Jens Eggers The role of singularities in hydrodynamics

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Jens Eggers The role of singularities in hydrodynamics

Jens Eggers The role of singularities in hydrodynamics

A shock wave a jump in density occurs at some finite time t 0

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 Harold Edgerton neck radius shrinks to

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

Universality experiment by Shi et al.

Universality experiment by Shi et al.

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

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

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

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

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

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

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

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

Making small things

Making small things

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

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

Why singularities? • Crucial events in the evolution of the flow-describe changes in topology,

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

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

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

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

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

Similarity solution regular at

Similarity solution regular at

Matching condition size of critical region:

Matching condition size of critical region:

Approach to the similarity solution is fixed point! stability?

Approach to the similarity solution is fixed point! stability?

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

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

Bubble breakup: beyond simple self-similarity

Bubble breakup: beyond simple self-similarity

Bubble breakup 101 surface tension-inertia S. T. Thoroddsen bubble Longuet-Higgins et al. , JFM

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

Slender body joint with M. Fontelos, D. Leppinen, J. Snoeijer. fluid xxxxx air z

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

Self-similarity

Self-similarity

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

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

Singularities: Outlook are the building blocks of PDEs Øform small things Øare seeds for

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

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