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. air Introduction: liquid Static contact angle θ o

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

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

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

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

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

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

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

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

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

13. Part two: 2-phase hydrodynamic model

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

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

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

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

18. 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 (C. Huh & L.E. Scriven, Journal of Colloid and Interface Science, 1971). U air liquid Stream lines θ 2-phase model:

19. 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. Our idea is… …… 2-phase model:

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

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

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

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

24. Part three: Results

25. e.g. fixed θ o =50 o, fixed R =0.1 Δ How is critical Ca Bc found? air liquid Static profile θ o =50 o :static contact angle (wettability) Control parameters:

26. Δ How is critical Ca Bc found? air liquid :static contact angle (wettability) Control parameters: Uniform speed U e.g. fixed θ o =50 o, fixed R =0.1

27. Δ How is critical Ca Bc found? air liquid :static contact angle (wettability) Control parameters: e.g. fixed θ o =50 o, fixed R =0.1 Uniform speed U

28. Δ How is critical Ca Bc found? air liquid :static contact angle (wettability) Control parameters: e.g. fixed θ o =50 o, fixed R =0.1 Uniform speed U

29. Δ How is critical Ca Bc found? air liquid :static contact angle (wettability) Control parameters: e.g. fixed θ o =50 o, fixed R =0.1 Uniform speed U Ca c

30. Critical capillary no. (Ca c ) fixed θ o =50 o :static contact angle (wettability) Control parameters: How does Ca Bc depend on R ?

31. U Fluid A Fluid B (fixed θ o =50 o )

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

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

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

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

36. How does Ca Bc depend on R ?

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

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

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

40. How does Ca Bc depend on θ o (wettability)? (fixed R = 0.01) Critical speed decreases significantly for hydrophobic surface ! Ca Bc

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

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 Ca Bc 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 Ca Bc for small R? Dewetting regime (-1 scaling) ? Funded by: Thank you!