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.