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Bubble phenomena in sould fields: part two
Acoustic
cavitation
series: part three
Bubble phenomena part two
in sound fields:
A. PROSPERETTI
Large-amplitude radial oscillations of gas bubbles are briefly illustrated with the aid of numerical examples. The origin and possible effects of pressure-radiation forces are considered and an estimate of the coalescence time under their action is given. Non-spherical oscillations, the related problem of the fragmentation of oscillating bubbles by instability of the spherical shape, and sound propagation in bubbly liquids conclude the review.
KEYWORDS:
Large-amplitude
ultrasonics, cavitation, bubble dynamics
are not expected to be seriously affected by this omission. Figs 9 and 10 show two sets of results obtained by Cramer” by numerical integration of the Gilmore equation (19). The graphs show the normalized maximum radius (R,,, - Ro)/Ro during steady oscillations as a function of the normalized frequency for R, = low3 cm (Fig. 9) and as a function of the equilibrium radius for a sound frequency of 7.208 kHz. The different curves correspond to different values of the sound pressure amplitude, and the fraction m/n labelling the peaks indicates that m (local) maxima in the radius-time curve occur during the n periods of the sound-field necessary to complete one cycle of the bubble oscillation. Thus a strong mth order harmonic of the fundamental, of the first subharmonic. of the second subharmonic, and so on, will be present in the oscillations of the peak of order mll, m/2,m/3, respectively. This component will be accompanied by its multiples and submultiples. as can be seen in Fig. 11 where an example of a complete oscillation together with its Fourier spectrum is shown.
oscillations
Large-amplitude bubble oscillations are closely related to acoustic cavitation noise. which will form the subject of another article in this series (see also Refs l-l 1). For completeness included here is a very brief account of this topic referring to the future article for further details.
Equations (15) and (17) for radial motion of a bubble, in part one of this paper. are so highly non-linear that not much progress can be expected on large-amplitude oscillations by analytical means. Some results are nevertheless available both for steady oscillations’*-I4 and for transient motionls for the incompressible case (17) with the polytropic relation (32) and sinusoidal forcing. Limited as they are, these results are also useful in the interpretation of numerical computations, since the behaviour of the observed stable solutions is often determined by the presence of unstable solutions that can be computed analytically much more readily than numerically.
In Figs 9 and 10 the dashed lines correspond to jumps in the steady state amplitude as the frequency, or the equilibrium radius. is increased past the instability threshold of the lower solutionr2Js. A notable feature of the numerical results is the impossibility of obtaining a steady solution at large excitation amplitudes for all frequencies or radii. For instance, in Fig 10 only a few such solutions, indicated by dots, could be obtained in the region of equilibrium radii between 150 and 250 pm for E = 0.8. For neighbouring values of the radius a non-steady, apparently chaotic behaviour is found; the Fourier analysis of which contains a broadband component in addition to lines. The thresholds for the onset of this behaviour are close to those for the appearance of subharmonic components at a frequency
A detailed discussion of these findings is lengthy, and rather than dwelling on them a brief account of some of the extensive numerical results obtained by Lauterborn and co-workers16-20 will be given. These studies have been based on the incompressible equation of motion (17) or on the Gilmore equation (19). The internal pressure was approximated by the polytropic relation (32) with K = 1.33. Thermal losses are therefore not included in the mathematical model, but the qualitative features of the solutions obtained
The
author
Celona this
16,
paper
is at the 20133 appeared
Dipartlmento Milano. in the
Italy.
di F~s~ca. Univewta Paper
previous
received
0041-624X/84/0301 ULTRASONICS
degli
1 June
Studi.
1983.
Via
Part one
of
issue.
. MAY 1984
15-l
O/$03.00
0
1984
Butterworth
& Co (Publishers)
Ltd 115
2.0
I
I.0
i
1.6
1.4
0.6
0.4
0.2
-I
0
0.1
0.15
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.9
I
1.5
2
3
f f F&g. 9
Numewal
frequency R,
values
to the
bubble
= 1 0m3 cm, the
(from
Ref.
of the
normaltzed
resonance
polytropic
frequency,
exponent
maximum
radius
for several
is K = 1.33
the
(Rmax n Ro)/Ro in steady of the sound pressure
values driving
frequency
is w = 2~
forced oscillations as a function of w/we. Rattos of the drwing amplitude indicated in bars on the curves. The bubble radius
x 340.4
kHz,
the amblent
pressure
IS 1 bar and the liquid
sound is
is water.
17)
2.0
I.0
-
1.6
-
0.6
0.4
0.2
0 30
40
50
60
70
80
90
100
150
200
300
400
500
600
700
000
900
Ho CpmI Fig.
10
function
Numerical of the
x 1 0e2 cm, the
values
of the
bubble
radius
liquid
is water,
R.
normalized for several
K = 1.33
maximum values
radius
of the
and the ambient
(R,,,
sound pressure
- Ro)/Ro in steady forced oscillations at the frequency w = 271 x 7.200 kHz as a pressure amplitude indicated in bars on the curves. The resonance radius is 4.305 IS 1 bar (from
a situation very reminiscent of the well known results of Feigenbaum*‘~**, according to which a series of period-doubling (that is. subharmonic) bifurcating oscillations precedes the onset of chaotic motion. (n/2)w.
The integration of these observations into a coherent picture of acoustic cavitation noise still lies in the future. It will undoubtedly require a consideration of
116
Fief. 17)
the role of rectified diffusion in the change of bubble population with time, of the instability of the spherical shape which causes the break-up of bubbles in largeamplitude oscillation, of bubble-bubble interaction forces, and of the radial dynamics that determine the appearance of transient events near the onset of the chaotic behaviour.
ULTRASONICS
. MAY
1984
I (X
Xc,);nj dS = V Sji
(49)
s
Upon taking the average over a cycle the following expression is obtained for the average radiation force F on the bubble F=
I
I
I
I
I
I
Ii
III
0.5
I
Time,
0
I
2
III
III
1.0
3
4
5
I 2.0
1.5
t/t,
6
7
6
9
IO
II
Frequency, f/G Fig.
11
Complete
steady
oscillation
of a bubble
of radtus
8 x 1 0v3
cm
The order of the resonance IS 3/2, meaning that the amplitude exhibits three local maxima during the two pressure cycles necessary to complete the oscillation. a - the upper curve is the pressure cycle and the lower curve the radius; b - the Fouler components of the oscillation are shown (from driven
Ref.
by a sound
pressure
of 0.8
bar at a frequency
of 23.56
kHz.
17)
Steady forces on bubbles
in sound-fields
A droplet. bubble. or particle present in a sound-wave is subjected. in addition to an alternating force, to a smaller (second-order) time-independent force, which is referred to as radiation force. In the case of a bubble the evaluation of this force is considerably simplified by the enormous compressibility of the gas compared with that of the liquid, which allows one to treat the latter as incompressible. The total pressure force acting on the bubble is given by -
Ppna
(48) s where n is the unit normal outwardly directed from the bubble surface S. The pressure p contains a spatially slowly varying contribution due to the sound-wave and a rapidly varying component caused by the relative motion of the bubble with respect to the liquid. If the liquid has a small compressibility this component contributes to the alternating force23-2s. but not to the steady one and can be neglected, so that p in (48) can be regarded as just the sound pressure. If the wavelength is large compared with the radius we can therefore write p(x; t> = p(q), t) + (x - x,,) * v p + . . where x0 is a point in the bubble. The first term does not contribute to the integral. while the second one gives - VVp, where V is the bubble volume, in view of the identity
ULTRASONICS
MAY 1984
-
where the angle brackets indicate the time average. This result is clearly valid for any shape of the bubble. Its physical origin becomes clear once it is noticed that the bubble is subject to an acceleration against the pressure gradient. which is proportional to its volume. In an oscillating pressure field the gradient reverses periodically so that the bubble executes an oscillatory translational motion, the amplitude of which, in the expansion phase, is greater than in the contraction phase. Therefore the bubble tends to drift in the direction opposite to that of the pressure gradient when the bubble volume is at its greatest. (We take this opportunity to correct an oversight in Ref. 26 in which the same argument leads to the opposite (!) conclusion: the word ‘greatest’ there should in fact be ‘smallest’.) This explanation clearly shows the second-order nature of the effect. For bubbles driven below resonance. sound pressure and volume increase are out of phase, the volume is largest during the expansion period and the bubble is therefore attracted to the pressure antinodes which are pressure minima during expansion. The opposite will be true for bubbles driven above resonance. and in this case the drift will be towards the pressure nodes. The preceding derivation neglects the compressibility of the liquid and hence two terms are absent from (50). The first is a correction term similar to (50), which subtracts from F the force that would act on an equal liquid volume. so as to make F = 0 if the bubble were replaced by liquid. The second term is due to the interaction between the relative motion of the bubble and the volume change. Both these terms turn out to be proportional to the compressibility of the liquid (pc2)-‘, and hence are overshadowed by the term retained in (50). which is proportional to the compressibility of the bubble24.2s. The first application that we make of (50) is the equilirium position of a bubble in a sound-field with a vertical gradient. In this case the bubble is acted upon by the force (50) and by the buoyancy force p g V. If we write the pressure field asp = p,[l - E(X) cos wr] where E(X) is the dimensionless space-dependent amplitude. the balance of the average buoyancy force and of (50) gives <
v coswt>
= _ P&f
I VQLE)I
(51)
The right-hand side of this relation (which may be termed the levitation number _F)2’ is a function of the bubble position. and therefore a measurement of this position in a known pressure field enables one to obtain a value for the left-hand side. In experiments of this type the bubble is said to be ‘levitated’ in the liquid. If the left-hand side is evaluated theoretically. one can subject the theory to experimental verification in this way. For instance. in the case of smallamplitude oscillations, using (28) we find
117
<%”- W2)CYf
_Y2.
pv,z (52)
2 (0,’ -cd2)2 t4p2w2
For w Q w,,. using
tension
contribution.
