Breakup of bubbles, drops and jets with surfactant

1. Breakup of bubbles, drops and jets with surfactant M. Siegel Collaborators: M. Booty, M. Hameed, D. Papageorgiou, Y. Young Mathematical Sciences, New Jersey Institute of Technology, J. Li Cambridge University

2. Outline 1. Pinch off of inviscid (or slightly viscous) bubble or thread -Experiment -Numerical simulation of full free boundary problem -Slender body theory (simple example) -Effect of surfactant 2. Break up of inviscid bubble with surfactant in extensional flow -Tip streaming -Briefly review 2D result -Slender body theory for axisymmetric extensional flow

3. Previous work (breakup of drop or thread with surfactant) Milliken, (1993) Kwak and Pozrikidis (2001) -Numerical calculations, mainly focus on viscocity ratio 1 or greater Ambbravaneswaran, Basaran (1999) Liao et al (2006) -Numercal investigation of effect of insoluble surfactant on pinch off of liquid bridge in inviscid surrounding Jin et al (2006) -Soluble surfactant effects on detachment of viscous drop from nozzle

4. Bubble breakup • Experiment showing the breakup of a water drop in silicone oil • (viscosity ratio ) • Parabolic profile near the minimum collapses radially, persists until • --this is inviscid collapse • Interior viscosity `cuts off’ radial collapse • Inviscid collapse will continue to molecular scale for • (air bubble in thick syrup) Doshi, Cohen, Zhang, Siegel, Howell, Basaran, Nagel Science(2003)

5. Numerical simulation of Navier-Stokes free boundary problem • ALE method with body fitted grid Doshi et al Science 2003 Hameed, Siegel, Young, Li,,Booty, Papageorgiou (JFM,to appear 2007) • Constant velocity at pinch • Numerics and analysis show that pinch off is `nonuniversal’, • e.g., interface shape at pinch off depends on initial and boundary • conditions Doshi, Cohen, Zhang, Siegel, Howell, Basaran, • Nagel (2003)

6. Arbitrary Lagrangian-Eulerian method(ALE) • Body-fitted grid: fluid interface coincides with a grid line • Finite-volume scheme for the insoluble surfactant, and finite-difference for the soluble surfactant -Can set surface diffusion=0 -Total amount of insoluble surfactant is exactly conserved • Validated by comparison with previous simulations of clean bubble in uniaxial flow and against analytical stagnant cap solution for spherical rising bubble.

7. Surfactant • Long chain molecules, hydrophobic and hydrophillic ends • -alcohols, fatty acids,detergents • Preferentially absorbed at interface • -insoluble: at interface only • -soluble: also in bulk fluid • Lowers surface tension • Introduces force (Marangoni force) • -Force directed from low to high surface tension U

8. Mathematical Formulation for inviscid bubble Fluid flow: • Axisymmetric flow • Inviscid (passive) interior fluid • Stress balance at interface (Last term is Marangoni stress due to surfactant gradient) • Kinematic condition Volume of bubble is fixed (for inviscid bubble this must be imposed as a constraint)

9. Equation for insoluble surfactant • Soluble surfactant: later in talk

10. Movie of pinch off with surfactant Why the difference with clean case?

11. Formation of thin thread due to surfactant DNS result for Re=0.17, insoluble surfactant with no diffusion

12. Slender body theory: Electrical capitance example

13. Slender body 1 -1

14. Slender body theory (cont’d) • Boundary condition

15. Slender body theory (cont’d) • Q(z) only weakly dependent on shape

16. Summing the logs

17. Slender body analysis (Stokes flow) • The bubble causes a modification of the flow in the exterior fluid b l slenderness ratio where Evaluate at

18. Slender thread equations • Evolution equations for slender (periodic) inviscid thread, • no flow at infinity • Marangoni force term does not arise at leading order

19. Solution to slender body equations • Velocity is radial, i.e., • Tangential stress balance decouples from leading order equations (hence • Marangoni term is unimportant) • No pinch off!

20. Comparison with the slender body theory

21. Clean (surfactant-free) bubble

22. Mechanism of thread formation Recall Quasi-steady thread formation

24. Slender thread equations with viscous internal fluid • Viscous thread (For clean thread, see also Sierou and Lister 2003) • Marangoni force term does not arise at leading order

25. Additional effects required for pinch off No Marangoni Thread formation with/without Marangoni stress term, from full N-S simulation

26. What happens at the constriction?

27. Thread formation and pinch off

28. Surfactant evolution equations Surfactant flux from bulk to surface

29. Longwave equations for thread with surfactant (large Pe) • Have generalized equations to include small interior viscosity

