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During its brief existence, drop goes through many stages.

Topological transitions and singularities in fluids: Life of a Drop Manne Siegbahn Memorial Lecture, Oct. 18, 2007. During its brief existence, drop goes through many stages. Shape changes often accompanied by dynamic singularities*.

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During its brief existence, drop goes through many stages.

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  1. Topological transitions and singularities in fluids:Life of a DropManne Siegbahn Memorial Lecture, Oct. 18, 2007 During its brief existence, drop goes through many stages. Shape changes often accompanied by dynamic singularities*. *Dynamic singularities: infinitesimally small, very short time Test-bed for understanding broad class of phenomena.

  2. How do drops fall, break apart? Topology changes - real transition Neck radius  0 (curvature ) Pressure  Cannot do simulation to get to other side of snapoff ? Is there understanding for these transitions like that for thermodynamic phase transitions?

  3. PILLARS OF CREATION IN STAR-FORMING REGION Gas Pillars in M16 - Eagle Nebula Hubble Space Telescope 4/1/95 Similar behavior: Star formation

  4. Similar behavior: Breakup of bacteria colonies Elena Budrene - Harvard Dynamic singularities appear everywhere in physics - celestial  microscopic  nuclear fission…

  5. Who Did ItExperimentTheory/SimulationOsman Basaran Itai CohenMichael BrennerNathan Keim Pankaj Doshi Xiangdong ShiJens EggersLei XuLaura SchmidtWendy ZhangFunding: NSF

  6. Birth and Childhood A happy childhood! Surface tension + gravity

  7. R Rayleigh-Plateau Instability L • Surface area decreases if L 2R. • Unstable to perturbations. • Pressure greatest at minimum thickness •  liquid squeezed out.

  8. Bolas Spider

  9. Midlife Crisis

  10. Midlife Crisis

  11. Water into Air Singularity is same even though gravity points in opposite direction Xiangdong Shi and Michael Brenner

  12. Water Drops

  13. How to Think About Shapes: Scale invariance(borrowed from statistical mechanics) Breakup  radius smaller than any other length. Dynamics insensitive to all other lengths. Flow depends onlyon shrinking radius.

  14. How to Think About Shapes: Scale invariance(borrowed from statistical mechanics) • Breakup  radius smaller than any other length. • Dynamics insensitive to all other lengths. • Flow depends onlyon shrinking radius. • But: Radius depends on flow • (which depends on radius • (which depends on flow • (which depends on radius • (which depends on flow • (which depends on radius • (which depends on flow • (which depends on radius • (which depends on flow • (which depends on radius • (which depends on flow • . . . • Self-similar structure: • Blow up any part  regain original. • Universal shapes

  15. h z Similarity Solution h(z,t) = f(t) H[(z-zo)/f(t)ß] Similarity solutions same AT DIFFERENT TIMES with different magnifications along h and z. Scaling determined by force balance at singularity. PDE  ODE Keller and Miksis (83); Eggers, RMP (97)

  16. Unhappy drops are all unhappy in their own way Explore different asymptotic regimes by tuning parameters. Depends on: viscosity of inner fluid,  viscosity ratio of fluids, (air is a fluid) density of inner fluid, density difference,  surface tension,  nozzle diameter, D

  17. Scaling Profiles - Glycerol into OilStretching axes by different amounts at different times produces master curve = h(z,t)/f(t) = (z-zo)/f(t)ß Itai Cohen, Michael Brenner Jens Eggers, Wendy Zhang

  18. Singularities tame non-linearity of Navier-Stokes Eqs. Role of scale invariance - borrowed from critical phenomena Near singularity, dynamics insensitive to all other lengths Emphasize what is Universal But . . .

  19. Water into OilNot so simple Persistence of memory. No similarity solution. No universality! Separation of scales but also of axial and radial length scales. I. Cohen, W. Zhang P. Doshi, O. Basaran P. Howell, M. Siegel

  20. Water into OilContinued… Viscosity of water begins to matter creates very fine thread.

  21. Remember water drop in air?

  22. Remember water drop in air? What about air drop in water (i.e., a bubble)? N. Keim, W. Zhang

  23. Perturbations oblong nozzle slight nozzle tilt N. Keim, W. Zhang

  24. The drop falls  splashes Is splash interesting?Break-up localizes energy from the kinetic energy into singular points as surface ruptures. How?Coronal splash Lei Xu, Wendy Zhang

  25. Drop splashes Drop of alcohol hitting smooth, dry slide Lei Xu, Wendy Zhang

  26. Drop splashes atmospheric pressure 1/3 atmospheric pressure (Mt. Everest) Lei Xu, Wendy Zhang

  27. Non Monotonic (in all cases) Impact Velocity vs. Threshold Pressure V0 (m/s) Splash No Splash Non-monotonic PT (kPa) Lei Xu, Wendy Zhang

  28. Singularity during splash t0 Vexp (t - t0)-0.5 Drop rim expands infinitely rapidly at moment of impact.

  29. At high viscosity, does air matter? 5 cSt High pressure Low pressure Lei Xu, Casey Stevens, Nathan Keim

  30. At high viscosity, does air matter? 10 cSt 100 kPa 43 kPa Air still matters. Does compressibility?

  31. 1000 cSt

  32. Last stage of the drop: What remains? Why are drops always ring-shaped?

  33. Black

  34. Drop pinned at contact line How does evaporation bring everything to edge? Analogy with Electrostatics   (steady state diffusion) vapor saturated at surface Equations for potential of a charged conductor: at points, electric field (evaporation rate) diverges.

  35. Every stage of drop’s life arouses astonishment. Ideas used to treat singularities appear in different variations - from thermodynamic to topological transitions A great idea “is like a phantom ocean beating upon the shores of human life in successive waves of specialization.”A. N. Whitehead Surprises and beauty await us… even in the most familiar phenomena!

  36. END

  37. Navier-Stokes Equation • [v/t + (v .)]v = -P + 2v + F • inertial terms = internal pressure + viscous force + body force • + Incompressibility Equation • + Laplace pressure equation for surface

  38. Remember water drop in air? What about air drop in water (i.e., a bubble)? slight nozzle tilt Singularity sensitive to small perturbations - remembers axial asymmetry N. Keim, W. Zhang

  39. Memory of all initial amplitudes Modes oscillate: an()  e i cnln n=3 Nathan Keim Laura Schmidt & Wendy Zhang

  40. SF6 V0 (m/s) PT (kPa) V0 vs. PT for different gases SF6

  41. SF6 V0 (m/s) √M PT (arb. unit) Data Collapse for Different Gases

  42. V0 Ve d Destabilizing stress:SG ~rG CG Ve ~ P M/kT √gkT/M√RV0/2t Stabilizing stress:SL= s/d = s/ √nLt The ratio of SGandSLdetermines splashing: SG /SL ~ 1 at threshold Model

  43. V0 (m/s) PT (kPa) Different liquid viscosities

  44. SG/SL V0(m/s) Threshold values of G/L all liquids, gases and velocities Turn over V0

  45. Glycerol/water 100 x viscosity of water

  46. Glycerol/water Xiangdong Shi, Michael Brenner

  47. Glycerol/water Xiangdong Shi, Michael Brenner

  48. Scaling for different

  49. Atmospheric pressure (100kPa)

  50. Reduced pressure (17kPa)

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