on the detachment of a bubble from an orifice
Post on 23-Feb-2016
24 Views
Preview:
DESCRIPTION
TRANSCRIPT
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
Production of Small Bubbles
Dietrich et al. 2013
Constant Gas Flow Rate
Zhang & Shoji 2001
Increasing Flow Rate Q
Q
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
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
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
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
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
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
Finite Element Mesh
rc=1, Re=1, Q=7.5
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
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
Increasing Gas Flow Rate
rc=1.0, Re=10,000
Q=0.01
Q=0.05
Q=0.1
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
Increasing Reynolds number
rc=1.0, Q=0.1
Re=1 Re=100
Re=10,000
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
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
Increasing Gas Flow Rate
rc=0.1, Re=10,000
Q=5 x 10-4
Q=10-3
Q=5 x 10-3
Bubble Pinch-off
Thoroddsen et al. 2007
ttr d ~min
t
ttr d min
rc=1, Re=10000, Q=10-5
SummaryDeveloped a framework for
bubble detachment phenomenon.
Results agree qualitatively with experiments.
Identify the accuracy of various scaling laws.
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.
top related