nonlinear bubble mit

8/3/2019 Nonlinear Bubble MIT

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Annu. Rev. Fluid. Mech. 1997. 29:20143

Copyright c 1997 by Annual Reviews Inc. All rights reserved

NONLINEAR BUBBLE DYNAMICS

Z. C. FengMassachusetts Institute of Technology, Department of Mechanical Engineering,

Cambridge, Massachusetts 02139

L. G. LealUniversity of California, Department of Chemical and Nuclear Engineering, Santa

Barbara, California 93106

KEY WORDS: bubble dynamics, nonlinear dynamics, shape oscillation, cavitation, boundaryintegral method

ABSTRACT

The inertia-dominated dynamics of a single gas or vapor bubble in an incompress-ible or nearly incompressible liquid has been the subject of intense investigation

for many years. Studies prior to 1976 were thoroughly reviewed by Plesset &

Prosperetti (1977) in Volume 9 of this series. Our review fills the gap between

Plesset & Prosperettis review and the present. We focus on new understandings

of bubble dynamics through a nonlinear dynamical systems approach.

1. INTRODUCTION

The inertia-dominated dynamics of a single gas or vapor bubble in an incom-

pressible or nearly incompressible liquid has been the subject of intense inves-

tigation for many years. Studies prior to 1976 were thoroughly reviewed by

Plesset & Prosperetti (1977) in Volume 9 of this series. The only other related

review in this series, written by Blake & Gibson (1987), considered the problem

of a bubble in a quiescent fluid near a plane wall.

The present review is focused on the dynamics of a single bubble in an

unbounded fluid. Thus, it is intended to fill the gap between Plesset & Pros-

peretti (1977) and the present. One main message is that the introduction of

ideas and techniques from theoretical studies of nonlinear dynamical systems

in more recent years has led to some major revisions in our understanding of

the dynamics of a gas or vapor bubble.

201

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202 FENG & LEAL

Much early work on the motion of gas bubbles was aimed at understanding

the bubble as a source of sound. For this problem, it is time-dependent changes

in bubble volume that are of dominant interest. Because shape deformations area weaker source of sound in linear theory (a dipole or higher-order singularity

versus a monopole source), and because the spherical bubble problem is easier,

most theoretical studies of bubble dynamics have focused on this problem. The

majority of this work was reviewed by Plesset & Prosperetti (1977). In the

present paper, our discussion of spherical bubbles is limited to the improved

understanding that results from applying modern theories of nonlinear dynamics

to the Rayleigh-Plesset equation.

Although gas or vapor bubbles in flow are very frequently nonspherical, or

become so as a consequence of instabilities of the oscillating spherical bubble,the dynamics of a nonspherical bubble are much more difficult to study. Until

recently, studies had been restricted to the growth or collapse of an axisymmetric

bubble in a quiescent fluid near a plane wall (Blake & Gibson 1987); and to the

parametric instability of an oscillating spherical bubble in an unbounded fluid

(Plesset & Prosperetti 1977).

A thorough review of the first topic was written by Blake & Gibson (1987).

In the meantime, there have been some important experimental and theoretical

studies of the motion and deformation of cavitation bubbles traveling through

water near a standard head form (Ceccio & Brennen 1991, Kumar & Brennen1993, Brennen 1994, Chahine 1994, Kuhn de Chizelle et al 1995). These studies

show that the asymmetry of shape produced for large bubbles by the shear near

the wall fundamentally alters the dynamics of the bubble during collapse and

rebound, largely suppressing the re-entrant jet that is generally accepted as the

source of most of the noise and all of the damage associated with a collapsing

bubble. In spite of these very important developments, we believe that it would

be premature to discuss additional aspects of the interactions between bubbles

and walls at this time.

The earlier studies of the linear stability of a spherical bubble to infinitesimal

perturbations of shape have been largely subsumed by recent work on resonant

coupling between small (but finite) oscillations of bubble volume and shape.

