The process of coalescence of two identical liquid drops is simulated numerically in the framework of two essentially different mathematical models, and the results are compared with experimental data on the very early stages of the coalescence process reported recently. The first model tested is the “conventional” one, where it is assumed that coalescence as the formation of a single body of fluid occurs by an instant appearance of a liquid bridge smoothly connecting the two drops, and the subsequent process is the evolution of this single body of fluid driven by capillary forces. The second model under investigation considers coalescence as a process where a section of the free surface becomes trapped between the bulk phases as the drops are pressed against each other, and it is the gradual disappearance of this “internal interface” that leads to the formation of a single body of fluid and the conventional model taking over. Using the full numerical solution of the problem in the framework of each of the two models, we show that the recently reported electrical measurements probing the very early stages of the process are better described by the interface formation/disappearance model. New theory-guided experiments are suggested that would help to further elucidate the details of the coalescence phenomenon. As a by-product of our research, the range of validity of different “scaling laws” advanced as approximate solutions to the problem formulated using the conventional model is established.

1.
C. T.
Bellehumeur
,
M. K.
Biaria
, and
J.
Vlachopoulos
, “
An experimental study and model assessment of polymer sintering
,”
Polym. Eng. Sci.
36
,
2198
2207
(
2004
).
2.
T. M.
Dreher
,
J.
Glass
,
A. J.
O'Connor
, and
G. W.
Stevens
, “
Effect of rheology on coalescence rates and emulsion stability
,”
AIChE J.
45
,
1182
1190
(
1999
).
3.
W. M.
Grissom
and
F. A.
Wierum
, “
Liquid spray cooling of a heated surface
,”
Int. J. Heat Mass Transfer
24
,
261
271
(
1981
).
4.
A.
Kovetz
and
B.
Olund
, “
The effect of coalescence and condensation on rain formation in a cloud of finite vertical extent
,”
J. Atmos. Sci.
26
,
1060
1065
(
1969
).
5.
T. M.
Squires
and
S. R.
Quake
, “
Microfluidics: Fluid physics at the nanoliter scale
,”
Rev. Mod. Phys.
77
,
977
1026
(
2005
).
6.
B.
Derby
, “
Inkjet printing of functional and structural materials: Fluid property requirements, feature stability and resolution
,”
Annu. Rev. Mater. Res.
40
,
395
414
(
2010
).
7.
M.
Singh
,
H.
Haverinen
,
P.
Dhagat
, and
G.
Jabbour
, “
Inkjet printing process and its applications
,”
Adv. Mater.
22
,
673
685
(
2010
).
8.
S.
Richardson
, “
Two-dimensional bubbles in slow viscous flows
,”
J. Fluid Mech.
33
,
475
493
(
1968
).
9.
S. T.
Thoroddsen
,
K.
Takehara
, and
T. G.
Etoh
, “
The coalescence speed of a pendent and sessile drop
,”
J. Fluid Mech.
527
,
85
114
(
2005
).
10.
J. D.
Paulsen
,
J. C.
Burton
, and
S. R.
Nagel
, “
Viscous to inertial crossover in liquid drop coalescence
,”
Phys. Rev. Lett.
106
,
114501
(
2011
).
11.
J.
Frenkel
, “
Viscous flow of crystalline bodies under the action of surface tension
,”
J. Phys. (USSR)
9
,
385
391
(
1945
).
12.
R. W.
Hopper
, “
Coalescence of two equal cylinders: Exact results for creeping viscous plane flow driven by capillarity
,”
J. Am. Ceram. Soc.
67
,
262
264
(
1984
).
13.
R. W.
Hopper
, “
Plane Stokes flow driven by capillarity on a free surface
,”
J. Fluid Mech.
213
,
349
375
(
1990
).
14.
R. W.
Hopper
, “
Coalescence of two viscous cylinders by capillarity: Part 1. Theory
,”
J. Am. Ceram. Soc.
76
,
2947
2952
(
1993
).
15.
R. W.
Hopper
, “
Coalescence of two viscous cylinders by capillarity: Part 2. Shape evolution
,”
J. Am. Ceram. Soc.
76
,
2953
2960
(
1993
).
16.
S.
Richardson
, “
Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension
,”
Eur. J. Appl. Math.
3
,
193
207
(
1992
).
17.
J.
Eggers
,
J. R.
Lister
, and
H. A.
Stone
, “
Coalescence of liquid drops
,”
J. Fluid Mech.
401
,
293
310
(
1999
).
