A calculation of the motion of two bubbles in a constant temperature gradient is presented for the limit of creeping flow and vanishing Péclet number. The results are valid for undeforming, spherical bubbles of arbitrary size and orientation with respect to the temperature gradient, and for separations up to near touching. Trajectories of two bubbles, as they approach one another in an infinite fluid, are then determined from these instantaneous velocity calculations. The limiting trajectories, where the bubbles ‘‘just miss’’ each other, are used to tabulate collision efficiencies over a wide range of bubble size ratios. On the basis of the two‐bubble results, a model for the evolution of a statistically homogeneous cloud of bubbles with a given initial size distribution is formulated in terms of a discrete stochastic collection equation. Using this simple model, the dependence of the rate of coalescence in a bubble cloud upon material properties, the bubble volume fraction, the average temperature gradient, and other characteristics of the system, is demonstrated. A significant conclusion is that the bubble collision rate increases with the standard deviation of the bubble size distribution.
REFERENCES
1.
N. O.
Young
, J. S.
Goldstein
, and M. J.
Block
, “The motion of bubbles in a vertical temperature gradient
,” J. Fluid Mech.
6
, 350
(1959
).2.
R. S.
Subramanian
, “Slow migration of a gas bubble in a thermal gradient
,” AIChE J.
27
, 646
(1981
).3.
R.
Balasubramaniam
and A.
Chai
, “Thermocapillary migration of droplets: An exact solution for small Marangoni numbers
,” J. Colloid Interface Sci.
119
, 531
(1987
).4.
H. S.
Kim
and R. S.
Subramanian
, “Thermocapillary migration of a droplet with insoluble surfactant. I. Surfactant cap
,” J. Colloid Interface Sci.
127
, 417
(1989
).5.
H. S.
Kim
and R. S.
Subramanian
, “Thermocapillary migration of a droplet with insoluble surfactant. II. General case
,” J. Colloid Interface Sci.
130
, 112
(1989
).6.
N.
Shankar
and R. S.
Subramanian
, “The Stokes motion of a gas bubble due to interfacial tension gradients at low to moderate Marangoni numbers
,” J. Colloid Interface Sci.
123
, 512
(1988
).7.
J.
Szymczyk
and J.
Siekmann
, “Numerical calculation of the thermocapillary motion of a bubble under microgravity
,” Chem. Eng. Commun.
69
, 129
(1988
).8.
J. C.
Chen
and Y. T.
Lee
, “Effect of surface deformation on thermocapillary bubble migration
,” AIAA J.
30
, 993
(1992
).9.
N.
Shankar
and R. S.
Subramanian
, “The slow axisymmetric thermocapillary migration of an eccentrically placed bubble inside a drop in zero gravity
,” J. Colloid Interface Sci.
94
, 258
(1983
).10.
M.
Meyyappan
and R. S.
Subramanian
, “Thermocapillary migration of a gas bubble in an arbitrary direction with respect to a plane surface
,” J. Colloid Interface Sci.
115
, 206
(1987
).11.
E. P.
Ascoli
and L. G.
Leal
, “Thermocapillary motion of a deformable drop toward a planar wall
,” J. Colloid Interface Sci.
138
, 220
(1990
).12.
M.
Meyyappan
, W. R.
Wilcox
, and R. S.
Subramanian
, “The slow axisymmetric motion of two bubbles in a thermal gradient
,” J. Colloid Interface Sci.
94
, 243
(1983
).13.
M.
Meyyappan
and R. S.
Subramanian
, “The thermocapillary motion of two bubbles oriented arbitrarily relative to a thermal gradient
,” J. Colloid Interface Sci.
97
, 291
(1984
).14.
J. L.
Anderson
, “Droplet interactions in thermocapillary motion
,” Int. J. Multiphase Flow
11
, 813
(1985
).15.
D. J.
Jeffrey
, “Conduction through a random suspension of spheres
,” Proc. R. Soc. London Ser. A
335
, 355
(1973
).16.
D. J.
Jeffrey
and Y.
Onishi
, “Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds number flow
,” J. Fluid Mech.
139
, 261
(1984
).17.
S.
Weinbaum
, P.
Ganatos
, and Z. Y.
Yan
, “Numerical multipole and boundary integral equation techniques in Stokes’ flow
,” Annu. Rev. Fluid Mech.
22
, 275
(1990
).18.
J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Martinus Nijhoff, Dordrecht, 1973).
19.
D. M.
Mattox
, H. D.
Smith
, W. R.
Wilcox
, and R. S.
Subramanian
, “Thermal-gradient-induced migration of bubbles in molten glass
,” J. Am. Ceram. Soc.
65
, 437
(1982
).20.
G. K.
Batchelor
, “Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory
,” J. Fluid Mech.
119
, 379
(1982
).21.
R.
Balasubramaniam
and J. E.
Lavery
, “Numerical simulation of thermocapillary bubble migration under microgravity for large Reynolds and Marangoni numbers
,” Num. Heat Transfer A
16
, 175
(1989
).22.
F.
Feuillebois
, “Thermocapillary migration of two equal bubbles parallel to their line of centers
,” J. Colloid Interface Sci.
131
, 267
(1989
).23.
A.
Acrivos
, D. J.
Jeffrey
, and D. A.
Saville
, “Particle migration in suspensions by thermocapillary or electrophoretic motion
,” J. Fluid Mech.
212
, 95
(1990
).24.
M. Meyyappan, “Interaction effects in thermocapillary bubble migration,” Ph.D. dissertation, Clarkson University, 1984.
25.
H. J.
Keh
and S. H.
Chen
, “The axisymmetric thermocapillary motion of two fluid droplets
,” Int. J. Multiphase Flow
16
, 515
(1990
).26.
S. G.
Yiantsios
and R. H.
Davis
, “Close approach and deformation of two viscous drops due to gravity and van der Waals forces
,” J. Colloid Interface Sci.
144
, 412
(1991
).27.
P. S.
Hahn
, J. D.
Chen
, and J. C.
Slattery
, “Effects of London-van der Waals forces on the thinning and rupture of a dimpled liquid film as a small drop or bubble approaches a fluid-fluid interface
,” AIChE J.
31
, 2026
(1985
).28.
L. M.
Hocking
, “The collision efficiency of small drops
,” Q. J. R. Meteorol. Soc.
66
, 44
(1959
).29.
X. Zhang and R. H. Davis, “The collision rate of small drops undergoing thermocapillary migration,” J. Colloid Interface Sci. (in press).
30.
J. F.
Brady
and G.
Bossis
, “Stokesian dynamics
,” Annu. Rev. Fluid Mech.
20
, 111
(1988
).31.
L.
Ratke
, “Simultaneous coarsening of dispersions by growth and coagulation
,” J. Colloid Interface Sci.
119
, 391
(1987
).32.
E. X.
Berry
, “Cloud droplet growth by collection
,” J. Atmos. Sci.
24
, 688
(1967
).33.
R. L. Drake, “A general mathematical survey of the coagulation equation,” in Topics in Current Aerosol Research (Part 2), edited by G. M. Hidy and J. R. Brock (Pergamon, New York, 1972).
34.
J. R.
Rogers
and R. H.
Davis
, “Modeling of collision and coalescence of droplets during microgravity processing of Zn-Bi immiscible alloys
,” Metall. Trans. A
21
, 59
(1990
).
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