We consider the spreading of a thin two-dimensional droplet on a planar substrate as a prototype system to compare the contemporary model for contact line motion based on interface formation of Shikhmurzaev [Int. J. Multiphase Flow19, 589

610
(1993)], to the more commonly used continuum fluid dynamical equations augmented with the Navier-slip condition. Considering quasistatic droplet evolution and using the method of matched asymptotics, we find that the evolution of the droplet radius using the interface formation model reduces to an equivalent expression for a slip model, where the prescribed microscopic dynamic contact angle has a velocity dependent correction to its static value. This result is found for both the original interface formation model formulation and for a more recent version, where mass transfer from bulk to surface layers is accounted for through the boundary conditions. Various features of the model, such as the pressure behaviour and rolling motion at the contact line, and their relevance, are also considered in the prototype system we adopt.

1.
P. G.
de Gennes
, “
Wetting: Statics and dynamics
,”
Rev. Mod. Phys.
57
,
827
863
(
1985
).
2.
T.
Blake
, “
The physics of moving wetting lines
,”
J. Colloid Interface Sci.
299
,
1
13
(
2006
).
3.
D.
Bonn
,
J.
Eggers
,
J.
Indekeu
,
J.
Meunier
, and
E.
Rolley
, “
Wetting and spreading
,”
Rev. Mod. Phys.
81
,
739
805
(
2009
).
4.
Y. D.
Shikhmurzaev
, “
The moving contact line on a smooth solid surface
,”
Int. J. Multiphase Flow
19
,
589
610
(
1993
).
5.
D.
Bedeaux
,
A.
Albano
, and
P.
Mazur
, “
Boundary conditions and non-equilibrium thermodynamics
,”
Physica A
82
,
438
462
(
1976
).
6.
Y. D.
Shikhmurzaev
,
Capillary Flows with Forming Interfaces
(
Taylor & Francis
,
London
,
2008
).
7.
Y. D.
Shikhmurzaev
, “
Singularities at the moving contact line: Mathematical, physical and computational aspects
,”
Physica D
217
,
121
133
(
2006
).
8.
H. K.
Moffatt
, “
Viscous and resistive eddies near a sharp corner
,”
J. Fluid Mech.
18
,
1
18
(
1964
).
9.
C.
Huh
and
L. E.
Scriven
, “
Hydrodynamic model of steady movement of a solid/liquid/fluid contact line
,”
J. Colloid Interface Sci.
35
,
85
101
(
1971
).
10.
D. E.
Weidner
and
L. W.
Schwartz
, “
Contact-line motion of shear-thinning liquids
,”
Phys. Fluids
6
,
3535
3538
(
1994
).
11.
L. M.
Pismen
and
Y.
Pomeau
, “
Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics
,”
Phys. Rev. E
62
,
2480
2492
(
2000
).
12.
P.
Colinet
and
A.
Rednikov
, “
On integrable singularities and apparent contact angles within a classical paradigm
,”
Eur. Phys. J. Spec. Top.
197
,
89
113
(
2011
).
13.
N.
Savva
and
S.
Kalliadasis
, “
Dynamics of moving contact lines: A comparison between slip and precursor film models
,”
Europhys. Lett.
94
,
64004
(
2011
).
14.
L. M.
Pismen
and
J.
Eggers
, “
Solvability condition for the moving contact line
,”
Phys. Rev. E
78
,
056304
(
2008
).
15.
C.-L.
Navier
, “
Mémoire sur les lois du mouvement des fluides
,”
Mem. Acad. Sci. Inst. Fr.
6
,
389
440
(
1823
).
16.
Y. D.
Shikhmurzaev
, “
Moving contact lines in liquid/liquid/solid systems
,”
J. Fluid Mech.
334
,
211
249
(
1997
).
17.
L. M.
Hocking
, “
Rival contact-angle models and the spreading of drops
,”
J. Fluid Mech.
239
,
671
681
(
1992
).
18.
T.
Lindner-Silwester
and
W.
Schneider
, “
The moving contact line with weak viscosity effects–an application and evaluation of Shikhmurzaev's model
,”
Acta Mech.
176
,
245
258
(
2005
).
19.
Y. D.
Shikhmurzaev
, “
Remark on ‘The moving contact line with weak viscosity effects—an application and evaluation of Shikhmurzaev's model' by T. Lindner-Silwester and W. Schneider (Acta Mech. 176, 245–258, 2005)
,”
Acta Mech.
182
,
141
143
(
2006
).
20.
T.
Lindner-Silwester
and
W.
Schneider
, “
Authors' reply to the Remark on ‘The moving contact line with weak viscosity effects—an application and evaluation of Shikhmurzaev's model’
,”
Acta Mech.
182
,
145
146
(
2006
).
21.
J.
Eggers
and
R.
Evans
, “
Comment on ‘Dynamic wetting by liquids of different viscosity,' by T. D. Blake and Y. D. Shikhmurzaev
,”
J. Colloid Interface Sci.
280
,
537
538
(
2004
).
22.
