1. J.E. Sprittles (University of Oxford, U.K.) Y.D. Shikhmurzaev(University of Birmingham, U.K.) International Society of Coating Science & Technology Symposium, Atlanta

2. Inkjet Printers: Microfluidic Technologies Key elements are the interaction of: Drops with a solid - Dynamic Wetting Drops with other drops - Coalescence

3. Dynamic Wetting Phenomena 50nm Channels 27mm Radius Tube 1 Million Orders of Magnitude! Millimetre scale Microfluidics Nanofluidics Emerging technologies Routine experimental measurement

5. Dynamic Wetting: Conventional Models 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. Model 1Model 2Model 3 has no solution. (Navier slip) (Young’s equation)

6. Conventional Modelling Dynamic Contact Angle Formula Assumption: A unique angle for each speed

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

8. Capillary Rise: Effect of Geometry Sobolev et al 01

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

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

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

13. Relevance of the Young Equation R σ 1e σ 3e - σ 2e Dynamic contact angle results from dynamic surface tensions. The angle is now determined by the flow field. Slip created by surface tension gradients (Marangoni effect) θeθe θdθd Static situationDynamic wetting σ1σ1 σ 3 - σ 2 R

14. In the bulk (Navier Stokes): On liquid-solid interfaces: At contact lines: On free surfaces: Interface Formation Model θdθd e2e2 e1e1 n n f (r, t )=0 Interface Formation Modelling

15. 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, 2012, Finite Element Simulation of Dynamic Wetting Flows as an Interface Formation Process, to J. Comp. Phy. (In Press)

16. Mesh Resolution is Critical

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

19. Capillary Rise: Models vs Experiments Interface formation & Lucas-Washburn ( ) vs experiments of Joos et al 90 Silicon oil of viscosity 12000cP for two capillary sizes (0.3mm and 0.7mm)

20. Lucas-Washburn vs Interface Formation Tube Radius = 0.36mm; Meniscus shape every 100secs Tube Radius = 0.74mm; Meniscus shape every 50secs After 100 secs LW IF After 50 secs LW IF

21. Comparison to Experiment Full Simulation Washburn JES & YDS 2012, J. Comp. Phy. (In press) Meniscus height h, in cm, as a function of time t, in seconds.

22. Speed-Angle Relationship New effect: angle decreases with increasing speed Asymptotic result (speed-angle formula) Equilibrium u=0 t = 0

23. ‘Hydrodynamic Resist’ Smaller Capillaries Dependence of the contact angle on geometry (capillary size) Sobolev et al 01

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

26. Microdrop Impact Velocity Scale Pressure Scale 25  m water drop impacting at 5m/s.

27. Surfaces of Variable Wettability 1 1.5 Hydrophilic Hydrophobic

28. Flow Control on Patterned Surfaces Green: hydrophobic. Grey: hydrophilic. JES & YDS 2012, PoF

30. Coalescence Conventional model: cusp becomes rounded in zero time -> infinite velocities Interface formation: cusp is rounded in finite time Forming interface Instant rounding Infinite bridge speed Gradual rounding Finite bridge speed

31. Coalescence of Liquid Drops 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 Experiment Simulation

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

33. Funding This presentation is based on work supported by:

35. Flow Characteristics

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

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

38. Influence of Viscosity 230cP48cP Nagel’s Experiments Thoroddsen’s Experiments 3.3cP Conventional Interface formation Widening gap

40. Periodically Patterned Surfaces No slip – No effect.No slip – No effect.

41. Interface Formation vs MDS Solid 2 less wettable Qualitative agreement JES & YDS 2007, PRE; JES &YDS 2009 EPJ