Q f i i c e of NCJ'J~D R e s e a r c h

D e p o r t m e n t of the W a v y

C o n t r a c t N o n r - 2231 2 8)

THEORY OF T H E ACOi iST IC ABSORPTION B Y A GAS BUBBLE IN A LIQUID

Office of Naval Research Department of the Navy Contract Nonr-220(28)

THEORY O F THE ACOUSTIC ABSORPTION

BY A GAS BUBBLE IN A LIQUID

by

Din-Yu Hsieh and Milton S. P lesse t

Reproduction in whole o r in pa r t is permit ted fo r any purpose of the United States Government

Engineering Division California Institute of Technology

Report No. 85-19 November, 1961

THEORY OF THE ACOUSTIC ABSORPTION

BY A GAS BUBBLE IN A LIQUID

by

Din-Yta Hsieh and Milton S. Plesset

Abstract

A complete analysis of acoustic absorption by a spherical gas

bubble i s developed by the application of the classical Rayleigh method.

The absorption considered i s that due to the viscosity and heat conduct-

ion of the gas bubble. Specific results a re presented for the S-wave

scatter and absorption for the case of an air bubble in water, and the

absorption effects of viscosity and heat conduction alone a re calculated

explicitly. The results found here a re of similar magnitude to those

found by Pfriem and Spitzer who used an approximate procedure.

Introduction

While the scattering of sound waves by a spherical gas bubble in a

1 liquid has received extensive treatment by the familiar Rayleigh procedure , a complete analysis of the acoustic absorption does not appear to be avail-

able. This absorption is a consequence of viscosity and heat conduction

2 in the gas bubble. Results for the absorption have been obtained by Pfriem

3 and Spitzer , among others, but these approaches were based on approxi-

mate models which a re limited to the case of spherically symmetric

bubble oscillations. It follows that the scatter and absorption is determined

only for the case in which the S-wave i s significant. It i s found here that

the long wavelength limit of the present analysis is in approximate agree-

ment with the results of previous calculations.

The calculation here will be made first of the absorption due t o the

viscosity of the gas, and then the absorption due to both the viscosity and

the heat conduction of the gas i s determined. It i s shown in this way that

the thermal absorption i s more important than the viscous absorption in

the range of bubble sizes and acoustic frequencies of general interest.

The numerical results which are given below show the scattering and

-absorption cross sections for an air bubble in water with viscosity only

compared with the corresponding cross sections with heat conduction only.

1 See, for example, Lord Rayleigh, IrTheory of Sound, Vol. 11, p. 282, Dover Publications (New York, 1945); V. C. Anderson, J.Acous. Soc.

-3 A m . , &2, 426-431 (1950).

L H. Pfriem, Akustische Zeitschrift, 2, 2.02- 21 2 (1 940). 2 -I

Lo Spitzer, J r . , OSRD Report M05, Sec. 6.1-sr20-918 (1943).

Viscous Absorption

The gas bubble will be taken to have the equilibrium radius - a , and

a spherical coordinate system will be used with origin at the center of the

bubble. The radial distance from the origin i s r and the polar angle be-

tween the radius vector and the z-axis i s It, . The disturbance represent-

ing the sound wave field outside the gas bubble can be described by the

velocity potential, 9 , which satisfies the wave equation

where c i s the sound speed in the liquid. The appropriate solution of W

Eq. ( I ) , in the presence of the gas bubble, will be composed of an

incident plane wave of unit amplitude and of angular frequency o and

scattered waves:

where kw i s the wave number given by kw = w/cw, hc is the spherical

Hankel function of the first kind of order C , Pd is the Legendre poly-

nomial of degree 8 , and the S a re constants to be determined. Using E

the familiar expansion of a plane wave

ikz ikrcos* e = e = iC (24 +l)jC (kr) PC (cos*)

where j i s the spherical Bessel function of order C , one may write E

the su~erposit ion of incident and scattered waves as

The acoustic'velocity field is given by Vy , and the acoustic pressure i s

given by -pway /at. If this pressure i s written in the form

where Pw is the mean, or unperturbed, pressure then

The mean pressure Pw is related to the equilibrium gas pressure Po

as follows:

where tr i s the surface tension constant. In particular, the perturbation

pressure, Pwp, at the bubble wall, r = a , i s given by

P w P = i u p w [ie (28+l)je(kWa)+sChC(kwa)] P 4 (cos$)e - iot , ( 6 )

and the radial component of the velocity a t r = a is

v r = [ ie ( 2 ~ + l ) ~ ~ ( k w a) + se hl(kwa)] kwPe (cos $) e - iwt (7)

Within the gas bubble, the motion of the fluid i s governed by the

equation of continuity

4 ' by the momentum equation

and by the equation of state

In these equations p is the density, Po i s the equilibrium value of the

density, v i s the fluid velocity, q is the coefficient of shear viscosity, *r

p i s the coefficient of bulk viscosity, and y i s the ratio of the specific

heat of the gas at constant pressure to the specific heat a t constant volume.

