Two-phase hydrodynamic model for air entrainment at moving contact line

1. Two-phase hydrodynamic model for air entrainment at moving contact line Tak Shing Chan and Jacco Snoeijer Physics of Fluids Group Faculty of Science and Technology University of Twente

2. Part one: Introduction

3. Introduction: air Static contact angle θo liquid

4. Introduction: Constant U Dewetting (receding contact line): air liquid

5. Introduction: U > Uc e.g. Landau-Levich-Derjaguin film Dewetting (receding contact line): air Cac~10-2 liquid Bonn et al. (Rev. Mod. Phys. 2009) Lubrication theory

6. Introduction: Wetting (advancing contact line): air liquid Constant U

7. Introduction: Wetting (advancing contact line): air liquid Air entrainment ? U > Uc

8. Instability of advancing contact line (experimental motivation) A fiber is pulled into a liquid bath. Pressurized liquid, Cac ~ 50 (P.G. Simpkins & V.J. Kuck, J. Colloid & Interface Sci. 263, 2003) A splash is observed when the speed of the bead is larger than a threshold value. (Duez, C. et al Nature Phys. 3, 2007) Dip coating: air bubbles are observed. Cac ~1 (H. Benkreira & M.I. Khan, Chem. Engineering Sci. 63, 2008)

9. Introduction: Questions: What is the mechanism for air entrainment? Can we compute the critical Cac theoretically? Wetting (advancing contact line): air liquid U > Uc

10. Introduction: Questions: What is the mechanism for air entrainment? Can we compute the critical Cac theoretically? Wetting (advancing contact line): air liquid Lubrication theory still valid ??? Air flow important ??? U > Uc

11. Analogy with free surface cusp: role of air flow Lorenceau, Restagno, Quere, PRL 2003 Eggers PRL 2001 air Increasing speed liquid critical Ca depends on viscosity ratio !!

12. Analogy with free surface cusp: role of air flow Lorenceau, Restagno, Quere, PRL 2003 Eggers PRL 2001 air Increasing speed liquid critical Ca depends on viscosity ratio !! What happens for flow with a contact line?

13. Part two: 2-phase hydrodynamic model

14. 2-phase model: Assume straight contact line (2D problem) We consider very small Re number (Re << 1)and stationary state ( ) only: Fluid A (e.g. air) interface h Fluid B (e.g. water) Constant speed U

15. 2-phase model: Assume straight contact line (2D problem) We consider very small Re number (Re << 1)and stationary state ( ) only: Fluid A (e.g. air) interface h Young-Laplace equation Fluid B (e.g. water) Constant speed U

16. 2-phase model: Assume straight contact line (2D problem) We consider very small Re number (Re << 1)and stationary state ( ) only: Fluid A (e.g. air) interface h Young-Laplace equation Fluid B (e.g. water) Stokes equation (Re<< 1) Constant speed U

17. 2-phase model: For standard lubrication theory (1 phase, small slope), we use Poiseuille flow to approximate the velocity field. h x

18. Stream lines U air liquid 2-phase model: For standard lubrication theory (1 phase, small slope), we use Poiseuille flow to approximate the velocity field. h x For two phase flow ??? Huh & Scriven’s solution in straight wedge problem θ (C. Huh & L.E. Scriven, Journal of Colloid and Interface Science, 1971).

19. 2-phase model: Our idea is… …… With the assumption that the curvature of interface is small, we approximate the flow in our wetting problem by the flow in straight wedge problem.

