1. Dr James Sprittles Mathematics Institute, University of Warwick Science of Inkjet and Printed Drops, November 2014

3. Microdrop Spreading Sponsored by Kodak European Research. To study dynamics of drop ejected from inkjet printers. Focus on models used for spreading dynamics. Contact angle dynamics, liquid-solid slip, etc. Computational framework developed for models. Gas dynamics neglected as: & Consider impact of water drop at …

4. Microdrop Spreading WettableNon-Wettable

5. Microdrop Spreading Velocity Scale Pressure Scale

6. Microdrop Spreading ? 25  m water drop impacting at 5m/s. Experiments: Dong et al 06 Do you see a gas bubble trapped under the drops?

8. Gas Cushion Dynamics van Dam & Le Clerc 2004, PoF

9. Gas Cushion Dynamics

10. De Ruiter et al 2012, PRL Bouwhuis et al 2012, PRL Gas Film

11. Influence of the Ambient Gas

12. Ambient gas pressure is key to the drop’s behaviour What physical mechanism causes this and how does it enter a mathematical model?

13. Gas Effects: Which Mechanism? Wetting Gas Film Impact

15. Coating Experiments Advantages: Flow is steady making experimental analysis more tractable. Parameter space is easier to map: Speeds over 6 orders Viscosities over 3 orders Liquid GasSolid

16. Air Entrainment Courtesy of Jacco Snoeijer, University of Twente Critical speed of wetting => gas pulled into the liquid

17. Effect of Gas Pressure on Wetting Speed Benkreira & Khan 2008, Air Entrainment in Dip Coating Under Reduced Pressures, Chemical Engineering Science Reduced Gas Pressure Increased Coating Speed

18. Different Ambient Gases Benkreira & Ikin 2010, Dynamic Wetting and Gas Viscosity Effects, Chemical Engineering Science

20. The ‘Classical’ Model

21. 1) Impact Phase When will the gas film rupture? Never! Gas Film

22. 2) Wetting Phase No Solution!! ‘Moving contact line problem’

23. Wetting Models: Liquid Phase A. A `slip’ condition: Slip region of size ~ l B. Dynamic contact angle formula: No-slip ( u=0) u=U A. ‘Classical’ formulation B. Dynamic contact angle must be specified. has no solution. (Navier slip) (Young’s equation)

24. L.E.Scriven (1971), 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 der Zanden (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) Wetting Models: Liquid Phase L.E.Scriven (1971)

25. Wetting Models: Gas Phase A. A `slip’ condition: Slip region of size ~ l No-slip ( u=0) A. ‘Classical’ formulation has no solution (Navier slip)

27. Non-Equilibrium Gas Dynamics Slip at solid-gas interface is due to finite mean free path. Mean free path (hence Kn) depends on gas density/pressure.

28. Find where & At the gas-solid boundary we have: Whilst at the gas-liquid free-surface: ‘turns off’ free-surface Maxwell-slip Maxwell Slip Conditions

29. Dynamic Wetting Model Simplest possible dynamic wetting model: Navier-slip on the liquid-solid interface with Fixed equilibrium contact angle

30. JES & YDS 2012, Finite Element Framework for Simulating Dynamic Wetting Flows, International Journal for Numerical Methods in Fluids. JES & YDS, 2013, Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process, Journal of Computational Physics

31. Multiscale Mesh: For Coating Flows Gas Liquid x1 x10 8 2 4 Resolution: Bulk Scale Slip Lengths

32. Sprittles 2014, Air Entrainment in Dynamic Wetting: Knudsen Effects and the Influence of Ambient Air Pressure, Submitted

33. Free Surface Profiles Consider silicone oil with as a base state Gas Liquid

34. Effect of Gas Pressure Maxwell-slip at solid and liquid is critical

35. Flow Field Atmospheric Pressure Reduced PressureAtmospheric Pressure Velocity Continuous Maxwell Slip

36. Comparison to Experiment

37. Gas Film’s Dynamics Liquid Gas

38. Gas Film’s Dynamics Liquid Gas

39. A Local Knudsen Number Calculating a local Knudsen number based on gas film’s height. 0.4310.011 0.470.160.1 0.860.01213 1.290.009460

40. Implications for Drop Impact Xu et al 05: threshold pressure required to suppress splashing

41. Threshold Pressures Threshold Pressure vs Impact Speed for Different Gases Air Helium Krypton SF 6

42. Non-Equilibrium Gas Effects Note where &

43. Open Problems Alternative flow configurations. Theory-driven experimental analysis. Navier-Stokes Boltzmann Coupling: Continuum Mechanics Navier Stokes ? Statistical Mechanics Boltzmann Equation

45. Impact Phenomena Classical Model Maxwell-Slip Model Knudsen Effects Drop actually impacts solid!

47. 1. Model: Prevents film ever breaking + 2. Computation: Poor resolution of film initiates mesh- dependent breakup Failure of Commercial Software 6 different ‘answers’! Hysing et al, 2009, IJNMF

48. Formation of Drops Compound Drops: Mr J.A. Simmons Drop Breakup: Dr Y. Li

49. (Post-Impact) Coalescence of Liquid Drops Coalescence of Liquid Drops: Different Models vs Experiments, Physics of Fluids 2012 Experiments: Dr J.D. Paulsen Our Simulation: Green Lines

51. (Lack of) Influence of Inertia  Bulk flow can’t be responsible for the effect. Re = 0 Re = 100

52. A Local Knudsen Number Dependence of film height on capillary number