In Eq. (9) it has been assumed that there a re no body forces, and it has

been assumed further that the coefficients of viscosity a re constants,

Eq. (1 0) states the gas expands and contracts adiabatically. It may also

be remarked that the s t ress on a surface element with outward unit normal

The motion will be taken to be irrotational and then the system of

equations will be linearized. If one writes

4 ~ . Goldstein, "Lectures on Fluid Mechanics", pp. 89-90, Interscience Publishers, Inc. (New York, 1960).

d

5 L. M. Milne- Thomson, "Theoretical Hydrodynamics1', 3rd. ed. , p. 538, MacMillan Co. (New York, 1955).

the linearized. equations obtained by treating r , p, 0 and v as small H

quantities are

and

p = y r (1 0 ' )

Equations (8'), ( 9 ' ) , and (10') lead directly to the relation

where

With the substitution

- iot P = P e

Eq. (1 2) becomes

where

If one defines

then the complex wave number of Eq. (14) may be written as 3

Since p i s finite at r = 0, the general solution of Eq. (1 3) must have the

form

From Eq. (9') one finds for the radial component of the velocity

It follows that a t r = a

where

Equation (11) gives the perturbed radial stress as

which can also be written

where

This perturbed radial stress at r = a may be put in the form

where by Eqs. (15) and (17)

The boundary conditions at r = a a re that the radial component of

the velocity and the radial component of the s t ress be continuous. Eq, (7)

and (1 7) gjve the velocity relation

A f = S k hl(kwa) + iC (24+l) kw jl(kwa) , e e e w

and Eqs. (6) and (20) give the stress relation

These equations determine A4 and St . Solving for the scattering

amplitudes S one gets e

where

and

Since % and f involve spherical Bessel functions with complex argu- C

ments, it is difficult to perform an evaluation of S for the general case. C

Simplifications can be made, however, when the absolute values of these

arguments a re small in comparison with unity in which case the apherical

Bessel functions may be expanded in rapidly converging power series.

For bubble sizes of general interest and for frequencies in the ordinary

acoustic range, both ( k a 1 and 1 kwa( a re small compared with unity and

the expansion procedure may be employed. In this way scattering

amplitudes after some algebraic manipulations a re found to be in the first

approximation

where

The total scattering cross section i s determined by a familiar procedure 6

in terms of the S as follows

and the absorption cross section i s

6 See, for example, P. Me Morse and H. Feshbach, "Methods of Theoretical Physics" Par t 11, p. 1488, McGraw-Hill Book Co. .(New York, -195 3).

For the extinction cross section one has

The scattering and absorption cross sections which are obtained are thus

Some features of these expressions may be pointed out. The

resonance effect in the cross sections is observed with the different resonant

frequencies for the various modes of excitation. It may also be noted that

in the absorption cross section only the bulk viscosity contributes to the

S -wave absorption. There is , of course, no absorption for the P -wave

since in the present approximation the t = 1 disturbance corresponds to

a translatory os~cillation of the whole bubble without change of shape. There

i s , finally, the familiar result that for 1 x 1 FC 1 Q ~ ( O ) and %(o) give

' essentially the entire scattering and absorption cross sections.

Some numerical results have been computed for the case of an air

bubble in water for Po = 1 atm and T = 20°c. The bulk viscosity of a ir

7 has been taken to be given by P = 0 . 5 7 ~ . The results a re shown in

Figs. 1-4.

Viscous and Thermal Absorption

When the effect of heat conduction in the gas i s included, one can

no longer use the adiabatic equation of state. One has instead an energy

equation and and equation of state connecting pressure, density, and

temperature. The gas will be taken here to be a perfect gas so that the

equation of state i s

where M i s the molar weight and R i s the universal gas constant. The

energy equation is 4

where

------- - 7 ~ . S. Sherman, NACA Tech. Note 3298 p. 2l(July, 9 5 5 ) C Truesdell, J . Rat. Mech. and Anal. - 2, p. 649 (1 95 3).

In these equations K i s the coefficient of heat conduction and c i s the v

specific heat a t constant volumo-. The perfect gas equation of state when

' .linearized becomes

where p, 6 , and 0 have been defined in the previous section. The

linearization of Eq. (30) gives

where D = K / P ~ C ~ is the thermal diffusivity. Equations (8') and ( 9 ' ) to-

gether with Equations (31) and (32) lead to

In Eq. (33) ci = (pol Po) ' I2 i s the isothermal sound speed and ca = ( yF a / p o )If2

i s the adiabatic sound speed. Far periodic oscillations

Eq. (33) becomes

where

2 and as before k = u / ~ .

v

Equation (34) may also be written in the form

with

and

Both 0 and vr a re finite at the origin so that one may write the expansion

of 0 in the form

0 = [ACjC(klr) + Be jC(k2r)] PL (COS*) , for r E a . (39)

At r = a one has the boundary conditions, as before, which ensure the

continuity of the radial components s f the s t ress and of the velocity. It is

now necessary to impose an additional boundary condition to d e t e r d n e the

temperature field. This boundary condition will be taken here to be that

the perturbed temperature, TOO, a t the bubble wall i s zero at all times.

