cabook.DVI - chap3.pdf

BUBBLE OR DROPLET TRANSLATION

3.1 INTRODUCTION

In the last chapter it was assumed that the particles were rigid and therefore

were not deformed, fissioned or otherwise modified by the flow. However,

there are many instances in which the particles that comprise the disperse

phase are radically modified by the forces imposed by the continuous phase.

Sometimes those modifications are radical enough to, in turn, affect the

flow of the continuous phase. For example, the shear rates in the continuous

phase may be sufficient to cause fission of the particles and this, in turn,

may reduce the relative motion and therefore alter the global extent of phase

separation in the flow.

The purpose of this chapter is to identify additional phenomena and is-

sues that arise when the translating disperse phase consists of deformable

particles

, namely bubbles, droplets or fissionable solid grains.

3.2 DEFORMATION DUE TO TRANSLATION

3.2.1 Dimensional analysis

Since the fluid stresses due to translation may deform the bubbles, drops

or deformable solid particles that make up the disperse phase, we should

consider not only the parameters governing the deformation but also the

consequences in terms of the translation velocity and the shape. We con-

centrate here on bubbles and drops in which surface tension,

,actsasthe

force restraining deformation. However, the reader will realize that there

would exist a similar analysis for deformable elastic particles. Furthermore,

the discussion will be limited to the case of

steady

translation, caused by

gravity,

. Clearly the results could be extended to cover translation due

to fluid acceleration by using an effective value of

as indicated in section

2.4.2.

The characteristic force maintaining the sphericity of the bubble or drop

is given by

. Deformation will occur when the characteristic anisotropy

in the fluid forces approaches

; the magnitude of the anisotropic fluid

force will be given by

∞

for

∞

R/ν

1orby

∞

for

∞

R/ν

1. Thus defining a Weber number,

∞

R/S

,defor-

mation will occur when

We/Re

approaches unity for

1orwhen

approaches unity for

1. But evaluation of these parameters requires

knowledge of the terminal velocity,

∞

, and this may also be a function

of the shape. Thus one must start by expanding the functional relation of

equation 2.87 which determines

∞

to include the Weber number:

(

Re, W e, F r

) = 0

(3.1)

This relation determines

∞

where

is given by equations 2.85. Since all

three dimensionless coefficients in this functional relation include both

∞

and

, it is simpler to rearrange the arguments by defining another nondi-

mensional parameter, the Haberman-Morton number (1953),

,thatisa

combination of

,and

but does not involve

∞

.TheHaberman-

Morton number is defined as

gμ

−

(3.2)

In the case of a bubble,

and therefore the factor in parenthesis

is usually omitted. Then

becomes independent of the bubble size. It

follows that the terminal velocity of a bubble or drop can be represented by

functional relation

(

Re, Hm, F r

)=0 or

∗

(

Re, Hm, C

) = 0

(3.3)

and we shall confine the following discussion to the nature of this relation

for bubbles (

Some values for the Haberman-Morton number (with

/ρ

=0) for

various saturated liquids are shown in figure 3.1; other values are listed in

table 3.1. Note that for all but the most viscous liquids,

is much less

than unity. It is, of course, possible to have fluid accelerations much larger

than

; however, this is unlikely to cause

values greater than unity in

practical multiphase flows of most liquids.

Figure 3.1.

Values of the Haberman-Morton parameter,

, for various

pure substances as a function of reduced temperature where

is the triple

point temperature and

is the critical point temperature.

Table 3.1.

Values of the Haberman-Morton numbers,

gμ

/ρ

,for

various liquids at normal temperatures.

Filtered Water

−

Turp entine

−

Methyl Alcohol

−

Olive Oil

−

Mineral Oil

−

Syrup

3.2.2 Bubble shapes and terminal velocities

Having introduced the Haberman-Morton number, we can now identify

the conditions for departure from sphericity. For low Reynolds numbers

(

1) the terminal velocity will be given by

∝

. Then the shape

will deviate from spherical when

≥

or, using

∝

and

−

,when

≥

−

(3.4)

Thus if

Hm <

1 all bubbles for which

1 will remain spherical. How-

ever, there are some unusual circumstances in which

Hm >

1 and then there

will be a range of

,namely

−

<Re<

1, in which significant depar-

ture from sphericity might occur.

