J.E. Sprittles (University of Oxford, U.K.) Y.D. Shikhmurzaev(University of Birmingham, U.K.) Mathematics of Splashing Workshop, ICMS, Edinburgh May 2013

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

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

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 The ‘apparent angle’

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

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

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

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 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)

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

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

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.

Mesh Resolution Key

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

Free Surface Profiles With:

Computations vs Experiments Water-glycerol solutions of & Asymptotics

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

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.”

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

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

Fibre Coating: Effect of Geometry Simpkins & Kuck 03

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

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

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

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

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

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

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

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

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

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 Simulation Experiment

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 Forming interface Instant rounding Infinite bridge speed Gradual rounding Finite bridge speed

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

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

Funding This presentation is based on work supported by:

‘Hydrodynamic Resist’ Smaller Capillaries New effect: contact angle depends on capillary size Sobolev et al 01

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

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

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

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