Modelling and simulation of high-speed dynamic wetting phenomena

1. Modelling and simulation of high-speed dynamic wetting phenomena J.E. Sprittles (University of Oxford, U.K.) Y.D. Shikhmurzaev (University of Birmingham, U.K.) Mathematics of Splashing Workshop, ICMS, Edinburgh May 2013

2. Duez et al 07 Wettability effects splash thresholds even for Re, We >>1

3. Rein & Delplanque 08 Splash threshold for drop impact can be expressed in terms of the capillary number.

4. Coating Experiments Solid Gas Liquid The ‘apparent angle’ • Advantages: • Flow is steady making experimental analysis more tractable. • Parameter space is easier to map: Speeds over 6 orders Viscosities over 3 orders

5. Coating Results Increasing μ Apparent angle measured at resolution of 20microns for water-glycerol solutions with μ=1, 10, 100 mPas.

6. Interpretation A: Static Contact Angle The ‘actual angle’ You only observe the ‘apparent angle’. The actual one is fixed. Free surface bends below the experiment’s resolution (20μm)

7. Interpretation B: Dynamic Contact Angle Dynamics of angle cause change in apparent angle Dynamic contact angle is a function of speed

8. Modelling of Dynamic Wetting Phenomena

9. The ‘Moving Contact Line Problem’ L.E.Scriven& C.Huh(1971), A.W.Neumann(1971), S.H. Davis (1974), E.B.Dussan (1974), E.Ruckenstein (1974), A.M.Schwartz (1975), M.N.Esmail (1975), L.M.Hocking (1976), O.V.Voinov (1976), C.A.Miller (1976), P.Neogi (1976), S.G.Mason (1977), H.P.Greenspan (1978), F.Y.Kafka (1979), L.Tanner (1979), J.Lowndes (1980), D.J. Benney (1980), W.J.Timson (1980), C.G.Ngan (1982), G.F.Telezke (1982), L.M.Pismen (1982), A.Nir (1982), V.V.Pukhnachev (1982), V.A.Solonnikov (1982), P.-G. de Gennes (1983), V.M.Starov (1983), P.Bach (1985), O.Hassager (1985), K.M.Jansons (1985), R.G.Cox (1986), R.Léger (1986), D.Kröner (1987), J.-F.Joanny (1987), J.N.Tilton (1988), P.A.Durbin (1989), C.Baiocchi (1990), P.Sheng (1990), M.Zhou (1990), W.Boender (1991), A.K.Chesters (1991), A.J.J. van derZanden (1991), P.J.Haley (1991), M.J.Miksis (1991), D.Li (1991), J.C.Slattery (1991), G.M.Homsy (1991), P.Ehrhard (1991), Y.D.Shikhmurzaev (1991), F.Brochard-Wyart (1992), M.P.Brenner (1993), A.Bertozzi (1993), D.Anderson (1993), R.A.Hayes (1993), L.W.Schwartz (1994), H.-C.Chang (1994), J.R.A.Pearson (1995), M.K.Smith (1995), R.J.Braun (1995), D.Finlow (1996), A.Bose (1996), S.G.Bankoff (1996), I.B.Bazhlekov (1996), P.Seppecher (1996), E.Ramé (1997), R.Chebbi (1997), R.Schunk (1999), N.G.Hadjconstantinou (1999), H.Gouin (10999), Y.Pomeau (1999), P.Bourgin (1999), M.C.T.Wilson (2000), D.Jacqmin (2000), J.A.Diez (2001), M.&Y.Renardy (2001), L.Kondic (2001), L.W.Fan (2001), Y.X.Gao (2001), R.Golestanian (2001), E.Raphael (2001), A.O’Rear (2002), K.B.Glasner (2003), X.D.Wang (2003), J.Eggers (2004), V.S.Ajaev (2005), C.A.Phan (2005), P.D.M.Spelt (2005), J.Monnier (2006)

10. A) Slip Models(‘The Hydrodynamic Model’)

11. Slip Models B: No-slip => No solution l s A: Equilibrium contact angle B: Slip - typically Navier-slip

12. Asymptotics for the Apparent Angle Often, we have In Cox 86, it was shown that in this case: And for Voinov (76) has shown:

13. A Finite Element Based Computational Framework JES &YDS 2011, Viscous Flows in Domains with Corners, CMAME JES & YDS 2012, Finite Element Framework for Simulating Dynamic Wetting Flows, Int. J. Num. Meth Fluids. JES & YDS, 2012, The Dynamics of Liquid Drops and their Interaction with Surfaces of Varying Wettabilities, Phy. Fluids. JES & YDS, 2013, Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process, to J. Comp. Phy.