(34) and neglecting the surface this expression becomes
_!? = E/2K.
(53)
Hence a measurement of Yat low frequency gives the polytropic exponent directly. The measurements of this quantity shown in Fig. 8 were obtained using this procedure. As a second application of (50) we consider the mutual force between two pulsating bubbles arising because each is immersed in the pressure field of the other. The pressure field around a bubble of radius R’ in radial motion in an incompressible liquid is given by (see, for example, Refs 23. 28-34) p= k
+pc
_ ;pkt2
(R’&2$2)
where r =I x - x ‘I. Since the bubbles move towards each other, &/dt = -2~. Integrating from r = r(0) to r N 0 we have the following estimate of the coalescence time t, t, =
16 n2r~Ror3(0)
If we limit the analysis to small-amplitude motion, in terms of the volume V’ this expression may be written
This result cannot be trusted to be more accurate than to within an order of magnitude in view of the complex phenomena that occur when the bubbles are close, and which we have neglected 36.37.The volume pulsation amplitude 6 is three times the radius pulsation amplitude X. in terms of which the previous result is then =
wr3
We may note that if the bubble does not pulsate as a sphere, the correction terms to this expression involve higher powers of r-’ which are small far from the bubble. Upon insertion of (55) into (50) we find the following expression for the so-called Bjerknes force between two pulsating bubbles3s located at x and x’. =
_4b
5-x’ Ix - x
‘13
For,periodic mo$n an integration by parts gives = - . which explicitly shows the symmetric nature of this force. The dependence upon the inverse square of the distance is a characteristic feature of this result. If we set V = V, [I + 6 cos(wr + qb)], V’ = V,‘[l + 6’ cos(wt + @‘)I. where @ and @’ are the phases relative to the sound pressure that drives the oscillations. the preceding expression becomes F = -
&
cos(@-4’)
v, v;66’w2
-xz Ix-x
‘13
(57)
This force is attractive if cos(@ - @‘) > 0. which will be the case if both bubbles are driven below or above resonance. whereas it is repulsive if one is below and the other above resonance. The effectiveness of the attractive force will be increased because the bubbles tend to collect near the nodes or the anti-nodes of the field. This result can be used to obtain an estimate of the coalescence time between two bubbles. which for simplicity we may take to be of equal size. We further assume that inertial effects are negligible and that the force (56) is balanced by a drag force FD proportional to the velocity I* F,, = 47ry R, p
(w
y = 1 and y = 3/2 for the Stokes-Rybczynski and Stokes small-Reynolds number cases respectively. and y = 3 for the large-Reynolds number Lcvich drag law. Equating (57) and (5X) we find that where
118
(0)
(61)
Using for X the linear becomes t
c
equation
(28) this expression
= ytipr3 (0) [(w’ - ~02)~ + 4P2 a2 3R,oZ~’
(55)
F
(60)
3p v(j%%_?
3pR;02X2 (54)
r
(59)
32 ~~~yR,/.ir~
t,
4
fi2w2
v=
1
p:
(62)
where. as in (25). E is the dimensionless sound amplitude. For sound frequencies in the range of tens of kHz and small bubbles (R, < -IO-’ cm) w0 % w and. using (34). the previous expression simplifies to t
_ ~2-ypr3(0)[1 c -
+ (1 -
~/SK)
2u/R0 p-l2 (63)
pR;02c2
To give an idea of some orders of magnitude, for K = I. y = I, r(O) = I cm. o = 2rr x 20 kHz. E = 0.5 in water, this relation gives t, = 7600 s (more than 2 h) for R, = 10m3 cm but 2,. = 2.4 s for R, = 5 x IF3 cm. This extreme sensitivity to the value of the radius makes this mechanism significant for relatively large bubbles. The time t, is a minimum for resonance conditions and has the value 4 rppf12 r3 (0) tc = 3 Roe2 p?,
Distortion
(64)
from spherical
shape
Surface tension tends to maintain the bubble in a spherical configuration, which may be distorted however. by asymmetrical pressure fields and dynamic effects. For example, if a pressure difference Ap exists across the bubble, its effect on the spherical shape will be small provided that the pressure difference due to surface tension, 2u,,R,, is much greater. We can estimate Ap by R,I Vpl , which leads to R, < (2a/j Vp( )I”
(65)
For a sound-wave of wave-number k, I Vp 1 - k~p m and the previous relation is R, < (2u/kcp,,)‘12. For pm = I bar and k = 0.84 cm-‘. corresponding to a frequency of about 20 kHz in water. the preceding relation gives R,, < 0.02 cm for E = 0.5 and R, < 0.04 cm for E = 0.1. The resonance radius at this frequency is 0.015 cm approximately. When this limitation is not met the
ULTRASONICS.
MAY 1984
decay proportional to r-r associated with changes of volume. Hence, the acoustic energy loss in the nonspherical motion is much less than that in volume pulsations, and this mode of damping is accordingly also small. The main contribution to energy dissipation comes now from viscosity. The theoretical analysis of non-spherical motion is naturally much more difficult than for the spherical case, except in the case of small distortions of the spherical shape. In this case, if the bubble shape is written as a sum of spherical harmonicsy/,
r(e) @;t) = R(t) + -y
0
a
(66)
0) Y/m (0,$1
aI,
< It is possible to a first approximation to neglect terms of second and higher orders in the al, and in fact all the interactions between the different modes disappear. The problem therefore simplifies in that the effect of each single mode can be studied separately, writing
r(& q5; t) = R(r) + a,(r) P,,(cos 8)
(67)
where P,, is a Legendre polynomial. In this expression we have taken the bubble shape to be axisymmetrical, which is permissible since it is found that the equation of motion for the coefficients al,,, is independent of m and hence is the same as that obtained using (67). For the case of small viscosity and free oscillations this equation is found to be3*J9 0
b
’
c
-!-
wf
Fig. 12 Transient oscillations of gas bubbles in a sound-field, the pressure amplitude of which is specified in bars on the curves. The lower line is the pressure cycle. a - the equilibrium radius is 1O* cm and W/W,, = 0.107; b - it is 5 x 1 O4 cm and w/w0 = 0.706. The sound frequency is w = 2n x 500 Hz, the ambient pressure 1 atm and the liquid is water
sound-wave will excite radial pulsations. The a relative velocity with and, more importantly, radial motion that can
shape oscillations in addition to bubble may also be distorted by respect to the liquid, buoyancy, by an interaction with its own lead to its breakup.
In approaching the problems associated with the loss of spherical symmetry it is useful to keep in mind some general physical features which make them very different from their radial counterparts. In the first place a small distortion of the spherical shape has a much smaller (in fact, of second order) effect on the volume, so that the restoring force due to the compressibility of the gas is unimportant, the only restoring mechanism being provided by surface tension. For the same reason thermal dissipation processes are negligible. If the volume of the bubble is unchanged the pressure field caused by its motion has a dipole character if the centre executes oscillatory motion, of the quadrupole type if the bubble undergoes oscillations from a prolate spheroidal shape to an oblate one, and of higher multipole types for other forms of oscillations. Accordingly, the rate of decay with distance of the pressure perturbations is proportional to r-*, rs3 or higher powers, which are to be compared with the slow
MAY
1984
1
f&
PR’
I
ULTRASONICS.
. 3 ;t2(nt2)(2n+l)
ii,+
.
t 2(n t2)
I.IR a,=O.
pR3 1 The value n = 1 corresponds to (small-amplitude) translatory motion for which the restoring force (the last term) obviously vanishes. In this case the equation may be written = -12n/.~Ra’, The left-hand side is the rate of change of the liquid momentum, while the right-hand side is Levich’s drag force on a spherical bubble at large Reynolds numbe14°-42. When the bubble ;i,
+2(n t2)(2n
radius +1)
t(n-l)(ntl)(nt2) We can read directly frequency won 02,,=
(n-l)(?ztl)(n+-2)
is constant 5
(68) reduces
to
&
+
a,=0
from this equation
p+
the natural
(71)
119
and the damping
constant
p,,
/$I = (n + 2) (2n + 1) /dpR’
(72)
Both of these results are classicaP. The prolate-oblate oscillation corresponds to n = 2 and for water we find wo2/2rr = 0.15, 4.7. 149 kHz, fi, = 0.02. 2. 200 kHz for R = 10-r. IO-*, IO--’ cm respectively. The coupling between the radial motion and the shape oscillations present in (68) has some important consequences which we cow discuss. In the first place it is seen that the term 3R/R. the qualitative effect of which will be similar to damping. tends to dampen the shape oscillation on expansion (R > 0). but acts as. ‘negative damping’ that is, a destabilizing force. on contraction (R < 0). The physical origin of this effect is the diverging or converging nature of the streamlines which stretch or compress the bubble surface. Secondly. the acceleration of the radial motion is seen to act as a restoring (stabilizing) force when it is negative - for instance in the later stage of expansion. However. this term will have a strongly destabilizing effect when R > 0. which will occur at the beginning of the expansion or. upon collapse. near the minimum radius. The mechanism of this effect is the same as that of the ordinary Rayleigh-Taylor instability for plane interfaces44.
example Refs 45. 46). The first unstable range occurs for k = 1 and gives rise to growing oscillations at a frequency close to the subharmonic value won = Y20. (The unstable ranges for larger values of k have widths of order Xok and hence cannot consistently be considered in a theory that is correct to order X,,, as is the present one.) The rate of growth of the oscillations in 6, is proportional to exp(Xr) where X = [l/4 8*X, (n,, - 1/4)*]‘/*. but, from (74) it can be seen that the rate of growth of the oscillations in a, is WA-P,. Requiring this to be positive gives the following approximate condition on X,, for instability
The physical meaning of this instability cannot be completely explained within the framework of the linear theory described. If the threshold is not exceeded by a large amount it is likely that non-linear effects will stabilize the shape oscillation to some finite valm?‘. For more violent motion, however, the amplitude of the oscillation may become so large as to result in a breakup of the bubble. With this interpretation in mind it is of interest to discuss (79) further. The minimum of the threshold x0 occurs for w = 2w,, and has the value
Of particular interest in the present context is the solution of (68) when the bubble radius executes an oscillatorv motion. For simnlicitv we shall consider the linear. small-amplitude case R=R,(l
+X,,cosot)
( 73)
Furthermore. for small viscosity. we can neglect the variations of R in the terms containing p. With the substitution Q,I = R-‘I* exp(-/?,,f)
b,,
74)
with p,, given by (72). we find i;,, + G,,(f) b,, = 0
(75)
where
t(n-l)(ntl)(n+2)
/?-pR3
(76) To simplify this form of G,,. terms proportional to f?*. p,: and ~.LRhave been dropped. We can now use (73) and let r = tit to find d2b, __ dr2
t
[7)” +28,X,
cos271 b,=O
(77)
where (78)
This is Mathieu’s equation which is known to possess exponentially growing solutions in the neighbourhood . that 1s q,,, = !A kw (see for of tl,, = kZ/4. k = 1.2. .
