Transcript Slide 1

Ribbing instability
Non Newtonian effects
Marta Rosen
Facultad de Ingenieria
Universidad de Buenos Aires
PASI
Mar del Plata
Argentina
August 2007
A planar interface between two fluids (air
and a liquid) leads to a well-defined
patterned surface above a certain threshold.
Such an interface occurs over curved
surfaces in many industrial processes.
Different distributions of rollers used in industry
a
Laboratory models
b
Sketch of experimental models used to study the
problem in laboratory
Experimental set-up
Figure a (previous)
A plane-cylinder device:
-cylinder made of stainless steel, 380 mm long; 37.50
-DC controlled motor
-10mm thick Plexiglass plate
-Gap, h0=0.20mm
-Aspect ratio: 5.33 10-3
-T=20 0C

0.02 mm
Liquid-air interface
System details
Interfases liquido- aire
(meniscos)
reflux
Reflujo
g
w
w2
Canal convergente - divergente
Convergent-divergent channel
w1
“roll coating”

Gap
R
Aspect ratio
System variables
Objective: the control of the thickness “t”.
Ca 
U
T
lc 
T
g
V tangencial velocity
lc capillary length (it takes into account the effect of surface tension)
Ca  Ca *
The loss of stability of the bidimensional flow is
related to pressure conditions at both sides of the
meniscus.
Pressure profile:
continuous line, lubrication
hypothesis.
Dots, experimental
measurements.
Reynolds Eq.
For a given tangential velocity, the application of a
lubrication condition allows us to establish a
relationship between the pressure gradient, the gap
and the variable thickness.
Pattern formation
stabilizing T
destabilizing dp/dx >0
Ribbing instability evolution obtained with a
Newtonian fluid (vaseline)  is a measures of the
threshold distance.
 = 0.02
l=∞
 = 0.17
l = 7.05 mm
 = 0.31
l = 6.86 mm
 = 0.46
l = 5.36 mm
 = 0.75
l = 3.63 mm
 = 0.89
l = 3.88 mm
 = 1.04
l = 3.66 mm
 = 2.13E-03
Ca* = 0.118
Linear Analysis
Approximation
Approach to the problem: slow viscous flow + lubrication
theory.
Velocity field in x direction:
Capillary effects
Boundary conditions
for
Kinetic conditions
q is fraction of film dragged by the moving wall
Perturbations on the interface
With solutions
We can thus transform the eqs. system of partial derivatives
into a system of ordinary differential eqs.
We obtain an self-values eq.
With solution
b.c.
What do we have to analyze?
• Geometric influence
•Thickness control
•Fluids and their properties
•Instability
•Pressure distributions
Geometric influence (Newtonian case)
Homsy obtained
Critical wave number
when
is
Geometry
Ca* 
V *


Threshold drop in a viscoelastic case
h0
R
Adimensional thickness
In general
r ranges between 2/3 when
and ½ when
Newtonian case
Dip coating (immersion)
Thicknesses for different fluids and geometrical conditions
Considering the elasticity effect, the thickness of the
film dragged by the cylinder is reduced .
Ro, Homsy
where
is defined as elastic number
lT
N
h0
l is a relaxation time
It only combines fluid properties.
Instability evolution as a function of Ca number.
Ca
Pictures of the interface, below and above threshold (increasing Ca, from up to down). On each
picture, air is on the upper side (minimum gap h0 is exactly located at the lower boundary of
each frame).
(a) Glycerol (Ca = 0.210, 0.226, 0.262, 0.425)
(b) Xanthan 1000ppm (Ca = 0.076, 0.123, 0.155, 0.338)
(c) PAAm 1000ppm (Ca = 0.110, 0.124, 0.154, 0.165)
The fluids
Rheological properties
Deformation velocity
.

Shear rate
Three types of rheological behaviors.
Xanthane solutions
PAAm solutons
PIB solutions
Industrial examples
====================
Paints
Shear –thinning polymer
solutions
. n 1
  k
Carreau model
0
and

