–394. [Google Scholar] [CrossRef] - Rebrov, E.V. Two-phase flow regimes in microchannels. Theor. Found. Chem. Eng.
**2010**, 44, 355–367. [Google Scholar] [CrossRef] - Yan, P.; Jin, H.; Tao, F.; He, G.; Guo, X.; Ma, L.; Yang, S.; Zhang, R. Flow characterization of gas-liquid with different liquid properties in a Y-type microchannel using electrical resistance tomography and volume of fluid model. J. Taiwan Inst. Chem. Eng.
**2022**, 136, 104390. [Google Scholar] [CrossRef] - Waelchli, S.; Rudolf Von Rohr, P. Two-phase flow characteristics in gas-liquid microreactors. Int. J. Multiph. Flow
**2006**, 32, 791–806. [Google Scholar] [CrossRef] - Essimoz, A.-L.; Cavin, L.; Renken, A.; Kiwi-Minsker, L. Liquid-liquid two-phase flow patterns and mass transfer characteristics in rectangular glass microreactors. Chem. Eng. Sci.
**2008**, 63, 4035–4044. [Google Scholar] [CrossRef] [Green Version] - Cao, Z.; Wu, Z.; Sundén, B. Dimensionless analysis on liquid-liquid flow patterns and scaling law on slug hydrodynamics in cross-junction microchannels. Chem. Eng. J.
**2018**, 344, 604–615. [Google Scholar] [CrossRef] - Garstecki, P.; Fuerstman, M.J.; Stone, H.A.; Whitesides, G.M. Formation of droplets and bubbles in a microfluidic T-junction-scaling and mechanism of break-up. Lab. Chip
**2006**, 6, 437. [Google Scholar] [CrossRef] - Wang, K.; Xie, L.; Lu, Y.; Luo, G. Generating microbubbles in a co-flowing microfluidic device. Chem. Eng. Sci.
**2013**, 100, 486–495. [Google Scholar] [CrossRef] - Parhizkar, M.; Edirisinghe, M.; Stride, E. Effect of operating conditions and liquid physical properties on the size of monodisperse microbubbles produced in a capillary embedded T-junction device. Microfluid. Nanofluid.
**2013**, 14, 797–808. [Google Scholar] [CrossRef] - Sobieszuk, P.; Cygański, P.; Pohorecki, R. Bubble lengths in the gas-liquid Taylor flow in microchannels. Chem. Eng. Res. Des.
**2010**, 88, 263–269. [Google Scholar] [CrossRef] - Rodríguez-Rodríguez, J.; Sevilla, A.; Martínez-Bazán, C.; Gordillo, J.M. Generation of microbubbles with applications to industry and medicine. Annu. Rev. Fluid Mech.
**2015**, 47, 405–429. [Google Scholar] [CrossRef] - Peng, Z.; Gai, S.; Barma, M.; Rahman, M.M.; Moghtaderi, B.; Doroodchi, E. Experimental study of gas-liquid-solid flow characteristics in slurry Taylor flow-based multiphase microreactors. Chem. Eng. J.
**2021**, 405, 126646. [Google Scholar] [CrossRef] - Taylor, J.R. Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, 2nd ed.; University Science Books: Boulder, CO, USA, 1997. [Google Scholar]
- Lei, L.; Zhao, Y.; Wang, X.; Xin, G.; Zhang, J. Experimental and numerical studies of liquid-liquid slug flows in micro channels with Y-junction inlets. Chem. Eng. Sci.
**2022**, 252, 117289. [Google Scholar] [CrossRef] - Kucuk, I.; Yilmaz, N.F.; Sinan, A. Effects of junction angle and gas pressure on polymer nanosphere preparation from microbubbles bursted in a combined microfluidic device with thin capillaries. J. Mol. Struct.
**2018**, 1173, 422–427. [Google Scholar] [CrossRef]

**Figure 3.**A schematic of slug formation process corresponding with forces: (

**a**) The pressure field around the gas thread; (

**b**) The force analysis of the gas slug formation.

**Figure 4.**The flow pattern maps of the three Y-junctions with (

**a**) φ = 60°; (

**b**) φ = 90°; (

**c**) φ = 120°.

**Figure 5.**The comparisons of experimental data and proposed flow transition model: (

**a**,

**b**) The transition from full of liquid to dispersed bubble flow; (

**c**,

**d**) The transition from dispersed bubble flow to slug bubble flow; (

**e**,

**f**) The transition from slug bubble flow to annular flow.

**Figure 6.**Effects of air pressure and liquid flow rate on microbubble generation frequency: (

**a**) Variation of the generation frequency with air pressure; (

**b**) Variation of the generation frequency with liquid flow rate.

**Figure 7.**The variation of the microbubble size (diameter) with air pressure and angle factor at constant liquid flow rates: (

**a**) Q

_{L}= 36 μL/min; (

**b**) Q

_{L}= 60 μL/min; (

**c**) Q

_{L}= 77 μL/min; (

**d**) Q

_{L}= 24~77 μL/min.

**Figure 8.**The variation of the microbubble size (diameter) with liquid flow rate and angle factor at constant air pressures: (

**a**) P

_{g}= 30 kPa; (

**b**) P

_{g}= 40 kPa; (

**c**) P

_{g}= 50 kPa; (

**d**) P

_{g}= 20~50 kPa.

**Figure 9.**The variations of the microbubble size(diameter) with Capillary number Ca and dimensionless parameter (We·Eu) in the Y-60°, Y-90° and Y-120° systems.

**Figure 10.**Microbubble size d as a function of the dimensionless parameters: (

**a**) φ = 60°; (

**b**) φ = 90°; (

**c**) φ = 120°; (

**d**) function with angle factor coefficients.

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## Share and Cite

**MDPI and ACS Style**

Han, Y.; Xu, X.; Liu, F.; Wei, W.; Liu, Z.
Dimensionless Analysis of the Effects of Junction Angle on the Gas-Liquid Two-Phase Flow Transition and the Scaling Law of the Microbubble Generation Characteristics in Y-Junctions. *Sustainability* **2022**, *14*, 8592.
https://doi.org/10.3390/su14148592

**AMA Style**

Han Y, Xu X, Liu F, Wei W, Liu Z.
Dimensionless Analysis of the Effects of Junction Angle on the Gas-Liquid Two-Phase Flow Transition and the Scaling Law of the Microbubble Generation Characteristics in Y-Junctions. *Sustainability*. 2022; 14(14):8592.
https://doi.org/10.3390/su14148592

**Chicago/Turabian Style**

Han, Yu, Xiaofei Xu, Fengxia Liu, Wei Wei, and Zhijun Liu.
2022. "Dimensionless Analysis of the Effects of Junction Angle on the Gas-Liquid Two-Phase Flow Transition and the Scaling Law of the Microbubble Generation Characteristics in Y-Junctions" *Sustainability* 14, no. 14: 8592.
https://doi.org/10.3390/su14148592