120
030) For n = 2 in water we have y0 = 0.0085, 0.027, 0.085 for R, = lo-‘, lo-*, lo-) cm respectively. For ftxed R. this is a decreasing function of n, which seems to imply that all sufficiently large modes are unstable. This prediction however is invalidated by the fact that (68) is actually only valid for n(p/po)“*
phenomenon
frequently
observed
ULTRASONICS.
in acoustic
MAY
1984
cavitation experiments, which is presumably connected with the stability of the spherical shape, is the rectilinear motion in apparently random directions that bubbles execute once they attain a certain sizeJ2. A possible explanation for this behaviour is that a suitable coupling of different shape oscillations produces asymmetries in the bubble surface velocity which propel it by reaction. This explanation has been offered by Benjamin and Strasberg52. and it rests on the somewhat indirect evidence that thresholds for the rectilinear motion are in agreement with those for shape oscillations53.
Let us consider a homogeneous mixture of bubbles with equal radius R, in a liquid contained within a pipe, thus making the problem effectively onedimensional. The mixture can be characterized by the void fraction 8, the volume occupied by the bubbles in a unit volume of the mixture. If n is the number of bubbles per unit volume. each having a volume vt,, it is evident that
Important effects of acoustic cavitation such as erosion and cleaning may be connected to very large amplitude deformations of the spherical shape in the vicinity of a solid boundary. which result in a high-speed jet piercing the bubble in the direction of the wall. There is no theoretical information on this effect for the oscillating pressure conditions typical of acoustic cavitation, but the phenomenon has been investigated numerically54-58 and experimentally59-64 for the constant pressure case characteristic of flow cavitation.
The average density
For a pressure in excess of 1 bar the jet velocities are found to be of the order of 100 m s-l, and this result is predicted to be proportional to (p_ - pi)“2. It can therefore be expected that very strong impulses will be communicated to solid boundaries by bubbles undergoing violent collapses in their vicinity. These impulses can detach impurities from the solid. but they may also cause local damage to the wall structure leading to fatigue and erosion. In this connection Fig. 13 shows a very beautiful picture by LA. Crum in which the jet piercing the bubble is clearly visible. The bubble shown contains gas and vapour and is oscillating at 60 Hz at reduced pressure (4 X 10) - 5 X 10’ Pa) and its size is approximately 0.2 cm. The liquid is water with 25% by volume of glycerine. and the temperature is approximately 25”C.65
Sound
propagation
in a bubbly mixture
The presence of even a few bubbles substantially alters the compressibility of the medium and hence has a strong effect on the speed of sound and its attenuation. This is just one of the many aspects of the cooperative effects of bubbles in liquids, of which an elementary discussion is given here.
F=nvb=+R;
(81) of the mixture
is
P=6PG+S6)PL
(82)
and. if the bubbles and the liquid are taken to move with the same velocity u. conservation equations for mass and momentum can readily be derived in the standard way. Their linearized form is
-a; ,Fg=O at
- au Pat+==0
ap
If a relation of the type p = p(p) is assumed, the wave equation follows from this system with a speed of propagation c given by cm2 = dp/dp. or, from (82). 1 _= c2
6+-_ c&
l-6 (PL
c;
-
(85)
PG) g
where c20,~ = dpldp~, L. To evaluate daldp we consider a volume V of the mixture consisting of a gas volume VG, and a liquid volume. Vt_. By definition of 8 we have
g= dp
-d dp
~--
;
d$
(86)
Since VG = nVvb and the number of bubbles in V is constant the first derivative is simply dVG/dp = nV x dvddp = (V&+-J dvtJdp. The total mass of liquid in K VLpL, is also constant and therefore
Upon substitution into (85) we find 1 _=-+, c2 Fig. 13
Jet
formation
bubble
at low
bubble
size
Professor
pressure
durmg
Crum,
ULTRASONICS
collapse
(0.04-0.05
IS approximkely
LA.
the 0.2
University
. MAY
bars) cm.
of an oscillating in a 60
(Unpubkhed
of Mlsslsslppi)
1984
of these results into (86) and then
62
(1 - S)”
c&
CL
(
1+ p”
J-
PL
l-6
i
gas-vapour
Hz sound-field. photograph
The by
-PL
6(1 y
-6)
dub ~ dP
(87)
121
(To obtain the first term of this relation we have used the approximation PGdvb N -V&IG, which would be exact only if the gas density were uniform in the bubble.) The final step is to observe that dvddp may be written as (dvddf)/(dpldf). From (25) and (26) we then have for harmonic oscillations
dut, -= dt
-hR;
p$
l
-
b-
or, upon substituting the expression for X derived (28) and neglecting the small term p~/p~ in (87), 1 -=_-+ c2
62
(1 -&I2 c;
c&
R; (08
- o2
- -
R=0.250cm
from
36(1-S)
+
R=O.l89cm
(88)
+ 2i@)
The last term is found to be much larger than the other two when S is not too small (nor too close to one, but then we would be dealing with droplets in a gas rather than with a bubbly liquid) so that (88) can be approximated by c2 =
R ‘0(ui
- o2
t 2iflw) (89)
36Q-6) A wave equation consistent with this dispersion relation can be written when the frequency is sufficiently low that o, and p are approximately constant; it is given by
v---
R=0.364cm
l-
R= 0.260cm
.
R= 0.268cm
(90) IO-' t
The last two terms are negligible at extremely low frequencies and, with K = 1 and the neglect of surface tension, by (34). (89) reduces to the well known resulP
,J, ,A
(91)
IO
b
IO2
i,l , , , IO’
1
-I
IO’
Frequency, v Cs-’1
independent of bubble size. For 6 = 5%, pm = 1 bar. pL = 1 g crnm3 this expression gives c = 46 m s-l, a value to be compared with the typical ones for the speed of sound in the gas, CG N 340 m s-l, and in the liquid, cL 2: 1500 m s-l.
Fig. 14 Theoretxal attenuation of sound-waves in water containing air bubbles (Ref. 67) compared with experimental data (Ref. 68) as a function of the sound frequency. The volume fraction occupied by the bubbles is 5.84 x 1 0T2% in a and 1% in b. The values of the bubble radius are Indicated in the figures. In both cases the temperature was 20°C. The arrows mark the position of the resonance. (From Ref. 26)
In (88) and (89) the imaginary part of c corresponds to attenuation of the pressure waves. This quantity, calculated on the basis of a more precise theory26.67, is shown in Fig. 14 expressed in units of dB cm-‘. (1 dB cm-* means that the wave intensity is attenuated by a factor 10-O.’ z 0.794 and the wave amplitude by 10-0.05 C=0.891 in 1 cm.) Some classical experimental measurements by Silberman6* are also shown in the figure. The peaks correspond to the resonance region of the bubbles, and the corresponding enormous attenuation is worthy of notice. The bubbly medium acts essentially as a low-pass filter, the cut-off frequency of which is determined by the natural frequency of the bubbles and hence by their size.
in itself very broad and here we must confine ourselves to the preceding simple treatment. The reader is referred to Refs 66-72 for further information and developments.
The subject
122
of sound
propagation
in bubbly
liquids
is
Esche, R. Untersuchung der Schwingungskavitation in Fliissigkeiten, Acusricu AB2 (1952) 208-218 Bohn, I, Schalldruckverlauf und Spectrum bei der Schwingungskavitation Acustica 7 (1957) 201-216 Neppirss, E.A. Subharmonic and Other Low-frequency Emission from Bubbles in Sound Irradiated Liquids. 1 Acoust Sot Am 46 (1969) 587-601 Neppiras, E.A. Subharmonic and Other Low-frequency Signals from Sound-irradiated Liquids, .I Sound Vib. 10 (1969) 176-186
ULTRASONICS.
MAY
1984
5
De Santis, P., Sette, D., Wanderlingh,
Detection: the use of the Subharmonic. 6
(1967) 514-516 Vaughan, P.W. Investigation
Thresholds 7 8 9
by Observation
F. Cavitation 1 Acoust Sot. Am
35 42
36 of Acoustic Cavitation of the First Subharmonic.
1 Muhiihase
37
J.
