Transcript The role of singularities in hydrodynamics Jens Eggers

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
   : 0.0065t '0.661
Burton et al, PRL `04
t '  t0  t (nanoseconds)
L
Harold Edgerton
neck radius shrinks to
zero in finite time
Universality
experiment
by Shi et al.
Drops and bubbles
water drop in air
Shi et al.
very different!
air bubble in water
Thoroddsen et al.
Corner singularity
U
Quéré, Fermigier, Clanet
Huh and Scriven’s paradox:
no motion for
micro
 0!
Pouring a viscous liquid
fluid jet
Eggers, PRL `01
Lorenceau,
Restagno,
Quéré,
PRL `03
Lorenceau,
Quéré,
Eggers, PRL `04
viscous fluid
cusp
cusp forms
Charged drop
experiment:
Leisner et al.
theory:
Fontelos et al.
Making small things
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, 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!
Scale invariance: Self-similarity
glycerol
drop center
t0  t  350 s
experiment by
Tomasz
Kowalewski
t0  t  198 s
t0  t  46 s
h( z, t )  t ' ( z '/ t ' )
1/ 2
z '  ( z  z0 ) /

1mm
t '  (t0  t ) / t
Weak shock wave
power laws, self-similarity,
and all that...
u
u
u
0
t
x
Similarity solution
u
u
u
0
t
x
t   t0  t
x 


uu tt UU  1  xxt t1
t t  

 1
2  





 00
tU
UU
U)U t  UU
U(1
1

11/ 
, i 
, i  0,1, 2,
U  CU
 
2i  2
U ,
 =0
regular at
 0
Matching condition
u ( x  0) finite!
as t   0
x 

u  t  U   1    x t  1
 t 

size of critical region:
x
  0: u   
t
  U  CU
11/  i
x
t   t0  t
t
 1
infinite for t  0 !
1
, i 
, i  0,1, 2,
2i  2
Approach to the similarity solution
u  t  U   , 

  x t  1

   ln t 
u
u
u
0
t
x
U  U  (1   )U   UU 
similarity solution is
fixed point!
stability?
Fixed point: stability
( i  ) P  (1   i ) P  PU i  PU i  0
eigenvalue problem
eigenvalue:

(i )
j
u ( x, t )  t 1/ 2U  x / t 3/ 2 
only stable solution!
2i  4  j

2i  2
  U  U
3
Bubble breakup:
beyond simple self-similarity
Bubble breakup 101
surface tension-inertia
h0
S.T. Thoroddsen
L
bubble p0

2h0
pout  p0
Longuet-Higgins et al., JFM 1991
Oguz and Prosperetti, JFM 1993
 t   
1/ 3
2
&2


h
2
L
2
0
&
& ln
h&

h
h

 
0
0 0
2
h0  At  h0 
h0 :

t '1/ 2
 ln t '
1/ 4
  0.56
  1/ 2
t ' ( s)
Keim et al. PRL `06
Slender body
;

fluid
xxxxxxxxxx
air
z
joint with M. Fontelos,
D. Leppinen, J. Snoeijer.
C ( ) d 
(z   )  r
2
:exp. by Burton +
Taborek

a( , t )d
a2

2
( z   )  a ( z , t ) 2a
ah
2
2
Self-similarity
  2a0 / a0
a( z, t )  a0 ( ) A( )

z
  
   ln t '
a0
4
a0 ln

a0
2a0
2
2
 8 
a0 a0 a0 a a02 a0
a0 ln  3   2

 2
a0
a0
2a0
 a0e 
a  a0 g ( )

z/
Approach to the fixed point
2  (ln a0 )
 a0 
2    ln 2 
  
1
linearize:    u ( ),
2
  v( )
define:
   ln(t0  t )
u  u  v
3
v  8v
  1/ 2 
1
4 

cubic
 equation!
 0.57
very slow
approach!
Thoroddsen
Singularities:
Outlook
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
…may possess
complex
according
to dynamics
close to fixed point
inner structure