30. Soluble surfactant • Grid is generated according to the surface curvature,and is Relatively insensitive to the gradient in C (which is very large near the neck) • Distribution of C before pinch-off for J=0.001, Pe=100, and K=1 • Contour plot of log( C ): C varies from large value (yellow) to unity (red)

31. Direct numerical simulation: soluble surfactant Pe=10, K=100 ‘clean-like’ ‘insoluble-like’ • The exchange coefficient J and adsorption/desorption ratio K determine whether thread will pinch-off in a ‘clean-like’ or ‘insoluble-like’ manner

32. Inviscid bubble in extensional flow • Four roller mill experiments (GI Taylor 1934) • Taylor observed rounded at low Q and pointed shapes • at higher Q

33. Tip streaming DeBruijn 1993 • First noticed by Taylor (1934) • De Bruijn (1992) attributed to • presence of surfactant

34. First noticed by Taylor (1934) • De Bruijn (1992) attributed to • presence of surfactant Tipstreaming From review of Eggers (1997) .5mm

35. Tipstreaming End pinching • Eggleton, Tsai and Stebe (2001) • Boundary integral simulation • Viscosity ratio=0.1, insoluble surfactant • 2 months computer time • Other simulations: James et al (2004), • Renardy, Renardy, Cristini (2001), Milliken, Stone,Leal • (1993)

36. Clean inviscid bubble in extensional flow • Steady, stable slender solutions exist for arbitrary strain rate (Buckmaster 1972, Acrivos and Lo 1978) • Pointed shape Deformation vs. Q D 1 0 Q • Bursting solutions for viscosity ratio > 0, • How is the inviscid bubble solution branch affected by • presence of surfactant?

37. 2D Exact solutions • 2D nonlinear far-field straining flow • Time dependent generalization Siegel(2000), Crowdy and Siegel (2005) Clean bubble • Exact conformal map solution Antanovskii (1995)

38. Steady State, diffusionless surfactant

39. Bubble evolution, with • surfactant • An unsteady cusped • shape is formed in • finite time

40. Slender body analysis (with M. Booty JFM 2005) • The bubble causes a modification of the flow from the imposed state of • uniaxial extension where Evaluate at • BC provide equations for f, g R etc.

41. Steady bubble covered with a nonuniform distribution of surfactant • As for a solid, solution is developed as expansion in powers of • Solution to all integer powers of Scaled capillary number • Consistent with slender body solution for flow around ellipsoidal solid (Tillett 1972), • and exact solution (Jeffery 1922) for a rigid ellipsoid of arbitrary aspect ratio • Here the ellipsoidal shape occurs fortuitously for an inviscid bubble covered with a • nonuniform distribution of immobile surfactant

42. Surfactant cap bubble z=a is cap edge • For sufficiently large strain, get stagnant cap bubble • -Mixed boundary conditions, cap edge is free boundary curve • -Similar to stagnant cap in spherical rising bubble (Sadhal and Johnson 1983) • Leading order equations for + and – functions are coupled, e.g. TSB is

43. Steady state response: inviscid bubble with surfactant • Unlike the solution branch for the clean bubble (Buckmaster 1972), for which steady • states exist for arbitrarily large capillary number Q, with surfactant there is a critical • above which solutions no longer exist. • Nonmonotone dependence of critical Q on • Unsteady dynamics for

45. Increasing initial surfactant

46. Unsteady evolution: asymptotic equations Scaled capillary number • When is constant this system is equivalent to Hinch (1980) • System is not closed since bubble halflength L(t) is undetermined

47. Endpoint expansions Moran (1962) Handelsman and Keller (1967) Potential flow around a slender solid Jeffery (1922), Tillett (1972) Chwang and Wu (1975) Stokes flow around a solid • Steady surfactant covered ends are rounded, in contrast to pointed • clean bubble • Unsteady evolution: , i.e., ‘no-slip’ bubble tips pulled along by imposed flow

48. Tipstreaming solution • Surfactant is swept to poles; stagnant surface is pulled along by imposed flow • Main body of bubble is similar to pointed shapes of Buckmaster (1972) • The ‘o’ markers denote the time dependent `Buckmaster’ bubble (Hinch 1980) Covered initial slender bubble No rounding at tip

49. Conclusions • Considered effect of surfactant on steady and time-evolving inviscid • bubbles and jets • In the presence of surfactant, we find a critical capillary number above • which steady bubble solutions no longer exist • Equations governing the evolution of a slender inviscid bubble with • surfactant, valid for large capillary number, are derived. The solutions • exhibit spindle shapes with tipstreaming filaments • Surfactant retards or prevents pinch off during surface tension • driven instability of a inviscid jet • We have also analyzed the effect of surfactant in the • breakup of slightly viscous jets in a viscous surrounding, and • the effect of soluble surfactant