Jens Eggers The role of singularities in hydrodynamics.

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Presentation transcript:

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 L neck radius shrinks to zero in finite time Harold Edgerton Burton et al, PRL `04

Universality experiment by Shi et al.

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

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

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

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

Making small things

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

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

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

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

Similarity solution regular at

Matching condition size of critical region:

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

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

Bubble breakup: beyond simple self-similarity

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

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

Self-similarity

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

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