18.
L.
Duchemin
,
J.
Eggers
, and
C.
Josserand
, “
Inviscid coalescence of drops
,”
J. Fluid Mech.
487
,
167
178
(
2003
).
19.
A.
Menchaca-Rocha
,
A.
Martínez-Dávalos
,
R.
Núńez
,
S.
Popinet
, and
S.
Zaleski
, “
Coalescence of liquid drops by surface tension
,”
Phys. Rev. E
63
,
046309
(
2001
).
20.
J. D.
Paulsen
,
J. C.
Burton
,
S. R.
Nagel
,
S.
Appathurai
,
M. T.
Harris
, and
O.
Basaran
, “
The inexorable resistance of inertia determines the initial regime of drop coalescence
,”
Proc. Natl. Acad. Sci. U.S.A.
109
,
6857
6861
(
2012
).
21.
H. N.
Oguz
and
A.
Prosperetti
, “
Surface-tension effects in the contact of liquid surfaces
,”
J. Fluid Mech.
203
,
149
171
(
1989
).
22.
A.
Jagota
and
P. R.
Dawson
, “
Micromechanical modeling of powder compacts - I. Unit problems for sintering and traction induced deformation
,”
Acta Metall.
36
,
2551
2561
(
1988
).
23.
J. I.
Martínez-Herrera
and
J. J.
Derby
, “
Viscous sintering of spherical particles via finite element analysis
,”
J. Am. Ceram. Soc.
78
,
645
649
(
1995
).
24.
D. G. A. L.
Aarts
,
H. N. W.
Lekkerkerker
,
H.
Guo
,
G. H.
Wegdam
, and
D.
Bonn
, “
Hydrodynamics of droplet coalescence
,”
Phys. Rev. Lett.
95
,
164503
(
2005
).
25.
M.
Wu
,
T.
Cubaud
, and
C.
Ho
, “
Scaling law in liquid drop coalescence driven by surface tension
,”
Phys. Fluids
16
,
51
54
(
2004
).
26.
It should be pointed out here that the measurements are taken for relatively large bridge radii, so that a comparison with the inertial scaling (2) is valid, but the theory of Ref. 18 is well past its limits of applicability, so that one cannot expect good agreement for the proposed prefactor.
27.
S. C.
Case
and
S. R.
Nagel
, “
Coalescence in low-viscosity liquids
,”
Phys. Rev. Lett.
100
,
084503
(
2008
).
28.
S. C.
Case
, “
Coalescence of low-viscosity fluids in air
,”
Phys. Rev. E
79
,
026307
(
2009
).
29.
J. C.
Burton
,
J. E.
Rutledge
, and
P.
Taborek
, “
Fluid pinch-off dynamics at nanometer length scales
,”
Phys. Rev. Lett.
92
,
244505
(
2004
).
30.
Y. D.
Shikhmurzaev
, “
Coalescence and capillary breakup of liquid volumes
,”
Phys. Fluids
12
,
2386
2396
(
2000
).
31.
D. D.
Joseph
,
J.
Nelson
,
M.
Renardy
, and
Y.
Renardy
, “
Two-dimensional cusped interfaces
,”
J. Fluid Mech.
223
,
383
409
(
1991
).
32.
Y. D.
Shikhmurzaev
, “
Singularity of free-surface curvature in convergent flow: Cusp or corner?
,”
Phys. Lett. A
345
,
378
(
2005
).
33.
Y. D.
Shikhmurzaev
, “
Capillary breakup of liquid threads: A singularity-free solution
,”
IMA J. Appl. Math.
70
,
880
907
(
2005
).
34.
Y. D.
Shikhmurzaev
, “
Macroscopic mechanism of rupture of free liquid films
,”
C. R. Mec.
333
,
205
210
(
2005
).
35.
Y. D.
Shikhmurzaev
, “
The moving contact line on a smooth solid surface
,”
Int. J. Multiphase Flow
19
,
589
610
(
1993
).
36.
Y. D.
Shikhmurzaev
, “
Moving contact lines in liquid/liquid/solid systems
,”
J. Fluid Mech.
334
,
211
249
(
1997
).
37.
Y. D.
Shikhmurzaev
, “
Singularities at the moving contact line. Mathematical, physical and computational aspects
,”
Physica D
217
,
121
133
(
2006
).
38.