The latter work was largely motivated by the suggestion of Longuet-Higgins

(1989a,b) and others that nonspherical bubbles produced by such phenomena

as breaking waves or air entrainment in the impact of raindrops on the ocean

surface (Oguz & Prosperetti 1990a, 1990b, 1991) may play a major role in

the production of ambient sound in the upper ocean, and may also provide an

explanation for the phenomenon of bubble dancing (cf Mei & Zhou 1991).

The problem of resonant coupling between the bubble volume and one (or

more) deformation mode(s) is a main focus of this review. This work differs in

one important respect from the stability analyses that preceded it; the emphasis

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NONLINEAR BUBBLE DYNAMICS 203

is on the coupling between oscillations of bubble volume and shape, whereas

in the earlier work it was presumed that the purely radial oscillations were

fixed/invariant to the stability or instability of the shape perturbations. Althoughthis is still an evolving field of research, enough is now known to provide the

necessary framework for future studies, and thus it seems appropriate to review

here.

2. DYNAMICS OF A SPHERICAL BUBBLE

One of the most significant new developments in bubble dynamics since the

review of Plesset & Prosperetti (1977) is the realization that the bubble response

to a time-periodic pressure field can be chaotic, even when the bubble is as-sumed to remain spherical. Indeed it is fair to say that research on the nonlinear

dynamics of spherical bubbles has contributed significantly to a better under-

standing of chaos physics. Lauterborn & Parlitz (1988) have shown that the

notions of strange attractors, subharmonic and superharmonic resonances, and

period-doubling bifurcations can all be illustrated using the Rayleigh-Plesset

framework. It is quite likely that chaos had already been observed in numeri-

cal work by Lauterborn and coworkers as early as the mid-1970sfor exam-

ple, Lauterborn (1976). However, extensive study of chaotic bubble dynamics

did not begin until the 1980s, and a comprehensive review has not previouslyappearedthough some mention may be found in the reviews by Prosperetti

(1984a,b).

Research on dynamical systems in the past two decades has provided a much

more comprehensive framework for understanding the dynamics of a spherical

bubble. In this respect, one clear message is that most of the important dynam-

ical features are inherent in the form of the nonlinear terms that apply for any

radially oscillating spherical body at high Reynolds number. Hence the most

interesting behavior is already revealed in the dynamics of a vapor or ideal gas

bubble in an unbounded, incompressible, inviscid fluid. In that spirit, the earlywork of Ma & Wang (1962) is now seen to provide extremely important insights

into the nonlinear nature of bubble dynamics.

The Hamiltonian Structure of the Simplified Model

A simplified model for a gas bubble in an incompressible liquid under adiabatic

or isothermal conditions is the well-known Rayleigh-Plesset equation

R R

+3

2 R2

+4

R R

=1

Pv (T) P + Pg

2

R (1)

where R is the spherical bubble radius, the liquid density, the liquid visco-

sity, and the surface tension coefficient. The vapor pressure of the liquid as

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204 FENG & LEAL

a function of temperature is denoted as Pv (T), while Pg is the partial pressure

of gas in the bubble and P

is the far-field background pressure. Although

an explicit solution for the bubble radius as a function of time can only beobtained numerically, a qualitative understanding can be obtained by examining

the solution trajectories in the phase space (R, R), which Ma & Wang (1962)have shown to be derivable from a Hamiltonian function in the inviscid limit.

The existence of this Hamiltonian function tremendously simplifies the analysis

for free oscillations of the gas bubble. In fact, Chang & Chen (1986) used the

Hamiltonian structure discovered by Ma & Wang to give a complete overview of

the dynamics of free oscillations as described by the Rayleigh-Plesset equation.

