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Armed Services Technical Information Agency

AN EXPERIMENTAL INVESTIGATION
OF THE DRAG AND SHAPE
OF AIR BUBBLES RISING
IN VARIOUS LIQUIDS
W.L. Haberman
&amp;
R.K. Morton
Report 802
NS 715-102

NAVY DEPARTMENT
THE DAVID W. TAYLOR MODEL BASIN
WASHINGTON 7, D.C.
SEPTEMBER 1953

Abstract
In connection with other investigations at the David Taylor Model Basin, detailed information became necessary on the motion of air bubbles in variable pressure fields.
Since no information on the subject was available, a fundamental study of the motion of
bubbles was undertaken. As an initial step, experiments were conducted to determine
the drag and shape of single air bubbles rising freely in various liquids. The results of the
experiments show, that a complete description of the motion of air bubbles is not possible
by use of dimensionless parameters containing the usual physical properties of the liquid
(viscosity, surface tension, density). Three types of bubble shapes were observed in each
liquid, namely spherical, ellipsoidal, and spherical cap. For a specific liquid, the shape of
the bubble was a function of its volume. For tiny spherical bubbles, the drag coefficients
coincide with those of corresponding rigid spheres. With increase in bubble size, a decrease in the drag as compared to that of rigid spheres occurs in some liquids. Thus, the
drag curves of the spherical bubbles rising in various liquids fall between two limiting
curves, namely the drag curve of rigid and fluid spheres, respectively. It was not possible
to determine a criterion for the transition of the bubbles from ”rigid” to fluid spheres.
The region of ellipsoidal bubbles extends over different ranges of Reynolds numbers for
the various liquids. The drag coefficients of spherical cap bubbles are independent of
bubble size and have a constant value of 2.6. For bubbles (equivalent radius 0.08 to
0.80 cm) rising in tap water or in water containing certain surface-active substances,
experiments show an increase in drag as compared to bubbles in pure water. Results of
tests to determine the effect of the container walls on the velocity of rise are presented.
A description of the experimental apparatus is given. A summary of the theoretical and
experimental work of other investigators is also included.

Contents
1. Introduction

1

2. Theoretical Solutions

2

3. Dimensional Analysis

5

4. Previous Experimental Work
4.1. Rate of Rise of Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Wall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Cylindrical Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8
8
9
10

5. Scope of the Present Investigation

11

6. Experimentation
6.1. Test Tanks . . . . . . . . . . . . . . .
6.2. Test Liquids . . . . . . . . . . . . . .
6.3. Bubble Generation . . . . . . . . . .
6.4. Determination of Bubble Size . . . .
6.5. Other Experimental Techniques . . .
6.6. Motion Pictures and their Evaluation

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56

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7. Results
7.1. Terminal Velocity of Air Bubbles . . . . . . .
7.2. Wall and Proximity Effects . . . . . . . . . . .
7.3. Non-dimensional Presentation of Bubble Data
7.4. Spherical Bubbles . . . . . . . . . . . . . . . .
7.5. Ellipsoidal Bubbles . . . . . . . . . . . . . . .
7.6. Spherical Cap Bubbles . . . . . . . . . . . . .
7.7. Path of Bubbles . . . . . . . . . . . . . . . . .
7.8. Bubbles in Filtered and Tap Water . . . . . .
7.9. Effect of Surface-Active Substances . . . . . .
7.10. Bubbles as Rigid Bodies . . . . . . . . . . . .

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8. Summary

57

9. Acknowledgements

58

Contents

4

Appendix

58

A. Rate of Rise of Gas Bubbles in Distilled Water

59

B. Some Details and Expansions to Equation (2.1)

60

C. Illustration of a Bubble Moving through a Liquid at high Re

62

List of Symbols

63

Literature

68

List of Figures
6.1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3. Evaluation of Velocity of Rise of Bubbles . . . . . . . . . . . . . . . . . .
7.1. Terminal Velocity of Air Bubbles in Filtered or Distilled Water as a Function of Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Terminal Velocity of Air Bubbles in Tap Water as a Function of Bubble Size
7.3. Terminal Velocity of Air Bubbles in Cold Filtered Water as a Function of
Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4. Terminal Velocity of Air Bubbles in Hot Tap Water as a Function of
Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5. Terminal Velocity of Air Bubbles in Water Containing Glim as a Function
of Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6. Terminal Velocity of Air Bubbles in Mineral Oil as a Function of Bubble
Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7. Terminal Velocity of Air Bubbles in Varsol as a Function of Bubble Size .
7.8. Terminal Velocity of Air Bubbles in Turpentine as a Function of Bubble
Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9. Terminal Velocity of Air Bubbles in Methyl Alcohol as a Function of
Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10. Terminal Velocity of Air Bubbles in Corn Syrup-Water Mixtures as a
Function of Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.11. Terminal Velocity of Air Bubbles in Liquids as Obtained from Bryn’s Data
7.12. Comparison of Terminal Velocities of Air Bubbles in Various Liquids . .
7.13. Drag Coefficient as a Function of Reynolds number for Air Bubbles Rising
at their Terminal Velocity in Various Liquids . . . . . . . . . . . . . . . .
7.14. Drag Coefficient as a Function of Weber number for Air Bubbles Rising
at their Terminal Velocity in Various Liquids . . . . . . . . . . . . . . . .
7.15. Typical Shapes of Bubbles of Several Volumes in the Various Liquids . .
7.16. Drag Coefficient as a Function of Reynolds Number for Spherical Air
Bubbles Rising at Their Terminal Velocity in Various Liquids . . . . . .
7.17. Drag Coefficient as a Function of Reynolds Number for Ellipsoidal and
Spherical Cap Bubbles in Various Liquids . . . . . . . . . . . . . . . . . .
7.18. Drag Coefficient as a Function of Weber Number for Ellipsoidal and Spherical Cap Bubbles in Various Liquids . . . . . . . . . . . . . . . . . . . . .
7.19. Path and Corresponding Shapes of Air Bubbles Rising in Methyl Alcohol

