Jens Eggers

The role of singularities in

hydrodynamics

A shock wave

a jump in density occurs at some finite time t0 !

W.C. Griffith, W. Bleakney

Pinch-off singularity

L neck radius shrinks to zero in finite time

Harold Edgerton

Burton et al, PRL `04

0' (nanoseconds)t t t

0.661: 0.0065 't

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: no motion for 0!micro

b_33_m_4037.mov

UQuéré, 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

0 198t t s

glyceroldrop center0 350t t s

1mm

experiment byTomasz Kowalewski

0 46t t s

Scale invariance: Self-similarity

0' ( ) /z z z 0' ( ) /t t t t

1/ 2( , ) ' ( '/ ' )h z t t z t

power laws, self-similarity,

and all that...

Weak shock wave

0u uut x

Similarity solution

0u uut x

1 2 0t U U t UU

xu t Ut

x t

0t t t

11 xu t U x t

t

(1 ) 0U U UU

0 regular at 1 1/

i1, , 0,1,2,

2 2, =0

U CU ii

U

Matching condition 0t t t

11 xu t U x t

t

1x t size of critical region:

finit( 0) as

e!0

u xt

xut

0 : infinite f 0or !t

1 1/i

1, , 0,1, 2,2 2

iU CU ii

Approach to the similarity solution

similarity solution is fixed point!

(1 )U U U UU

stability?

0u uut x

1

, lnx t

u t Ut

Fixed point: stability( ) (1 ) 0i i i iP P PU P U

eigenvalue problem

( ) 2 42 2

ij

i ji

eigenvalue:

only stable solution!

1/ 2 3/ 2( , ) /u x t t U x t 3U U

Bubble breakup:beyond simple self-similarity

Bubble breakup 101

S.T. Thoroddsen

Longuet-Higgins et al., JFM 1991 Oguz and Prosperetti, JFM 1993

1/ 2

0 1/ 4

'ln 'tht

: 1/ 2

0p

0outp p

02hbubble

22 0

0 0 00

2ln2

L hh h hh

&& &&L

surface tension-inertia

1/320h t

Keim et al. PRL `06

0.56

0h At

' ( s)t

Slender body

z

fluid

x x x x x x x x x x

2 2

( )

( )

C d

z r

;

air

2a h

2

2

( , )2( ) ( , )

a t d aaz a z t

:exp. by Burton +

Taborek

joint with M. Fontelos,

D. Leppinen, J. Snoeijer.

Self-similarity

0 ( )a a g

/z

0( , ) ( ) ( )a z t a A

z

ln 't

220

00 0

4ln2aa

a a

20 0 0 0 0

0 3 20 0 0 0

8ln 22

a a a a a aaa e a a a

0 02 /a a

Approach to the fixed point

02 (ln )a 022 lna

1 ( ),2u v( ) linearize:

define:

very slow approach!11/ 2

4

0ln( )t t

3v 8v vu u

cubic equation!0.57

Thoroddsen

OutlookSingularities:

form small thingsare seeds for new structures

are scale-invariantuniversallink micro-and macroworld

are the building blocks ofPDEs

…may possess complex inner structure

A catalogue of singularities: classify singularities according to dynamics close to fixed point