For high Reynolds numbers (

1) the terminal velocity is given by

≈

(1) and distortion will occur if

We >

1. Using

=1 and

−

it follows that departure from sphericity will occur when

−

(3.5)

Consequently, in the common circumstances in which

Hm <

1, there exists a

range of Reynolds numbers,

Re < Hm

−

, in which sphericity is maintained;

nonspherical shapes occur when

Re > Hm

−

.For

Hm >

1 departure from

sphericity has already occurred at

Re <

1 as discussed above.

Experimentally, it is observed that the initial departure from sphericity

causes ellipsoidal

bubbles that may osc

illate in shape and have oscillatory

trajectories (Hartunian and Sears 1957). As the

bubble size is further in-

creased to the point at which

≈

20, the bubble acquires a new asymp-

totic shape, known as a

spherical-cap bubble.

A photograph of a typical

spherical-cap bubble is shown in figure 3.2; the notation used to describe

the approximate geometry of these bubbles is sketched in the same figure.

Spherical-cap bubbles were first investigated by Davies and Taylor (

1950),

who observed that the terminal velocity is simply related to the radius of

curvature of the cap,

, or to the equivalent volumetric radius,

,by

∞

(

)

(3.6)

Assuming a typical laminar drag coefficient of

5, a spherical solid

particle with the same volume would have a terminal velocity,

∞

=(8

)

(3.7)

that is substantially higher than the spherical-cap bubble. From equation

3.6 it follows that the effective

for spherical-cap bubbles is 2

67 based

on the area

πR

Wegener and Parlange (1973) have reviewed the literature on spherical-

cap bubbles. Figure 3.3 is taken from their review and shows that the

value of

∞

(

)

reaches a value of about 1 at a Reynolds number,

∞

/ν

, of about 200 and, thereafter, remains fairly constant. Vi-

sualization of the flow reveals that, for Reynolds numbers less than about

360, the wake behind the

bubble is laminar and takes the form of a toroidal

vortex (similar to a Hill (1894) spherical vortex) shown in the left-hand pho-

tograph of figure 3.4. The wake undergoes transition to turbulence about

= 360, and

bubbles at higher

have turbulent wakes as illustrated

in the right side of figure 3.4. We should add that scuba divers have long

observed that spherical-cap bubbles rising in the ocean seem to have a max-

Figure 3.2.

Photograph of a spherical cap bubble rising in water (from

Davenport, Bradshaw, and Richardson 1967) with the notation used to

describe the geometry of spherical cap bubbles.

imum size of the order of 30

in diameter. When they grow larger than

this, they fission into two (or more) bubbles. However, the author has found

no quantitative study of this fission process.

In closing, we note that the terminal velocities of the bubbles discussed

here may be represented according to the functional relation of equations 3.3

as a family of

(

) curves for various

. Figure 3.5 has been extracted

from the experimental data of Haberman and Morton (1953) and shows the

dependence of

(

)on

at intermediate

. The curves cover the

spectrum from the low

spherical bubbles to the high

spherical cap

bubbles. The data demonstrate that, at higher values of

, the drag coef-

ficient makes a relatively smooth transition from the low Reynolds number

result to the spherical cap value of about 2

7. Lower values of

result in

a deep minimum in the drag coefficient around a Reynolds number of about

200.

3.3 MARANGONI EFFECTS

Even if a bubble remains quite spherical, it can experience forces due to

gradients in the surface tension,

, over the surface that modify the sur-

face boundary conditions and therefore the translational velocity. These are

called Marangoni effects. The gradients in the surface tension can be caused

by a number of different factors. For example, gradients in the temperature,

solvent concentration, or electric potential can create gradients in the surface

tension. The

thermocapillary

effects due to temperature gradients have been

Figure 3.3.

Data on the terminal velocity,

∞

(

)

1

2

,andtheconi-

cal angle,

, for spherical-cap bubbles studied by a number of different

investigators (adapted from Wegener and Parlange 1973).

Figure 3.4.

Flow visualizations of spherical-cap bubbles. On the left is

a bubble with a laminar wake at

≈

180 (from Wegener and Parlange

1973) and, on the right, a bubble with a turbulent wake at

≈

000

(from Wegener, Sundell and Parlange 1971, reproduced with permission of

the authors).