16
18
20
23
24
25
26
27
28
29
30
31
32
33
34
36
37
39
40
42
43
46

List of Figures

6

7.20. Path and Corresponding Shapes of Air Bubbles Rising in Varsol . . . . .
7.21. Path and Corresponding Shapes of Air Bubbles Rising in Cold Water . .
7.22. Path and Corresponding Shapes of Air Bubbles Rising in Mineral Oil . .
7.23. Drag Coefficient as a Function of Reynolds Number for Air Bubbles Rising
at Their Terminal Velocity in Filtered or Distilled Water at Different
Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.24. Drag Coefficient as a Function of Reynolds Number for Air Bubbles Rising
at Their Terminal Velocity in Tap Water at Different Temperatures . . .
7.25. Drag Coefficient as a Function of Reynolds Number for Air Bubbles Rising
at Their Terminal Velocity in Water at Room Temperature as Obtained
from Data of Various Investigators . . . . . . . . . . . . . . . . . . . . .
7.26. Drag Coefficient as a Function of Reynolds Number for Bubbles Rising
at Their Terminal Velocity in Water Containing Various Surface-Active
Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
48
49

51
52

53

55

A.1. Terminal Velocity of Gas Bubbles in Distilled Water as a Function of
Bubble Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

C.1. Bubble moving through liquid at high Reynolds number

62

. . . . . . . . .

List of Tables
6.1. Summary of Liquid Properties . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15
19

1. Introduction
The tests described in this report are a continuation of experiments given in a previous
Taylor Model Basin report. These experiments were initiated in conjunction with the
work under projects NS713-201 and NE051-237. The present tests, continued under
NS715-102, were conducted for the purpose of investigating the motion of bubbles rising
under the influence of gravity as an initial step for obtaining information on the behavior
of bubbles in variable pressure fields.
A body rising or falling under the influence of gravity reaches a constant velocity (terminal velocity) when all forces acting on it are in equilibrium:
Drag + Buoyant force + Weight = 0 .
For rigid bodies, the drag will, in general, be a complicated function of the geometry
of the body, the velocity, and the physical properties of the medium, i.e., the density
and viscosity. For fluid bodies, such as drops and gas bubbles, the function is further
complicated by the fact that the body may be of changeable shape and that properties
of the fluid inside the globule, such as density and viscosity, and interfacial effects may
also be important factors. In general, the shape that the fluid globule assumes is some
complicated function of the hydrodynamic, viscous, and interfacial forces.
The drag of fluid bodies may either be equal to (as is the case for small bubbles) or less
than that of the corresponding rigid body depending upon the conditions at the interface.
In the former case there exists, effectively, a rigid surface at the interface. In the latter
case, the fluid particles at the boundary have, in contrast to rigid bodies, non-vanishing
tangential velocities. The circulation inside the fluid body thus reduces the drag of the
body. The experiments described in this report consisted of the determination of the
terminal velocity, shape, and path of single air bubbles rising freely in various liquids
as a function of bubble size. The possible effect of the walls of the container on the
velocity of rise of the bubble was also investigated. A summary of pertinent theoretical
and experimental work of other investigators is included.

2. Theoretical Solutions
Theoretical solutions for the drag of rigid and fluid spheres, moving slowly in an infinite
medium, have been obtained for the following boundary conditions at the surface of the
sphere:
1. Rigid Spheres
a) Stokes’ solution 
i. Velocity
A. The velocity vanishes
2. Fluid Spheres
a) Hadamard-Rybczynski’s solution [26, 30, 46]
i. Velocity
A. The normal velocity component vanishes.
B. The tangential velocity components at both sides of the surface are
equal.
ii. Stresses
A. The normal and tangential stresses at both sides of the surface are
equal.1
b) Boussinesq’s solution [7–9, 15]
i. Velocity
A. The normal velocity component vanishes.
B. The tangential velocity components at both sides of the surface are
equal
ii. Stresses

1


The pressure increase across the surface due to surface tension = 2σ
r
analysis. Inclusion of this pressure drop in the boundary condition for the normal stress does not
change the results. That is to say, surface tension as manifested only in a pressure increase inside
the fluid sphere does not affect its motion. (This result is also obtained by putting, in Boussinesq’s
analysis, the coefficient of surface viscosity, see subsequent footnote, equal to zero.)

2. Theoretical Solutions

3

A. The normal stress at the inside of the surface is larger than the stress
at the outside due to the dynamic surface tension.2
B. The tangential stress at the surface is increased across the surface due
to the dynamic increment of surface tension.
The drag of a sphere in an infinite medium of uniform velocity U thus becomes:
1. Rigid sphere: D = 6 π µ r U
2. Fluid sphere:
a) D = 6 π µ r U ·

2 µ+3 µ8
3 µ+3 µ8

b) D = 6 π µ r U ·

 + r·(2 µ+3 µ8 )
 + 3 r·(3 µ+3 µ8 )

Using the condition of equilibrium for a sphere rising under the influence of gravity, we
obtain for the rigid case:
4
4
6 π µ r U = πr 3 ρg − πr 3 ρ8 g
3
3
2
2r g
U=
· (ρ − ρ8 )
9 µ

(2.1)

which is Stokes’ Law.
For the fluid sphere we get
2 r 2g
· (ρ − ρ8 ) ·
9 µ
2 r 2g
U=
· (ρ − ρ8 ) ·
9 µ
U=

3µ8 + 3µ
3µ8 + 2µ
 + 2r · (µ + µ8 )
Boussinesq’s Law.
 + r · (2µ + 3µ8 )

(2.2)
(2.3)

For the case of bubbles, where µ8  µ and ρ8  ρ, the last two expressions reduce to:
U=

2

1 r 2 gρ
3 µ

(2.4)

Boussinesq assumed that a dynamic surface tension exists at interfaces in motion. Its magnitude
is given by the sum of the usual (static) surface tension and the dynamic increment. The dynamic
increment varies over the surface of the sphere and at a given point is proportional to the rate
of dilatation at that point. The constant of proportionality is called (due to its similarity to
the viscosity coefficient) the coefficient of surface viscosity. (Surface viscosity has the dimensions
mass/time, while the dimensions for viscosity are mass/length · time.)

