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Jens Eggers

Jens Eggers. The role of singularities in hydrodynamics. A shock wave. a jump in density occurs at some finite time t 0 !. W.C. Griffith, W. Bleakney. Pinch-off singularity. Burton et al, PRL `04. L. neck radius shrinks to zero in finite time. Harold Edgerton. Universality.

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Jens Eggers

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  1. Jens Eggers The role of singularities in hydrodynamics

  2. A shock wave a jump in density occurs at some finite time t0 ! W.C. Griffith, W. Bleakney

  3. Pinch-off singularity Burton et al, PRL `04 L neck radius shrinks to zero in finite time Harold Edgerton

  4. Universality experiment by Shi et al.

  5. Drops and bubbles very different! water drop in air Shi et al. air bubble in water Thoroddsen et al.

  6. Corner singularity Huh and Scriven’s paradox: U Quéré, Fermigier, Clanet

  7. Pouring a viscous liquid fluid jet Eggers, PRL `01 Lorenceau, Restagno, Quéré, PRL `03 viscous fluid cusp cusp forms Lorenceau, Quéré, Eggers, PRL `04

  8. Charged drop experiment: Leisner et al. theory: Fontelos et al.

  9. Making small things

  10. Boundary layer separation Re=500 Ut/d=1 Ut/d=3 Coutanceau, Bouard finite time singularity of boundary layer equations!

  11. Why singularities? • Crucial events in the evolution of the flow-describe • changes in topology, seedsfor new structures • Universality determines structure of flow, independent of boundary conditions • Building blocks of a partial differential equation • Points where computers stop Main mathematical ingredient: self-similarity!

  12. Scale invariance: Self-similarity glycerol drop center experiment by Tomasz Kowalewski 1mm

  13. power laws, self-similarity,and all that... Weak shock wave

  14. Similarity solution regular at

  15. Matching condition size of critical region:

  16. Approach to the similarity solution similarity solution is fixed point! stability?

  17. Fixed point: stability eigenvalue problem eigenvalue: only stable solution!

  18. Bubble breakup:beyond simple self-similarity

  19. Bubble breakup 101 Keim et al. PRL `06 bubble surface tension-inertia S.T. Thoroddsen Longuet-Higgins et al., JFM 1991 Oguz and Prosperetti, JFM 1993

  20. Slender body joint with M. Fontelos, D. Leppinen, J. Snoeijer. x x x x x x x x x x :exp. by Burton + Taborek fluid air z

  21. Self-similarity

  22. Approach to the fixed point linearize: cubic equation! very slow approach! Thoroddsen define:

  23. Outlook …may possess complex inner structure Singularities: are the building blocks of PDEs • form small things • are seeds for new structures • are scale-invariant • universal • link micro-and macroworld A catalogue of singularities: classify singularities according to dynamics close to fixed point

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