The first step is to determine the number of equilibria (or fixed points) as a

function of the physical parameters. Assume that the gas behaves as an idealgas with a partial pressure that is given by

Pg =GT

R3

where G is assumed to be a constant proportional to both the specific gas con-

stant and the mass of the gas. This assumption excludes mass diffusion of the

gas through the boundary (Plesset & Prosperetti 1977, Fyrillas & Szeri 1994,

Lofstedt et al 1995, Asaki & Marston 1995a). The critical parameter in the be-

havior of Equation 1 is the difference between the vapor and ambient pressures,Pv P. The equilibrium solutions are the roots of a cubic polynomial. Thecubic polynomial has repeated roots when Pv P is the following criticalvalue:

( Pv P)c =

323

27GT.

The corresponding equilibrium is

R0=

3GT

2.

If we then let

R = R0x , and t=

3R302

,

the following dimensionless form of the Rayleigh-Plesset equation is obtained

in the inviscid limit:

xd2x

d2+ 3

2

d x

d= 1

x 3+ 2P 3

x, (2)

where

P = Pv P(Pv P)c

.

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Hence, if we set

q = x , and p = x 3 d xd

,

Equation 2 can be written as

dq

d= p

q3,

d p

d= 3p

2

2q4+ 1

q 3q + 2Pq 2,

which are derivable from the following Hamiltonian function

H= p2

2q3 log q + 3

2q2 2

3Pq3. (3)

The Hamiltonian function depends on the single dimensionless parameter P ,

which thus determines the number of equilibria of the dynamic system, as shown

in Figure 1. In particular, we see that there are zero, one, or two equilibrium

solutions for R depending on whether P > 1, P < 0, or 0 < P < 1. The

inviscid bubble dynamics corresponding to these three regimes are shown in the

Figure 1 Dependence of the equilibrium radius upon the pressure driving force.

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206 FENG & LEAL

phase space (q, p) in Figure 2(a)(c). When PvP < 0, as is typical of a puregas bubble, the bubble oscillates periodically for any initial condition. When

the vapor pressure of the bubble dominates so that P > 1, the bubble radiusgrows to unbounded values for any initial condition. Finally, for intermediate

values of the dimensionless pressure difference P , there exists a homoclinic

orbit (separatrix) dividing the phase space into regions of periodic motion and

regions of unbounded growth. At the critical pressure difference, i.e. P = 1so that Pv =

323

27GT, a saddle-node bifurcation takes place. In the sub-critical

regime 0 < P < 1, the bubble may exhibit unbounded growth or bounded

oscillations depending upon the initial bubble radius and velocity.

The above picture clearly distinguishes an ideal gas bubble ( Pv is negligible,

hence Pv P is negative) from a vapor bubble (PvP can be either positiveor negative). An ideal gas bubble will undergo periodic oscillation for any initial

perturbation of the bubble radius. For a vapor bubble, however, there exists

a critical pressure difference, Pv P, above which stable oscillation is notpossible. Even below this value of Pv P, periodic oscillation of a vaporbubble is only possible for perturbations of radius below some critical value.

This is because expansion of the radius of a vapor bubble leads to a diminished

restoring surface tension force, but with an unabated vapor pressure inside the

bubble. For a gas bubble, on the other hand, the gas pressure inside the bubble

decreases as the volume increases; and this eventually limits the bubble growth.

Although the above results are obtained from the simplified Rayleigh-Plesset

model for an inviscid fluid, they provide the basic framework for a perturbation

analysis to include weak viscous effects. Using such an approach, Chang &

Chen (1986) analyzed the correlation between the critical value ofPvP (alsoknown as the cavitation pressure) and the fluid viscosity. Although ( Pv P)cis independent of the fluid viscosity, when unbounded bubble growth occurs

subcritically it does depend on the fluid viscosity as well as the initial conditions.

Using the well-known Melnikov method, Chang & Chen studied the breakup of

the homoclinic orbits [shown in Figure 2(b)] due to perturbations in the external

pressure, P, and confirmed the correlation between the cavitation pressure andthe fluid viscosity that had previously been obtained experimentally (Bull 1956).