2. Theoretical Solutions

4

and
U=

2 r2 g ρ  + 3r µ
·
.
9 µ
 + 2rµ

(2.5)

3rµ
In the last equation, the factor  +
approaches 1 for r approaching zero or  very large.
+ 2rµ
3
It further approaches 2 for   rµ, i.e., for large r or for  approaching zero. Hence, for
very small bubbles, Boussinesq’s solution approaches Stokes’ law as a limit, while the
From the boundary conditions as stated above, it is obvious that, for the HadamardRybczynski and Boussinesq solutions, circulation exists inside the bubble. For the Stokes
solution, of course, there is no circulation inside the sphere.

3. Dimensional Analysis
Since an analytical solution for the drag of fluid bodies over a large range of sizes is
hardly attainable, dimensional analysis of the phenomenon may serve to correlate the
experimental results. In the case of fluid bodies the following physical variables are
usually considered as pertinent:

U
g
ρ
ρ8
L
µ
µ8
σ

Velocity of the body
Acceleration due to gravity
Density of the fluid medium
Density of the fluid body
Length parameter of the fluid body
Coefficient of dynamic viscosity of the fluid medium
Coefficient of dynamic viscosity of the fluid inside the fluid body
Interfacial tension1

The complete set of dimensionless products will therefore contain five such products.
In principle, it is immaterial which complete set is chosen for the representation of the
phenomenon. For example, we may use


µ ρ
f1 cD , Re, We, 8 , 8 = 0
µ ρ
or


8 ρ
f2 cD , Re, Mo, Re , 8 = 0
ρ
and so on, where cD is the drag coefficient.
Re is the Reynolds number and is defined according to
Re =

2L U ρ
,
µ

(3.1)

We is the Weber number defined according to
We =

2L U 2 ρ
,
σ

(3.2)

3. Dimensional Analysis

6

Re8 is the Reynolds number inside the fluid body defined as
Re8 =

2L U ρ8
µ8

(3.3)

and Mo is a dimensionless parameter, called the Morton number, which is defined as
Mo =

gµ 4
.
ρσ 3

(3.4)

If the density and viscosity of the gas inside a bubble are considered negligible, the
physical variables are reduced to six. The dimensionless products then take the form of
f3 (cD , Re, We) = 0
or
f4 (cD , Re, Mo) = 0
or
f5 (cD , We, Mo) = 0
etc..
When experimental data on gas bubbles are plotted in terms of dimensionless products,
complete correlation will be obtained provided the variables chosen are all the variables
upon which the phenomenon depends. In the case of bubbles, it is most convenient to use
a length parameter which is based on its volume rather than a physical dimension as is
customary for rigid bodies. The length parameter chosen is the equivalent radius re where
s
Volume
re = 3
.
(3.5)
4
π
3
For bubbles rising at their terminal velocity the drag coefficient cD can then be written as
8
gre
3
U2

cD =
(3.6)

instead of its usual form of D/ 12 ρ U 2 A where A is the projected area. For the analytical
solutions of Stokes and Hadamard-Rybczynski the drag coefficient becomes, respectively:
cD =

24
Re

(3.7)

for a rigid sphere, and
16
(3.8)
Re
in case of a fluid sphere. For the special case in which only four variables, namely the
velocity, the acceleration of gravity, the density of the fluid medium, and the equivalent
cD =

3. Dimensional Analysis

7

radius, are taken as pertinent, only one dimensionless group, the drag coefficient, is
obtained, i.e.,
cD = constant

(3.9)

This solution will be shown to apply to the region of very large (spherical cap) bubbles.

4. Previous Experimental Work
4.1. Rate of Rise of Bubbles
Interest in the motion of air bubbles has existed for many years. The work on bubbles has,
however, been mostly experimental in nature. Exceptions are an attempted theoretical
analysis by Theremin [52,53] in the year 1829, the analytical solution of Hadamard [26,30],
Rybczynski , and Boussinesq [7–9,15], and the dimensional analysis of Schmidt [48,49]
and Rosenberg .
The early experimental work on bubbles was largely concerned with very small bubbles
and was carried out for the purpose of determining the extent of Stokes region. Allen
[1, 2] determined the rate of rise of air bubbles in water and in aniline up to bubble radii
of 0.04 and 0.06 cm, respectively. Arnold  measured velocities of small air bubbles in
olive oil and in aniline. Bond and Newton  investigated air bubbles in syrup and in
water glass (sodium silicate).
The range of bubble sizes was extended by other investigators [19, 33] who were mainly
interested in the problem in connection with air-lift pumps, [28,37,38,55] gas absorption,
[24, 31, 36] or propagation of sound in liquids . These experiments were carried out
in water. In subsequent years, some investigations were also made in liquids other
than water. Davies and Taylor  used nitrobenzene as well as water and measured
velocities of large bubbles. Temperley and Chambers  extended the range of Taylor’s
experiments in water to bubbles of equivalent radii up to approximately 6 cm. Bryn 
made tests in various water-glycerine and water-ethyl alcohol mixtures. Robinson 
measured the rate of rise of small air bubbles in lubricating oils. His results, however,
show considerable scatter. Reports by Pickert,  Pekeris,  Worster,  and Datta
et al.  give summaries of the results of experiments of other investigators.
Gorodetskaya  investigated the effect of surface-active substances on the rate of
rise of air bubbles in water. Further tests on air bubbles are reported in References
[4, 5, 12, 20, 34, 42, 50]. In addition, a limited number of tests using gas bubbles of oxygen,
nitrogen, and a mixture of carbon dioxide and oxygen have been carried out in artificial
sea water [43, 57]. A number of tests with oxygen bubbles were also conducted in water
 and in aqueous solutions of sodium hydroxide . Recently, Stuke  investigated
the rate of rise of oxygen bubbles in pure (presumably distilled) water and in water
containing surface-active substances. The results of the tests with gas bubbles (given in
the Appendix) show no significant change in the rate of rise of the bubbles with change
in the gas inside the bubble. Because of the scatter of previous results of experiments on
the rate of rise of air bubbles in water, Rosenberg  repeated these tests for a large

4. Previous Experimental Work

9

range of air bubble sizes. He showed the geometric similarity between large bubbles of
spherical cap shape and suggested the use of three dimensionless parameters, the drag
coefficient, the Reynolds number, and the parameter Mo for describing bubble motion
in liquids.