14. Mesh Resolution Key

15. Arbitrary Lagrangian Eulerian Mesh Based on the ‘spine method’ of Scriven and co-workers Microdrop simulation with impact, spreading and rebound

16. Results: Slip Models

17. Free Surface Profiles With:

18. & Asymptotics Computations vs Experiments • Water-glycerol solutions of

19. Influence of the Gas Asymptotic formulae (Cox) including finite viscosity ratio.

20. Limitations of Cox’s Formula Chen, Rame & Garoff 95: “Aspects of the unique hydrodynamics acting in the inner region, not included in the model, project out and become visible in the imaged region.”

21. Computations vs Asymptotics Ca=0.5 Ca=0.05 Ca=0.005 Computations resolve all scales and confirm failure of asymptotic approaches at high capillary number.

22. B) Evidence for a Dynamic (Flow Dependent) Angle

23. U, cm/s Hydrodynamic Assist Vary Flow Rate Blake et al 99 Effect is not due to free surface bending (Wilson et al 06)

24. Fibre Coating: Effect of Geometry Simpkins & Kuck 03

25. ) Drop Spreading: Effect of Impact Speed Bayer & Megaridis 06

26. Dynamic Wetting:An Interface Formation Process

27. Physics of Dynamic Wetting Liquid-solid interface Solid Forming interface Formed interface • Make a dry solid wet. • Create a new/fresh liquid-solid interface. • Class of flows with forming interfaces.

28. Relevance of the Young Equation Static situation Dynamic wetting σ1e σ1 θe θd σ3 - σ2 σ3e - σ2e R R Dynamic contact angle results from dynamic surface tensions. Theangle is now determined by the flow field. Slip created by surface tension gradients (Marangoni effect)

29. f (r, t )=0 e1 n n θd e2 Interface Formation Modelling In the bulk (Navier Stokes): Interface Formation Model Liquid-solid interface On free surfaces: At contact lines:

30. Asymptotic Formula for Actual Angle in IFM When there is no ‘hydrodynamic assist’, for small capillary numbers the actual angle is dynamic: Moffat 64

31. & Asymptotics IFM vs Experiments Shikhmurzaev 93 • Actual angle varies and free surface bends. Shikhmurzaev 93 + Cox 86

32. IFM: Influence of the Gas Which alters both the actual and apparent angles.

33. Microdrop Dynamics

34. Microdrop Impact ? 25mm water drop impacting at 5m/s. Experiments: Dong et al 06

35. Dynamic Wetting Phenomena Routine experimental measurement Significant differences between models 1 Million Orders of Magnitude! 50nm Channels 27mm Radius Tube Millimetre scale Microfluidics Nanofluidics

36. Coalescence of Liquid Drops:Models vs Experiments

37. Coalescence of Liquid Drops Experiment Simulation Developed framework can be adapted for coalescence. Thoroddsen’s Group: Ultra high-speed imaging Nagel’s Group: Sub-optical electrical measurements Thoroddsen et al 2005

38. Coalescence • Conventional model: singular as initial cusp is • rounded in zero time -> infinite velocities • Interface formation: singularity-free as cusp is rounded in finite time that it takes internal interface to disappear Instant rounding Infinite bridge speed Gradual rounding Finite bridge speed Forming interface

39. Coalescence: Models vs Experiments Thoroddsen’s Optical Experiments Conventional Interface formation Nagel’s Electrical Measurements Bridge radius versus time: 2mm drops of 220cP water-glycerol.

40. Microfluidic Dynamic Wetting Phenomena

41. Microscale Dynamic Wetting Ultra high speed imaging of microfluidic wetting phenomenon, with Dr E. Li & Professor S.T. Thoroddsen

42. Funding • Funding • This presentation is based on work supported by:

43. Thanks

44. ‘Hydrodynamic Resist’ New effect: contact angle depends on capillary size Smaller Capillaries Sobolevet al 01

45. Effect of Gas Pressure Air entrainment speed in Dip Coating Benkreira & Ikin 10 Splashing in Drop Impact: Xu, Zhang & Nagel 05

46. Microdrop Impact 25 micron water drop impacting at 5m/s on left: wettable substrate right: nonwettable substrate

47. Coalescence: Free surface profiles Time: 0 < t < 0.1 Conventional theory Interface formation theory Water- Glycerol mixture of 230cP

48. Appearance of V’s Impact of a solid sphere: Duez, Ybert, Clanet & Bocquet 07 Dip coating experiments Courtesy of Terry Blake