The unsteady axisymmetric problem of a liquid drop impacting onto a rigid vibrating substrate is studied. Initially, the drop is spherical and touches the flat substrate at a single point. Then, the substrate starts to move toward the drop and vibrate with a small amplitude and high frequency. The early stage of the impact is studied by using the potential flow theory and the Wagner approach in dimensionless variables. The effect of the substrate vibration on the drop impact is described by a single parameter. It is shown that the vibration of the substrate leads to oscillations of the pressure in the contact region. The low-pressure zone periodically appears in the wetted part of the substrate. The low-pressure zone can approach the contact line, which may lead to ventilation with separation of the liquid from the substrate. The magnitude of the low pressure grows in time. The acceleration of the contact line oscillates with time, leading to splashing of the droplet with the local increase of the thickness of the spray jet sheet at a distance from the contact line. The phase shift of the substrate vibration with respect to the impact instant is not studied. Splashing can be produced only by a forced vibration of the substrate. The impact onto an elastically supported rigid plate does not produce splashing. The obtained results and the theoretical model of the initial stage of drop impact are valid for certain ranges of parameters of the problem.

1.
Courbin
,
L.
,
Bird
,
J. C.
, and
Stone
,
H. A.
, “
Splash and anti-splash: Observation and design
,”
Chaos
16
(
4
),
041102
(
2006
).
2.
Chen
,
J. H.
, “
Characteristics of drop impact on elastic and compliant surfaces
,”
J. Mar. Sci. Technol.
13
,
156
161
(
2005
).
3.
Faltinsen
,
O. M.
, “
Hydroelastic slamming
,”
J. Mar. Sci. Technol.
5
(
2
),
49
65
(
2000
).
4.
James
,
A. J.
,
Vukasinovic
,
B.
,
Smith
,
M. K.
, and
Glezer
,
A.
, “
Vibration-induced drop atomization and bursting
,”
J. Fluid Mech.
476
,
1
28
(
2003
).
5.
Howison
,
S. D.
,
Ockendon
,
J. R.
, and
Wilson
,
S. K.
, “
Incompressible water-entry problems at small deadrise angles
,”
J. Fluid Mech.
222
(
1
),
215
230
(
1991
).
6.
Howland
,
C. J.
,
Antkowiak
,
A.
,
Castrejon-Pita
,
J. R.
,
Howison
,
S. D.
,
Oliver
,
J. M.
,
Style
,
R. W.
, and
Castrejon-Pita
,
A. A.
, “
It’s harder to splash on soft solids
,”
Phys. Rev. Lett.
117
(
18
),
184502
(
2016
).
7.
Khabakhpasheva
,
T. I.
and
Korobkin
,
A. A.
, “
Liquid drop impact on a vibrating substrate
,” in
Proceedings of 3rd International Conference on Violent Flows (VF-2016)
,
Osaka, Japan
,
9–11 March 2016
.
8.
Khabakhpasheva
,
T. I.
and
Korobkin
,
A. A.
, “
Oblique elastic plate impact on thin liquid layer
,”
Phys. Fluids
32
(
6
),
062101
(
2020
).
9.
Korobkin
,
A. A.
,
Liquid-Solid Impact
(
Siberian Branch of the Russian Academy of Sciences
,
Novosibirsk
,
1997
).
10.
Korobkin
,
A. A.
, “
Wave impact on the center of an Euler beam
,”
J. Appl. Mech. Tech. Phys.
39
(
5
),
134
147
(
1998
).
11.
Korobkin
,
A. A.
and
Khabakhpasheva
,
T. I.
, “
Regular wave impact onto an elastic plate
,”
J. Eng. Math.
55
(
1-4
),
127
150
(
2006
).
12.
Korobkin
,
A.
and
Khabakhpasheva
,
T.
, “
Effect of vibrations on liquid drop impact
,” in
Book of Abstracts BAMC, Oxford, United Kingdom, 5–8 April 2016
p.
139
, available at https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxiYW1jMjAxNm94Zm9yZHxneDoyOTM2ZGE5Yjk2OWJhNjE3.
13.
Korobkin
,
A. A.
,
Khabakhpasheva
,
T. I.
, and
Maki
,
K. J.
, “
Hydrodynamic forces in water exit problems
,”
J. Fluids Struct.
69
,
16
33
(
2017
).
14.
Korobkin
,
A. A.
,
Khabakhpasheva
,
T. I.
, and
Wu
,
G. X.
, “
Coupled hydrodynamic and structural analysis of compressible jet impact onto elastic panels
,”
J. Fluids Struct.
24
(
7
),
1021
1041
(
2008
).
15.
Korobkin
,
A. A.
and
Pukhnachov
,
V. V.
, “
Initial stage of water impact
,”
Annu. Rev. Fluid Mech.
20
(
1
),
159
185
(
1988
).
16.
Korobkin
,
A. A.
and
Scolan
,
Y.-M.
, “
Three-dimensional theory of water impact. Part 2. Linearized Wagner problem
,”
J. Fluid Mech.
549
,
343
373
(
2006
).
17.
Moradi
,
M.
,
Rahimian
,
M. H.
, and
Chini
,
S. F.
, “
Numerical simulation of droplet impact on vibrating low-adhesion surfaces
,”
Phys. Fluids
32
(
6
),
062110
(
2020
).
18.
Oliver
,
J. M.
, “
Water entry and related problems
,” Ph.D. thesis,
University of Oxford
,
2002
.
19.
Pegg
,
M.
,
Purvis
,
R.
, and
Korobkin
,
A. A.
, “
Droplet impact onto an elastic plate: A new mechanism for splashing
,”
J. Fluid Mech.
839
,
561
(
2018
).
20.
Pepper
,
R. E.
,
Courbin
,
L.
, and
Stone
,
H. A.
, “
Splashing on elastic membranes: The importance of early-time dynamics
,”
Phys. Fluids
20
(
8
),
082103
(
2008
).
21.
Rein
,
M.
, “
Phenomena of liquid drop impact on solid and liquid surfaces
,”
Fluid Dyn. Res.
12
(
2
),
61
(
1993
).
22.
Scolan
,
Y.-M.
and
Korobkin
,
A. A.
, “
Energy distribution from vertical impact of a three-dimensional solid body onto the flat free surface of an ideal fluid
,”
J. Fluids Struct.
17
(
2
),
275
286
(
2003
).
23et al..
Tsai
,
P. H.
,
Wang
,
C. H.
,
Wang
,
A. B.
,
Korobkin
,
A.
,
Purvis
,
R.
, and
Khabakhpasheva
,
T.
, “
Investigation of droplet oscillation on a vibrating elastic plate
,” in
13th Asian Symposium on Visualization, Novosibirsk, Russia, June 22-26 2015
(
Parallel, Novosibirsk
,
2015
), p.
9
.
24.
Wagner
,
H.
, “
Über stoß- und gleitvorgänge an der oberfläche von flüssigkeiten
,”
ZAMM
12
(
4
),
193
215
(
1932
).
25.
Yarin
,
A. L.
, “
Drop impact dynamics: Splashing, spreading, receding, bouncing...
,”
Annu. Rev. Fluid Mech.
38
,
159
192
(
2006
).
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