4.2. Wall Effect
Previous investigations on the motion of air bubbles in liquids were, with a few exceptions, conducted in containers of limited dimensions. Only for very large bubbles did
Exner  and Bryn  make their measurements in lakes. Inasmuch as the effect of
the walls of the container on the rate of rise of bubbles was unknown, it was generally
neglected. Miyagi  conducted a few tests in containers of different sizes and found
that a reduction of 4 percent in the rate of rise occurred for the range of bubble sizes
investigated. Dubs [16, 17] derived, from energy considerations, an analytical expression
for the wall effect and concluded that a bubble of the same radius as its cylindrical
container has a velocity of rise of zero. It is clear that this conclusion is in error, as
experiments on cylindrical bubbles have shown.
As indicated previously, no analytical solution has yet been obtained for flows beyond
the region of slow flow. Consequently, the much more difficult problem of also including
a finite boundary in the equations of motion becomes less capable of solution. For very
slow flow about rigid spheres moving in an infinite cylindrical container, Ladenburg 
obtained an analytical solution for the effect of the boundary on the drag and, consequently, the velocity of the sphere. McNown et al.  arrived experimentally at a
wall correction coefficient for rigid spheres descending in a cylindrical container. Since
air bubbles of small volume rising in water behave essentially like rigid spheres, this
correction factor may be applied to such bubbles as long as the flow is still in the Stokes
region.
With the exception of a number of tests [37, 55] for large bubbles, no data concerning
the effect of the boundary on the rate of rise of gas bubbles beyond the region of slow
flow are available.1 However, Möller  showed, by means of a dye technique, that the
flow about a rigid sphere at a lower Reynolds number and larger boundary dimensions
was identical to that at a higher Reynolds number and smaller boundary dimensions and
therefore that the effect of the walls was to stabilize the flow about the sphere. These
results for rigid spheres beyond the Stokes region of flow at least suggested the possibility
of a similar effect for the motion of gas bubbles.

1

A paper by Coppock and Meiklejohn  has recently come to the attention of the authors. From
tests conducted with air bubbles in water, they conclude that no wail effect exists for bubbles
ranging in equivalent radius from 0.01 to 0.1 cm rising in a tube of 5 cm diameter.

4. Previous Experimental Work

10

4.3. Cylindrical Bubbles
At this point, it may be of interest to mention experiments on a special form of finite
boundary dimensions, i.e., the case of cylindrical bubbles. This term was first used by
Gibson  and applies to the type of bubble formed when a long cylindrical tube filled
with liquid is emptied from below or when a large amount of air is introduced through
the bottom of the tube.
Gibson investigated the velocity and shape of these bubbles in water. Ward and Kessler
 conducted tests in pipes of various diameters. Hattori  was interested in the
problem in connection with the possibility of evaluating the surface tension of a liquid.
Hence, he was concerned with tubes of small diameter, since the so-called critical tube
diameter (below which the bubble no longer rises but remains stationary) is a function of
the surface tension of the liquid. Dumitrescu  obtained an analytical expression for
the velocity of a cylindrical bubble by neglecting viscous and surface tension forces, thus
reducing the problem to one of potential flow. The differential equation for the velocity
potential together with the existing boundary condition yields a solution for the velocity
of rise as a function of the tube diameter only. His experimental tests in water show
that for a tube of sufficiently large diameter (3 cm for water at room temperature) the
measured velocities agree very closely with the theoretical values. Therefore, for large
bubbles, the physical properties of the liquid no longer have any effect on the flow about
the bubble and the bubbles are geometrically similar. Davies and Taylor  investigated
the shape and rate of rise of cylindrical bubbles in order to obtain a better understanding
of the pressure distribution of spherical cap bubbles in an infinite medium.

5. Scope of the Present Investigation
The present investigation was initiated in connection with a program of study of the
behavior of air bubbles in water at variable pressure gradients. Since extensive experimentation was required for direct experimental study of the motion of air bubbles in such
pressure fields,1 an alternate approach appeared more feasible. It consists of calculating
the motion of the bubbles in variable pressure fields from a knowledge of bubble drag at
various constant pressure gradients. Experimental data on the drag of air bubbles at various pressure gradients are essential in this procedure. However, only data on the motion
of bubbles in pressure gradients produced by gravity were available. Therefore, information on bubble motion in water at pressure gradients other than gravity became necessary.
This information could be obtained by investigating the rise of bubbles in liquids having
the same physical properties as water with the exception of the density. With all other
properties of the liquids identical, varying the density would be equivalent to varying the
pressure gradient. This approach, however, is not practicable, since there are no liquids
available which possess such properties. The other approach is to investigate the rise of
air bubbles in various liquids having different physical properties and then to attempt to
correlate the results in terms of non-dimensional parameters. The available information
on the rise of bubbles in different liquids was too meager to allow definite conclusions
regarding the significance of the parameters suggested by . The present investigation
was therefore initiated with the purpose of determining the non-dimensional parameters
for bubble rise by investigating bubble motion in a number of liquids of different physical properties. If it were found that the motion of air bubbles rising freely in a liquid,
that is to say the motion in the pressure field produced by gravity, could be described,
for example, in terms of the drag coefficient, the Reynolds number, and the parameter
4
Mo = ρg σµ 3 ,2 the results thus obtained could be used in evaluating the drag of bubbles
in water at pressure gradients other than that produced by gravity. To do this it would
have to be shown that the non-dimensional parameters used for the freely rising bubbles
are also applicable to other pressure fields. This might be accomplished, for example, by
comparing the results of a bubble experiment in water in a non-gravity pressure gradient
with the results of bubbles rising freely in various liquids at identical Morton number
Mo. By conducting the tests on the rise of bubbles in various liquids in a large tank,
the possibility of the effect of the tank walls on the velocity of the bubbles is eliminated.
Since the high cost of many desirable liquids makes the use of a large tank impractical,
1
2

Exploratory experiments of such a nature are reported in .
µ4
This parameter is given in a more general form as Vρ·p
2 σ 3 , where V · p is the pressure gradient. For a
gravity field V · p = ρ · g). Therefore, for a specific liquid, it is proportional to the pressure gradient.

