contact line dynamics part 2: hydrodynamics dynamic contact angle? lubrication: Cox-Voinov theory maximum speed for instability
corner shape? dimensional analysis: speed U position r viscosity η pressure p, same dimension as shear stress τ p ~ 1/ r
corner shape? dimensional analysis: speed U position r viscosity η pressure p, same dimension as shear stress τ p ~ 1/ r corner inconsistent with Laplace pressure: p = p 0 - γh
interface shape corner is not a solution, interface curved!! θ (x)
interface shape Voinov 1976, Cox 1986: " 3 = " e 3 # 9Caln(x /l micro ) corner is not a solution, interface curved!! θ (x)
interface shape Voinov 1976, Cox 1986: " 3 = " e 3 # 9Caln(x /l micro ) - interface is curved θ (x) - angle variation ~ Ca - singularity: ln(x)
dynamic contact angle? Le Grand et al. J. Fluid Mech. 2005
dynamic contact angle? Le Grand et al. J. Fluid Mech. 2005
dynamic contact angle Le Grand et al. J. Fluid Mech. 2005 θ ( ) Ca = Uη/γ
dynamic contact angle Le Grand et al. J. Fluid Mech. 2005 θ ( ) but attention: θ(x) depends on scale of measurement Ca = Uη/γ
scale dependence Marsh et al. Phys. Rev. Lett. 1993
scale dependence Marsh et al. Phys. Rev. Lett. 1993 θ x (µm)
Cox-Voinov using lubrication theory, let us derive: " 3 = " e 3 # 9Caln(x /l micro )
lubrication equation h << 10-3 m h >> 10-9 m balance between viscosity η and surface tension γ Ca = Uη/γ
lubrication equation h << 10-3 m h >> 10-9 m h'''= 3Ca h 2 receding h'''= " 3Ca h 2 advancing
lubrication equation h'''= 3Ca h 2 analytical solution due to Duffy & Wilson 1997:
lubrication equation h'''= 3Ca h 2 analytical solution due to Duffy & Wilson 1997: h(x) in implicit form: x(s), h(s)
lubrication equation h'''= 3Ca h 2 take a deep breath...
lubrication equation h'''= 3Ca h 2 first, verify corner solution for h(x) does not exist
lubrication equation h'''= 3Ca h 2 first, verify corner solution for h(x) does not exist h(x) = x" h''= h'''= 0 not a solution!!
varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = x"(x)
varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = x"(x) h'= " + x"'(x)
varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = x"(x) h'= " + x"'(x) h''= "' h'''= "''
Cox-Voinov " 2 "''= 3Ca x 2 approximate solution: Cox-Voinov law " 3 (x) = " e 3 # 9Caln(x /l micro )
application (i): spreading drop perfectly wetting θ e =0 : R(t)??
application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)??
application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)?? inner solution "(x) 3 = 9Ca ln(x /l micro )
application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)?? inner solution "(x) 3 = 9Ca ln(x /l micro ) see: Bonn, Eggers, Indekeu, Meunier, Rolley, to appear Rev. Mod. Phys. (2009) 3 " app matching: # R & = 9Ca ln% ( $ 2e 2 l micro '
application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ " 3 " ~ Vol R 3
application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ " 3 " ~ Vol R 3 exercise: what is R(t)?
application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ " 3 " ~ Vol R 3 Tanner s law: R(t) ~ t 1/10
application (ii): forced wetting low Ca high Ca film air liquid
application (ii): forced wetting low Ca high Ca film air liquid what is critical Ca?
contact line stability low speeds: contact line equilibrates at height z cl z cl Ca (lubrication + slip length to treat contact line)
contact line stability low speeds: contact line equilibrates at height z cl z cl z cl Ca Ca Snoeijer, Delon, Fermigier, Andreotti, J. Fluid Mech. 2007
evolution towards film dimple Snoeijer, Delon, Fermigier, Andreotti, J. Fluid Mech. 2007
evolution towards film z cl t U p -U cl Snoeijer, Delon, Fermigier, Andreotti, J. Fluid Mech. 2007
apparent contact angle maximum Ca achieved at z cl = 1.408... z cl Ca
apparent contact angle maximum Ca achieved at z cl = 1.408... z cl static bath: Ca z cl = 2(1" sin#)
apparent contact angle maximum Ca achieved at z cl = 1.408... z cl θ app 0 static bath: Ca z cl = 2(1" sin#)
application (ii): forced wetting Ca θ app =0 Matching static bath to Duffy & Wilson solution Eggers, Phys. Rev. Lett. 2004 Ca
application (ii): forced wetting Ca θ app =0 Cox-Voinov: does not describe Ca c!! Ca (due to approximations)
inclined plate Ca c Ca Ca c depends on inclination angle: not universal! α / θ eq Ziegler, Snoeijer, Eggers, EPJST 2009
drops sliding on incline U partial wetting θ e = 45
drops sliding on incline increasing speed Ca=ηU/γ U partial wetting θ e = 45 Podgorski et al. PRL 01 Le Grand et al. JFM 05
drops sliding on incline Podgorski et al. PRL 01 U some basic questions: critical speed? 3D hydrodynamics in corner? how sharp is the tip-singularity?
corner geometry Φ Ω geometry: h(x,y) «Cone-like»
corner geometry force balance: viscosity vs surface tension y φ Φ Ω high pressure low pressure x 2D cone model experiment: tracer particles Limat & Stone, Europhys. Lett. 04 Snoeijer, Rio, Le Grand, Limat, Phys. Fluids 05
tip? how sharp is the corner tip?
tip? small but finite radius of curvature R
recent measurements Peters, Snoeijer, Daerr, Limat, to be submitted 1/R Ca
distance << R: straight c.l. theory for tip
theory for tip distance << R: straight c.l. h'''= 3Ca h 2 $ h' 3 = " 3 e # 9Ca ln& % x l micro Cox-Voinov : divergence small length ' ) (
theory for tip y distance ~ R: two principal curvatures "'= 3Ca h 2 " # h''$ 2h w 2 x
theory for tip Peters, Snoeijer, Daerr, Limat, to be submitted 1/R Cox-Voinov like result: Ca % R ( " 3 = # 3 e $ 9Ca ln' * & lmicro ) R " l micro e # e 3 / 9Ca
theory for tip 1/R best fit: l micro = 8 nm Cox-Voinov like result: % R ( " 3 = # 3 e $ 9Ca ln' * & lmicro ) R " l micro e # e 3 / 9Ca 1/Ca
conclusion contact line dynamics: multi-scale (micro to macro) interface curved Cox-Voinov critical speed of wetting is beyond Cox-Voinov
open questions... rapidly advancing contact line: - 2 phase flow problem - inertial effects?
open questions... air entrainment U liquid rapidly advancing contact line: - 2 phase flow problem - inertial effects?