Sound vib. 7 (1968) 236-246 CorIdey, W.T. Acoustical Detection
of Single Cavitation Events in a Focused Field in Water at 1 MHz. J Acoust Sot Am 49 (1971) 792-801 Neppiras, E.A., Coakley, W.T. Acoustic Cavitation in a Focused Field in Water at 1 MHz. J Sound Mb. 45 (1976) 341-373 Lauterhom,
38
Fluids, 21 (1978) 14651470 Ceschia,
G. Digital
Spectral Analysis
of
40
Physicochemical Hydrodynamics, PrenticeHall. Englewood Cliffs (1962) 445 Landau, L.D., Lifshitz, E.M. Fluid Mechanics, Pergamon Press, London (1959) 172 Haruer. J.F. The Motion of Bubbles and Droos Throueh Liquids.Adv. Appl. Me& 12 (1972) 59-131 ’ Lamb, H. Hydrodynamics, Cambridge University Press, Cambridge (1932): reprinted by Dover. New York (1945) 475. 641 Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stabilitv. Oxford Universitv Press. Oxford (1961): reorinted by Dover. New York (1981) Ch. X ’ . Cole, J.D. Perturbation Methods in Applied Mathematics. Blaisdell. Waltham (1968) 94 Bender, C.M., Orszag, S.A. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill. New York (1978) 560 Hall, P., Seminara, G. Nonlinear Oscillations of Nonspherical Cavitation Bubbles in Acoustic Fields. 1 Fluid Mech. 101 (1980) 423-444 Prosperetti, A. On the Stability of Spherically Symmetric Flows. Rend Accad Nar Lincei 52 (1977) 196-203 Hullin, C. Stabilitatsgrenze Pulsierender Luftblasen in Wasser. Acustica 37 (1977) 64-73 Francescutto, A., Nabergoj, R. Pulsation Amplitude Threshold for Surface Waves on Oscillating Bubbles.
41
Lauterhorn, W., Haussman, G. Digital Analysis of Acoustic Cavitation Noise, 9th International Congress on Acoustics
42
Madrid (1977) 670-671
43
11
Cramer, E., Lauterhorn, W. Acoustic Cavitation Noise Soectra. Aool. Sci. Rex 38 (1982) 209-214. See Ref. 73 Prosperetti A. Nonlinear bsciliations of Gas Bubbles in Liquids: Steady-state Solutions, J Acoust Sot Am. 56 (1974) 878-885
44
13
Samek, I_..Nonlinear Oscillations of Gas Bubbles in Liquids. Czech. .J Phys B30 (1980) 1210-1226
14
Francescutto, Gas Bubbles
16
A., Nabergoj,
R Steady-state
Oscillations
45 of
48
tauterhorn, Oscillations
49
W. Numerical Investigation of Nonlinear of Gas Bubbles in Liquids. 1 Acousr. Sot Am.
17 18
Cramer, E., Lauterhorn,
W. Zur Dynamik
Lauterborn, W. Cavitation Bubble Dynamics-new Tools for an Intricate Problem. Appl. Sci Res. 38 (1982) 165-178 see
21
M.J. The Onset Spectrum of Turbulence, Phys L.ett A74 (1979) 375-378 Feigenbaum, M.J. The Transition to a Periodic Behaviour in Turbulent Systems, Comm. Mark Phys. 77 (1980) 65-86 Coakley, W.T., Nyborg, W.L Cavitation: Dynamics of Gas Bubbles; Applications. Ultrasound: Its Applications in Medicine and Biology. Fry. F.J., editor. Elsevier. New York (1978) 77-159 Vosidka, K., Kawasima, Y. Acoustic Radiation Pressure on a Comnressible Sohere. Acustica 5 (1955) 167-173 Weiser, M.A.H.; Apfel, RE. Extension’of Acoustic Levitation to Include the Study of Micron-sized Particles in a More Compressible Host Liquid. 1 Acoust Sot. Am. 71 (1982)
24 25
32 33
251 ApfeI, RA. Acoustic Cavitation,
27 28 29 30 31
34
ULTRASONICS
. MAY
1984
Strube,
H.W.
Numerische Untersuchungen zur Stabilitlt Schwingender Blasen. Acusricu, 25 (1971)
289-303
52
T. Surface Effects in Non-spherical Motion of Small Cavities. Cavitation in Real Liquids. Davies. R.. ed. American Elsevier. New York (1964) 164-180 Eller, A.I., Crum, L.A. Instability of the Motion of a Pulsating Bubble in a Sound Field. J. Acoust Sot. Am. 47
Brooke Benjamin,
(1970)762-767 54 55
56
A.
Methods of Experimental Physics, 19, Ultrasonics, Edmonds, P.D.. ed. Academic Press, New York (1981) 355-411 Prosperetti, A. A Generalization of the Rayleigh-Plesset Equation of Bubble Dynamics. Phys. Fluids 25 (1982) 409-410
Levich, V.G.
nichtsphlrisch
57
Bubble Dynamics: a Review and some Recent Results. Appl. Sci Res 38 (1982) 145-164 See Ref. 73 Crum, LA., ProsperettI, A. Nonlinear Oscillations of Gas Bubbles in Liquids, an Interpretation of Some Experimental Results, 1 Acoust Sot Am. 73 (1983) 121-127 Flynn, H. Physics of Acoustic Cavitation in Liquids, Physical Acoustics, Mason, W.P.. ed. Academic Press. New York, (1964). IB, 57-172 Plesset, M.S. Bubble Dynamics, Cavitation in Real Liquids, Davies. R ed. American Elsevier, New York (1964) 1-18 Piesset, M.S., Prosperetti, A. Bubble Dynamics and Cavitation. Ann Rev. Fluid Mech. 9 (1977) 145-185 Prosperetti, A. Current Topics in the Dynamics of Gas and Vapor Bubbles, Meccanica 12 (1977) 214-235 Neppiras, E.A. Acoustic Cavitation. Phys. Rep. 61 (1980) 159Prosperetti,
M., Nabergoj, R. On the Motion of a Nearly Spherical Bubble in a Viscous Liquid Phys Fluids 21 (1978)
Acusricu 41 (1978) 215-220
51
Feigenbaum,
1261-1268
26
50
53
Ret 74
23
47
und
Schallabstrahlung Kiigelformiger Kavitationsblasen in einem Schallfeld. Acusrica 49 (1981) 226-237 Lauterhorn, W. Resonanzkurven von Gasblasen in Fliissigkeiten. Acusrica, 23 (1970) 73-81
20
22
46
in Liquids: Explicit Formulas for Frequency Response Curves, J Acoust Sot Am 73 (1983) 457-460 Prosperetti, A. Nonlinear Oscillations of Gas Bubbles in Liquids: Transient Solution and the Connections Between Subharmonic Signal and Cavitation, J Acoust Sot Am. 57 (1975) 810-821 59 (1976) 283-293 Cramer,‘E. The Dynamics and Acoustic Emission of Bubbles Driven bv a Sound Field. in Ret 69. 54-63
19
AX.,
39
10
I5
Flow, 7 (1981) 709-717 Hofman, G. Bubble Coalescence in Pure Liquids. Appl. Sci Res. 38 (1982) 353-361 See Ret 73 Prosperetti, A., Seminara, G. Linear Stability of a Growing or Collapsing Bubble in a Slightly Viscous Liquid, Phys. Chesters,
140-142 W., Haussman,
Acoustic Cavitation Noise, 7th International Symposium on Nonlinear Acoustic, Nayfeh. AH.. Kaiser, J.E. eds. Virginia Polytechnic Institute (1976) 191-194
12
LA. Bierknes Forces on Bubbles in a Stationary Sound Field.2 Acoust Sot. Am 57 (1975) 1363-1370 _ Thomas. RM. Bubble Coalescence in Turbulent Flows. Int. Crum,
58
59
60
61
62
M.S., Chapman, R.B. Collapse of an Initially Spherical Vapor Cavity in the Neighbourhood of a Solid B0undary.J Fluid Mech. 47 (1971) 283-290 Bevir, M.K., Fielding, P.J. Numerical Solution of Incompressible Bubble Collapse with Jetting Moving Boundary Problems in Heat Flow and Diffusion. Ockendon. J.R., Hodgkins. W.R. eds. Clarendon Press, Oxford (1975) 286-294 Lenoir, M. Calcul Numerique de I’Implosion dune Bulle de Cavitation au Voisinage dune Paroi ou dune Surface Libre, J Mkanique 15 (1976) 725-751 Shima, A., Sato, Y. The Collapse of a Spheroidal Bubble Near a Solid Wall. 1 Mkanique 20 (198 1) 253-27 1 Guerri, L, Lucca, G., Prosperetti, A. A Numerical Method for the Dynamics of Non-spherical Cavitation Bubbles Proceedings of the Second International Colloquium on Drons and Bubbles. Le Croissette. D.H.. ed Jet Prooulsion Labbratory, Pasadena (1982) 175-181 Fujikawa, S., Akamatsu, T. Effects of the Non-equlibrium Condensation of Vapor on the Pressure Wave Produced by the Collapse of a Bubble in a Liquid, J Fluid Mech. 97 (1980) 481-512 Naud& C.F., Ellis, A.T. On the Mechanism of Cavitation Damage by Non-hemispherical Cavities Collapsing in Contact with a Solid Boundary, J Basic Eng. 83 (1961) 648656 Shutler, N.D., Mesler, RB. A Photographic Study of the Dynamics and Damage Capabilities of Bubbles Collaosina Near Solid Boundaries. J &sic Eng. 87 (1965) Sll-51; Brooke Benjamin, T., Ellis, A.T. The Collapse of Cavitation Bubbles and the Pressures thereby Produced Against Solid Boundaries, Trans. Roy. Sot. Ser. A 260 (1966) 221-240 Plesset,
123
63
Kling, C.L., Hammitt, F.G. A. Photographic Study of Sparkinduced Cavitation Bubble Collapse, J. Basic Eng. 94 (1972) 825-833 64 Gibson, D.C., Blake, J.R. The Growth and Collapse of Bubbles near Deformable Surfaces, Appl. Sci. Rex 38 (1982) 215-224 See Ref. 73 65 Rath, H.J. On the Stability of Gas Bubbles Oscillating Nonspherically in a Compressible Liquid J. Eng. Murh. 15 (1982) 161-170 66 Van Wijngaarden, L. One Dimensional Flow of Liquids Containing Small Gas Bubbles, Ann. Rev. Fluid Mech. 4 (1972) 369-396 67 Fanelli, M., Prosperetti, A., Realii M. Sound Propagation in Bubbly Liquids, to appear
124
68 69 70 71
72 73
Silberman, E. Sound Velocity and Attenuation in Bubbly Mixtures Measured in Standing Wave Tubes, J. ACOUSL Sot. Am 29 (1957) 925-933 Lauterbom, W. ed, Cavitation and Inhomogeneities in Underwater Acoustics, Springer Series in Electrophysics. 4 Springer-Verlag Berlin (1980) Drumheller, D.S., Bedford, A. A Theory of Bubbly Liquids, 1 Acoust Sot Am 66 (1979) 197-208 Anderson, A.L., Hampton, LD. Acoustics of Gas-bearing Sediments I. Background, J Acoust Sot Am 67 (1980) l8651889 Bedford, A., Stew M. A Model for Wave Propagation in _ Gassy Sediments,.J Acoust. Sot. Am 73 (1983) 409-417 Van Wijngaarden, L ed, Mechanics and Physics of Bubbles in Liquids. Appl Sci Res. 38 (1982)
ULTRASONICS.