5. Scope of the Present Investigation

12

it became necessary to determine the possible effect of the walls on the velocity over the
range of bubble sizes to be tested. This investigation consequently acquired two purposes:
1. The determination of the effect of variation of liquid properties on the motion of
air bubbles.
2. The evaluation of wall effect.

6. Experimentation
The experimental study consisted of measuring the terminal velocity of individual bubbles
of various sizes rising in eight liquids. It also included the determination of the effect of
the walls of the container on the bubble velocity. The experimental apparatus consisted of
test tanks, means for determining the physical properties of the liquids, a regulated bubble
supply, and means for measuring bubble size and velocity. Details of the experimental
apparatus and procedure, the generation of the bubbles, and the test liquids are given
in the following paragraphs.

6.1. Test Tanks
In order to obtain free bubble rise and to reduce the effects of such boundaries as the
bottom of the tank and the free liquid surface, the containers for the liquids had to be
at least 2 ft high. The tests were performed in three transparent wall tanks The large
one was of 3 x 3 ft cross section and 5 ft height, the medium one was of 1 x 1 ft cross
section and 3 1/2 ft height, and the small one was of 6 x 6 in. cross section and 2 ft
height. In addition, tests were also performed in an insert of 6 x 6 in. cross section
and 20 in. height, placed in the center of the medium tank. Since the large tank was of
sufficiently large dimensions, no significant wall effect was expected.1 The dimensions of
the medium tank were chosen large enough to reduce wall effects, yet small enough to
allow use of a variety of liquids. The small tank and the insert provided an additional
tank size. It was intended that if wall effect existed, the results obtained in the finite
containers would be extrapolated to the case of an infinite medium.

1

A few of the previous Taylor Model Basin tests  were repeated in the large tank to observe any
change in results. These previous tests were conducted in a tank of 4 1/2 × 25 ft cross section and
9 ft height with 8 ft depth of filtered water at room temperature, using one end of the tank for the
tests.

6. Experimentation

14

6.2. Test Liquids
The eight test liquids used were water (at three different temperatures), Varsol,2 methyl
alcohol, turpentine,3 water containing 0.42 percent (by volume) Glim,4 mineral oil, and
two corn syrup-water mixture. Turpentine was selected as one of the test liquids because
at room temperature it has the same viscosity as cold water. One of the corn syrup
mixtures had approximately the same viscosity as the mineral oil (see Table 6.1). The
viscosity of the liquids was measured by means of ordinary and modified Ostwald viscosimeters. The accuracy of measurement of viscosity was 1.5 percent and 0.5 percent,
respectively. The surface tension was determined by the capillary-rise method (accuracy
of measurement: 3 percent) and the specific gravity of the liquids was obtained by means
of hydrometers (accuracy of measurement: 1 percent). These physical properties were
measured following the completion of each test. They are summarized in Table 6.1
together with those of liquids used by several other investigators. [3, 6, 10].
For Varsol and water (room temperature and hot), tests were conducted in all three
tanks. For cold water and mineral oil in the medium tank and insert, and for all other
liquids, in the small tank only.

6.3. Bubble Generation
Small bubbles were generated by means of hypodermic needles and glass nozzles of various
sizes. The larger bubbles were obtained by use of a dumping cup, which was inverted to
release the air bubble. The nozzles and needles were connected to a brass tube which
was fastened to a sliding mechanism (Figure 6.1). This sliding mechanism allowed the
tips of the various nozzles to be placed at the identical position.5 The air was supplied
from a compressed air bottle. A needle valve regulated the air flow so that bubbles were
released at the interval desired.

2
3

4

5

A trade name (Standard Oil Company) for mineral spirits (heavy naphtha), a petroleum distillate.
Turpentine (also called spirit of turpentine, oil of turpentine, wood turpentine) is a fluid obtained
by the distillation of resin obtained from live trees, mainly pines. It is mainly used as a solvent
and as a source of materials for organic synthesis. Turpentine is composed of terpenes, mainly the
monoterpenes alpha-pinene and beta-pinene with lesser amounts of carene, camphene, dipentene,
and terpinolene.
Glim (Antarox A-480), a surface-active agent, is the trade name (B.T. Babbitt, Inc.) of a non-ionic,
liquid detergent, a condensation product of ethylene oxide and lauryl alcohol.
This device eliminated focusing of the camera after each change of nozzle. The camera was used
to determine velocity, path, and shape of the bubble.

6. Experimentation

15

Table 6.1.
Summary of Liquid Properties

Liquid

Water
Water
Cold Water
Hot Water
Glim Solution
Mineral Oil
Varsol
Turpentine
Methyl Alcohol
62% Corn Syrup
68% Corn Syrup
56% Glycerine and
Water 
42% Glycerine and
Water 
13% Ethyl Alcohol
and Water 
Olive Oil 
Syrup 

Temperature
deg C

Viscosity µ

Density ρ

Morton
number

gm/cc

Surface
Tension σ
dynes/cm

poises

19
21
6
49
19
27.5
28
23
30
22
21
18

0.0102
0.0098
0.0147
0.0056
0.0103
0.580
0.0085
0.0146
0.0052
0.550
1.090
0.0915

0.998
0.998
0.999
0.989
1.000
0.866
0.782
0.864
0.782
1.262
1.288
1.143

72.9
72.6
74.8
68.1
32.8
20.7
24.5
27.8
21.8
79.2
79.9
69.9

0.26·10 −10
0.24·10 −10
1.08·10 −10
0.307·10 −11
2.78·10 −10
1.45·10 −2
4.3·10 −10
24.1·10 −10
0.89·10 −10
0.155·10 −3
0.212·10 −2
1.75·10 −7

18

0.043

1.105

71.1

4.18·10 −8

22

0.0176

0.977

43.5

1.17·10 −8

22
17

0.73
180

0.925
1.48

34.7
91

0.716·10 −2
0.92·10 −6

g µ4
ρσ3

6.4. Determination of Bubble Size
The bubble size was determined by ”weighing” a sufficient number of bubbles in the
inverted funnel (Figure 6.1) by means of an analytical balance. Since the density of air
is negligible in comparison to that of a liquid, the difference in balance reading equals
the buoyancy of the bubbles (i.e., it equals the volume of the bubbles times the density
of the liquid). The change in balance reading was always at least 0.2 gm, resulting in an

