Drops sliding down an incline: Singular corners.

Transcription

1 Drops sliding down an incline: Singular corners. Laurent Limat Laboratoire Matière et Systèmes Complexes, MSC, Paris Diderot University with: -Jean-Marc Flesselles, Thomas Podgorski (initial experiments) -Adrian Daerr, Nolwenn Le Grand Bruno Andreotti, Emmanuelle Rio, Ivo Peters (3D visualization + PIV+optics) - Jacco Snoeijer, Howard Stone, Jens Eggers (corner models)

2 Partial wetting What happens on a tilted plate?

3 V Hysteresis: drop can remain pinned...! ("cos#) ~ $g V 2/3 or begins to slide U -viscous effects -> shape changes? -wetting dynamics with 3D aspects ->? -Inclined, curved contact line possibly developing a singular point

4 3D Sliding drops: -Bikerman, JCS 1950, Furmidge, JCS Dussan (and Chow), JFM 1983, Kim, Lee + Kang, JCIS > onset of motion -> calculations for rounded drops yield condition -> sliding velocities of oval drops related to viscous dissipation -Podgorski, Flesselles, LL, PRL Le Grand, Daerr, LL, JFM Rio, Daerr, Andreotti, LL, PRL > Singularity at drop rear -> 3D structure of interface -> flow structure, contact angle distribution -Stone + LL, Europhys. Lett > similarity solution of hydrodynamics -Snoeijer, Le Grand, Rio, LL, Phys. Fl > flow structure, rounded corners -Snoeijer, Le Grand, LL, Stone, Eggers, Phys. Fl > opening angle selection, pearling transition Other model: Cummins, Ben Amar, Pomeau, Phys. Fl > model based on Laplace +directional Young condition Numerical simulations: -Schwartz et al., Physica D, Thiele et al --Gaskell et al. (Leeds)

5 Ca=&U/! sin%~1/ca Silicon oil drops on fluoropolymers

6 Cohen Nagel Courrech-du-Pont Eggers Daerr, Le Grand, LL, 2005 Singularities at interfaces (Rutland and Jameson!71) Quéré, Lorenceau

7 U air liquid Analogy with Landau-Levich Blake, Ruschak (Nature 1979) Snoeijer, Delon, Andreotti, Fermigier PRL 2006

8 Interpretation: # d =F(Ca) --> 0 (or a critical value) # # a # r! d 0 U! U e Ca=&U/! U n = U sin "! d >0 " U U! U e ; U n " U e! d U < U e # d =F(Ca sin%)! 0 => sin! = Ca c Ca Blake, Ruschak(Nature 1979)

9 Other point of view: Drops avoid wetting by tilting contact lines plate withdrawn from bath Ca sin! U>U C! Blake and Ruschak!79 Ca normal velocity: maximum speed=u c //Mach cone => sin% ~ 1/Ca

10 Outline: Experiment - Observations 3D structure and flow in sliding drops 3D geometry of sliding drops flow structure - autosimilarity Matching with contact lines opening angle selection pearling transition Curvature at the tip curved contact lines divergence of curvature U>U c U n =U>U c U n =Usinf =U c

11 Pipette EXPERIMENT Syringe VideoCamera Miror Bécher a FAST CAM ' iiyama Tilted glass plate + 3M fluoropolymers A. Daerr, N. Le Grand, LL, J. Fl. Mech. (2005)

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13 sliding silicon oil drops... increasing Ca="U0/# Podgorski, Flesselles Limat, PRL!01 Le Grand, Daerr, Limat JFM!05 Side views rounded corner cusp conical singularity pearling U0 Stone, LL, Europhys. "04 Glass plate + fluoropolymer coating Partial wetting: contact angle 45º

14 80 # d =# c # d = 0 1 sin(") # d adv # d rcd # d ( ) # c advancing receding rounded drops corners cusps! ( ) pearling drops 0,8 0,6 0,4 0,2 % ,002 0,004 0,006 0,008 Ca? Non-zero // De Gennes // Raphaël, Golestanian ( -Sin% ~1/Ca, but - non-zero critical angle # c -Conical interface : #, ( non-zero Daerr, Le Grand, LL, J.Fl. Mech. 2005

15 Voïnov-Cox model 80 sin(") 0,8 # d ( ) adv 60 0,6 40 0,4 # d ( ) rcd 20 0,2! ( ) ,002 0,004 0,006 0,008 rounded drops corners Ca cusps pearling drops 1 - Mass conservation: Uh = h 3 3" # (h xx) x - To be completed with : # = # e for x=a and a matching with an outer solution for x -> ) 9Log(b/a)=130 9Log(b/a)=130 3 Ca h 2 = h xxx macroscopic microscopic " 3 (b) = " 3 (a) ± 9CaLog( b a) # e (equilibrium angle) everything is singular for h=0

16 ! =104 cp 1! =1040 cp 1 80 Rounded drops sin$ 0,8 80 " a sin$ 0,8 60 " a 40 " r 0,6 Rounded corners " a, " r, # ( ) sin$ Corners Cusp 0,4 60 Rounded drop " a, " r, # ( ) sin$ Rounded corner 40 Corner " r 0,6 0,4 20 # 0,2 20 # Cusp 0, ,001 0,002 0,003 0,004 0,005 0,006 0,007 Ca ,002 0,004 0,006 0,008 0,01 Ca Same with two other viscosities.