Y. D.
Shikhmurzaev
and
T. D.
Blake
, “
Response to the comment on [J. Colloid Interface Sci. 253 (2002) 196] by J. Eggers and R. Evans
,”
J. Colloid Interface Sci.
280
,
539
541
(
2004
).
23.
D.
Bedeaux
, “
Nonequilibrium thermodynamic description of the three-phase contact line
,”
J. Chem. Phys.
120
,
3744
3748
(
2004
).
24.
J.
Billingham
, “
On a model for the motion of a contact line on a smooth solid surface
,”
Eur. J. Appl. Math.
17
,
347
382
(
2006
).
25.
Y. D.
Shikhmurzaev
, “
Some dry facts about dynamic wetting
,”
Eur. Phys. J. Spec. Top.
197
,
47
60
(
2011
).
26.
J. R.
Henderson
, “
Discussion Notes on ‘Some dry facts about dynamic wetting,' by Y. D. Shikhmurzaev
,”
Eur. Phys. J. Spec. Top.
197
,
61
62
(
2011
).
27.
L. M.
Pismen
, “
Discussion Notes on ‘Some dry facts about dynamic wetting,' by Y. D. Shikhmurzaev
,”
Eur. Phys. J. Spec. Top.
197
,
63
65
(
2011
).
28.
U.
Thiele
, “
Discussion Notes: Thoughts on mesoscopic continuum models
,”
Eur. Phys. J. Spec. Top.
197
,
67
71
(
2011
).
29.
Y. D.
Shikhmurzaev
, “
Discussion notes
,”
Eur. Phys. J. Spec. Top.
197
,
73
74
(
2011
).
30.
Y. D.
Shikhmurzaev
, “
Discussion Notes on ‘Some singular errors near the contact line singularity, and ways to resolve both,' by L. M. Pismen
,”
Eur. Phys. J. Spec. Top.
197
,
75
80
(
2011
).
31.
Y.
Pomeau
, “
Discussion Notes: More (and last remarks) on the debate on capillarity
,”
Eur. Phys. J. Spec. Top.
197
,
81
83
(
2011
).
32.
Y. D.
Shikhmurzaev
, “
Discussion Notes: On capillarity and slightly beyond
,”
Eur. Phys. J. Spec. Top.
197
,
85
87
(
2011
).
33.
Y. D.
Shikhmurzaev
, “
Discussion Notes on ‘Disjoining pressure of planar adsorbed films,' by J. R. Henderson
,”
Eur. Phys. J. Spec. Top.
197
,
125
127
(
2011
).
34.
U.
Thiele
, “
Note on thin film equations for solutions and suspensions
,”
Eur. Phys. J. Spec. Top.
197
,
213
220
(
2011
).
35.
Y. D.
Shikhmurzaev
, “
Discussion Notes on ‘Note on thin film equations for solutions and suspensions,' by U. Thiele
,”
Eur. Phys. J. Spec. Top.
197
,
221
225
(
2011
).
36.
T. D.
Blake
, “
Discussion notes: A more collaborative approach to the moving contact-line problem?
Eur. Phys. J. Spec. Top.
197
,
343
345
(
2011
).
37.
J.
Billingham
, “
Gravity-driven thin-film flow using a new contact line model
,”
IMA J. Appl. Math.
73
,
4
36
(
2008
).
38.
E. B.
Dussan V.
and
S. H.
Davis
, “
On the motion of a fluid-fluid interface along a solid surface
,”
J. Fluid Mech.
65
,
71
95
(
1974
).
39.
Q.
Chen
,
E.
Ramé
, and
S.
Garoff
, “
Experimental studies on the parametrization of liquid spreading and dynamic contact angles
,”
Colloids Surf., A
116
,
115
124
(
1996
).
40.
Q.
Chen
,
E.
Ramé
, and
S.
Garoff
, “
The velocity field near moving contact lines
,”
J. Fluid Mech.
337
,
49
66
(
1997
).
41.
M. J.
Savelski
,
S. A.
Shetty
,
W. B.
Kolb
, and
R. L.
Cerro
, “
Flow patterns associated with the steady movement of a solid/liquid/fluid contact line
,”
J. Colloid Interface Sci.
176
,
117
127
(
1995
).
42.
Y. D.
Shikhmurzaev
, “
Singularity of free-surface curvature in convergent flow: Cusp or corner?
Phys. Lett. A
345
,
378
385
(
2005
).
43.
L. M.
Pismen
, “
Diffuse-interface effects near a cusp singularity on a free surface
,”
Phys. Rev. E
70
,
051604
(
2004
).
44.
Y. D.
Shikhmurzaev
, “
Spreading of drops on solid surfaces in a quasi-static regime
,”
Phys. Fluids
9
,
266
275
(
1997
).
45.
Y. D.
Shikhmurzaev
, “
Capillary breakup of liquid threads: a singularity-free solution
,”
IMA J. Appl. Math.
70
,
880
907
(
2005
).
46.
Y. D.
Shikhmurzaev
, “
Macroscopic mechanism of rupture of free liquid films
,”
C. R. Mec.
333
,
205
210
(
2005
).
47.
T. D.
Blake
,
M.
Bracke
, and
Y. D.
Shikhmurzaev
, “
Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle
,”
Phys. Fluids
11
,
1995
2007
(
1999
).