Figure 3.5.

Drag coefficients,

, for bubbles as a function of the

Reynolds number,

, for a range of Haberman-Morton numbers,

as shown. Data from Haberman and Morton (1953).

explored by a number of investigators (for example, Young, Goldstein, and

Block 1959) because of their importance in several technological contexts.

For most of the range of temperatures, the surface tension decreases linearly

with temperature, reaching zero at the critical point. Consequently, the con-

trolling thermophysical property,

dS/dT

, is readily identified and more or

less constant for any given fluid. Some typical data for

dS/dT

is presented

in table 3.2 and reveals a remarkably uniform value for this quantity for a

wide range of liquids.

Surface tension gradients affect free surface flows because a gradient,

dS/ds

, in a direction,

, tangential to a surface clearly requires that a shear

stress act in the negative

direction in order that the surface be in equilib-

rium. Such a shear stress would then modify the boundary conditions (for

example, the Hadamard-Rybczynski conditions used in section 2.2.2), thus

altering the flow and the forces acting on the bubble.

As an example of the Marangoni effect, we will examine the steady mo-

tion of a spherical bubble in a viscous fluid when there exists a gradient

of the temperature (or other controlling physical property),

dT /dx

,inthe

direction of motion (see figure 2.1). We must first determine whether the

temperature (or other controlling property) is affected by the flow. It is il-

lustrative to consider two special cases from a spectrum of possibilities. The

first and simplest special case, that is not so relevant to the thermocapillary

phenomenon, is to assume that

dT /dx

)

throughout the flow field

so that, on the surface of the bubble,

dθ

−

sin

(3.8)

Much more realistic is the assumption that thermal conduction dominates

the heat transfer (

∇

= 0) and that there is no heat transfer through the

surface of the bubble. Then it follows from the solution of Laplace’s equation

for the conductive heat transfer problem that

dθ

−

sin

(3.9)

The latter is the solution presented by Young, Goldstein, and Block (1959),

but it differs from equation 3.8 only in terms of the effective value of

dS/dT

Here we shall employ equation 3.9 since we focus on thermocapillarity, but

other possibilities such as equation 3.8 should be borne in mind.

For simplicity we will continue to assume that the bubble remains spher-

ical. This assumption implies that the surface tension differences are small

Table 3.2.

Values of the temperature gradient of the surface tension,

−

dS/dT

, for pure liquid/vapor interfaces (in

kg/s

Water

−

Methane

−

Hydrogen

−

Butane

−

Helium-4

−

Carbon Dioxide

−

Nitrogen

−

Ammonia

−

Oxygen

−

Toluene

−

Sodium

−

Freon-12

−

Mercury

−

Uranium Dioxide

−

compared with the absolute level of

and that the stresses normal to the

surface are entirely dominated by the surface tension.

With these assumptions the tangential stress boundary condition for the

spherical bubble becomes

∂u

∂r

−

dθ

= 0

(3.10)

and this should replace the Hadamard-Rybczynski condition of zero shear

stress that was used in section 2.2.2. Applying the boundary condition

given by equations 3.10 and 3.9 (as well as the usual kinematic condition,

(

)

= 0) to the low Reynolds number solution given by equations 2.11,

2.12 and 2.13 leads to

−

;

(3.11)

and consequently, from equation 2.14, the force acting on the bubble becomes

−

πρ

−

πR

(3.12)

In addition to the normal Hadamard-Rybczynski drag (first term), we can

identify a Marangoni force, 2

πR

(

dS/dx

), acting on the bubble in the di-

rection of

decreasing

surface tension. Thus, for example, the presence of a

uniform temperature gradient,

dT /dx

, would lead to an additional force

on the bubble of magnitude 2

πR

(

−

dS/dT

)(

dT /dx

) in the direction of the

warmer fluid since the surface tension decreases with temperature. Such

thermocapillary effects have been observed and measured by Young, Gold-

stein, and Block (1959) and others.

