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

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 …

Microdrop Spreading WettableNon-Wettable

Microdrop Spreading Velocity Scale Pressure Scale

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?

Gas Cushion Dynamics van Dam & Le Clerc 2004, PoF

Gas Cushion Dynamics

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

Influence of the Ambient Gas

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

Gas Effects: Which Mechanism? Wetting Gas Film Impact

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

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

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

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

The ‘Classical’ Model

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

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

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)

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)

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)

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.

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

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

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

Multiscale Mesh: For Coating Flows Gas Liquid x1 x Resolution: Bulk Scale Slip Lengths

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

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

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

Flow Field Atmospheric Pressure Reduced PressureAtmospheric Pressure Velocity Continuous Maxwell Slip

Comparison to Experiment

Gas Film’s Dynamics Liquid Gas

Gas Film’s Dynamics Liquid Gas

A Local Knudsen Number Calculating a local Knudsen number based on gas film’s height

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

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

Non-Equilibrium Gas Effects Note where &

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

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

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

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

(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

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

A Local Knudsen Number Dependence of film height on capillary number