6. Experimentation

16

Figure 6.1.
Experimental setup
accuracy of measurement of volume of 1 percent. The volume of the individual bubble
was obtained by dividing the total volume by the number of bubbles collected in the
funnel. A comparison of photographs of different bubbles showed, that the bubble size
did not vary if the frequency of bubble generation remained constant. Large bubbles
from the dumping cup were weighed individually. The volume of the bubble was adjusted
for the change in pressure due to difference in depth between the level at which the rate
of rise is determined and the level of the inverted funnel. This was done by use of the
general gas law at constant temperature, taking into account the partial pressure of the
saturated vapor at test temperature6 . Tiny spherical bubbles could not be generated at
a frequency to allow a sufficient number to be collected in the funnel, hence their size
was determined from the photographic record. No correction for change in depth is then
needed.

6

Details of this correction are as follows
V1 =

p0 − p8
· V0
p1 − p8

6. Experimentation

17

6.5. Other Experimental Techniques
To avoid any changes in the volume of the bubble due to air interchange with the liquid,
the latter was saturated with air prior to actual testing. This was accomplished by
stirring the liquid and by blowing air through it.
The temperature of the liquids, with the exception of cold and hot water, was room
temperature, which varied little throughout the day. Water was cooled by circulation
through a water cooler. The water was heated with immersion heaters or obtained
directly from the hot water faucet. Both filtered and tap water were used in the tests.
Uniformity of liquid temperature was achieved by means of mechanical stirring before
each test. Frequent checks of temperature at various locations inside the tank were
made by means of immersion thermometers. In the process of stirring, small bubbles
appeared in the liquid. The irregularity of motion of these small bubbles, which were
still present after completion of the stirring, served as an indication of the presence of
residual turbulence in the liquid. In sufficient time, the motion of the small bubbles
always became regular and hence indicated that the residual turbulence, if still present,
was not large enough to affect the motion of the bubbles. The actual tests were not
begun until all the small bubbles reached the surface of the liquid.
The rate of bubble flow was then regulated by the needle valve so as to release bubbles
with a minimum spacing of 24 in. This reduced the effect of the wake created by the
passage of a bubble on the motion of a bubble following.7 At higher bubble rates, the
velocity of the individual bubble is increased.
The same precaution was observed for the larger bubbles that were formed by dumping.
An additional precaution was to rotate the dumping cup with steady speed in order to
avoid splitting of the bubble or the formation of satellites upon release. The slow passage
of the air through the brass tube inside the tank allowed the air to reach the temperature
of the liquid. Contact of the air at the nozzle tip or inside the dumping cup with the
liquid allowed saturation of the air with liquid vapor, so that the air bubble can, in each
instance, be assumed to be saturated with the vapor of the liquid in which it rises.
where

V1 = is the volume of the bubble at the camera level,
V0 = is the volume of the bubble as determined by weighing,
p0 = is the absolute pressure at the funnel,
p1 = is the absolute pressure at the camera level and
p8 = is the vapor pressure of the liquid at the test temperature.

7

Napier  showed the absence of proximity effect for air bubbles in water, ranging in equivalent
radius from 0.14 to 0.38 cm, if the frequency was below 30 bubbles per minute.

6. Experimentation

18

Figure 6.2.
Test conditions

6.6. Motion Pictures and their Evaluation
The velocity, path, and shape of the bubbles were obtained from motion pictures made
with a Mitchell 35 mm camera using a special lens attachment to permit close-ups. Film
speeds of 25 to 35 frames per second and back lighting from a white reflector were used.
For the first few tests, the film speed was obtained by photographing a rotating clock
dial. Subsequently a neon timing light with a 60-cycle voltage source was utilized. The
film speed was determined from the marks of the timing light on the film. The field
of the camera varied from 1.4 × 1.8 to 1.75 × 2.3 in. depending upon the refractive
index and horizontal depth of liquid. A transparent scale photographed in the plane of
the bubble provided the distance scale factor for the evaluation of displacement and size.
The camera lens was placed at approximately the midpoint between the liquid level and
the bottom of the tank for all tests. The camera location was in each instance sufficiently
above the nozzle tip so that the bubbles reached their terminal velocity before passing
in front of the camera. A summary of camera location and depth of liquids in the tanks
is given in Figure 6.2 and Table 6.2. Changes in bubble volume due to differences
in liquid depth were minimized by making velocity measurements over a very short
vertical displacement (less than 2 1/2 in.). The rate of rise of bubbles was determined by
measuring the displacement of a bubble from a reference point on successive frames of the
film by means of a Bausch and Lomb contour-measuring projector using a magnification
of twenty-five (Figure 6.3). These displacements were then plotted against the frame
number. The straight-line plot indicates that the velocity of the bubble remained constant
during the time it passed the field of the camera. From the slope of the line, the frame
speed, the scale factor, and the velocity of the bubble is computed.

6. Experimentation

19

Table 6.2.
Test conditions
Aver.
Temp.
deg C

Tank

Width
of Tank
W, inches

Height
of Liquid
h, inches

h/W

Height of Liquid
above camera
h1 , inches

Water (filtered)
Water (Tap)

19
20
22
20

Large
Large
Medium
Insert

36
36
12
6

48
48
31.5
36.5

1.33
1.33
2.62
3.33

22.5
25
15.5
20

Water (Filtered)

6
5

Large
Medium

36
12

48
30

1.33
2.50

23
14

Water (Tap)

