Analytical solutions to the heat convection from two bubbles were obtained. These solutions were applied to the two bubbles with the flow along their line of centers and perpendicular to their line of centers. The method of images has been used to give a solution to the velocity distribution around the two bubbles. The derived solutions apply to fluid spheres. In general the derived solutions were compared well with the other available analytical and numerical results. Some avenues for further research were pointed out.

1.
Ruzicka
,
M. C.
, 2000, “
On Bubbles Rising in Line
,”
Int. J. Multiphase Flow
0301-9322,
26
, pp.
1141
1181
.
2.
Ramachandran
,
R. S.
,
Kleinstreuer
,
C.
, and
Yang
,
T.-Y.
, 1989, “
Forced Convection Heat Transfer of Interacting Spheres
,”
Numer. Heat Transfer, Part A
1040-7782,
15
, pp.
471
487
.
3.
Zydney
,
A. L.
, and
Colton
,
C. K.
, 1988, “
Augmented Solute Transport in the Shear Flow of a Concentrated Suspension
,”
PCH, PhysicoChem. Hydrodyn.
0191-9059,
10
, pp.
77
96
.
4.
Yuan
,
H.
, and
Prosperetti
,
A.
, 1994, “
On the In-Line Motion of Two Spherical Bubbles in a Viscous Fluid
,”
J. Fluid Mech.
0022-1120,
278
, pp.
325
349
.
5.
Aminzadeh
,
K.
,
Al-Taha
,
T. R.
, and
Cornish
,
A. R. H.
, 1974, “
Mass Transport Around Two Spheres at Low Reynolds Numbers
,”
Int. J. Heat Mass Transfer
0017-9310,
17
, pp.
1425
1436
.
6.
Ramachandran
,
R. S.
, and
Kleinstreuer
,
C.
, 1985, “
Mass Transfer with Chemical Reaction in Two Interacting Drops
,”
Chem. Eng. Commun.
0098-6445,
36
, pp.
197
210
.
7.
Tal
,
R.
,
Lee
,
D. N.
, and
Sirignano
,
W. A.
, 1984, “
Heat and Momentum Transfer Around a Pair of Spheres in Viscous Flow
,”
Int. J. Heat Mass Transfer
0017-9310,
27
, pp.
1953
1962
.
8.
Chen
,
W. C.
, and
Tong
,
A. Y.
, 1987, “
Numerical Study on Hydrodynamic and Heat Transfer of Droplets Arrays Using Grid Generation
,”
AIChE Symp. Ser.
0065-8812,
83
, pp.
230
236
.
9.
Chiang
,
H.
, and
Kleinstreuer
,
C.
, 1992, “
Computational Analysis of Interacting Vaporizing Fuel Droplets on a One-Dimensional Trajectory
,”
Combust. Sci. Technol.
0010-2202,
86
, pp.
289
309
.
10.
Leal
,
L. G.
, 1992,
Laminar Flow and Convective Transport Processes
,
Butterworth-Heinemann
,
Boston
.
11.
Paidoussis
,
M. P.
,
Mavriplis
,
D.
, and
Price
,
S. J.
, 1984, “
A Potential Flow Theory for the Dynamics of Cylinder Arrays in Cross-Flow
,”
J. Fluid Mech.
0022-1120,
146
, pp.
227
240
.
12.
Balsa
,
T. F.
, 1977, “
Potential Flow Interactions in an Array of Cylinders in Cross-Flow
,”
J. Sound Vib.
0022-460X,
50
, pp.
285
303
.
13.
Mulholand
,
J. A.
,
Srivastava
,
R. K.
, and
Wendt
,
J. O. L.
, 1988, “
Influence of Droplet Spacing on Drag Coefficient in Nonevaporating, Monodisperse Streams
,”
AIAA J.
0001-1452,
26
, pp.
1231
1245
.
14.
Katz
,
J.
, and
Meneveau
,
C.
, 1996, “
Wake-Induced Relative Motion of Bubbles Rising in Line
,”
Int. J. Multiphase Flow
0301-9322,
22
, pp.
239
245
.
15.
Kok
,
J. B. W.
, 1993, “
Dynamics of a Pair of Gas Bubbles Moving Through Liquid. Part I. Theory
,”
Eur. J. Mech. B/Fluids
0997-7546,
12
, pp.
515
540
.
16.
Sanada
,
T.
, 2005, “
Interaction and Coalescence of Bubbles in Quiescent Liquid
,” Ph.D. thesis, Kyushu University, Japan.
17.
Clift
,
R.
,
Grace
,
R. J.
, and
Weber
,
M. E.
, 1978,
Bubbles, Drops and Particles
,
Academic
,
New York
.
18.
Milne-Thomson
,
L. M.
, 1972,
Theoretical Hydrodynamics
,
Macmillan
,
London
.
19.
Kendoush
,
A. A.
, 1995, “
Low Prandtl Number Heat Transfer to Fluids Flowing Past an Isothermal Spherical Particle
,”
Int. J. Heat Fluid Flow
0142-727X,
16
, pp.
291
297
.
20.
Durst
,
F.
,
Schonung
,
B.
,
Selanger
,
K.
, and
Winter
,
M.
, 1988, “
Bubble-Driven Liquid Flow
,”
J. Fluid Mech.
0022-1120,
170
, pp.
53
82
.
21.
de Varies
,
A. W. G.
,
Biesheuvel
,
A.
, and
van Wijngaarden
,
L.
, 2002, “
Notes on the Path and Wake of a Gas Bubble Rising in Pure Water
,”
Int. J. Multiphase Flow
0301-9322,
28
, pp.
1823
1835
.
22.
Mougin
,
J.
, and
Magnaudet
,
J.
, 2002, “
Path Instability of a Rising Bubble
,”
Phys. Rev. Lett.
0031-9007,
88
, pp.
14
19
.
23.
Boussinesq
,
M. J.
, 1905, “
Calcul du pouvoír refroídissant des courants fluides
,”
J. Math. Pures Appl.
0021-7824,
1
, pp.
285
390
.
24.
Kleinstreuer
,
C.
, and
Chiang
,
H.
, 1993, “
Convection Heat Transfer of Closely-Spaced Spheres With Surface Blowing
,”
Waerme- Stoffuebertrag.
0042-9929,
28
, pp.
285
293
.
25.
Kendoush
,
A. A.
, 1989, “
The Delay Time During Depressurization of Saturated Water
,”
Int. J. Heat Mass Transfer
0017-9310,
32
, pp.
2149
2157
.
26.
Harper
,
J. F.
, 1970, “
On Bubbles Rising in Line at Large Reynolds Numbers
,”
J. Fluid Mech.
0022-1120,
41
, pp.
751
758
.
27.
Tong
,
A. Y.
, and
Chen
,
S. J.
, 1986, “
Heat Transfer Correlations for Vaporizing Liquid Droplet Array in a High Temperature Gas at Intermediate Reynolds Number
,”
Proceedings Fall Technical Meeting
,
Eastern Section of the Combustion Institute
,
San Juan, Puerto Rico
.
28.
Legendre
,
D.
,
Magnaudet
,
J.
, and
Mougin
,
G.
, 2003, “
Hydrodynamic Interactions Between Two Spherical Bubbles Rising Side by Side in a Viscous Liquid
,”
J. Fluid Mech.
0022-1120,
497
, pp.
133
166
.
29.
Kendoush
,
A. A.
, 2006, “
The Virtual Mass of Two Spheres Moving in Fluids
,”
ASME J. Fluids Eng.
0098-2202, submitted.
This content is only available via PDF.
You do not currently have access to this content.