Jens Eggers

Polymer threads: formationand instability

Drop breakup

…createsextensional flow

and produces multiplesingularities

water drop

Flexible polymers

characteristic time λ

extension rate

stress ~ extension

maximal extension

50 ppm PEO in water

Y. Amarouchene et al. PRL 86, 3558 (2001)

water

1.2mm, T = 0,1 sec

PolymersWagner +

Sattler

64 10WM amu

polymer (Ch. Clasen)

beads on a string

Polymer breakup: overview

blistering

water jet

2 k ( , ) cos( )th z t R e kz

kR

0

3 1/ 20 ( /( ))R

surface tension-inertial balance:

W. Strutt(1873)fastest growing mode: max 9.01R

Newtonian breakup: linear stability

0 198t t s

glyceroldrop center0 350t t s

1mm

experiment byTomasz Kowalewski

0 46t t s

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

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

Newtonian breakup: similarity solution

Eggers, PRL `93

3

0.12R

R

1R

Bead formationWagner et al.,

PRL ‘05pure waterPEO in water

One-dimensional approximationEntov et al. `80

Renardy `82

2 2( v) 0zh ht

2

2

3 v+v v v

hz s z z

z z

hpt

z zv 2 v zz z zt

stretching

z

Thread formation Ch. Clasen and G. McKinley

v=2z / zh p

p

0 exp( / 3 )z t

0 0( ) ?h

0 exp( / 3 )threadh h t

Bazilevskii et al, `81:thinning dominated by end effects!

Corner region

thread drop

0 exp( / 3 )h t ( , ) cz zh z t H

Clasen et al.,

JFM ‘06

H

similarity

solution

string tension: 2 ( )T th h

Experiment: rescale drop

thread

Clasen et al. JFM ‘06

( , ) exp( / 3 ) / exp( / 3 )h z t t H z t similarity form:

…but experimental curvature greater x 2

Boger fluid

Blistering instability

30 m

min

zzE h

increases by 5

orders of magnitude!

full strech

Fontelos + Li:

Rayleigh instability

with effective E

Sattler et al.,

PRL ‘08

PEO in water, 2000 ppm

Rayleigh instability?

tamp e

9.3ms

9eff Pas

600E Pas time (ms)

sub-pixel

resolution!

Nonlinear instabilities

1) Iterated thread instability

(bulk)

2) Breathing instability

(end)

Weird instabilities

PEO, 4Mio MW, 2kppm, overall duration 0.016s

Does it ever break?

drops do not wet the fiber!

A solid fiber!

Mayonnaise, ketchup, yoghurt

movies by D. Bonn

Remarks, open problems

• No predictive model for flow at high deformations!

FENE-P: have to be treated as adjustable, , ,s p b

• thinning rate of filament depends significantly

on concentration (factor 10), and even on nozzle

radius!

• no significant quantitative statement about saturation!

• filament thinning still open (stability, full 3D model)

• mechanism for demixing ???!

• extensional behavior of suspensions; is there

extensional thinning?

Wagner, Amarouchene,Bonn, Eggers:PRL (`05)

1-D simulation

Polymeric fluid (100 ppm of PEO)

experiment