On the origin of the vorticity-banding instability
constant shear rate throughout the system
multi-valued flow curve
 band 1

2 cm
5 cm
isotropic and nematic branch
different concentrations
 band 2

 applied
shear-induced viscous phase
not clear what the origin of the
banding instability is
rolling flow within the bands
normal stresses along the gradient direction
normal streses generated within the
interface of a gradient-banded flow
low

high

( S. Fielding, Phys. Rev. E 2007 ; 76 ; 016311 )
L = 880 nm
D = 6.7 nm
P = 2200 nm
fd virus :
Critical
point
Binodal
. 2

-1
1
[[s
s ]]
Spinodals
( P. Lettinga )
Vorticity
banding
11
Tumbling
wagging
0
0
0.0
0.2
concentration
concentration
1
0.4
0.6 j
0.8
nem 1.0
j
almost crossed polarizers distinguish
orientational order
vorticity
direction
P
100  m
A
2
3
4
5
~ 1 mm
1
6
7
H[m]
8 120
stretching of
100 inhomogeneities
vorticity
direction
80
A
growth of bands
60
1
0
2
10
3
4
20
5
30
6
7
8
40 50 60
Time [min]

 t  t0  
H (t )  H 0  A 1  exp  





Gapwidth 2.0 mm
Shear flow
N 
 
 
heterogeneous vorticity banding
band width
growth rate
interconnected
A
disconnected
H0

 t  t0  
H (t )  H 0  A 1  exp  





23 % : A  finite;  
35 % : A  0;  finite
spinodal decomposition :
j nem  0.23
nucleation and growth :
j nem  0.75
100 m
( with Didi Derks, Arnout Imhof and Alfons van Blaaderen )
tracking of a seed particle
( counter-rotating couette cell )
with Bernard Pouligny (Bordeaux)
elastic instability for polymers :
equidistant
velocity lines
non-uniform deformation
100
H [m]
90
80
70
60
1.0
1.5
2.0
2.5
G [mm]
Weissenberg or rod-climbing effect
K. Kang, P. Lettinga, Z. Dogic, J.K.G. Dhont
Phys. Rev. E 74, 2006, 026307-1 – 026307-12
New viscous phases can be induced by the flow
(under controlled shear-rate conditions )
inhomogeneous
new phase
stress
shear rate
homogeneous
personal communication with John Melrose
Stability analysis :
discreteness of inhomogeneities along the flow direction is of minor importance :
 u y ( y, z , t )
 u y ( y, z , t ) 
  u y ( y, z , t )
m 
 uy
 uz
  By ( y , z , t )

t

y

z


mass density
z-dependence
gradient component of the
body force
exp i k z   t with   2 / k the typical distance between inhomogeneities
z
 u y ( y, z, t )   u ( y ) expik z   t
 By ( y, z, t )   B( y) exp ik z   t
By  F   (r , uˆ, t )
“Brownian contributions”
+”rod-rod interactions”
probability density for
the position r
and orientation û
of a rod
linear
bi-linear
+“flow-structure coupling”
linear
û
y
z
r
x
y
J.K.G. Dhont and W.J. Briels
J. Chem Phys. 117, 2002, 3992-3999
J. Chem Phys. 118, 2003, 1466-1478
 ( r , uˆ, t )  A  0 (r , uˆ )   A ( y )  1 ( r , uˆ, t )
“renormalized base flow probability”
By
 
0
2

large 
small
 
4
1   
2
linear contributions
 By ( y )
bi-linear contributions
 By ( y )
rod-rod interactions
   A( y)
4
1   
2
2
   
A  A( y )

4
1    
2
2
2

    

 By ( y)  C1  C2 A
 A( y)
4
4
1     1   

2
2


      A( y)
 m   u ( y )  C1  C2 A
4
4
1


1








2
2

    

0
C1  C2 A
4
4
1     1   

A  0  A 0
 
4
1   
u  0
2
C1  0 C2  0
A
C 0
A
( ) 2
A
1  ( ) 4
 
2
1   
4
C 0
depends on the microstructural
properties of the inhomogeneities
 l
 u
C
A  4C
unstable
stable
AC
0.0

0.2
concentration
0.4
0.6
Bonn D, Meunier J, Greffier O, Al-Kahwaji A, Kellay H,
Bistability in non-Newtonian flow : rheology and lyotropic liquid
crystals, Phys. Rev. E 1998 ; 58 ; 2115-2118.
Wilkins GMH, Olmsted PD, Vorticity banding
during the lamellar-to-onion transition in a lyotropic surfactant Micellar worms
solution in shear flow, Eur. Phys. J. E 2006 ; 21 ; 133-143.
-Worms
Fischer P, Wheeler EK, Fuller GG, Shear-banding
structure oriented in the vorticity direction observed for
equimolar micellar solution, Rheol. Acta 2002 ; 41 ; 35-44.
- Entanglements
- Shear-induced phase
Lin-Gibson S, Pathak JA, Grulke EA, Wang H,
Hobbie EK, elastic flow instability in nanotube suspensions, Nanotube bundles
Phys. Rev. Lett. 2004 ; 92, 048302-1 - 048302-4.
Vermant J, Raynaud L, Mewis J, Ernst B, Fuller GG,
Colloidal aggregates
Large-scale bundle ordering in sterically stabilized latices,
J. Coll. Int. Sci. 1999 ; 211 ; 221-229.
Kyongok Kang
Pavlik Lettinga
Wim Briels