48
49
48

Large
Medium
Small

36
12
6

49
39
19

1.36
3.25
3.17

23.5
22.75
9

Glim Solution

19

Small

6

20

3.33

9.5

Mineral Oil

28
27

Medium
Insert

12
6

36
36.5

3.00
3.33

19
19.5

Varsol

28
28
25

Large
Medium
Insert

36
12
6

39.5
38.5
36

1.10
3.21
3.33

24.5
21.5
19

Turpentine

23

Small

6

19.5

3.25

9

Methyl Alcohol

30

Medium

12

39

3.25

24

Water and 62% 22
Corn Syrup

Small

6

20

3.33

9.5

Water and 68% 21
Corn Syrup

Small

6

19

3.17

9

Liquid

6. Experimentation

Figure 6.3.
Evaluation of Velocity of Rise of Bubbles

20

7. Results
7.1. Terminal Velocity of Air Bubbles
The results of tests to determine the velocity of rise of air bubbles in various liquids
are most conveniently presented as a function of the equivalent radius of the bubble,
defined as the radius of a sphere having the same volume as the bubble. Figure 7.1 till
Figure 7.10 show the terminal velocity of air bubbles rising freely in tap (unfiltered)
and in filtered water (including data from other investigators), in water containing Glim,
in mineral oil, Varsol, turpentine, methyl alcohol, and two corn syrup-water mixtures
as a function of the equivalent radius. Figure 7.11 presents Bryn’s results in an ethyl
alcohol-water mixture and two glycerine-water mixtures.1 Figure 7.12 summarizes all
velocity curves (except those for tap water). A compilation of the properties of the liquids
is given in Table 6.1. In general, the results as seen from Figure 7.12 indicate that
for small (spherical) air bubbles of given volume, the viscosity of the liquid is the most
important property determining the rate of rise. Very large bubbles (spherical caps) rise
independently of the properties of the liquid.

7.2. Wall and Proximity Effects
As indicated previously, the effect of the container walls on the velocity of a bubble
had to be determined if the results of tests conducted in a tank of limited dimensions
were to be applied to bubble motion in an infinite medium. Tests were, therefore, conducted in tanks of different sizes in water, Varsol, and mineral oil. Figure 7.1 gives
the results of tests conducted with filtered water by several investigators including the
Taylor Model Basin. The cross sections of the containers used are also indicated in the
figure. No wall effect is noticeable from these results. For example, Gorodetskaya’s
results for bubbles ranging from 0.01 to 0.07 cm rising in a tube of 5 cm diameter show
no wall effect when compared with results of tests conducted in larger containers. The
results of the present experiments, given in Figure 7.2, 7.3, 7.4, 7.6, and 7.7, show
1

The results of the 81 percent (by weight) glycerine-water mixture have been omitted. Bryn presented
these results in terms of drag coefficient and Reynolds number, from which the terminal velocity
can be computed. In the region of bubble size where the rate of rise is shown to be a function of
size only (hence a common velocity curve for all liquids; see Figure 7.12), the velocity curve for
the 81 percent mixture falls appreciably above the common curve. The discrepancy is probably
due to erroneous evaluation of the two dimensionless parameters for the 81 percent glycerine-water
mixture.

7. Results

22

within experimental accuracy, the absence of any wall effect for the range of bubble sizes
tested. Subsequent tests in the other liquids were made in the small tank only and the
results may be applied to the case of an infinite medium. No systematic investigation
was made of vertical proximity effect, i.e., the effect of the wake created by the passage
of a bubble on the motion of a bubble rising at a distance below. Such effects were
avoided in the experiments by sufficient spacing between the bubbles. However, the
results of a few special observations indicate that proximity effects may be appreciable.
For example, tests in mineral oil show an increase of 9 percent and 89 percent for bubbles
of equivalent radius of 0.17 cm, rising 7.7 cm and 3.2 cm apart, respectively. Napier 
observed an increase of 6 percent for bubbles of 0.14 cm radius in water, rising 6 cm apart.
The presence of the wake in the liquid thus results in higher velocities of rise of the bubble.

7. Results

Figure 7.1.
Terminal Velocity of Air Bubbles in Filtered or Distilled Water as a Function of Bubble Size

23

7. Results

Figure 7.2.
Terminal Velocity of Air Bubbles in Tap Water as a Function of Bubble Size

24

7. Results

Figure 7.3.
Terminal Velocity of Air Bubbles in Cold Filtered Water as a Function of Bubble Size

25

7. Results

Figure 7.4.
Terminal Velocity of Air Bubbles in Hot Tap Water as a Function of Bubble Size

26

7. Results

Figure 7.5.
Terminal Velocity of Air Bubbles in Water Containing Glim as a Function of Bubble Size

27

7. Results

Figure 7.6.
Terminal Velocity of Air Bubbles in Mineral Oil as a Function of Bubble Size

28

7. Results

Figure 7.7.
Terminal Velocity of Air Bubbles in Varsol as a Function of Bubble Size

29

7. Results

Figure 7.8.
Terminal Velocity of Air Bubbles in Turpentine as a Function of Bubble Size

30

7. Results

Figure 7.9.
Terminal Velocity of Air Bubbles in Methyl Alcohol as a Function of Bubble Size

31

7. Results

Figure 7.10.
Terminal Velocity of Air Bubbles in Corn Syrup-Water Mixtures as a Function of Bubble Size

32

7. Results

Figure 7.11.
Terminal Velocity of Air Bubbles in Liquids as Obtained from Bryn’s Data

33

7. Results

Figure 7.12.
Comparison of Terminal Velocities of Air Bubbles in Various Liquids

34

7. Results

35

7.3. Non-dimensional Presentation of Bubble Data
In a previous section it was pointed out that presentation of the experimental data on
air bubbles in terms of dimensionless products gives complete correlation provided the
variables considered in the analysis are complete and pertinent. The results of the Taylor
Model Basin bubble tests and those of Arnold,  Bond and Newton,  and Bryn 
are given in terms of the drag coefficient, Reynolds number and the parameter Mo in
Figure 7.13.2 Figure 7.14 presents the bubble data in terms of the drag coefficient,
Weber number, and the parameter Mo. The curve for filtered or distilled water at a
temperature of 19 deg C was drawn through points obtained from the experiments of
Bryn and the Taylor Model Basin tests.
Examination of Figure 7.13 or 7.14 shows no systematic arrangement of the curves
with change in the parameter Mo, which is constant for a specific liquid. It can, therefore,
be concluded that neither of the non-dimensional sets presented nor any other complete
set using the same six variables (namely, velocity, acceleration of gravity, density and
viscosity of the liquid, surface tension, and equivalent radius) is sufficient for a complete
description of bubble motion.
The question now arises whether correlation of bubble data could be obtained by using
two additional dimensionless parameters, for example, the liquid to air viscosity and
density ratios or the Reynolds number inside the bubble and the density ratio, etc.
The results of the experiments conducted do not permit conclusions regarding the
importance of these parameters. A short discussion of the significance of the internal
Reynolds number will be given in subsequent sections.