48.
S. P.
Decent
, “
Hydrodynamic assist and the dynamic contact angle in the coalescence of liquid drops
,”
IMA J. Appl. Math.
71
,
740
767
(
2006
).
49.
S. P.
Decent
, “
The spreading of a viscous microdrop on a solid surface
,”
Microfluid. Nanofluid.
2
,
537
549
(
2006
).
50.
J. E.
Sprittles
and
Y. D.
Shikhmurzaev
, “
Viscous flow over a chemically patterned surface
,”
Phys. Rev. E
76
,
021602
(
2007
).
51.
L. M.
Hocking
, “
The spreading of a thin drop by gravity and capillarity
,”
Q. J. Mech. Appl. Math.
36
,
55
69
(
1983
).
52.
P. J.
Haley
and
M. J.
Miksis
, “
The effect of the contact line on droplet spreading
,”
J. Fluid Mech.
223
,
57
81
(
1991
).
53.
J. C.
Slattery
,
L.
Sagis
, and
E. -S.
Oh
,
Interfacial Transport Phenomena
(
Springer
,
New York
,
2007
).
54.
P. G.
de Gennes
,
F.
Brochard-Wyart
, and
D.
Quéré
,
Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves
(
Springer
,
New York
,
2004
).
55.
J. R.
Henderson
, “
Statistical mechanics of fluids adsorbed in planar wedges: Finite contact angle
,”
Phys. Rev. E
69
,
061613
(
2004
).
56.
A.
Pereira
and
S.
Kalliadasis
, “
Equilibrium gas-liquid-solid contact angle from density-functional theory
,”
J. Fluid Mech.
692
,
53
77
(
2012
).
57.
G. K.
Batchelor
,
An Introduction to Fluid Dynamics
(
Cambridge University Press
,
Cambridge
,
2000
).
58.
A.
Pereira
and
S.
Kalliadasis
, “
On the transport equation for an interfacial quantity
,”
Eur. Phys. J. Appl. Phys.
44
,
211
214
(
2008
).
59.
J.
Monnier
and
P.
Witomski
, “
Analysis of a local hydrodynamic model with marangoni effect
,”
J. Sci. Comput.
21
,
369
403
(
2004
).
60.
S.
Kalliadasis
,
C.
Ruyer-Quil
,
B.
Scheid
, and
M. G.
Velarde
,
Falling Liquid Films
,
Springer Series on Applied Mathematical Sciences
Vol.
176
(
Springer
,
London
,
2012
).
61.
N.
Savva
and
S.
Kalliadasis
, “
Two-dimensional droplet spreading over topographical substrates
,”
Phys. Fluids
21
,
092102
(
2009
).
62.
L. H.
Tanner
, “
The spreading of silicone oil drops on horizontal surfaces
,”
J. Phys. D: Appl. Phys.
12
,
1473
(
1979
).
63.
G.
McHale
,
M. I.
Newton
,
S. M.
Rowan
, and
M.
Banerjee
, “
The spreading of small viscous stripes of oil
,”
J. Phys. D: Appl. Phys.
28
,
1925
1929
(
1995
).
64.
J.
Eggers
, “
Toward a description of contact line motion at higher capillary numbers
,”
Phys. Fluids
16
,
3491
3494
(
2004
).
65.
J. E.
Sprittles
and
Y. D.
Shikhmurzaev
, “
Viscous flow in domains with corners: Numerical artifacts, their origin and removal
,”
Comput. Methods Appl. Mech. Eng.
200
,
1087
1099
(
2011
).
66.
J.
Kierzenka
and
L. F.
Shampine
, “
A BVP solver based on residual control and the Matlab PSE
,”
ACM Trans. Math. Softw.
27
,
299
316
(
2001
).
67.
E.
Ruckenstein
and
C. S.
Dunn
, “
Slip velocity during wetting of solids
,”
J. Colloid Interface Sci.
59
,
135
138
(
1977
).
68.
R.
Vellingiri
,
N.
Savva
, and
S.
Kalliadasis
, “
Droplet spreading on chemically heterogeneous substrates
,”
Phys. Rev. E
84
,
036305
(
2011
).
69.
N.
Savva
,
S.
Kalliadasis
, and
G. A.
Pavliotis
, “
Two-dimensional droplet spreading over random topographical substrates
,”
Phys. Rev. Lett.
104
,
084501
(
2010
).
70.
N.
Savva
,
G. A.
Pavliotis
, and
S.
Kalliadasis
, “
Contact lines over random topographical substrates. Part 1. Statics
,”
J. Fluid Mech.
672
,
358
383
(
2011
).
71.
N.
Savva
,
G. A.
Pavliotis
, and
S.
Kalliadasis
, “
Contact lines over random topographical substrates. Part 2. Dynamics
,”
J. Fluid Mech.
672
,
384
410
(
2011
).
72.
T. D.
Blake
and
Y. D.
Shikhmurzaev
, “
Dynamic wetting by liquids of different viscosity
,”
J. Colloid Interface Sci.
253
,
196
202
(
2002
).
You do not currently have access to this content.