On the Detachment of a Bubble from an Orifice

1. 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 9th April 2013

2. Production of Small Bubbles Dietrich et al. 2013

3. Constant Gas Flow Rate Q Increasing Flow Rate Q Zhang & Shoji 2001

4. Influence of Gas Flow Rate td Vd Q Q Gerlach et al. 2007 Corchero et al. 2007 Low gas flow rate – static regime High gas flow rate – dynamic regime

5. Modelling Assumptions z • 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. φ Axis of Symmetry ta na Liquid za tg ng Gas r rc Solid Surface

6. Dimensionless Problem Formulation z 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 φ Axis of Symmetry ta na Liquid za tg ng Gas r rc Solid Surface

7. Dimensionless Problem Formulation BulkBubble Apex Contact Line Free SurfaceFar Field Axis of Symmetry Solid SurfaceOther

8. Parameter Regime Three parameters: Re, rc, Q Consider: rc=1, 0.1 Re=1, 100, 10000

9. Numerical Method • Finite Element Method • ‘Far field’ set far from bubble so as not influence bubble growth • Method of spines z φ ta Axis of Symmetry na Liquid Liquid tg za ng Gas Free Surface r rc Solid Surface

10. Finite Element Mesh rc=1, Re=1, Q=7.5

11. Quasi-static Approximation Young-Laplace equation (Fordham 1948) rc=1, Re=1, Q=10-5 rc=0.1, Re=100, Q=10-6

12. rc=1.0 • Low Gas Flow Rate • As Q 0, Vd approaches a limit. • Re has negligible influence on Vd.

13. Increasing Gas Flow Rate Q=0.01 Q=0.05 Q=0.1 rc=1.0, Re=10,000

14. rc=1.0 • Low Gas Flow Rate • As Q 0, Vd approaches a limit. • Re has negligible influence on Vd.

15. Increasing Reynolds number Re=1 Re=100 Re=10,000 rc=1.0, Q=0.1

16. rc=1.0 • Low Gas Flow Rate • As Q 0, Vd approaches a limit. • Re has negligible influence on Vd. High Gas Flow Rate As Q increases, -td tends to a limit, which increases with Re. -Vd increases with Q and Re. -Good agreement with scaling laws.

17. rc=0.1 • Low Gas Flow Rate • As Q 0, Vd approaches a limit. • Re has negligible influence on Vd. High Gas Flow Rate As Q increases, -Vd increases with Q and Re. -Not so good agreement with scaling laws.

18. Increasing Gas Flow Rate Q=5 x 10-4 Q=10-3 Q=5 x 10-3 rc=0.1, Re=10,000

19. Bubble Pinch-off t rc=1, Re=10000, Q=10-5 Thoroddsen et al. 2007

20. Summary • Developed a framework for bubble detachment phenomenon. • Results agree qualitatively with experiments. • Identify the accuracy of various scaling laws.

21. 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.