2

The results for tap water and for water containing Glim are not shown. They will be discussed in
subsequent sections.

7. Results

Figure 7.13.
Drag Coefficient as a Function of Reynolds number for Air Bubbles Rising at their Terminal Velocity in Various Liquids

36

7. Results

Figure 7.14.
Drag Coefficient as a Function of Weber number for Air Bubbles Rising at their Terminal Velocity in Various Liquids

37

7. Results

38

7.4. Spherical Bubbles
It was observed in the experiments that, as the bubble size was increased, a change in
bubble shape from spherical to ellipsoidal to spherical cap shape occurred in all liquids.
Very small bubbles are spherical. Larger bubbles are flattened, i.e., ellipsoidal in shape,
whereas very large bubbles assume a spherical cap shape. Of course, the volumes at
which these transitions occur vary with the liquids. Photographs of typical shapes are
shown in Figure 7.15. It should be noted here that some of the shapes shown in these
photographs are instantaneous shapes, since the shape of large bubbles does not remain
constant during the ascent. An exception are bubbles rising in a highly viscous medium
(e.g., mineral oil and corn syrup). The results for the spherical bubbles only are plotted
in terms of the drag coefficient and the Reynolds number in Figure 7.16, with Reynolds
numbers ranging up to about 400. The drag curve for rigid spheres is also included
. From it, the following can be observed: The drag curves of spherical bubbles in
the various liquids fall between two limiting curves. As upper limit, the drag curve
of rigid spheres is obtained, while the lower limit is the drag curve for fluid spheres.
With decreasing Reynolds number, the rigid sphere curve connects with the straight
line of Stokes’ Law, while the fluid sphere curve connects with the line of HadamardRybczynski’s Law. The curve for the fluid spheres was obtained by drawing the lower
envelope to the experimental curve. Its accuracy can be confirmed by additional tests
in other liquids or by extension of the theoretical solution into regions beyond that
of very slow flow. It will be noted from Figure 7.16, that the curve for mineral oil,
for example, follows the straight line of Hadamard-Rybczynski’s Law over a certain
region of Reynolds numbers. This indicates that the boundary conditions assumed in
the analytical solution for fluid spheres are actually fulfilled and that circulation exists
inside the bubble. Circulation inside bubbles has been observed experimentally . The
experimental curves of Figure 7.16 also indicate an interesting aspect of the phenomenon
of bubble motion, namely that with decreasing Reynolds number, the drag coefficient of
the bubbles becomes equal to the drag of rigid spheres. This transition may occur at a
Reynolds number of about 40 (as for filtered and distilled water) or may not take place
until very low Reynolds numbers are reached, i.e., well within the region of slow flow (as
for olive oil  or very viscous syrup ). Thus, from the experimental data available,
it appears certain, that tiny air bubbles rising in any liquid follow Stokes’ Law. For
bubbles behaving like rigid bodies, thus indicating absence of motion inside the bubble,
the internal Reynolds number (although non-vanishing) is of no significance in describing
the rising motion of the bubbles. Likewise, the internal Reynolds number cannot be used
to predict the transition point at which the drag of the bubbles becomes less than that of
corresponding rigid spheres. Beyond this transition point, the internal Reynolds number
might be of importance in describing the motion of the bubbles. Surface tension tends
to make the surface area of the bubble as small as possible. For a given volume, the
configuration of minimum surface area is a sphere. This effect of surface tension would
be most pronounced for bubbles of small radii.

7. Results

39

(a)

(b)

Figure 7.15.
Typical Shapes of Bubbles of Several Volumes in the Various Liquids

7. Results

Figure 7.16.
Drag Coefficient as a Function of Reynolds Number for Spherical Air Bubbles Rising at Their Terminal Velocity in Various Liquids

40

7. Results

41

7.5. Ellipsoidal Bubbles
For larger bubble sizes, the surface forces, which are essential in maintaining the spherical
shape of a bubble, become smaller in comparison to the viscous and hydrodynamic
forces, and flattening of the bubble occurs. This flattening to approximately an oblate
spheroid results in higher drag as compared to a sphere of the same volume. Figure
7.17 shows the drag curves of ellipsoidal and spherical cap bubbles in terms of the
Reynolds number. The estimated extent of the regions of ellipsoidal and spherical cap
bubbles is indicated in the figure. It will be observed that the region of ellipsoidal
bubbles for the various liquids occurs at different ranges of Reynolds number, that for
liquids of low Morton number a minimum in the drag curves is reached at Reynolds
numbers of the order of 250, and that these minima occur near the transition from
spherical to ellipsoidal shape. Such minima are not obtained for liquids of high Morton
number. The drag coefficients of bubbles in such liquids decrease until a constant value
for the drag coefficient (spherical caps) is attained. In the ellipsoidal region the curves
are arranged according to the magnitude of the Morton number, indicating that the
liquid properties contained in this parameter, namely surface tension, viscosity, and
density, are of primary importance in the motion of these bubbles. Thus, in this region
the Reynolds number inside the bubble is of no importance in the description of bubble
rise, since a correlation was obtained in terms of drag coefficient, Reynolds number, and
Morton number.
In Figure 7.18 the drag coefficient of the bubbles was plotted as a function of Weber
number. It is seen that, for the liquids tested, transition to a constant value of drag
coefficient (spherical cap region) is reached at a Weber number of about 20.

7. Results

Figure 7.17.
Drag Coefficient as a Function of Reynolds Number for Ellipsoidal and Spherical Cap Bubbles in Various Liquids

42

7. Results

Figure 7.18.
Drag Coefficient as a Function of Weber Number for Ellipsoidal and Spherical Cap Bubbles in Various Liquids

43

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