MAY
1984
cavitation
series: part three
Bubble phenomena part two
in sound fields:
A. PROSPERETTI
Large-amplitude radial oscillations of gas bubbles are briefly illustrated with the aid of numerical examples. The origin and possible effects of pressure-radiation forces are considered and an estimate of the coalescence time under their action is given. Non-spherical oscillations, the related problem of the fragmentation of oscillating bubbles by instability of the spherical shape, and sound propagation in bubbly liquids conclude the review.
KEYWORDS:
Large-amplitude
ultrasonics, cavitation, bubble dynamics
are not expected to be seriously affected by this omission. Figs 9 and 10 show two sets of results obtained by Cramer” by numerical integration of the Gilmore equation (19). The graphs show the normalized maximum radius (R,,, - Ro)/Ro during steady oscillations as a function of the normalized frequency for R, = low3 cm (Fig. 9) and as a function of the equilibrium radius for a sound frequency of 7.208 kHz. The different curves correspond to different values of the sound pressure amplitude, and the fraction m/n labelling the peaks indicates that m (local) maxima in the radius-time curve occur during the n periods of the sound-field necessary to complete one cycle of the bubble oscillation. Thus a strong mth order harmonic of the fundamental, of the first subharmonic. of the second subharmonic, and so on, will be present in the oscillations of the peak of order mll, m/2,m/3, respectively. This component will be accompanied by its multiples and submultiples. as can be seen in Fig. 11 where an example of a complete oscillation together with its Fourier spectrum is shown.
oscillations
Large-amplitude bubble oscillations are closely related to acoustic cavitation noise. which will form the subject of another article in this series (see also Refs l-l 1). For completeness included here is a very brief account of this topic referring to the future article for further details.
Equations (15) and (17) for radial motion of a bubble, in part one of this paper. are so highly non-linear that not much progress can be expected on large-amplitude oscillations by analytical means. Some results are nevertheless available both for steady oscillations’*-I4 and for transient motionls for the incompressible case (17) with the polytropic relation (32) and sinusoidal forcing. Limited as they are, these results are also useful in the interpretation of numerical computations, since the behaviour of the observed stable solutions is often determined by the presence of unstable solutions that can be computed analytically much more readily than numerically.
In Figs 9 and 10 the dashed lines correspond to jumps in the steady state amplitude as the frequency, or the equilibrium radius. is increased past the instability threshold of the lower solutionr2Js. A notable feature of the numerical results is the impossibility of obtaining a steady solution at large excitation amplitudes for all frequencies or radii. For instance, in Fig 10 only a few such solutions, indicated by dots, could be obtained in the region of equilibrium radii between 150 and 250 pm for E = 0.8. For neighbouring values of the radius a non-steady, apparently chaotic behaviour is found; the Fourier analysis of which contains a broadband component in addition to lines. The thresholds for the onset of this behaviour are close to those for the appearance of subharmonic components at a frequency
A detailed discussion of these findings is lengthy, and rather than dwelling on them a brief account of some of the extensive numerical results obtained by Lauterborn and co-workers16-20 will be given. These studies have been based on the incompressible equation of motion (17) or on the Gilmore equation (19). The internal pressure was approximated by the polytropic relation (32) with K = 1.33. Thermal losses are therefore not included in the mathematical model, but the qualitative features of the solutions obtained
The
author
Celona this
16,
paper
is at the 20133 appeared
Dipartlmento Milano. in the
Italy.
di F~s~ca. Univewta Paper
previous
received
0041-624X/84/0301 ULTRASONICS
degli
1 June
Studi.
1983.
Via
Part one
of
issue.
. MAY 1984
15-l
O/$03.00
0
1984
Butterworth
& Co (Publishers)
Ltd 115
2.0
I
I.0
i
1.6
1.4
0.6
0.4
0.2
-I
0
0.1
0.15
0.2
0.3
0.4
0.5
0.6
0.7
0.0
0.9
I
1.5
2
3
f f F&g. 9
Numewal
frequency R,
values
to the
bubble
= 1 0m3 cm, the
(from
Ref.
of the
normaltzed
resonance
polytropic
frequency,
exponent
maximum
radius
for several
is K = 1.33
the
(Rmax n Ro)/Ro in steady of the sound pressure
values driving
frequency
is w = 2~
forced oscillations as a function of w/we. Rattos of the drwing amplitude indicated in bars on the curves. The bubble radius
x 340.4
kHz,
the amblent
pressure
IS 1 bar and the liquid
sound is
is water.
17)
2.0
I.0
-
1.6
-
0.6
0.4
0.2
0 30
40
50
60
70
80
90
100
150
200
300
400
500
600
700
000
900
Ho CpmI Fig.
10
function
Numerical of the
x 1 0e2 cm, the
values
of the
bubble
radius
liquid
is water,
R.
normalized for several
K = 1.33
maximum values
radius
of the
and the ambient
(R,,,
sound pressure
- Ro)/Ro in steady forced oscillations at the frequency w = 271 x 7.200 kHz as a pressure amplitude indicated in bars on the curves. The resonance radius is 4.305 IS 1 bar (from
a situation very reminiscent of the well known results of Feigenbaum*‘~**, according to which a series of period-doubling (that is. subharmonic) bifurcating oscillations precedes the onset of chaotic motion. (n/2)w.
The integration of these observations into a coherent picture of acoustic cavitation noise still lies in the future. It will undoubtedly require a consideration of
116
Fief. 17)
the role of rectified diffusion in the change of bubble population with time, of the instability of the spherical shape which causes the break-up of bubbles in largeamplitude oscillation, of bubble-bubble interaction forces, and of the radial dynamics that determine the appearance of transient events near the onset of the chaotic behaviour.
ULTRASONICS
. MAY
1984
I (X
Xc,);nj dS = V Sji
(49)
s
Upon taking the average over a cycle the following expression is obtained for the average radiation force F on the bubble F=
I
I
I
I
I
I
Ii
III
0.5
I
Time,
0
I
2
III
III
1.0
3
4
5
I 2.0
1.5
t/t,
6
7
6
9
IO
II
Frequency, f/G Fig.
11
Complete
steady
oscillation
of a bubble
of radtus
8 x 1 0v3
cm
The order of the resonance IS 3/2, meaning that the amplitude exhibits three local maxima during the two pressure cycles necessary to complete the oscillation. a - the upper curve is the pressure cycle and the lower curve the radius; b - the Fouler components of the oscillation are shown (from driven
Ref.
by a sound
pressure
of 0.8
bar at a frequency
of 23.56
kHz.
17)
Steady forces on bubbles
in sound-fields
A droplet. bubble. or particle present in a sound-wave is subjected. in addition to an alternating force, to a smaller (second-order) time-independent force, which is referred to as radiation force. In the case of a bubble the evaluation of this force is considerably simplified by the enormous compressibility of the gas compared with that of the liquid, which allows one to treat the latter as incompressible. The total pressure force acting on the bubble is given by -
Ppna
(48) s where n is the unit normal outwardly directed from the bubble surface S. The pressure p contains a spatially slowly varying contribution due to the sound-wave and a rapidly varying component caused by the relative motion of the bubble with respect to the liquid. If the liquid has a small compressibility this component contributes to the alternating force23-2s. but not to the steady one and can be neglected, so that p in (48) can be regarded as just the sound pressure. If the wavelength is large compared with the radius we can therefore write p(x; t> = p(q), t) + (x - x,,) * v p + . . where x0 is a point in the bubble. The first term does not contribute to the integral. while the second one gives - VVp, where V is the bubble volume, in view of the identity
ULTRASONICS
MAY 1984
-
where the angle brackets indicate the time average. This result is clearly valid for any shape of the bubble. Its physical origin becomes clear once it is noticed that the bubble is subject to an acceleration against the pressure gradient. which is proportional to its volume. In an oscillating pressure field the gradient reverses periodically so that the bubble executes an oscillatory translational motion, the amplitude of which, in the expansion phase, is greater than in the contraction phase. Therefore the bubble tends to drift in the direction opposite to that of the pressure gradient when the bubble volume is at its greatest. (We take this opportunity to correct an oversight in Ref. 26 in which the same argument leads to the opposite (!) conclusion: the word ‘greatest’ there should in fact be ‘smallest’.) This explanation clearly shows the second-order nature of the effect. For bubbles driven below resonance. sound pressure and volume increase are out of phase, the volume is largest during the expansion period and the bubble is therefore attracted to the pressure antinodes which are pressure minima during expansion. The opposite will be true for bubbles driven above resonance. and in this case the drift will be towards the pressure nodes. The preceding derivation neglects the compressibility of the liquid and hence two terms are absent from (50). The first is a correction term similar to (50), which subtracts from F the force that would act on an equal liquid volume. so as to make F = 0 if the bubble were replaced by liquid. The second term is due to the interaction between the relative motion of the bubble and the volume change. Both these terms turn out to be proportional to the compressibility of the liquid (pc2)-‘, and hence are overshadowed by the term retained in (50). which is proportional to the compressibility of the bubble24.2s. The first application that we make of (50) is the equilirium position of a bubble in a sound-field with a vertical gradient. In this case the bubble is acted upon by the force (50) and by the buoyancy force p g V. If we write the pressure field asp = p,[l - E(X) cos wr] where E(X) is the dimensionless space-dependent amplitude. the balance of the average buoyancy force and of (50) gives <
v coswt>
= _ P&f
I VQLE)I
(51)
The right-hand side of this relation (which may be termed the levitation number _F)2’ is a function of the bubble position. and therefore a measurement of this position in a known pressure field enables one to obtain a value for the left-hand side. In experiments of this type the bubble is said to be ‘levitated’ in the liquid. If the left-hand side is evaluated theoretically. one can subject the theory to experimental verification in this way. For instance. in the case of smallamplitude oscillations, using (28) we find
117
<%”- W2)CYf
_Y2.
pv,z (52)
2 (0,’ -cd2)2 t4p2w2
For w Q w,,. using
tension
contribution.
