This work considers a nearly spherical bubble and a nearly flat free surface interacting under buoyancy at vanishing Bond number Bo. For each perturbed surface, the deviation from the unperturbed shape is asymptotically obtained at leading order on Bo. The task appeals to the normal traction exerted on the unperturbed surface by the Stokes flow due to a spherical bubble translating toward a flat free surface. The free surface problem is then found to be well-posed and to admit a solution in closed form when gravity is still present in the linear differential equation governing the perturbed profile through a term proportional to Bo. In contrast, the bubble problem amazingly turns out to be over-determined. It however becomes well-posed if the requirement of horizontal tangent planes at the perturbed bubble north and south poles is discarded or if the term proportional to Bo is omitted. Both previous approaches turn out to predict for a small Bond number, quite close solutions except in the very vicinity of the bubble poles. The numerical solution of the proposed asymptotic analysis shows in the overlapping range Bo = O ( 0.1 ) and for both the bubble and the free surface perturbed shapes, a good agreement with a quite different boundary element approach developed in Pigeonneau and Sellier [“Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface,” Phys. Fluids 23, 092102 (2011)]. It also provides approximated bubble and free surface shapes whose sensitivity to the bubble location is examined.

1.
L.
Pilon
, “
Foams in glass manufacture
,” in
Foam Engineering: Fundamentals and Aplications
, edited by
P.
Stevenson
(
John Wiley & Sons, Ltd.
,
Chichester, UK
,
2012
), Chap. 16, pp.
355
409
.
2.
S. H.
Lee
and
L. G.
Leal
, “
The motion of a sphere in the presence of a deformable interface. II. A numerical study of the translation of a sphere normal to an interface
,”
J. Colloid Interface Sci.
87
,
81
106
(
1982
).
3.
C.
Berdan
and
L. G.
Leal
, “
Motion of a sphere in the presence of a deformation interface. I. Perturbation of the interface from flat: The effects on drag and torque
,”
J. Colloid Interface Sci.
87
,
62
80
(
1982
).
4.
S. G.
Yiantsios
and
R. H.
Davis
, “
On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface
,”
J. Fluid Mech.
217
,
547
573
(
1990
).
5.
G. K.
Youngren
and
A.
Acrivos
, “
Stokes flow past a particle of arbitrary shape: A numerical method of solution
,”
J. Fluid Mech.
69
,
377
403
(
1975
).
6.
S.
Kim
and
S. J.
Karrila
,
Microhydrodynamics: Principles and Selected Applications
(
Martinus Nijhoff Publishers
,
The Hague
,
1983
).
7.
C.
Pozrikidis
,
Boundary Integral and Singularity Methods for Linearized Viscous Flow
(
Cambridge University Press
,
Cambridge
,
1992
).
8.
A.
Sellier
, “
Boundary element technique for slow viscous flows about particles
,” in
Boundary Element Methods in Engineering and Sciences
(
World Scientific
,
2010
), Vol.
4
, Chap. VII, pp.
239
281
.
9.
H.
Kočárková
,
F.
Rouyer
, and
F.
Pigeonneau
, “
Film drainage of viscous liquid on top of bare bubble: Influence of the bond number
,”
Phys. Fluids
25
,
022105
(
2013
).
10.
F.
Pigeonneau
and
A.
Sellier
, “
Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface
,”
Phys. Fluids
23
,
092102
(
2011
).
11.
G.
Hetsroni
and
S.
Haber
, “
The flow in and around a droplet or bubble submerged in an unbounded arbitrary velocity field
,”
Rheol. Acta
9
,
488
496
(
1970
).
12.
E.
Chervenivanova
and
Z.
Zapryanov
, “
On the deformation of two droplets in a quasisteady Stokes flow
,”
Int. J. Multiphase Flow
11
,
721
738
(
1985
).
13.
E.
Chervenivanova
and
Z.
Zapryanov
, “
The slow motion of droplets perpendicular to a deformable flat fluid interface
,”
Q. J. Mech. Appl. Math.
41
,
419
444
(
1988
).
14.
R.
Aris
,
Vectors, Tensors and the Basic Equation of Fluid Mechanics
(
Dover Publications, Inc.
,
New York
,
1962
).
15.
D. D.
Joseph
and
Y. Y.
Renardy
,
Fundamentals of Two-Fluid Dynamics. Part I: Mathematical Theory and Applications
(
Springer
,
New York
,
1993
).
16.
As shown in Ref. 32, the integration of Sn over the closed surface S0 gives no contribution.
S1σndS1ρlVbg=0,Vb=4πa3/3.
Clearly, the above equation means that the flow (u, p + ρlgx + pa) exerts on the bubble a zero force.
17.
E.
Bart
, “
The slow unsteady settling of a fluid sphere toward a flat fluid interface
,”
Chem. Eng. Sci.
23
,
193
210
(
1968
).
18.
M.
Meyyappan
,
W. R.
Wilcox
, and
R. S.
Subramanian
, “
Thermocapillary migration of a bubble normal to a plane surface
,”
J. Colloid Interface Sci.
83
,
199
208
(
1981
).
19.
J.
Happel
and
H.
Brenner
,
Low Reynolds Number Hydrodynamics
(
Martinus Nijhoff Publishers
,
The Hague
,
1983
).
20.

Misprints occurring in the Appendix of the Reference 18 have been however corrected when used in the present analysis.

21.
Z.
Zapryanov
and
S.
Tabakova
,
Dynamics of Bubbles, Drops and Rigid Particles
(
Kluwer Academic Publishers
,
Dordrecht
,
1999
).
22.
M.
Guémas
, “
Low-Reynolds-number gravity-driven migration and deformation of bubble(s) and/or solid particle(s) near a deformable free surface
,” Ph.D. thesis,
École Polytechnique, Université Paris-Saclay
,
2014
.
23.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions
(
Dover Publications, Inc.
,
New York
,
1965
).
24.

For l = 6a it is found that λ0,s ∼ 1.142 and λ0 ∼ 1.091. Since Bo/Ca ∼ 1.67 one has λ0≠Bo/Ca, i.e., the bubble is not force free in that case.

25.

Those conditions at the poles are not clearly required in Ref. 13.

26.
L.
Landau
and
E.
Lifchitz
,
Mécanique Des Fluides
(
Mir
,
Moscou
,
1994
).
27.
M.
Guémas
,
A.
Sellier
, and
F.
Pigeonneau
, “
Slow viscous gravity-driven interaction between a bubble and a free surface with unequal surface tensions
,”
Phys. Fluids
27
,
043102
(
2015
).
28.
A.
Sellier
, “
Thermocapillary motion of a two-bubble cluster near a plane solid wall
,”
C. R. Méc.
333
(
8
),
636
641
(
2005
).
29.
L.
Pasol
and
A.
Sellier
, “
Migration of a solid particle in the vicinity of a plane fluid–fluid interface
,”
Eur. J. Mech., B: Fluids
30
,
76
88
(
2011
).
30.
I. S.
Gradshteyn
and
I. M.
Ryzbik
,
Table of Integrals, Series, and Products
(
Academic Press
,
San Diego
,
1965
).
31.
M.
Stimson
and
G. B.
Jeffery
, “
The motion of two spheres in a viscous fluid
,”
Proc. R. Soc. London, Ser. A
111
,
110
116
(
1926
).
32.
L. G.
Leal
,
Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes
(
Cambridge University Press
,
New York
,
2007
).
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