Analytical solutions were obtained for the virtual mass of a Taylor bubble rising in a liquid confined by a circular pipe under transient conditions. The solution of the virtual mass coefficient was based on potential inviscid flow. The present solution is applicable to low viscosity liquids and to Capillary number (Ca)<0.005. The virtual mass solution showed dependence on bubble geometry. The present solution was validated by comparison with the available numerical solutions and experimental data of other investigators.

References

References
1.
Chen
,
Y.
, and
Jepson
,
W. P.
,
1999
, “
EIS Measurement for Corrosion Monitoring Under Multiphase Flow Conditions
,”
Electrochimica Acta
,
44
(24), pp.
4453
4464
.
2.
Prothero
,
J.
, and
Burton
,
A. C.
,
1961
, “
The Physics of Blood Flow in Capillaries—I: The Nature of the Motion
,”
Biophys. J.
,
1
, pp.
565
579
.
3.
James
,
M. R.
,
Lane
,
S. J.
,
Chouet
,
J. S.
, and
Gilbert
,
J. S.
,
2004
, “
Pressure Changes Associated With the Ascent and Bursting of Gas Slugs in Liquid-Filled Vertical and Inclined Conduits
,”
J. Volcanol. Geoth. Res.
,
129
(1–3), pp.
61
82
.
4.
U. S. NRC, 2017, “
Water Hammer at San Onofre
,” United States Nuclear Regulatory Commission, Washington, DC, accessed Dec. 14, 2017, https://www.nrc.gov/docs/ML1121/ML11216A244.pdf
5.
Campos
,
J. B. L. M.
, and
Guedes De Carvalho
,
J. R. F.
,
1988
, “
An Experimental Study of the Wake of Gas Slugs Rising in Liquids
,”
J. Fluid Mech.
,
196
, pp.
27
37
.
6.
Dumitrescu
,
D.
,
1943
, “
Stromung an Einer Luftblase Im Senkrechten
,”
Rohr. Z. Für Angew. Mathematik Und Mechanik.
,
23
(3), pp.
139
149
.
7.
Bretherton
,
F. P.
,
1961
, “
The Motion of Long Bubbles in Tubes
,”
J. Fluid Mech.
,
10
(2), pp.
166
188
.
8.
Batchelor
,
G. K.
,
1967
,
An Introduction to Fluid Dynamics
,
Cambridge University Press
,
Cambridge, UK
.
9.
Pinto
,
A. M. F. R.
, and
Campos
,
J. B. L. M.
,
1996
, “
Coalescence of Two Gas Slugs Rising in a Vertical Column of Liquid
,”
Chem. Eng. Sci.
,
51
(1), pp.
45
54
.
10.
Milne-Thomson
,
L. M.
,
1960
,
Theoretical Hydrodynamics
,
5th ed.
,
Macmillan
,
London
.
11.
Kendoush
,
A. A.
,
2007
, “
The Virtual Mass of an Oblate-Ellipsoidal Bubble
,”
Phys. Lett. A
,
366
(3), pp.
253
255
.
12.
Kendoush
,
A. A.
,
2003
, “
The Virtual Mass of a Spherical-Cap Bubble
,”
Phy. Fluids
,
15
(
9
), pp.
2782
2785.
13.
Kendoush
,
A. A.
,
2004
, “
Erratum: The Virtual Mass of a Spherical-Cap Bubble
,”
Phy. Fluids
,
16
(7), p.
2713
.
14.
Kendoush
,
A. A.
,
2005
, “
The Virtual Mass of a Rotating Sphere in Fluids
,”
ASME J. Appl. Mech.
,
72
(5), pp.
801
802
.
15.
Angeli
,
P.
, and
Gavriilidis
,
A.
,
2008
, “
Hydrodynamics of Taylor Flow in Small Channels: A Review
,”
J. Mech. Eng. Sci.
,
222
(5), pp.
737
751
.
16.
Ha-Ngoc
,
H.
, and
Fabre
,
J.
,
2006
, “
A Boundary Element Method for Calculating the Shape and Velocity of Two-Dimensional Long Bubble in Stagnant and Flowing Liquid
,”
Eng. Anal. Bound. Elem.
,
30
(7), pp.
539
552
.
17.
Chang
,
H. C.
,
2002
, “
Bubble/Drop Transport in Microchannels
,”
The MEMS Handbook
,
Mohamed Gad-el-Hak
, ed.,
CRC Press
,
Boca Raton, FL
, Chap. 11.
18.
Fabre
,
H.
, and
Line
,
A.
,
1992
, “
Modeling of Two-Phase Slug Flow
,”
Annu. Rev. Fluid Mech.
,
24
, pp.
21
46
.
19.
Ramdin
,
M.
, and
Henkes
,
R.
,
2012
, “
Compuational Fluid Dynamics Modeling of Benjamin and Taylor Bubbles in Two-Phase Flow in Pipes
,”
ASME J. Fluids Eng.
,
134
(4), p.
041303
.
20.
Ishii
,
M.
, and
Mishima
,
K.
,
1984
, “
Two-Fluid Model and Hydrodynamic Constitutive Relations
,”
Nucl. Eng. Des.
,
82
(2–3), pp.
107
126
.
21.