17 U Front and rear curvatures Ca=&U/!

18 Structure of the singularity h(x,y)? U(x,y)=<v(x,y,z)> /z? U(x,y) r U (x, y) = " h 2 3# $P Flow driven by capillary pressure! Stone + LL, Europhys 04 Snoeijer et al., Phys. Fluids 05

19 corner model r U (x, y) = h 2 3" #($h) + steady state solutions: y h( x - U 0 t, y) x 3Ca "h "x = #. h 3 #($h) [ ] Ca="U 0 / #

20 corner geometry 3Ca "h "x = #. h 3 #($h) [ ] y equation allows similarity solutions: 1/3 h(x,y) = Ca x H(y/x) x Note: Ca has disappeared

21 corner geometry y 3Ca "h "x = #. h 3 #($h) [ ] equations allow similarity solutions: x 1/3 h(x,y) = Ca x H(y/x) 1-parameter family: H b.c: H = H = 0 (symmetry) H0 H (continuity) Note: Ca has disappeared y/x

22 corner geometry y % ( x 1/3 h(x,y) = Ca x H(y/x) H tan( H0 = tan $/ Ca 1/3 tan% Infinite number of solutions, irrespectively of Ca (we will need later an extra-condition) y/x

23 relation ( and % % % ( H0 = tan $ / Ca 1/3 (no adjustable parameters) dashed: (tan$) 3 35 = 16 Ca ( tan!) 2

24 velocity field h(x,y) U(x,y)=U 0 F(y/x) F deduced from H PIV experiment: tracers Rio, Andreotti, Daerr

25 velocity field Theory: velocity self-similar U = U(y/x) 52 µm 312 µm # flow direction * " Snoeijer et al. Phys Fluids!05

26 velocity field Theory: velocity self-similar U = U(y/x) components 52 mm Ur 312 mm Uf % Snoeijer et al. Phys. Fluids!05

27 But. - Selection of %? - Prediction of pearling instability? How to manage with the singularity at contact line?

28 Classical argument: # # d =F(Ca) --> 0 or a critical value # r! d 0 U! U e Ca=&U/! U n = U sin "! d >0 " U U! U e ; U n " U e! d U < U e Blake, Ruschak(Nature 1979) Podgorski et al (PRL 2000) # d =F(Ca sin%)! 0 => sin! = Ca c Ca

29 Low % paradox: f! W pressure gradients become weaker... velocity + 0! + 0 implies drop speed + 0 maximum speed!! Breakdown of 1/Ca law Ca %~1/Ca %

30 %-selection : mixing contact line and bulk flow 70 rounded drop corner cusp pearling drop 1 tan 3 (!(35/16)Ca tan 2 % and: 60 50! a Cox-Voïnov 0,8! a,! r, " ( ) 40 30! r (! r =! c ) sin# 0,6 0,4 sin# 20! c #! 2 (/sin% and: " 3 = " 3 e # 9(Casin$)Log( b a) 10 Similarity solution 0 0 0,002 0,004 0,006 0,008 0,01 Ca Ca " = e sin# $ & 9Log(b/a) + %& 70 ' ) ( sin2# ) 2 () 0,2 0

31 Snoeijer, Eggers Ca=&U/! d dx = " d dx small - Lubrication + parabolic approx. Ca[ HY " H # Y # ] $ " H & 3 Y% H ) ( + ' Y 2 * X Flowing down Left behind Viscous capillary flux - Directional Cox-Voïnov law: " 3 # (2 H Y )3 # " e 3 $ 9Ca(%Y X )Log(b /a) Consistency: Ca, Log(b/a) ~Ca/,~1 Ca sin%

32 About directional Cox-Voïnov law E. Rio, A. Daerr, B. Andreotti, LL, Phys Rev Lett 2004

33 Lubrification: < r u >= " h 2 3# $P P " #$%h Singular at contact line " /"x # >> " /"x // Mass conservation: u - >>u // <u - >=u c-line u c-line -> local 2D situation

34 Ca=&U/! d dx = " d dx small - Lubrication + parabolic approx. Ca[ HY " H # Y # ] $ " H & 3 Y% H ) ( + ' Y 2 * X Flowing down Left behind Viscous capillary flux - Directional Cox-Voïnov law: " 3 # (2 H Y )3 # " e 3 $ 9Ca(%Y X )Log(b /a) Consistency: Ca, Log(b/a) ~Ca/,~1 Ca sin%

35 For, = 1/[Ln(b/a)] 1/2 small there exist corner solutions (H~Y~X) with: 3Ca "# $ 6(% /") 3 e (% /") 2 3Ca/(,# r3 ) %!25 unstable stable %/, Oval drops -> Log(b/a)~10,~0.3

36 3Ca/(,# r3 ) %/,

37 - beyond maximum speed: rivulet solutions Y! U - rivulet width Y! vs U f W

38 Curvature at the tip... Le Grand et al. JFM!05 1/R (1/mm) Problem: U 0 sin% = U c => U 0 >U c!! Ca Idea: U c =f(r) => U 0 sin% = U c (R=)) et U 0 =U c (R) -Critical speed Ca(1/R)?=> calc. for «large» R -> Snoeijer et al. Phys. Fl Other question: R(Ca)? Scaling laws for tip equilibrium

39 An open issue: divergence of 1/R Ivo Peters Adrian Daerr 1/R~Aexp(B Ca) Similar to Lorenceau

40 Difficult to understand: F cap ~ L# 2! ~F visc ~L & (U/#) Log(R/a) L -> 1/R ~(1/a) exp(# 3 /Ca) To get 1/R ~(1/a) exp(ca/# 3 ) One would need F visc ~L & (U/#) / Log(R/a).

41 Discussion drops have singular shapes to avoid wetting inclined: reduces normal speed curved: additional capillary forces Rio et al. PRL!05 3D structure governed by similarity solutions. Matching with contact line (where the flow is perpendicular) selects opening angle. Matching becomes impossible above a critical Capillary number -> pearling Behavior of 1/R at the tip remains to be Understood