Jens Eggers

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Jens Eggers

• The role of singularities in
• hydrodynamics

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A shock wave
W.C. Griffith, W. Bleakney
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Pinch-off singularity
Harold Edgerton
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Universality
experiment by Shi et al.
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Drops and bubbles
water drop in air
Shi et al.
air bubble in water
Thoroddsen et al.
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Corner singularity
Quéré, Fermigier, Clanet
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Pouring a viscous liquid
fluid jet
Eggers, PRL 01
Lorenceau, Restagno, Quéré, PRL 03
viscous fluid
cusp
cusp forms
Lorenceau, Quéré, Eggers, PRL 04
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Charged drop
experiment Leisner et al.
theory Fontelos et al.
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Making small things
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Boundary layer separation
Re500
Ut/d1
Ut/d3
Coutanceau, Bouard
finite time singularity of boundary layer
equations!
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Why singularities?

• 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

• Building blocks of a partial differential
  equation

• Points where computers stop

Main mathematical ingredient self-similarity!
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Scale invariance Self-similarity
experiment by Tomasz Kowalewski
1mm
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power laws, self-similarity,and all that...
Weak shock wave
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Similarity solution
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Matching condition
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Approach to the similarity solution
similarity solution is fixed point!
stability?
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Fixed point stability
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Bubble breakupbeyond simple self-similarity
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Bubble breakup 101
surface tension-inertia
S.T. Thoroddsen
Longuet-Higgins et al., JFM 1991
Oguz and Prosperetti, JFM 1993
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Slender body
joint with M. Fontelos, D. Leppinen, J. Snoeijer.
fluid
air
z
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Self-similarity
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Approach to the fixed point
define
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Outlook
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