(34) and neglecting the surface this expression becomes
_!? = E/2K.
(53)
Hence a measurement of Yat low frequency gives the polytropic exponent directly. The measurements of this quantity shown in Fig. 8 were obtained using this procedure. As a second application of (50) we consider the mutual force between two pulsating bubbles arising because each is immersed in the pressure field of the other. The pressure field around a bubble of radius R’ in radial motion in an incompressible liquid is given by (see, for example, Refs 23. 28-34) p= k
+pc
_ ;pkt2
(R’&2$2)
where r =I x - x ‘I. Since the bubbles move towards each other, &/dt = -2~. Integrating from r = r(0) to r N 0 we have the following estimate of the coalescence time t, t, =
16 n2r~Ror3(0)
If we limit the analysis to small-amplitude motion, in terms of the volume V’ this expression may be written
This result cannot be trusted to be more accurate than to within an order of magnitude in view of the complex phenomena that occur when the bubbles are close, and which we have neglected 36.37.The volume pulsation amplitude 6 is three times the radius pulsation amplitude X. in terms of which the previous result is then =
wr3
We may note that if the bubble does not pulsate as a sphere, the correction terms to this expression involve higher powers of r-’ which are small far from the bubble. Upon insertion of (55) into (50) we find the following expression for the so-called Bjerknes force between two pulsating bubbles3s located at x and x’. =
_4b
5-x’ Ix - x
‘13
For,periodic mo$n an integration by parts gives
&
cos(@-4’)
v, v;66’w2
-xz Ix-x
‘13
(57)
This force is attractive if cos(@ - @‘) > 0. which will be the case if both bubbles are driven below or above resonance. whereas it is repulsive if one is below and the other above resonance. The effectiveness of the attractive force will be increased because the bubbles tend to collect near the nodes or the anti-nodes of the field. This result can be used to obtain an estimate of the coalescence time between two bubbles. which for simplicity we may take to be of equal size. We further assume that inertial effects are negligible and that the force (56) is balanced by a drag force FD proportional to the velocity I* F,, = 47ry R, p
(w
y = 1 and y = 3/2 for the Stokes-Rybczynski and Stokes small-Reynolds number cases respectively. and y = 3 for the large-Reynolds number Lcvich drag law. Equating (57) and (5X) we find that where
118
(0)
(61)
Using for X the linear becomes t
c
equation
(28) this expression
= ytipr3 (0) [(w’ - ~02)~ + 4P2 a2 3R,oZ~’
(55)
F
(60)
3p v(j%%_?
3pR;02X2 (54)
r
(59)
32 ~~~yR,/.ir~
t,
4
fi2w2
v=
1
p:
(62)
where. as in (25). E is the dimensionless sound amplitude. For sound frequencies in the range of tens of kHz and small bubbles (R, < -IO-’ cm) w0 % w and. using (34). the previous expression simplifies to t
_ ~2-ypr3(0)[1 c -
+ (1 -
~/SK)
2u/R0 p-l2 (63)
pR;02c2
To give an idea of some orders of magnitude, for K = I. y = I, r(O) = I cm. o = 2rr x 20 kHz. E = 0.5 in water, this relation gives t, = 7600 s (more than 2 h) for R, = 10m3 cm but 2,. = 2.4 s for R, = 5 x IF3 cm. This extreme sensitivity to the value of the radius makes this mechanism significant for relatively large bubbles. The time t, is a minimum for resonance conditions and has the value 4 rppf12 r3 (0) tc = 3 Roe2 p?,
Distortion
(64)
from spherical
shape
Surface tension tends to maintain the bubble in a spherical configuration, which may be distorted however. by asymmetrical pressure fields and dynamic effects. For example, if a pressure difference Ap exists across the bubble, its effect on the spherical shape will be small provided that the pressure difference due to surface tension, 2u,,R,, is much greater. We can estimate Ap by R,I Vpl , which leads to R, < (2a/j Vp( )I”
(65)
For a sound-wave of wave-number k, I Vp 1 - k~p m and the previous relation is R, < (2u/kcp,,)‘12. For pm = I bar and k = 0.84 cm-‘. corresponding to a frequency of about 20 kHz in water. the preceding relation gives R,, < 0.02 cm for E = 0.5 and R, < 0.04 cm for E = 0.1. The resonance radius at this frequency is 0.015 cm approximately. When this limitation is not met the
ULTRASONICS.
MAY 1984
decay proportional to r-r associated with changes of volume. Hence, the acoustic energy loss in the nonspherical motion is much less than that in volume pulsations, and this mode of damping is accordingly also small. The main contribution to energy dissipation comes now from viscosity. The theoretical analysis of non-spherical motion is naturally much more difficult than for the spherical case, except in the case of small distortions of the spherical shape. In this case, if the bubble shape is written as a sum of spherical harmonicsy/,
r(e) @;t) = R(t) + -y
0
a
(66)
0) Y/m (0,$1
aI,
< It is possible to a first approximation to neglect terms of second and higher orders in the al, and in fact all the interactions between the different modes disappear. The problem therefore simplifies in that the effect of each single mode can be studied separately, writing
r(& q5; t) = R(r) + a,(r) P,,(cos 8)
(67)
where P,, is a Legendre polynomial. In this expression we have taken the bubble shape to be axisymmetrical, which is permissible since it is found that the equation of motion for the coefficients al,,, is independent of m and hence is the same as that obtained using (67). For the case of small viscosity and free oscillations this equation is found to be3*J9 0
b
’
c
-!-
wf
Fig. 12 Transient oscillations of gas bubbles in a sound-field, the pressure amplitude of which is specified in bars on the curves. The lower line is the pressure cycle. a - the equilibrium radius is 1O* cm and W/W,, = 0.107; b - it is 5 x 1 O4 cm and w/w0 = 0.706. The sound frequency is w = 2n x 500 Hz, the ambient pressure 1 atm and the liquid is water
sound-wave will excite radial pulsations. The a relative velocity with and, more importantly, radial motion that can
shape oscillations in addition to bubble may also be distorted by respect to the liquid, buoyancy, by an interaction with its own lead to its breakup.
In approaching the problems associated with the loss of spherical symmetry it is useful to keep in mind some general physical features which make them very different from their radial counterparts. In the first place a small distortion of the spherical shape has a much smaller (in fact, of second order) effect on the volume, so that the restoring force due to the compressibility of the gas is unimportant, the only restoring mechanism being provided by surface tension. For the same reason thermal dissipation processes are negligible. If the volume of the bubble is unchanged the pressure field caused by its motion has a dipole character if the centre executes oscillatory motion, of the quadrupole type if the bubble undergoes oscillations from a prolate spheroidal shape to an oblate one, and of higher multipole types for other forms of oscillations. Accordingly, the rate of decay with distance of the pressure perturbations is proportional to r-*, rs3 or higher powers, which are to be compared with the slow
MAY
1984
1
f&
PR’
I
ULTRASONICS.
. 3 ;t2(nt2)(2n+l)
ii,+
.
t 2(n t2)
I.IR a,=O.
pR3 1 The value n = 1 corresponds to (small-amplitude) translatory motion for which the restoring force (the last term) obviously vanishes. In this case the equation may be written = -12n/.~Ra’, The left-hand side is the rate of change of the liquid momentum, while the right-hand side is Levich’s drag force on a spherical bubble at large Reynolds numbe14°-42. When the bubble ;i,
+2(n t2)(2n
radius +1)
t(n-l)(ntl)(nt2) We can read directly frequency won 02,,=
(n-l)(?ztl)(n+-2)
is constant 5
(68) reduces
to
&
+
a,=0
from this equation
p+
the natural
(71)
119
and the damping
constant
p,,
/$I = (n + 2) (2n + 1) /dpR’
(72)
Both of these results are classicaP. The prolate-oblate oscillation corresponds to n = 2 and for water we find wo2/2rr = 0.15, 4.7. 149 kHz, fi, = 0.02. 2. 200 kHz for R = 10-r. IO-*, IO--’ cm respectively. The coupling between the radial motion and the shape oscillations present in (68) has some important consequences which we cow discuss. In the first place it is seen that the term 3R/R. the qualitative effect of which will be similar to damping. tends to dampen the shape oscillation on expansion (R > 0). but acts as. ‘negative damping’ that is, a destabilizing force. on contraction (R < 0). The physical origin of this effect is the diverging or converging nature of the streamlines which stretch or compress the bubble surface. Secondly. the acceleration of the radial motion is seen to act as a restoring (stabilizing) force when it is negative - for instance in the later stage of expansion. However. this term will have a strongly destabilizing effect when R > 0. which will occur at the beginning of the expansion or. upon collapse. near the minimum radius. The mechanism of this effect is the same as that of the ordinary Rayleigh-Taylor instability for plane interfaces44.
