Ribbing instability Non Newtonian effects

1. Ribbing instabilityNon Newtonian effects Marta Rosen Facultad de Ingenieria Universidad de Buenos Aires PASI Mar del Plata Argentina August 2007

2. 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.

3. Different distributions of rollers used in industry

4. a Laboratory models b

5. Sketch of experimental models used to study the problem in laboratory

6. 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

7. Liquid-air interface System details reflux w2 Convergent-divergent channel w1 Aspect ratio “roll coating”

8. System variables Objective: the control of the thickness “t”. V tangencial velocity lc capillary length (it takes into account the effect of surface tension)

9. The lossofstability 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.

10. 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.

11. stabilizing T Pattern formation destabilizing dp/dx >0

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

13. Linear Analysis Approximation

14. 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

15. Perturbations on the interface With solutions

16. 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.

17. What do we have to analyze? • Geometric influence • Thickness control • Fluids and their properties • Instability • Pressure distributions

18. Geometric influence (Newtonian case) Homsy obtained Critical wave number when is

19. Geometry Threshold drop in a viscoelastic case

20. Adimensional thickness r ranges between 2/3 when and ½ when In general Newtonian case Dip coating (immersion)

21. Thicknesses for different fluids and geometrical conditions

22. Considering the elasticity effect, the thickness of the film dragged by the cylinder is reduced. Ro, Homsy where is defined as elastic number l is a relaxation time It only combines fluid properties.

23. 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)

24. The fluids Rheological properties Deformation velocity Shear rate

25. Three types of rheological behaviors. Xanthane solutions PAAm solutons PIB solutions

26. Industrial examples ==================== Paints

27. Shear –thinning polymer solutions Carreau model low and high shear Newtonian viscosities and l is a characteristic time scale (when the shear thinning effect becomes important)

28. Viscoelastic effect Due to the anisotropy of the normal components of tension tensor, the normal stress difference becomes important. Where txx and tyy 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.

29. Rod- climbing

30. Microscopic behavior • Equilibrium configuration (spherical) • b) Configuration under movement • Its deformation causes anisotropies in the tension field.

31. N1 behavior as a function of shear stress PAAm PIB

32. Surface tension ================== Surface tension depends slightly on concentration.

33. Experimental set up Pressure measurements

34. The pressure gauge moves across the interface to obtain pressure distributions.

35. 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.

36. Pressure distribution for two types of fluids.

37. Results Newtonian Non Newtonian (PIB)

38. Lubrication theory (New) Best agreement for Ca<<Ca* (Ca*=0.24)

39. A) Newtoniano B) no-Newtoniano Identification of the threshold value (Ca*) for maxima and minima of the pressure profile

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. IAI2

41. A : wave amplitude σ : growth rate ℓ : Landau constant Amplitud Eq.: Af (A0 : initial amplitude) Amplitud A0 (independent of A0) t=0 tiempo Control parameter (R) ( if R > Rc ) lAl R ≡ Ca oscilación por excentricidad R=RC (umbral) σ (tasa de crecimiento) experimental ajuste de Landau (σ=1.17/s, Af =1.23mm)Ca = 0.274 gap = 0.2mm solución real solución aprox. R (Parámetro de control) Hydrodynamic instability • Landau Theory : Analytical solution

42. 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

43. Ca = 0.407 Ca = 0.382 Ca = 0.348 Ca = 0.287 Ca = 0.274 Ca = 0.268 Ca = 0.259 Hydrodynamic instability Results: agreement with Landau model • Vaseline (New.): • PIB (non New.):

44. Ca*PIB =0.153 Ca*VASELINA =0.231 experimental ajuste de Landau gap = 0.2 mm

45. 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.

46. Secondary Waves (with only one control parameter!) Ca*=0.204 PIB

47. Secondary Waves (with only one control parameter!)

48. References

49. 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). -“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.