Finally, we should comment on a related effect caused by surface contam-

inants that increase the surface tension. When a bubble is moving through

liquid under the action, say, of gravity, convection may cause contaminants

to accumulate on the downstream side of the bubble. This will create a posi-

tive

dS/dθ

gradient that, in turn, will generate an effective shear stress acting

in a direction opposite to the flow. Consequently, the contaminants tend to

immobilize the surface. This will cause the flow and the drag to change from

the Hadamard-Rybczynski solution to the Stokes solution for zero tangen-

tial velocity. The effect is more pronounced for smaller bubbles since, for a

given surface tension difference, the Marangoni force becomes larger rela-

tive to the buoyancy force as the bubble size decreases. Experimentally, this

means that surface contamination usually results in Stokes drag for spher-

ical bubbles smaller than a certain size and in Hadamard-Rybczynski drag

for spherical bubbles larger than that size. Such a transition is observed

in experiments measuring the rise velocity of bubbles and can be see in the

data of Haberman and Morton (1953) included as figure 3.5. Harper, Moore,

and Pearson (1967) have analyzed the more complex hydrodynamic case of

higher Reynolds numbers.

3.4 BJERKNES FORCES

Another force that can be important for bubbles is that experienced by a

bubble placed in an acoustic field. Termed the Bjerknes force, this non-linear

effect results from the the finite wavelength of the sound waves in the liquid.

The frequency, wavenumber, and propagation speed of the stationary acous-

tic field will be denoted by

and

respectively where

ω/c

.The

finite wavelength implies an instantaneous pressure gradient in the liquid

and, therefore, a

buoyancy

force acting on the bubble.

To model this we express the instantaneous pressure,

{

∗

sin(

κx

)

iωt

}

(3.13)

where

is the mean pressure level, ̃

∗

is the amplitude of the sound waves

and

is the direction of wave propagation. Like any other pressure gradient,

this produces an instantaneous force,

, on the bubble in the

direction

given by

−

πR

(3.14)

where

is the instantaneous radius of the spherical bubble. Since both

and

dp/dx

contain oscillating components, it follows that the combination

of these in equation 3.14 will lead to a nonlinear, time-averaged component

, that we will denote by

. Expressing the oscillations in the volume

or radius by

{

φe

iωt

}

(3.15)

one can use the Rayleigh-Plesset equation (see section 4.2.1) to relate the

pressure and radius oscillations and thus obtain

{

}

∗

(

−

)sin(

κx

)

(

−

)

+(4

ω/R

)

(3.16)

where

is the natural frequency of volume oscillation of an individual

bubble (see section 4.4.1) and

is the effective viscosity of the liquid in

damping the volume oscillations. If

is not too close to

, a useful approx-

imation is

{

}≈

∗

sin(

κx

)

/ρ

(

−

)

(3.17)

Finally, substituting equations 3.13, 3.15, 3.16, and 3.17 into 3.14 one

obtains

−

πR

{

}

∗

cos(

κx

)

≈−

πκR

( ̃

∗

)

sin(2

κx

)

(

−

)

(3.18)

This is known as the primary Bjerknes force since it follows from some of

the effects discussed by that author (Bjerknes 1909). The effect was first

properly identified by Blake (1949).

The form of the primary Bjerknes force produces some interesting bubble

migration patterns in a stationary sound field. Note from equation (3.18)

that if the excitation frequency,

, is less than the bubble natural frequency,

, then the primary Bjerknes force will cause migration of the bubbles

away from the nodes in the pressure field and toward the antinodes (points

of largest pressure amplitude). On the other hand, if

ω>ω

the bubbles will

tend to migrate from the antinodes to the nodes. A number of investigators

(for example, Crum and Eller 1970) have observed the process by which

small bubbles in a stationary sound field first migrate to the antinodes,

where they grow by rectified diffusion (see section 4.4.3) until they are larger

than the resonant radius. They then migrate back to the nodes, where they

may dissolve again when they experience only small pressure oscillations.

Crum and Eller (1970) and have shown that the translational velocities of

migrating bubbles are compatible with the Bjerknes force estimates given

above.

Figure 3.6.

