on the detachment of a bubble from an orifice
DESCRIPTION
On the Detachment of a Bubble from an Orifice. By Jonathan Simmons Prof. Yulii D. Shikhmurzaev Dr James Sprittles British Applied Mathematics Colloquium, University of Leeds, Tuesday 9 th April 2013. Production of Small Bubbles. Dietrich et al. 2013. Constant Gas Flow Rate. Q. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/1.jpg)
On the Detachment of a Bubble from an OrificeBy Jonathan Simmons
Prof. Yulii D. ShikhmurzaevDr James Sprittles
British Applied Mathematics Colloquium,University of Leeds,Tuesday 9th April 2013
![Page 2: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/2.jpg)
Production of Small Bubbles
Dietrich et al. 2013
![Page 3: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/3.jpg)
Constant Gas Flow Rate
Zhang & Shoji 2001
Increasing Flow Rate Q
Q
![Page 4: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/4.jpg)
Influence of Gas Flow Rate
Gerlach et al. 2007Q
Vd
Q
td
Corchero et al. 2007
Low gas flow rate – static regimeHigh gas flow rate – dynamic regime
![Page 5: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/5.jpg)
Modelling Assumptions Axisymmetric about the z-
axis in a cylindrical coordinate system.
Incompressible, viscous Newtonian liquid.
Submerged, smooth solid surface with a circular orifice.
Contact line remains pinned to the edge of the orifice.
Gas in inviscid and dynamically passive with negligible density and so gas pressure pg is spatially homogeneous.
Bubble inflates due to a constant gas flow rate.
z
na
r
ng
φ
za
rc Solid Surface
Axis of Symmetry
Gas
Liquidtg
ta
![Page 6: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/6.jpg)
Dimensionless Problem Formulation
z
na
r
ng
φ
za
rc Solid Surface
Axis of Symmetry
Gas
Liquidtg
ta
Scaling Lengths with L
Velocity u(r,z,t) with U
Time t with L/U
Flow Rate Q with L2U
Pressure p(r,z,t) with /L
12
gLBog
L
1
UCaU
![Page 7: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/7.jpg)
Bulk Bubble Apex
Contact Line
Free Surface Far Field
Axis of Symmetry
Solid Surface Other00 aaa nutPn
0u
220 zrasu
0,0, trf c
QtVVzr i 0u )0,,(
Dimensionless Problem Formulation
g
p
t
T
z
4
3
Re
)(Re0
uuIP
ePuuuu
0),,(),,(
tzrfttzrf
p
ggg
gggg
u
0nnIPn
nnPn
0 ga nn
![Page 8: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/8.jpg)
Parameter Regime
Consider: rc=1, 0.1Re=1, 100, 10000
Three parameters: Re, rc, QWater Silicone Oil Glycerol
Viscosity (Pa s)
0.001 0.01 1.4
Density ρ (kg/m3)
1000 800 1200
Surface tension (mN/m)
70 20 60
L (mm) 2.7 1.6 2.3Re 187,000 255 0.082L2 U (cm3/s) 500 5 0.22
![Page 9: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/9.jpg)
Numerical Method•Finite Element Method
•‘Far field’ set far from bubble so as not influence bubble growth
•Method of spines
Solid Surfacer
z
na
ng
φ
za
rc
Axis of Symmetry
Gas
Liquidtg
ta
Liquid
Free Surface
![Page 10: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/10.jpg)
Finite Element Mesh
rc=1, Re=1, Q=7.5
![Page 11: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/11.jpg)
Quasi-static Approximation
rc=1, Re=1, Q=10-5
rc=0.1, Re=100, Q=10-6
Young-Laplace equation (Fordham 1948)
21
11RR
gzpg 21
11RR
zpg
![Page 12: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/12.jpg)
rc=1.0
6.4dV
Qtd /6.4
Low Gas Flow Rate-As Q 0, Vd approaches a limit.-Re has negligible influence on Vd. 000,10Re,261 dt
100Re,31 dt1Re,7 dt
![Page 13: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/13.jpg)
Increasing Gas Flow Rate
rc=1.0, Re=10,000
Q=0.01
Q=0.05
Q=0.1
![Page 14: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/14.jpg)
rc=1.0
6.4dV
Qtd /6.4
Low Gas Flow Rate-As Q 0, Vd approaches a limit.-Re has negligible influence on Vd. 000,10Re,261 dt
100Re,31 dt1Re,7 dt
![Page 15: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/15.jpg)
Increasing Reynolds number
rc=1.0, Q=0.1
Re=1 Re=100
Re=10,000
![Page 16: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/16.jpg)
High Gas Flow RateAs Q increases,-td tends to a limit, which increases with Re.-Vd increases with Q and Re.-Good agreement with scaling laws.
rc=1.0
6.4dV
Qtd /6.4
Low Gas Flow Rate-As Q 0, Vd approaches a limit.-Re has negligible influence on Vd. 000,10Re,261 dt
100Re,31 dt1Re,7 dt
56
QVd
43
QVd
![Page 17: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/17.jpg)
rc=0.1Low Gas Flow Rate-As Q 0, Vd approaches a limit.-Re has negligible influence on Vd.
53.0dV
Qtd /53.0
High Gas Flow RateAs Q increases,-Vd increases with Q and Re.-Not so good agreement with scaling laws.
43
QVd
56
QVd
![Page 18: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/18.jpg)
Increasing Gas Flow Rate
rc=0.1, Re=10,000
Q=5 x 10-4
Q=10-3
Q=5 x 10-3
![Page 19: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/19.jpg)
Bubble Pinch-off
Thoroddsen et al. 2007
ttr d ~min
t
ttr d min
rc=1, Re=10000, Q=10-5
![Page 20: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/20.jpg)
SummaryDeveloped a framework for
bubble detachment phenomenon.
Results agree qualitatively with experiments.
Identify the accuracy of various scaling laws.
![Page 21: On the Detachment of a Bubble from an Orifice](https://reader035.vdocument.in/reader035/viewer/2022062323/56816337550346895dd3c54c/html5/thumbnails/21.jpg)
References Corchero, G., Medina, A., Higuera, F.J., Coll. Surf. A. 290:41-49,
2006.
Dietrich, N., Mayoufi, N., Poncin, S., Li, H. , Chem. Papers 67(3):313-325, 2013.
Fordham, S., Proc. R. Soc. Lond. A. 194:1-16, 1948.
Gerlach, D., Alleborn, N., Buwa, V., Durst, F., Chem. Eng. Sci. 62:2109-2125, 2007.
Kistler, S.F., Scriven, L.E., Coating Flows in Computational Analysis of Polymer Processing, Elsevier, New York, 1983.
Thoroddsen, S.T., Etoh, T.G., Takehara, K., Phys. Fluids 19:042101, 2007.
Zhang, L., Shoji, M., Chem. Eng. Sci. 56:5371-5381, 2001.