The study focuses on the numerical evolution of a droplet, which hangs from a horizontal plane and moves due to thermocapillary effects. It is assumed that the liquid completely wets the substrate, that the surface tension of the liquid decreases linearly with temperature, that the imposed thermal gradient on the substrate is uniform, and that heat transport within the droplet is such that the temperature of its surface replicates that of the substrate. These assumptions, along with the lubrication approximation, allow for obtaining a differential equation that governs the evolution of the droplet. By introducing appropriate scales, this equation has a single dimensionless parameter, which expresses the ratio of gravitational to thermocapillary forces. Numerical solutions show that at sufficiently large volumes or weak thermal gradients, the droplet moves while maintaining a steady, slightly decreasing its volume, and leaving behind a tail whose width is uniform. By contrast, if the droplet is small or the thermal gradient is strong, it advances and stretches in the direction of movement.

1.
S. J.
Weinstein
and
K. J.
Ruschak
, “
Coating flows
,”
Annu. Rev. Fluid Mech.
36
,
29
(
2004
).
2.
R. V.
Craster
and
O. K.
Matar
, “
Dynamics and stability of thin liquid films
,”
Rev. Mod. Phys.
81
,
1131
(
2009
).
3.
A. A.
Darhuber
,
J. P.
Valentino
,
J. M.
Davis
,
S. M.
Troian
, and
S.
Wagner
, “
Microfluidic actuation by modulation of surface stresses
,”
Appl. Phys. Lett
82
,
657
(
2003
).
4.
M. K.
Chaudhury
and
G. M.
Whitesides
, “
How to make water run uphill
,”
Science
256
,
1539
(
1992
).
5.
Q.
Dai
,
W.
Huang
,
X.
Wang
, and
M. M.
Khonsari
, “
Ringlike migration of a droplet propelled by an omnidirectional thermal gradient
,”
Langmuir
34
,
3806
(
2018
).
6.
A.
Dominguez Torres
,
J. R.
Mac Intyre
,
J. M.
Gomba
,
C. A.
Perazzo
,
P. G.
Correa
,
A.
Lopez-Villa
, and
A.
Medina
, “
Contact line motion in axial thermocapillary outward flow
,”
J. Fluid Mech.
892
,
A8
(
2020
).
7.
J. M.
Gomba
and
G. M.
Homsy
, “
Regimes of thermocapillary migration of droplets under partial wetting conditions
,”
J. Fluid Mech.
647
,
125
(
2010
).
8.
J. R.
Mac Intyre
,
J. M.
Gomba
,
C. A.
Perazzo
,
P. G.
Correa
, and
M.
Sellier
, “
Thermocapillary migration of droplets under molecular and gravitational forces
,”
J. Fluid Mech.
847
,
1
(
2018
).
9.
F.
Brochard
, “
Motions of droplets on solid surfaces induced by chemical or thermal gradients
,”
Langmuir
5
,
432
(
1989
).
10.
R.
Chebbi
, “
Dynamics of thermocapillary-driven motion of liquid drops
,”
ACS Omega
8
,
37196
(
2023
).
11.
M. L.
Ford
and
A.
Nadim
, “
Thermocapillary migration of an attached drop on a solid surface
,”
Phys. Fluids
6
,
3183
(
1994
).
12.
M. K.
Smith
, “
Thermocapillary migration of a two-dimensional liquid droplet on a solid surface
,”
J. Fluid Mech.
294
,
209
(
1995
).
13.
J. R.
Mac Intyre
,
J. M.
Gomba
,
C. A.
Perazzo
,
P. G.
Correa
, and
M.
Sellier
, in
IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics
, edited by
S.
Gutschmidt
,
J. N.
Hewett
, and
M.
Sellier
(
Springer International Publishing
,
Cham
,
2019
), pp.
85
95
.
14.
H.-B.
Nguyen
and
J.-C.
Chen
, “A numerical study of thermocapillary migration of a small liquid droplet on a horizontal solid surface,”
Phys. Fluids
22,
062102
.
15.
Y.
Sui
, “
Moving towards the cold region or the hot region? Thermocapillary migration of a droplet attached on a horizontal substrate
,”
Phys. Fluids
26
,
092102
(
2014
).
16.
Q.
Dai
,
M. M.
Khonsari
,
C.
Shen
,
W.
Huang
, and
X.
Wang
, “
Thermocapillary migration of liquid droplets induced by a unidirectional thermal gradient
,”
Langmuir
32
,
7485
(
2016
).
17.
G.
Karapetsas
,
N. T.
Chamakos
, and
A. G.
Papathanasiou
, “
Thermocapillary droplet actuation: Effect of solid structure and wettability
,”
Langmuir
33
,
10838
(
2017
).
18.
J.-X.
Wang
,
F.-Y.
Zhang
,
S.-Y.
Li
,
Y.-P.
Cheng
,
W.-C.
Yan
,
F.
Wang
,
J.-L.
Xu
, and
Y.
Sui
, “
Numerical studies on the controlled thermocapillary migration of a sessile droplet
,”
Ind. Eng. Chem. Res.
62
,
18792
(
2023
).
19.
A.
Oron
,
S. H.
Davis
, and
S. G.
Bankoff
, “
Long-scale evolution of thin liquid films
,”
Rev. Mod. Phys.
69
,
931
(
1997
).
20.
L. W.
Schwartz
and
R. R.
Eley
, “
Simulation of droplet motion on low-energy and heterogeneous surfaces
,”
J. C. Int. Sci
202
,
173
(
1998
).
21.
M. N.
Popescu
,
G.
Oshanin
,
S.
Dietrich
, and
A.-M.
Cazabat
, “
Precursor films in wetting phenomena
,”
J. Phys.: Condens. Matter
24
,
243102
(
2012
).
22.
A.
Hoang
and
H. P.
Kavehpour
, “
Dynamics of nanoscale precursor film near a moving contact line of spreading drops
,”
Phys. Rev. Lett.
106
,
254501
(
2011
).
23.
M. A.
Spaid
and
G. M.
Homsy
, “
Stability of Newtonian and viscoelastic dynamic contact lines
,”
Phys. Fluids
8
,
460
(
1996
).
24.
C. A.
Perazzo
and
J.
Gratton
, “
Bounds of waiting-time in nonlinear diffusion
,”
Appl. Math. Lett.
17
,
1253
(
2004
).
25.
C. A.
Perazzo
and
J.
Gratton
, “
Navier–Stokes solutions for parallel flow in rivulets on an inclined plane
,”
J. Fluid Mech.
507
,
367
379
(
2004
).
26.
C. A.
Perazzo
and
J.
Gratton
, “
Thin film of non-Newtonian fluid on an incline
,”
Phys. Rev. E
67
,
016307
(
2003
).
27.
J. F.
Padday
and
A. R.
Pitt
, “The stability of axisymmetric menisci,”
Proc. R. Soc. London, Ser. A
275
,
489
(
1973
).
28.
A.
Kumar
,
M. R.
Gunjan
,
K.
Jakhar
,
A.
Thakur
, and
R.
Raj
, “
Unified framework for mapping shape and stability of pendant drops including the effect of contact angle hysteresis
,”
Colloids Surf., A
597
,
124619
(
2020
).
29.
P.
Ehrhard
and
S. H.
Davis
, “
Non-isothermal spreading of liquid drops on horizontal plates
,”
J. Fluid Mech.
229
,
365
388
(
1991
).
You do not currently have access to this content.