Schematic of a bubble undergoing growth or collapse close to

a plane boundary. The associated translational velocity is denoted by

3.5 GROWING OR COLLAPSING BUBBLES

When the volume of a bubble changes significantly, that growth or collapse

can also have a substantial effect upon its translation. In this section we

return to the discussion of high

flow in section 2.3.3 and specifically

address the effects due to bubble growth or collapse. A bubble that grows or

collapses close to a boundary may undergo translation due to the asymmetry

induced by that boundary. A relatively simple example of the analysis of this

class of flows is the case of the growth or collapse of a spherical bubble near

a plane boundary, a problem first solved by Herring (

1941) (see also Davies

and Taylor 1942, 1943). Assuming that the only translational motion of

the bubble is perpendicular to the plane boundary with velocity,

,the

geometry of the bubble and its image in the boundary will be as shown

in figure 3.6. For convenience, we define additional polar coordinates, ( ̆

with origin at the center of the image bubble. Assuming inviscid, irrotational

flow, Herring (1941) and Davies and Taylor (1943) constructed the velocity

potential,

, near the bubble by considering an expansion in terms of

R/H

where

is the distance of the bubble center from the boundary. Neglecting

all terms that are of order

or higher, the velocity potential can be

obtained by superimposing the individual contributions from the bubble

source/sink, the image source/sink, the bubble translation dipole, the image

dipole, and one correction factor described below. This combination yields

−

cos

−

cos

2 ̆

−

cos

(3.19)

The first and third terms are the source/sink contributions from the bubble

and the image respectively. The second and fourth terms are the dipole

contributions due to the translation of the bubble and the image. The last

term arises because the source/sink in the bubble needs to be displaced

from the bubble center by an amount

normal to the wall in order

to satisfy the boundary condition on the surface of the bubble to order

. All other terms of order

or higher are neglected in this

analysis assuming that the bubble is sufficiently far from the boundary so

that

. Finally, the sign choice on the last three terms of equation

3.19 is as follows: the upper, positive sign pertains to the case of a solid

boundary and the lower, negative sign provides an approximate solution for

a free surface boundary.

It remains to use this solution to determine the translational motion,

(

), normal to the boundary. This is accomplished by invoking the condi-

tion that there is no net force on the bubble. Using the unsteady Bernoulli

equation and the velocity potential and fluid velocities obtained from equa-

tion (3.19), Davies and Taylor (1943) evaluate the pressure at the

bubble

surface and thereby obtain an expression for the force,

, on the bubble in

the

direction:

−

(3.20)

Adding the effect of buoyancy due to a component,

, of the gravitational

acceleration in the

direction, Davies and Taylor then set the total force

equal to zero and obtain the following equation of motion for

(

πR

= 0

(3.21)

In the absence of gravity this corresponds to the equation of motion first

obtained by Herring (1941). Many of the studies of growing and collapsing

bubbles near boundaries have been carried out in the context of underwater

explosions (see Cole 1948). An example illustrating the solution of equation

3.21 and the comparison with experimental data is included in figure 3.7

taken from Davies and Taylor (1943).

Another application of this analysis is to the translation of cavitation

bubbles near walls. Here the motivation is to understand the development

of impulsive loads on the solid surface. Therefore the primary focus is on

bubbles close to the wall and the solution described above is of limited value

since it requires

. However, considerable progress has been made in

recent years in developing analytical methods for the solution of the inviscid

Figure 3.7.

Data from Davies and Taylor (1943) on the mean radius and

central elevation of a bubble in oil generated by a spark-initiated explosion

of 1

ergs

situated 6

below the free surface. The two mea-

sures of the bubble radius are one half of the horizontal span (

)andone

quarter of the sum of the horizontal and vertical spans (

). Theoretical

calculations using Equation (3.21) are indicated by the solid lines.

free surface flows of bubbles near boundaries (Blake and Gibson

1987). One

of the concepts that is particularly useful in determining the direction of

bubble translation is based on a property of the flow first introduced by

Kelvin (see Lamb 1932) and called the Kelvin impulse. This vector property

applies to the flow generated by a finite particle or bubble in a fluid; it is

denoted by

anddefinedby

φn

(3.22)

where

is the velocity potential of the irrotational flow,

is the surface of

the bubble, and

is the outward normal at that surface (defined as positive

into the bubble). If one visualizes a bubble in a fluid at rest, then the Kelvin

impulse is the impulse that would have to be applied to the bubble in order

to generate the motions of the fluid related to the bubble motion. Benjamin

and Ellis (1966) were the first to demonstrate the value of this property in

determining the interaction between a growing or collapsing bubble and a

nearby boundary (see also Blake and Gibson

1987).