In this paper, the drainage and subsequent rebound of a liquid column in a cylindrical tube is examined experimentally and theoretically. When liquid is drawn up into a capillary and then released under gravity, inertia allows the meniscus to overshoot the equilibrium capillary rise height. The meniscus then rebounds up the tube, again overshooting the equilibrium height and undergoes oscillation. By varying both the immersion depth and radius of the tube, one can observe rich dynamical behavior, with the most dramatic being the formation of a fast liquid jet, barely visible to the naked eye but easily captured with high-speed video. In addition to the flow separation caused by the sudden expansion at the end of the tube, this jet serves as a mechanism of energy dissipation. Some qualitative differences between the works of Quere et al. [“Rebounds in a capillary tube,” Langmuir 15, 3679–3682 (1999)] and Lorenceau et al. [“Gravitational oscillations of a liquid column in a pipe,” Phys. Fluids 14(6), 1985–1992 (2002)] and the present experiment are observed and discussed. A critical condition for oscillatory behavior is derived theoretically and matches well with the experimental observation. Once in the oscillatory regime, both the maximum depth below and the maximum rebound height above the equilibrium level are investigated by performing a systematic sweep through the relevant parameter space, incorporating the initial meniscus height, immersion depth, tube radius, and fluid properties. Lastly, the characteristic period of oscillation, tp, is assessed and found to be largely independent of fluid viscosity, and could be reasonably well-collapsed by a single curve whereby tphi, where hi is the tube immersion depth.

1.
R.
Bergmann
,
D.
van der Meer
,
S.
Gekle
,
A.
van der Bos
, and
D.
Lohse
, “
Controlled impact of a disk on a water surface: Cavity dynamics
,”
J. Fluid Mech.
633
,
381
409
(
2009
).
2.
H.
Cao
,
C.
Amador
,
X.
Jia
, and
Y.
Ding
, “
Capillary dynamics of water/ethanol mixtures
,”
Ind. Eng. Chem. Res.
54
,
12196
12203
(
2015
).
3.
S.
Das
and
S. K.
Mitra
, “
Different regimes in vertical capillary filling
,”
Phys. Rev. E
87
,
063005
(
2013
).
4.
E.
Ghabache
,
T.
Seon
, and
A.
Antkowiak
, “
Liquid jet eruption from hollow relaxation
,”
J. Fluid Mech.
761
,
206
219
. (
2014
).
5.
S.-L.
Lee
and
H.-D.
Lee
, “
Evolution of liquid meniscus shape in a capillary tube
,”
J. Fluids Eng.
129
,
957
965
(
2007
).
6.
S.
Levine
,
J.
Lowndes
,
E. J.
Watson
, and
G.
Neale
, “
A theory of capillary rise of a liquid in a vertical cylindrical tube and in a parallel-plate channel
,”
J. Colloid Interface Sci.
73
,
136
151
(
1980
).
7.
E.
Lorenceau
,
D.
Quere
,
J.-Y.
Ollitraut
, and
C.
Clanet
, “
Gravitational oscillations of a liquid column in a pipe
,”
Phys. Fluids
14
(
6
),
1985
1992
(
2002
).
8.
J. O.
Marston
,
J. P. K.
Seville
,
Y.-V.
Cheun
,
A.
Ingram
,
S. P.
Decent
, and
M. J. H.
Simmons
, “
Effect of packing fraction on granular jetting from solid sphere entry into aerated and fluidized beds
Phys. Fluids
20
,
023301
(
2008
).
9.
R.
Masoodi
,
E.
Languri
, and
A.
Ostadhossein
, “
Dynamics of liquid rise in a vertical capillary tube
,”
J. Colloid Interface Sci.
389
,
268
272
(
2013
).
10.
D.
Quere
,
E.
Raphael
, and
J.-Y.
Ollitrault
, “
Rebounds in a capillary tube
,”
Langmuir
15
,
3679
3682
(
1999
).
11.
J. R.
Royer
,
E. I.
Corwin
,
A.
Flior
,
M.-L.
Cordero
,
M. L.
Rivers
,
P. J.
Eng
, and
H. M.
Jaeger
, “
Formation of granular jets observed by high-speed X-ray radiography
,”
Nat. Phys.
1
,
164
(
2005
).
12.
H. E.
Schulz
and
D. Z.
Zhu
, “
Oscillations of water levels in vertical semi-immersed tubes: Analytical solutions and experimental verification
,”
J. Appl. Fluid Mech.
10
(
6
),
1515
1525
(
2017
).
13.
P.
Wu
,
H.
Zhang
,
A.
Nikolov
, and
D.
Wasan
, “
Rise of the main meniscus in rectangular capillaries: Experiments and modeling
,”
J. Colloid Interface Sci.
461
,
195
202
(
2016
).
14.
B. V.
Zhmud
,
F.
Tiberg
, and
K.
Hallstensson
, “
Dynamics of capillary rise
,”
J. Colloid Interface Sci.
228
,
263
269
(
2000
).

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