That this assumption for the boundary condition i s a good approximation

may be shown by fhe following physical argument. If the thermal conductiv-

ity of the liquid is K~ and the temi>erature gradient in the liquid a t the

bubble boundary i s ( aT/a r)L. and if the corresponding quantities in the

bubble a re denoted by K~ and ( a/ F ) ~ , then the continuity of heat flow at

the interface requires that

One may observe that ordinarily K so that (=/a r ) c c (aT/ar)G. G 4 3 For example, for water K ~ N 5.9 x I0 (c.g. s . ) and K N 2 x 10 (c.g*so).

G

One may now argue that not only is the temperature gradient in the liquid

a small quantity, but in addition i t comes close to i ts constant value at

infinity in a short distance into the liquid from the bubble boundary. The

heat diffusion length in the liquid i s ( D ~ / w ) 'I2 and the heat diffusion length

in the bubble i s ( D ~ I ~ ) " ~ . Here w i s the angular oscillation frequency

and DL, DG a re the thermal diffusivities of the liquid and the g a s , respact-

ively . The diffusion length, ( D ~ / w ) 'I2, over which the temperature varies

1/2 in the liquid i s small compared with the diffusion length in the gas, ( D ~ / W )

This follows since D c C D For the example of water and a i r one has G'

D N 1.4 x ~ O - ~ ( C . ~ . S . ) and DGu 0.2 (c.g.s.) . Thus one may say L

that the temperature gradient in the liquid is small and further the distance

over which this gradient extends i s small. A s a consequence, the varia-

tian in the temperature at the bubble boundary must be small.

By a procedure similar to that followed to obtain Eqs. (1 7) and (30),

one obtains for r = a

and

where, with a = 1 or 2,

The boundary conditions can now be summarized a s follows:

A j ( k a ) t B j ( k a ) = O ; e c 1 e c 2

8 A f ( 1 ) + B f (2) = S k h ' (kwa) + i (28t l ) kwj; (k,a) ; c e t e e w e

iw Pw iwpw 1 A g ( 1 ) + B g (2) = - P S h (k a ) + -

t L 4 8 w i (2c+l)jt(kwa) . e e 0

one finds S a s c - I'F:

Sc - a- C

with

and

The c ross sections a r e thus determined in principle for any w in t e rms of

S by the general fo rml las for the scattering and absorption c ross sections C

already given in the previous section.

Thermal Absorption

One may determine the effect of heat conduction alone in the

scattering and absorption of sound waves, by omitting the cnqtributions of

the viscous t e rms in the general expressions just found. Of interest for

this case a r e the limits of Eqs. (42) and (43) a s k , ks, dnd k becorn3 v b

infinitely large:

2 2 where kl and k2 a r e now given by simplified expressions

with

and

Explicit expressions for the cross sections will be given only for

the case lk a1 1 in which the important contribution is from the W

S-wave alone. It may be observed that in the frequency range of general

interest one has I ka/kt l 2 cc 1, and the following approximations may be

made :

hen, after some manipulations, the cross sections a r e found to be:

and

,

where

] icosh d - cos d) + (sinh d - sind)

2 2 + I 8(y-2-- (cosh d + cos d) - 36(y;1)- (sinh d + sin d)

d (55)

d

N' = (cosh d - cos d) + 3(y-') (s inhd - s ind) , -d (56)

and

n' = 3(Y-1) [ d(sinhd + sind) - 2 (cosh d - cos d) - 3 . (58) d2(coshd - cos d) + 3 (y-l)d (sinhd - sind)

For d e 1, it i s found, approximately, that

and

n' = (kta) 2 73-

while for ed 1 it i s found that

and

It may be remarked that the resonance frequencies as obtained from

(54') and (5411), show a shift from the isothermal limlt to the adiabatic

limit a s the quantity k a increases. As kt i s actually the inverse of the t

diffusion length, this result certainly agrees with what one would expect

f rom physical arguments. Figure 1, 2, 3 and 4 show some numerical

results for the case of an a i r bubble in water. It can be seen that in the

range of frequencies shown the effect of thermal absorption i s appreciably

larger than the effect of viscosity.

Figure 1. The scattering cross section, Qs(H),

and the absorption cross section, Qa(H), take

the bubble heat conduction only into account. The

scattering cross section, Qs (V), and the absorp-

tion cross section, Q (V) take bubble viscosity a

only into account. The bubble radius i s crn.

Figure 2. The same cross sections shown in _ Figure 1 are shown for the bubble radius of 10 cm.

Figure 3. The cross sections of Figure 1 are shown

for the bubble radius of cm*

Figure 4. The cross sections of Figure 1 are shown

for the bubble radius of 10-I crn,

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