Hanau
,
V. D.
, and
Raithby
,
G. D.
,
1995
, “
A Transient Two-Fluid Model for the Simulation of Slug Flow in Pipelines—I: Theory
,”
Int. J. Multiphase Flow
,
21
(
3
), pp.
335
349
.
22.
Anglart
,
H.
, and
Podowski
,
M. Z.
,
2002
, “
Fluid Mechanics of Taylor Bubbles and Slug Flow in Vertical Channels
,”
Nucl. Eng. Des.
,
140
(2), pp.
165
171
.
23.
Bendiksen
,
K. H.
,
1985
, “
On the Motion of Long Bubbles in Vertical Tubes
,”
Int. J. Multiphase Flow
,
11
(6), pp.
797
812
.
24.
Baumbach
,
V.
,
Hopfinger
,
E. J.
, and
Cartelier
,
A.
,
2005
, “
The Transient Behavior of a Large Bubble in a Vertical Tube
,”
J. Fluid Mech.
,
524
, pp.
131
142
.
25.
Taha
,
T.
, and
Cui
,
Z. F.
,
2004
, “
Hydrodynamics of Slug Flow Inside Capillaries
,”
Chem. Eng. Sci.
,
59
(6), pp.
1181
1190
.
26.
Wang
,
Z.
, and
Tong
,
A. Y.
,
2008
, “
Deformation and Oscillations of a Single Gas Bubble Rising in a Narrow Vertical Tube
,”
Int. J. Therm. Sci.
,
47
(3), pp.
221
228
.
27.
Lu
,
X.
, and
Prosperetti
,
A.
,
2006
, “
Axial Stability of Taylor Bubbles
,”
J. Fluid Mech.
,
568
, pp.
173
192
.
28.
Serizawa
,
A.
,
Feng
,
Z.
, and
Zensaku
,
K.
,
2002
, “
Two-Phase Flow in Microchannels
,”
Exp. Therm. Fluid Sci.
,
26
(6–7), pp.
703
714
.
29.
Kendoush
,
A. A.
, and
Al-Khatab
,
S. A. W.
,
1994
, “
Experiments on Flow Characterization in Vertical Downward Two—Phase Flow
,”
Exp. Therm. Fluid Sci.
,
9
(1), pp.
34
38
.
30.
Mandal
,
T. K.
,
Das
,
G.
, and
Das
,
P. K.
,
2009
, “
Liquid Taylor Bubbles Rising in a Vertical Column of a Heavier Liquid: An Approximate Analysis
,”
ASME J. Fluids Eng.
,
131
(1), p.
011303
.
31.
Brannock
,
D.
, and
Kubie
,
J.
,
1996
, “
Velocity of Long Bubbles in Oscillating Vertical Pipes
,”
Int. J. Multiphase Flow
,
22
(5), pp.
1031
1034
.
32.
Clanet
,
C.
,
Heraud
,
P.
, and
Searby
,
G.
,
2004
, “
On the Motion of Bubbles in Vertical Tubes of Arbitrary Cross-Sections: Some Complements of the Dumitrescu-Taylor Problem
,”
J. Fluid Mech.
,
519
, pp.
359
376
.
33.
Madani
,
S.
,
Caballina
,
O.
, and
Souhar
,
M.
,
2009
, “
Unsteady Dynamics of Taylor Bubble Rising in Vertical Oscillating Tubes
,”
Int. J. Multiphase Flow
,
35
(4), pp.
363
375
.
34.
Talvy
,
C. A.
,
Shemer
,
L.
, and
Barnea
,
D.
,
2000
, “
On the Interaction Between Two Consecutive Elongated Bubbles in a Vertical Pipe
,”
Int. J. Multiphase Flow
,
26
(12), pp.
1905
1923
.
35.
Moissis
,
D.
, and
Griffith
,
P.
,
1962
, “
Entrance Effects in a Two-Phase Slug Flow
,”
ASME J. Heat Transfer
,
84
(
1
), pp.
29
38
.
36.
Lamb
,
H.
,
1932
,
Hydrodynamics
,
6th ed.
,
Cambridge University Press
,
Cambridge, UK
.
37.
Simcik
,
M.
,
Puncochar
,
M.
, and
Ruzicka
,
M. C.
,
2014
, “
Added Mass of a Spherical Cap Body
,”
Chem. Eng. Sci.
,
118
, pp.
1
8
.
38.
Taitel
,
Y.
,
Barnea
,
D.
, and
Dukler
,
A. E.
,
1980
, “
Modelling Flow Pattern Transition for Steady Upward Gas-Liquid Flow in Vertical Tubes
,”
AIChE J.
,
26
(3), pp.
345
354
.
39.
Clift
,
R.
,
Grace
,
J. R.
, and
Weber
,
M. E.
,
1978
,
Bubbles, Drops, and Particles
,
Academic Press
,
New York
.
40.
Klaseboer
,
E.
,
Gupta
,
R.
, and
Manica
,
R.
,
2014
, “
An Extended Bretherton Model for Long Taylor Bubbles at Moderate Capillary Numbers
,”
Phys. Fluids
,
26
(3), p.
032107
.
41.
Michaelides
,
M. M.
,
1997
, “
Review—The Transient Equation of Motion for Particles, Bubbles, and Droplets
,”
ASME J. Fluids Eng.
,
119
(2), pp.
233
247
.
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