Ribbing instabilityNon Newtonian effects

Marta RosenFacultad de Ingenieria

Universidad de Buenos Aires

PASIMar del Plata

ArgentinaAugust 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

b

Laboratory models

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

““roll coating”roll coating”

Canal convergente - divergente

Interfases liquido- aire(meniscos)

Reflujo

g

RGap

System details

Aspect ratio

Convergent-divergent channel

Liquid-air interface

reflux

System variables

Objective: the control of the thickness “t”.

V tangencial velocity

c capillary length (it takes into account the effect of surface tension)

TU

Ca

cg

T

Reynolds Eq.

*CaCa 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.

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.

destabilizing dp/dx >0

stabilizing T

Pattern formation

= 0.02 = ∞

= 0.17 = 7.05 mm

= 0.31 = 6.86 mm

= 0.46 = 5.36 mm

= 0.75 = 3.63 mm

= 0.89 = 3.88 mm

= 1.04 = 3.66 mm

= 2.13E-03Ca* = 0.118

= 0.02 = ∞

= 0.17 = 7.05 mm

= 0.31 = 6.86 mm

= 0.46 = 5.36 mm

= 0.75 = 3.63 mm

= 0.89 = 3.88 mm

= 1.04 = 3.66 mm

= 2.13E-03Ca* = 0.118

Ribbing instability evolution obtained with a Newtonian fluid (vaseline) is a measures of the threshold distance.

Linear Analysis

Approximation

Boundary conditionsfor

Approach to the problem: slow viscous flow + lubrication theory.

Velocity field in x direction:

Capillary effects

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.

b.c.

With solution

What do we have to analyze?

• Geometric influence

•Thickness control

•Fluids and their properties

•Instability

•Pressure distributions

Homsy obtained

Critical wave number

Geometric influence (Newtonian case)

when

is

Rh0

*

*V

Ca

Geometry

Threshold drop in a viscoelastic case

Adimensional thickness

In generalr ranges between 2/3 when

and ½ when

Newtonian case

Dip coating (immersion)

Thicknesses for different fluids and geometrical conditions

Ro, Homsy

Considering the elasticity effect, the thickness of the film dragged by the cylinder is reduced .

0h

TN

where

is defined as elastic number

It only combines fluid properties.

is a relaxation time

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)

Instability evolution as a function of Ca number.

Ca

Deformation velocity

The fluids

.

Shear rate

Rheological properties

Three types of rheological behaviors.

Xanthane solutions PAAm solutons

PIB solutions

Industrial examples

====================

Paints

Carreau model

Shear –thinning polymer solutions

1.

n

k

0and

low and high shear Newtonian viscosities

is a characteristic time scale (when the shear thinning effect becomes important)

yyxxN 1

Viscoelastic effect

Due to the anisotropy of the normal components of tension tensor, the normal stress difference becomes important.

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

xz

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.

(4) Placa de vidrio(plano)

(3) Válvula de 3 vías

(2) Transductor de membrana

(5) Dispositivo de traslación lineal

(1) Toma de presión

Experimental set up

Pressure measurements

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.

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

-20 -15 -10 -5 0 5 10

X [mm]

P-Pa

tm [P

a]

Ca= 0.311Ca= 0.274Ca= 0.255Ca= 0.235Ca= 0.212Ca= 0.199

Ca*= 0.24

Ca>Ca*

-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8-600

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

700

800

900

Ca=0,40

P-P

atm

[Pa

]

X [mm]

Ca=0,27 Ca=0,18

Ca=0,11

Fluido: 4,85% PIB

Ca=0,14 Ca<Ca*

18,0*Ca

-600

-400

-200

0

200

400

600

800

1000

-20 -15 -10 -5 0 5 10

X [mm]

P-Pa

tm [P

a]

Ca= 0.378Ca= 0.274Ca= 0.189Ca= 0.151Ca= 0.120

Ca*= 0.18

NewtonianNon Newtonian (PIB)

Results

-500

-400

-300

-200

-100

0

100

200

300

400

500

600

-20 -15 -10 -5 0 5 10

X [mm]

P-Pa

tm [P

a]

Ca= 0.311Lubric.Ca= 0.255Lubric.Ca= 0.212Lubric.

Lubrication theory (New)

Best agreement for Ca<<Ca* (Ca*=0.24)

-600

-400

-200

0

200

400

600

0.0 0.1 0.2 0.3 0.4

Ca

P-Pa

tm [P

a]

-600

-400

-200

0

200

400

600

800

1000

0.00 0.10 0.20 0.30 0.40

(A) (B)

Ca*= 0.24 Ca*= 0.18

A) Newtoniano

-600

-400

-200

0

200

400

600

800

1000

0.00 0.10 0.20 0.30 0.40

Ca

P-Pa

tm [P

a]

(A) (B)

Ca*= 0.24 Ca*= 0.18

B) no-Newtoniano

Identification of the threshold value (Ca*) for maxima and minima of the pressure profile

At Ca*, a pattern appears with a definite wavelength. The Amplitude Asatisfies a single mode of the Guinzburg-Landau Eq.

IAI2AAdt

dA

(supercritical transition)

Instability threshold

Hydrodynamic instability

•Landau Theory :

Amplitud Eq.:A : wave amplitude

σ : growth rate

ℓ : Landau constant

(A0 : initial amplitude)

Af

A0

t=0

lAl

Am

plitu

d

tiempo(independent of A0)

R=RC (umbral)

σ (

tasa d

e c

recim

ien

to)

R (Parámetro de control)

solución realsolución aprox.

Control parameter (R) ( if R > Rc )

R ≡ Ca

experimentalajuste de Landau(σ=1.17/s, Af =1.23mm)

Ca = 0.274 gap = 0.2mm

oscilación por excentricidad

Analytical solution

t =1.6s (perturbación)

~λ

~Af

t = 5.2s

t = 4.8s

t = 4.4s

t = 4.0s

t = 3.6s

t = 3.2s

t = 2.0s

Hydrodynamic instability

Results: agreement with Landau modelResults: agreement with Landau model

• PIB (non New.):• Vaseline (New.):

Ca = 0.268Ca = 0.274Ca = 0.287Ca = 0.348Ca = 0.382Ca = 0.407

Ca = 0.259

Ca*PIB =0.153 Ca*VASELINA =0.231

experimentalajuste de Landau

gap = 0.2 mm

153.049 CaCaPIB

231.05.25 CaCaVASELINA

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

CaCa

310*33.9

Ca*=0.204

PIB

Secondary Waves (with only one control parameter!)

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) 123-139.-“Rheological Effects in Roll Coating of Paints”. F.Varela López, M.Rosen. Latin American Applied Research 32:247-252 (2002).

References

-“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.