low and high shear Newtonian
viscosities
l is a characteristic time scale (when the shear thinning effect
becomes important)
Viscoelastic effect
Due to the anisotropy of the normal components of tension
tensor, the normal stress difference becomes important.
N1   xx   yy
 xz  
Where xx and yy are the normal components of
the stress tensor, parallel and transverse to the
flow respectively.
Examples of elastic behavior
*swell effect
*Weissemberg effect: the
circular flow induces radial
tensions that force the liquid to
climb up the rotor.
The Weissemberg number measures the
strength of the elastic effects in the flow.
.
We  l 
Rod- climbing
Microscopic behavior
a) Equilibrium configuration (spherical)
b) Configuration under movement
Its deformation causes anisotropies in the tension field.
N1 behavior as a function of shear stress
PAAm
PIB
Surface tension
==================
Surface tension depends slightly on concentration.
Experimental set up
(5) Dispositivo de
traslación lineal
(3) Válvula de
3 vías
(4) Placa de vidrio
(plano)
(1) Toma de
presión
Pressure measurements
(2) Transductor
de membrana
The pressure gauge moves across the interface to obtain
pressure distributions.
Pressure measurements.
•Pressure was obtained by a pressure gauge.
•The transducer was fixed exactly over the hole of the
pressure gauge, in order to avoid pressure losses and
to reduce hydraulic impedance.
•The membrane has a resolution of 1Pa with a
precision of about 1%.
•The spatial resolution is 0.1mm.
Pressure distribution for two types of fluids.
Results
600
Ca=
Ca=
Ca=
Ca=
Ca=
Ca=
500
400
1000
200
Ca=
Ca=
Ca=
Ca=
Ca=
800
100
600
0
-100 -20
-15
-10
-200
-300
-400
Ca*= 0.24
-5
P-Patm [Pa]
P-Patm [Pa]
300
0.311
0.274
0.255
0.235
0.212
0.199
400 0
5
10
200
0
-20
-15
-10
-5
0
-200
-500
0.378
0.274
0.189
0.151
0.120
X [mm]
-400
-600
Ca*= 0.18
X [mm]
Newtonian
Non Newtonian (PIB)
5
10
Lubrication theory (New)
600
500
400
P-Patm [Pa]
300
200
Ca= 0.311
Lubric.
Ca= 0.255
Lubric.
Ca= 0.212
Lubric.
100
0
-100 -20
-15
-10
-5
0
5
-200
-300
-400
-500
X [mm]
Best agreement for Ca<<Ca* (Ca*=0.24)
10
Identification of the threshold value (Ca*) for maxima
and minima of the pressure profile
(A)(A)
(B)(B)
A) Newtoniano
B) no-Newtoniano
1000 1000
600 600
800 800
400 400
600 600
0
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.3
-200 -200
-400 -400
0.4
400 400
P-Patm [Pa]
0
P-Patm [Pa]
P-Patm [Pa]
P-Patm [Pa]
200 200
200 200
0.4
0
0
0.00 0.00
-200 -200
0.10 0.10
0.20 0.20
-400 -400
Ca*=Ca*=
0.240.24
-600 -600
Ca
Ca
-600 -600
Ca*=Ca*=
0.180.18
Ca
Ca
0.30 0.30
0.40 0.40
Instability threshold
(supercritical transition)
At Ca*, a pattern appears with a definite wavelength. The Amplitude A
satisfies a single mode of the Guinzburg-Landau Eq.
dA
 A  A IAI2
dt
Hydrodynamic instability
•Landau Theory :
(A0 : initial amplitude)
Analytical
solution
Af
Amplitud
A : wave amplitude
σ : growth rate
ℓ : Landau constant
Amplitud Eq.:
A0
(independent of A0)
Control parameter
(R)
lAl
t=0
tiempo
( if R > Rc )
oscilación por
excentricidad
experimental
ajuste de Landau
(σ=1.17/s, Af =1.23mm)
Ca = 0.274
gap = 0.2mm
σ (tasa de crecimiento)
R ≡ Ca
R=RC (umbral)
solución real
solución aprox.
R (Parámetro de control)
t = 5.2s
t = 4.8s
t = 4.4s
t = 4.0s
~Af
t = 3.6s
t = 3.2s
t = 2.0s
t =1.6s
(perturbación)
~λ
Hydrodynamic instability
Results: agreement with Landau model
• Vaseline (New.):
• PIB (non New.):
Ca
Ca
Ca
Ca
Ca
Ca
Ca
=
=
=
=
=
=
=
0.407
0.382
0.348
0.287
0.274
0.268
0.259
VASELINA Ca   25.5Ca  0.231
 PIB Ca   49Ca  0.153
Ca*PIB =0.153
Ca*VASELINA =0.231
experimental
ajuste de Landau
gap = 0.2 mm
Relationship between σ and Ca for Vaseline and a
Non-Newtonian fluid :
Ca* is lower in the
PIB case, what showes
that viscoelastic
properties are
desestabilizers.
 The farther from the
threshold it is, the
worse is the Landau
agreement. However,
near the threshold, it
showed to be a valid
analytical tool.
Secondary Waves
(with only one control parameter!)

Ca  Ca *
Ca *
Ca*=0.204
  9.33*103
PIB
Secondary Waves
(with only one control parameter!)
References
References
-“Theshold of ribbing instability with Non Newtonian fluids”. F.Varela López,
L.Pauchard, M.Rosen, M.Rabaud. Proceed. In Advances in Coating and Drying
of Thin Films, Univ. Erlangen-Nurenberg, Alemania, 1999. (p.177-182)
- “On the effects of non newtonian fluids above the ribbing instability”.
L.Pauchard, F.Varela López, M.Rosen, C.Allain, P.Perrot, M.Rabaud. Proceed.
Advances in Coating and Drying of Thin Films, Univ. Erlangen-Nurenberg,
Alemania, 1999, 183-188.
-“Effect of polymer concentration on Ribbing Instability Threshold”. F.Varela
López, L. Pauchard, M.Rosen, M.Rabaud. Proceed. 4th. European Coating
Symposium 2001, Advances in Coating Processes, October 1-5, 2001, Brussels,
Belgium.
-“Non Newtonian effects on ribbing instability thershold”. F.Varela López,
L.Pauchard, M.Rosen, M.Rabaud. J.Non -Newtonian Fluids Mech.103 (2002) 123139.
-“Rheological Effects in Roll Coating of Paints”. F.Varela López, M.Rosen. Latin
American Applied Research 32:247-252 (2002).
-“Experimental pressure distribution in roll coating flows: Newtonian and non
Newtonian fluids”. F.Varela Lopez, C.Correa, M.Vazquez, M.Rosen. Proceed. 5
th European Coating Symposium, 95-102, 2003.
-”Secondary Waves in Ribbing Instability”, Marta Rosen, Mariano Vazquez,
American Institute of Physics, Proceed. # 913, 14-19. 2007.