This theoretical study proposes an analytical model to predict the maximum spread of single droplets on solid surfaces with zero or low Weber and Reynolds numbers. The spreading droplet is assumed as a spherical cap considering low impact velocities. Three spreading states are considered, which include equilibrium spread, maximum spontaneous spread, and maximum spread. Energy conservation is applied to the droplet as a control volume. The model equation contains two viscous dissipation terms, each of which has a defined coefficient. One term is for viscous dissipation in spontaneous spreading and the other one is for viscous dissipation of the initial kinetic energy of the droplet. The new model satisfies the fundamental physics of drop-surface interaction and can be used for droplets impacting on solid surfaces with or without initial kinetic energy.

1.
Worthington
,
A. M.
, 1877, “
On the Forms Assumed by Drops of Liquids Falling Vertically on a Horizontal Plate
,”
Proc. R. Soc. London
0370-1662,
25
, pp.
261
272
.
2.
Madejski
,
J.
, 1976, “
Solidification of Droplets on a Cold Surface
,”
Int. J. Heat Mass Transfer
0017-9310,
19
, pp.
1009
1013
.
3.
Bechtel
,
S. E.
,
Bogy
,
D. B.
, and
Talke
,
F. E.
, 1981, “
Impact of a liquid Drop Against a Flat Surface
,”
IBM J. Res. Dev.
0018-8646,
25
(
6
), pp.
963
971
.
4.
Delplanque
,
J. -P.
, and
Rangel
,
R. H.
, 1997, “
An Improved Model for Droplet Solidification on a Flat Surface
,”
J. Mater. Sci.
0022-2461,
32
, pp.
1519
1530
.
5.
Kim
,
H. -Y.
, and
Chun
,
J. -H.
, 2001, “
The Recoiling of Liquid Droplets Upon Collision With Solid Surfaces
,”
Phys. Fluids
1070-6631,
13
(
3
), pp.
643
659
.
6.
Roisman
,
I. V.
,
Rioboo
,
R.
, and
Tropea
,
C.
, 2002, “
Normal Impact of a Liquid Drop on a Dry Surface: Model for Spreading and Receding
,”
Proc. R. Soc. London, Ser. A
0950-1207,
458
, pp.
1411
1430
.
7.
Asai
,
A.
,
Shioya
,
M.
,
Hirasawa
,
S.
, and
Okazaki
,
T.
, 1993, “
Impact of an Ink Drop on Paper
,”
J. Imaging Sci. Technol.
1062-3701,
37
, pp.
205
207
.
8.
Scheller
,
B.
, and
Bousfield
,
D. W.
, 1995, “
Newtonian Drop Impact With Solid Surface
,”
AIChE J.
0001-1541,
41
, pp.
1357
1367
.
9.
Chandra
,
S.
, and
Avedisian
,
C. T.
, 1991, “
On the Collision of a Droplet With a Solid Surface
,”
Proc. R. Soc. London, Ser. A
0950-1207,
432
, pp.
13
41
.
10.
Pasandideh-Fard
,
M.
,
Qiao
,
Y. M.
,
Chandra
,
S.
, and
Mostaghimi
,
J.
, 1996, “
Capillary Effects During Droplet Impact on a Solid Surface
,”
Phys. Fluids
1070-6631,
8
(
3
), pp.
650
659
.
11.
Mao
,
T.
,
Kuhn
,
C. S. D.
, and
Tran
,
H.
, 1997, “
Spread and Rebound of Liquid Droplets Upon Impact on Flat Surfaces
,”
AIChE J.
0001-1541,
43
(
9
), pp.
2169
2179
.
12.
Park
,
H.
,
Carr
,
W. W.
,
Zhu
,
J. Y.
, and
Morris
,
J. F.
, 2003, “
Single Drop Impaction on a Solid Surface
,”
AIChE J.
0001-1541,
49
, pp.
2461
2471
.
13.
Fukai
,
J.
,
Tanaka
,
M.
, and
Miyatake
,
O.
, 1998, “
Maximum Spreading of Liquid Droplets Colliding With Flat Surfaces
,”
J. Chem. Eng. Jpn.
0021-9592,
31
, pp.
456
461
.
14.
Bennett
,
T.
, and
Poulikakos
,
D.
, 1993, “
Splat-Quench Solidification: Estimating the Maximum Spreading of a Droplet Impacting a Solid Surface
,”
J. Mater. Sci.
0022-2461,
28
, pp.
963
970
.
15.
Gong
,
S. -C.
, 2005, “
Spreading of Droplets Impacting on Smooth Solid Surface
,”
Jpn. J. Appl. Phys., Part 1
0021-4922,
44
(
5A
), pp.
3323
3324
.
16.
Bergeron
,
V. V.
,
Bonn
,
D.
,
Martin
,
J. Y.
, and
Vovelle
,
L.
, 2000, “
Controlling Droplet Deposition With Polymer Additives
,”
Nature (London)
0028-0836,
405
(
6788
), pp.
772
775
.
17.
Sikalo
,
S.
,
Marengo
,
M.
,
Tropea
,
C.
, and
Ganic
,
E. N.
, 2002, “
Analysis of Impact of Droplets on Horizontal Surfaces
,”
Exp. Therm. Fluid Sci.
0894-1777,
25
, pp.
503
510
.
18.
van Dam
,
D. B.
, and
Clerc
,
C. L.
, 2004, “
Experimental Study of the Impact of an Ink-Jet Printed Droplet on a Solid Substrate
,”
Phys. Fluids
1070-6631,
16
, pp.
3403
3414
.
19.
Clanet
,
C.
,
Béguin
,
C.
,
Richard
,
D.
, and
Quéré
,
D.
, 2004, “
Maximal Deformation of an Impacting Drop
,”
J. Fluid Mech.
0022-1120,
517
, pp.
199
208
.
20.
Dong
,
H.
,
Carr
,
W. W.
,
Bucknall
,
D. G.
, and
Morris
,
J. F.
, 2007, “
Temporally-Resolved Inkjet Drop Impaction on Surfaces
,”
AIChE J.
0001-1541,
53
(
10
), pp.
2606
2617
.
21.
Schiaffino
,
S.
, and
Sonin
,
A. A.
, 1997, “
Molten Droplet Deposition and Solidification at low Weber Numbers
,”
Phys. Fluids
1070-6631,
9
(
11
), pp.
3172
3187
.
22.
Quéré
,
D.
, 2008, “
Wetting and Roughness
,”
Annu. Rev. Mater. Res.
1531-7331,
38
, pp.
71
99
.
23.
Young
,
T.
, 1805, “
An Essay on the Cohesion of Fluids
,”
Philos. Trans. R. Soc. London, Ser. B
0962-8436,
95
, pp.
65
87
.
You do not currently have access to this content.