Y. D.
Shikhmurzaev
, “
Spreading of drops on solid surfaces in a quasi-static regime
,”
Phys. Fluids
9
,
266
275
(
1996
).
39.
Y. D.
Shikhmurzaev
,
Capillary Flows with Forming Interfaces
(
Chapman and Hall-CRC
,
Boca Raton
,
2007
).
40.
J. E.
Sprittles
and
Y. D.
Shikhmurzaev
, “
Finite element simulation of dynamic wetting flows as an interface formation process
,”
J. Comput. Phys.
233
,
34
65
(
2013
).
41.
T. D.
Blake
and
Y. D.
Shikhmurzaev
, “
Dynamic wetting by liquids of different viscosity
,”
J. Colloid Interface Sci.
253
,
196
202
(
2002
).
42.
T.
Young
, “
An essay on the cohesion of fluids
,”
Philos. Trans. R. Soc. London
95
,
65
87
(
1805
).
43.
J. E.
Sprittles
and
Y. D.
Shikhmurzaev
, “
A finite element framework for describing dynamic wetting phenomena
,”
Int. J. Numer. Methods Fluids
68
,
1257
1298
(
2012
).
44.
J. E.
Sprittles
and
Y. D.
Shikhmurzaev
, “
The dynamics of liquid drops and their interaction with solids of varying wettabilities
,”
Phys. Fluids
24
,
082001
(
2012
).
45.
S. F.
Kistler
, and
L. E.
Scriven
, “
Coating flows
,” in
Computational Analysis of Polymer Processing
, edited by
J. R. A.
Pearson
and
S. M.
Richardson
(
Applied Science
,
London/New York
,
1983
), pp.
243
299
.
46.
M.
Heil
, “
An efficient solver for the fully-coupled solution of large displacement fluid-structure interaction problems
,”
Comput. Methods Appl. Mech. Eng.
193
,
1
23
(
2004
).
47.
M. C. T.
Wilson
,
J. L.
Summers
,
Y. D.
Shikhmurzaev
,
A.
Clarke
, and
T. D.
Blake
, “
Nonlocal hydrodynamic influence on the dynamic contact angle: Slip models versus experiment
,”
Phys. Rev. E
73
,
041606
(
2006
).
48.
P.
Lötstedt
and
L.
Petzold
, “
Numerical solution of nonlinear differential equations with an algebraic constraints. I. Convergence results for backward differentiation formulas
,”
Math. Comput.
46
,
491
516
(
1986
).
49.
P. M.
Gresho
and
R. L.
Sani
,
Incompressible Flow and the Finite Element Method. Volume 2. Isothermal Laminar Flow
(
Wiley
,
New York
,
1999
).
50.
O. A.
Basaran
, “
Nonlinear oscillations of viscous liquid drops
,”
J. Fluid Mech.
241
,
169
198
(
1992
).
51.
We can see this from the data provided to us by Dr. J. D. Paulsen, Dr. J. C. Burton, and Professor S. R. Nagel, which was published in Ref. 10.
52.
Despite the drops in Ref. 9 having a different radius, R = 1.5 mm, and very slightly different viscosity, μ = 220 mPa s, our simulations show that at such high viscosity these alterations can be scaled out by an appropriate choice of viscous time-scale, as we use.
53.
S.
Fordham
, “
On the calculation of surface tension from measurements of pendant drops
,”
Proc. R. Soc. London, Ser. A
194
,
1
16
(
1948
).
54.
More precisely, it is the conventional model as at this stage the interface formation/disappearance dynamics have ended, so that the interfaces are in equilibrium and the interface formation model becomes equivalent to the conventional one.
55.
J.-T.
Jeong
and
H. K.
Moffatt
, “
Free-surface cusps associated with flow at low Reynolds number
,”
J. Fluid Mech.
241
,
1
22
(
1992
).
56.
P.
Attané
,
F.
Girard
, and
V.
Morin
, “
An energy balance approach of the dynamics of drop impact on a solid surface
,”
Phys. Fluids
19
,
012101
(
2007
).
57.
I. S.
Bayer
and
C. M.
Megaridis
, “
Contact angle dynamics in droplets impacting on flat surfaces with different wetting characteristics
,”
J. Fluid Mech.
558
,
415
449
(
2006
).
58.
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
).
59.
T. D.
Blake
,
A.
Clarke
, and
K. J.
Ruschak
, “
Hydrodynamic assist of wetting
,”
AIChE J.
40
,
229
242
(
1994
).
You do not currently have access to this content.