example Refs 45. 46). The first unstable range occurs for k = 1 and gives rise to growing oscillations at a frequency close to the subharmonic value won = Y20. (The unstable ranges for larger values of k have widths of order Xok and hence cannot consistently be considered in a theory that is correct to order X,,, as is the present one.) The rate of growth of the oscillations in 6, is proportional to exp(Xr) where X = [l/4 8*X, (n,, - 1/4)*]‘/*. but, from (74) it can be seen that the rate of growth of the oscillations in a, is WA-P,. Requiring this to be positive gives the following approximate condition on X,, for instability
The physical meaning of this instability cannot be completely explained within the framework of the linear theory described. If the threshold is not exceeded by a large amount it is likely that non-linear effects will stabilize the shape oscillation to some finite valm?‘. For more violent motion, however, the amplitude of the oscillation may become so large as to result in a breakup of the bubble. With this interpretation in mind it is of interest to discuss (79) further. The minimum of the threshold x0 occurs for w = 2w,, and has the value
Of particular interest in the present context is the solution of (68) when the bubble radius executes an oscillatorv motion. For simnlicitv we shall consider the linear. small-amplitude case R=R,(l
+X,,cosot)
( 73)
Furthermore. for small viscosity. we can neglect the variations of R in the terms containing p. With the substitution Q,I = R-‘I* exp(-/?,,f)
b,,
74)
with p,, given by (72). we find i;,, + G,,(f) b,, = 0
(75)
where
t(n-l)(ntl)(n+2)
/?-pR3
(76) To simplify this form of G,,. terms proportional to f?*. p,: and ~.LRhave been dropped. We can now use (73) and let r = tit to find d2b, __ dr2
t
[7)” +28,X,
cos271 b,=O
(77)
where (78)
This is Mathieu’s equation which is known to possess exponentially growing solutions in the neighbourhood . that 1s q,,, = !A kw (see for of tl,, = kZ/4. k = 1.2. .
120
030) For n = 2 in water we have y0 = 0.0085, 0.027, 0.085 for R, = lo-‘, lo-*, lo-) cm respectively. For ftxed R. this is a decreasing function of n, which seems to imply that all sufficiently large modes are unstable. This prediction however is invalidated by the fact that (68) is actually only valid for n(p/po)“*
phenomenon
frequently
observed
ULTRASONICS.
in acoustic
MAY
1984
cavitation experiments, which is presumably connected with the stability of the spherical shape, is the rectilinear motion in apparently random directions that bubbles execute once they attain a certain sizeJ2. A possible explanation for this behaviour is that a suitable coupling of different shape oscillations produces asymmetries in the bubble surface velocity which propel it by reaction. This explanation has been offered by Benjamin and Strasberg52. and it rests on the somewhat indirect evidence that thresholds for the rectilinear motion are in agreement with those for shape oscillations53.
Let us consider a homogeneous mixture of bubbles with equal radius R, in a liquid contained within a pipe, thus making the problem effectively onedimensional. The mixture can be characterized by the void fraction 8, the volume occupied by the bubbles in a unit volume of the mixture. If n is the number of bubbles per unit volume. each having a volume vt,, it is evident that
Important effects of acoustic cavitation such as erosion and cleaning may be connected to very large amplitude deformations of the spherical shape in the vicinity of a solid boundary. which result in a high-speed jet piercing the bubble in the direction of the wall. There is no theoretical information on this effect for the oscillating pressure conditions typical of acoustic cavitation, but the phenomenon has been investigated numerically54-58 and experimentally59-64 for the constant pressure case characteristic of flow cavitation.
The average density
For a pressure in excess of 1 bar the jet velocities are found to be of the order of 100 m s-l, and this result is predicted to be proportional to (p_ - pi)“2. It can therefore be expected that very strong impulses will be communicated to solid boundaries by bubbles undergoing violent collapses in their vicinity. These impulses can detach impurities from the solid. but they may also cause local damage to the wall structure leading to fatigue and erosion. In this connection Fig. 13 shows a very beautiful picture by LA. Crum in which the jet piercing the bubble is clearly visible. The bubble shown contains gas and vapour and is oscillating at 60 Hz at reduced pressure (4 X 10) - 5 X 10’ Pa) and its size is approximately 0.2 cm. The liquid is water with 25% by volume of glycerine. and the temperature is approximately 25”C.65
Sound
propagation
in a bubbly mixture
The presence of even a few bubbles substantially alters the compressibility of the medium and hence has a strong effect on the speed of sound and its attenuation. This is just one of the many aspects of the cooperative effects of bubbles in liquids, of which an elementary discussion is given here.
F=nvb=+R;
(81) of the mixture
is
P=6PG+S6)PL
(82)
and. if the bubbles and the liquid are taken to move with the same velocity u. conservation equations for mass and momentum can readily be derived in the standard way. Their linearized form is
-a; ,Fg=O at
- au Pat+==0
ap
If a relation of the type p = p(p) is assumed, the wave equation follows from this system with a speed of propagation c given by cm2 = dp/dp. or, from (82). 1 _= c2
6+-_ c&
l-6 (PL
c;
-
(85)
PG) g
where c20,~ = dpldp~, L. To evaluate daldp we consider a volume V of the mixture consisting of a gas volume VG, and a liquid volume. Vt_. By definition of 8 we have
g= dp
-d dp
~--
;
d$
(86)
Since VG = nVvb and the number of bubbles in V is constant the first derivative is simply dVG/dp = nV x dvddp = (V&+-J dvtJdp. The total mass of liquid in K VLpL, is also constant and therefore
Upon substitution into (85) we find 1 _=-+, c2 Fig. 13
Jet
formation
bubble
at low
bubble
size
Professor
pressure
durmg
Crum,
ULTRASONICS
collapse
(0.04-0.05
IS approximkely
LA.
the 0.2
University
. MAY
bars) cm.
of an oscillating in a 60
(Unpubkhed
of Mlsslsslppi)
1984
of these results into (86) and then
62
(1 - S)”
c&
CL
(
1+ p”
J-
PL
l-6
i
gas-vapour
Hz sound-field. photograph
The by
-PL
6(1 y
-6)
dub ~ dP
(87)
121
(To obtain the first term of this relation we have used the approximation PGdvb N -V&IG, which would be exact only if the gas density were uniform in the bubble.) The final step is to observe that dvddp may be written as (dvddf)/(dpldf). From (25) and (26) we then have for harmonic oscillations
dut, -= dt
-hR;
p$
l
-
b-
or, upon substituting the expression for X derived (28) and neglecting the small term p~/p~ in (87), 1 -=_-+ c2
62
(1 -&I2 c;
c&
R; (08
- o2
- -
R=0.250cm
from
36(1-S)
+
R=O.l89cm
(88)
+ 2i@)
The last term is found to be much larger than the other two when S is not too small (nor too close to one, but then we would be dealing with droplets in a gas rather than with a bubbly liquid) so that (88) can be approximated by c2 =
R ‘0(ui
- o2
t 2iflw) (89)
36Q-6) A wave equation consistent with this dispersion relation can be written when the frequency is sufficiently low that o, and p are approximately constant; it is given by
v---
R=0.364cm
l-
R= 0.260cm
.
R= 0.268cm
(90) IO-' t
The last two terms are negligible at extremely low frequencies and, with K = 1 and the neglect of surface tension, by (34). (89) reduces to the well known resulP
,J, ,A
(91)
IO
b
IO2
i,l , , , IO’
1
-I
IO’
Frequency, v Cs-’1
independent of bubble size. For 6 = 5%, pm = 1 bar. pL = 1 g crnm3 this expression gives c = 46 m s-l, a value to be compared with the typical ones for the speed of sound in the gas, CG N 340 m s-l, and in the liquid, cL 2: 1500 m s-l.
Fig. 14 Theoretxal attenuation of sound-waves in water containing air bubbles (Ref. 67) compared with experimental data (Ref. 68) as a function of the sound frequency. The volume fraction occupied by the bubbles is 5.84 x 1 0T2% in a and 1% in b. The values of the bubble radius are Indicated in the figures. In both cases the temperature was 20°C. The arrows mark the position of the resonance. (From Ref. 26)
In (88) and (89) the imaginary part of c corresponds to attenuation of the pressure waves. This quantity, calculated on the basis of a more precise theory26.67, is shown in Fig. 14 expressed in units of dB cm-‘. (1 dB cm-* means that the wave intensity is attenuated by a factor 10-O.’ z 0.794 and the wave amplitude by 10-0.05 C=0.891 in 1 cm.) Some classical experimental measurements by Silberman6* are also shown in the figure. The peaks correspond to the resonance region of the bubbles, and the corresponding enormous attenuation is worthy of notice. The bubbly medium acts essentially as a low-pass filter, the cut-off frequency of which is determined by the natural frequency of the bubbles and hence by their size.
in itself very broad and here we must confine ourselves to the preceding simple treatment. The reader is referred to Refs 66-72 for further information and developments.
The subject
122
of sound
propagation
in bubbly
liquids
is
Esche, R. Untersuchung der Schwingungskavitation in Fliissigkeiten, Acusricu AB2 (1952) 208-218 Bohn, I, Schalldruckverlauf und Spectrum bei der Schwingungskavitation Acustica 7 (1957) 201-216 Neppirss, E.A. Subharmonic and Other Low-frequency Emission from Bubbles in Sound Irradiated Liquids. 1 Acoust Sot Am 46 (1969) 587-601 Neppiras, E.A. Subharmonic and Other Low-frequency Signals from Sound-irradiated Liquids, .I Sound Vib. 10 (1969) 176-186
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Physicochemical Hydrodynamics, PrenticeHall. Englewood Cliffs (1962) 445 Landau, L.D., Lifshitz, E.M. Fluid Mechanics, Pergamon Press, London (1959) 172 Haruer. J.F. The Motion of Bubbles and Droos Throueh Liquids.Adv. Appl. Me& 12 (1972) 59-131 ’ Lamb, H. Hydrodynamics, Cambridge University Press, Cambridge (1932): reprinted by Dover. New York (1945) 475. 641 Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stabilitv. Oxford Universitv Press. Oxford (1961): reorinted by Dover. New York (1981) Ch. X ’ . Cole, J.D. Perturbation Methods in Applied Mathematics. Blaisdell. Waltham (1968) 94 Bender, C.M., Orszag, S.A. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill. New York (1978) 560 Hall, P., Seminara, G. Nonlinear Oscillations of Nonspherical Cavitation Bubbles in Acoustic Fields. 1 Fluid Mech. 101 (1980) 423-444 Prosperetti, A. On the Stability of Spherically Symmetric Flows. Rend Accad Nar Lincei 52 (1977) 196-203 Hullin, C. Stabilitatsgrenze Pulsierender Luftblasen in Wasser. Acustica 37 (1977) 64-73 Francescutto, A., Nabergoj, R. Pulsation Amplitude Threshold for Surface Waves on Oscillating Bubbles.