20. Fluid A (e.g. air) h interface θ Fluid B (e.g. water) U 2-phase model:

21. Fluid A (e.g. air) h interface θ Fluid B (e.g. water) U 2-phase model: Control parameters: :static contact angle (wettability)

22. Fluid A (e.g. air) h interface θ Fluid B (e.g. water) U 2-phase model: Control parameters: :static contact angle (wettability) Boundary conditions: 1. h (at the contact line) = 0 2. θ(at the contact line) = θo 3. θ(at the bath) = π/2 We use shooting method to find the solutions

23. Fluid A (e.g. air) h interface θ Fluid B (e.g. water) U Question: How CaBcdepends on R and θo ? 2-phase model: Control parameters: :static contact angle (wettability)

24. Part three: Results

25. Control parameters: How iscritical CaBc found? e.g. fixed θo =50o , fixed R =0.1 :static contact angle (wettability) Static profile θo =50o air Δ liquid

26. Control parameters: How iscritical CaBc found? e.g. fixed θo =50o , fixed R =0.1 :static contact angle (wettability) air Δ liquid Uniform speed U

27. Control parameters: How iscritical CaBc found? e.g. fixed θo =50o , fixed R =0.1 :static contact angle (wettability) air Δ liquid Uniform speed U

28. Control parameters: How iscritical CaBc found? e.g. fixed θo =50o , fixed R =0.1 :static contact angle (wettability) air liquid Δ Uniform speed U

29. Control parameters: How iscritical CaBc found? e.g. fixed θo =50o , fixed R =0.1 :static contact angle (wettability) air liquid Δ Uniform speed U Cac

30. How does CaBc depend on R ? Control parameters: Critical capillary no. (Cac) :static contact angle (wettability) fixed θo =50o

31. Fluid A Fluid B U How does CaBc depend on R ? (fixed θo =50o)

32. Fluid A Fluid B U How does CaBc depend on R ? (fixed θo =50o) Dewetting regime (-1 scaling)

33. Fluid A Fluid B U How does CaBc depend on R ? Wetting regime (fixed θo =50o) CaBc changes significantly with R, even for small air viscosity !

34. Fluid A Fluid B U How does CaBc depend on R ? What is the scaling ? Wetting regime (fixed θo =50o) CaBc changes significantly with R, even for small air viscosity !

35. Fluid A Fluid B U How does CaBc depend on R ? Wetting regime (fixed θo =50o) Special case : R = 0 (i.e. log(R) → -infinity)

36. How does CaBc depend on R ? Special case : R = 0 (i.e. log(R) → -infinity)

37. How does CaBc depend on R ? Special case : R = 0 (i.e. log(R) → -infinity) Outer region (balance between gravity and viscous force) Asymptotic solution when CaB very large

38. How does CaBc depend on R ? Special case : R = 0 (i.e. log(R) → -infinity) Outer region (balance between gravity and viscous force) Asymptotic solution when CaB very large Inner region (balance between surface tension and viscous force) Asymptotic solution when CaB very large

39. inner inner How does CaBc depend on R ? Special case : R = 0 (i.e. log(R) → -infinity) Outer region (balance between gravity and viscous force) Asymptotic solution when CaB very large Inner region (balance between surface tension and viscous force) Asymptotic solution when CaB very large Matching between inner region and outer region is always possible!

40. CaBc How does CaBc depend on θo (wettability)? (fixed R = 0.01) Critical speed decreases significantly for hydrophobic surface !

41. CaBc How does CaBc depend on θo (wettability)? (fixed R = 0.01) Critical speed decreases significantly for hydrophobic surface ! (consistent with Duez et al. Nature Physics)

42. Conclusion: 1. We developed a “lubrication-like” model for two- phase flow. 2. Air dynamics is crucial to find entrainment threshold. If air flow is neglected (i.e. R=0), there is no air entrainment no matter how large Ca is. 3. Asymptotic scaling of CaBc for small R? ? Dewetting regime (-1 scaling)

43. Conclusion: 1. We developed a “lubrication-like” model for two- phase flow. 2. Air dynamics is crucial to find entrainment threshold. If air flow is neglected (i.e. R=0), there is no air entrainment no matter how large Ca is. 3. Asymptotic scaling of CaBc for small R? Funded by: ? Dewetting regime (-1 scaling) Thank you!