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43
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Cramer, E., Lauterhorn, W. Acoustic Cavitation Noise Soectra. Aool. Sci. Rex 38 (1982) 209-214. See Ref. 73 Prosperetti A. Nonlinear bsciliations of Gas Bubbles in Liquids: Steady-state Solutions, J Acoust Sot Am. 56 (1974) 878-885
44
13
Samek, I_..Nonlinear Oscillations of Gas Bubbles in Liquids. Czech. .J Phys B30 (1980) 1210-1226
14
Francescutto, Gas Bubbles
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A., Nabergoj,
R Steady-state
Oscillations
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tauterhorn, Oscillations
49
W. Numerical Investigation of Nonlinear of Gas Bubbles in Liquids. 1 Acousr. Sot Am.
17 18
Cramer, E., Lauterhorn,
W. Zur Dynamik
Lauterborn, W. Cavitation Bubble Dynamics-new Tools for an Intricate Problem. Appl. Sci Res. 38 (1982) 165-178 see
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M.J. The Onset Spectrum of Turbulence, Phys L.ett A74 (1979) 375-378 Feigenbaum, M.J. The Transition to a Periodic Behaviour in Turbulent Systems, Comm. Mark Phys. 77 (1980) 65-86 Coakley, W.T., Nyborg, W.L Cavitation: Dynamics of Gas Bubbles; Applications. Ultrasound: Its Applications in Medicine and Biology. Fry. F.J., editor. Elsevier. New York (1978) 77-159 Vosidka, K., Kawasima, Y. Acoustic Radiation Pressure on a Comnressible Sohere. Acustica 5 (1955) 167-173 Weiser, M.A.H.; Apfel, RE. Extension’of Acoustic Levitation to Include the Study of Micron-sized Particles in a More Compressible Host Liquid. 1 Acoust Sot. Am. 71 (1982)
24 25
32 33
251 ApfeI, RA. Acoustic Cavitation,
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289-303
52
T. Surface Effects in Non-spherical Motion of Small Cavities. Cavitation in Real Liquids. Davies. R.. ed. American Elsevier. New York (1964) 164-180 Eller, A.I., Crum, L.A. Instability of the Motion of a Pulsating Bubble in a Sound Field. J. Acoust Sot. Am. 47
Brooke Benjamin,
(1970)762-767 54 55
56
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Methods of Experimental Physics, 19, Ultrasonics, Edmonds, P.D.. ed. Academic Press, New York (1981) 355-411 Prosperetti, A. A Generalization of the Rayleigh-Plesset Equation of Bubble Dynamics. Phys. Fluids 25 (1982) 409-410
Levich, V.G.
nichtsphlrisch
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Bubble Dynamics: a Review and some Recent Results. Appl. Sci Res 38 (1982) 145-164 See Ref. 73 Crum, LA., ProsperettI, A. Nonlinear Oscillations of Gas Bubbles in Liquids, an Interpretation of Some Experimental Results, 1 Acoust Sot Am. 73 (1983) 121-127 Flynn, H. Physics of Acoustic Cavitation in Liquids, Physical Acoustics, Mason, W.P.. ed. Academic Press. New York, (1964). IB, 57-172 Plesset, M.S. Bubble Dynamics, Cavitation in Real Liquids, Davies. R ed. American Elsevier, New York (1964) 1-18 Piesset, M.S., Prosperetti, A. Bubble Dynamics and Cavitation. Ann Rev. Fluid Mech. 9 (1977) 145-185 Prosperetti, A. Current Topics in the Dynamics of Gas and Vapor Bubbles, Meccanica 12 (1977) 214-235 Neppiras, E.A. Acoustic Cavitation. Phys. Rep. 61 (1980) 159Prosperetti,
M., Nabergoj, R. On the Motion of a Nearly Spherical Bubble in a Viscous Liquid Phys Fluids 21 (1978)
Acusricu 41 (1978) 215-220
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Feigenbaum,
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Schallabstrahlung Kiigelformiger Kavitationsblasen in einem Schallfeld. Acusrica 49 (1981) 226-237 Lauterhorn, W. Resonanzkurven von Gasblasen in Fliissigkeiten. Acusrica, 23 (1970) 73-81
20
22
46
in Liquids: Explicit Formulas for Frequency Response Curves, J Acoust Sot Am 73 (1983) 457-460 Prosperetti, A. Nonlinear Oscillations of Gas Bubbles in Liquids: Transient Solution and the Connections Between Subharmonic Signal and Cavitation, J Acoust Sot Am. 57 (1975) 810-821 59 (1976) 283-293 Cramer,‘E. The Dynamics and Acoustic Emission of Bubbles Driven bv a Sound Field. in Ret 69. 54-63
19
AX.,
39
10
I5
Flow, 7 (1981) 709-717 Hofman, G. Bubble Coalescence in Pure Liquids. Appl. Sci Res. 38 (1982) 353-361 See Ret 73 Prosperetti, A., Seminara, G. Linear Stability of a Growing or Collapsing Bubble in a Slightly Viscous Liquid, Phys. Chesters,
140-142 W., Haussman,
Acoustic Cavitation Noise, 7th International Symposium on Nonlinear Acoustic, Nayfeh. AH.. Kaiser, J.E. eds. Virginia Polytechnic Institute (1976) 191-194
12
LA. Bierknes Forces on Bubbles in a Stationary Sound Field.2 Acoust Sot. Am 57 (1975) 1363-1370 _ Thomas. RM. Bubble Coalescence in Turbulent Flows. Int. Crum,
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60
61
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M.S., Chapman, R.B. Collapse of an Initially Spherical Vapor Cavity in the Neighbourhood of a Solid B0undary.J Fluid Mech. 47 (1971) 283-290 Bevir, M.K., Fielding, P.J. Numerical Solution of Incompressible Bubble Collapse with Jetting Moving Boundary Problems in Heat Flow and Diffusion. Ockendon. J.R., Hodgkins. W.R. eds. Clarendon Press, Oxford (1975) 286-294 Lenoir, M. Calcul Numerique de I’Implosion dune Bulle de Cavitation au Voisinage dune Paroi ou dune Surface Libre, J Mkanique 15 (1976) 725-751 Shima, A., Sato, Y. The Collapse of a Spheroidal Bubble Near a Solid Wall. 1 Mkanique 20 (198 1) 253-27 1 Guerri, L, Lucca, G., Prosperetti, A. A Numerical Method for the Dynamics of Non-spherical Cavitation Bubbles Proceedings of the Second International Colloquium on Drons and Bubbles. Le Croissette. D.H.. ed Jet Prooulsion Labbratory, Pasadena (1982) 175-181 Fujikawa, S., Akamatsu, T. Effects of the Non-equlibrium Condensation of Vapor on the Pressure Wave Produced by the Collapse of a Bubble in a Liquid, J Fluid Mech. 97 (1980) 481-512 Naud& C.F., Ellis, A.T. On the Mechanism of Cavitation Damage by Non-hemispherical Cavities Collapsing in Contact with a Solid Boundary, J Basic Eng. 83 (1961) 648656 Shutler, N.D., Mesler, RB. A Photographic Study of the Dynamics and Damage Capabilities of Bubbles Collaosina Near Solid Boundaries. J &sic Eng. 87 (1965) Sll-51; Brooke Benjamin, T., Ellis, A.T. The Collapse of Cavitation Bubbles and the Pressures thereby Produced Against Solid Boundaries, Trans. Roy. Sot. Ser. A 260 (1966) 221-240 Plesset,
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Kling, C.L., Hammitt, F.G. A. Photographic Study of Sparkinduced Cavitation Bubble Collapse, J. Basic Eng. 94 (1972) 825-833 64 Gibson, D.C., Blake, J.R. The Growth and Collapse of Bubbles near Deformable Surfaces, Appl. Sci. Rex 38 (1982) 215-224 See Ref. 73 65 Rath, H.J. On the Stability of Gas Bubbles Oscillating Nonspherically in a Compressible Liquid J. Eng. Murh. 15 (1982) 161-170 66 Van Wijngaarden, L. One Dimensional Flow of Liquids Containing Small Gas Bubbles, Ann. Rev. Fluid Mech. 4 (1972) 369-396 67 Fanelli, M., Prosperetti, A., Realii M. Sound Propagation in Bubbly Liquids, to appear
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68 69 70 71
72 73
Silberman, E. Sound Velocity and Attenuation in Bubbly Mixtures Measured in Standing Wave Tubes, J. ACOUSL Sot. Am 29 (1957) 925-933 Lauterbom, W. ed, Cavitation and Inhomogeneities in Underwater Acoustics, Springer Series in Electrophysics. 4 Springer-Verlag Berlin (1980) Drumheller, D.S., Bedford, A. A Theory of Bubbly Liquids, 1 Acoust Sot Am 66 (1979) 197-208 Anderson, A.L., Hampton, LD. Acoustics of Gas-bearing Sediments I. Background, J Acoust Sot Am 67 (1980) l8651889 Bedford, A., Stew M. A Model for Wave Propagation in _ Gassy Sediments,.J Acoust. Sot. Am 73 (1983) 409-417 Van Wijngaarden, L ed, Mechanics and Physics of Bubbles in Liquids. Appl Sci Res. 38 (1982)
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1984