2.1 Dynamics of Gas and Vapor Bubbles under PeriodicForcing

For either a gas bubble or a vapor bubble, the free oscillation about a stable

equilibrium (if one exists) is nonlinear. The nonlinearity is manifested in the

amplitude dependence of the oscillation period. Take the orbits in Figure 2(b) as

an example. Orbits within the homoclinic manifold (the separatrix) are all time-

periodic motions. However, the periods of these orbits are different from orbit to

orbit; obviously the periodic orbit closest to the homoclinic orbit has a very long

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(a)

(b)

(c)

Figure 2 Three typical phase diagrams corresponding to different pressure driving force. (a)

P < 0, (b) 0 < P < 1, and (c) P > 1.

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208 FENG & LEAL

period. If thebubble radius oscillates close to thestable equilibrium, on theother

hand, the bubble dynamics can be approximated by a weakly nonlinear oscilla-

tor, and no qualitative difference exists between a gas bubble and a vapor bubble.For weakly nonlinear systems, perturbation techniques such as the method of

multiple time scales (Hinch 1991) have been employed to calculate the steady-

state forced responses, including superharmonic and subharmonic resonances.

See Nayfeh & Saric (1973) and Plesset & Prosperetti (1977) for a review of

articles in this area. A complete analysis of the main resonance as well as subhar-

monic and superharmonic resonances of the Rayleigh-Plesset equation has been

given by Francescutto & Nabergoj (1983). It is worth noting that the physics of

the subharmonic resonance suggest that the sound produced by the bubble has a

component at half of the frequency of the acoustic wave applied to the liquidaphenomenon that has been observed in experiments (Lauterborn 1991).

When the bubble radius is not close to the stable equilibrium radius, the

dynamics of a gas bubble can be significantly different from those of a vapor

bubble. If the periodic forcing and the fluid viscosity are regarded as small

perturbations, geometric methods for global bifurcation analysis such as the

Melnikov method (Wiggins 1988a) can be applied to both kinds of bubbles. The

Melnikov method relies upon an underlying integrable structure. The Hamilto-

nian function found by Ma & Wang provides a framework for such a purpose.

For a vapor bubble, the homoclinic orbit shown in Figure 2(b) encloses aregion in the phase plane inside which the bubble oscillation is stable. Outside

this region, the bubble radius undergoes unbounded growth. Under the action of

time-periodic forcing via P, the stable and unstable manifolds of the saddle in-tersect transversely at an infinite number of crossing points (provided the forcing

amplitude exceeds a critical value which goes to zero as 0) and the bubbleexhibits chaotic oscillation from any initial condition which lies inside the lob

structure formed by these intersecting manifolds (Wiggins 1988b, 1992). Using

these ideas, Szeri & Leal (1991) investigated the onset of chaoticoscillations and

rapid growth of a spherical vapor bubble at subcritical conditions, i.e. P < 1,

for both periodic and quasiperiodic background pressure oscillations (Wiggins

1988a). They found that multiple frequency components in the background

pressure oscillation cause the bubble to respond chaotically and grow explo-

sively at a lower forcing amplitude than bubbles forced with a single frequency.

For a gas bubble, Pv P < 0 is always true; the free oscillation of thebubble is always stable (Figure 2(a)) but the period of free oscillation increases

with amplitude. In this case, when weak periodic forcing is introduced via

oscillation of P

, orbits whose periods of oscillation are a rational multiple

of the driving period are selected for resonant interaction, and there is a well-

known route to chaotic dynamics via a sequence of period doubling bifurcations.

Indeed, Smereka et al (1987) have demonstrated how resonances occur and

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subharmonic orbits form, via a saddle-node bifurcation. Although their analysis

included a partial account of liquid compressibility, their conclusions would be

qualitatively unchanged even if an incompressible liquid model were used.When neither the bubble deviation from the equilibrium radius nor the forcing

amplitude is small, the bubble dynamics can only be studied using numerical

integration of the Rayleigh-Plesset equation. As far as we are aware, all previous

investigations of this nature have been limited to gas bubbles in water. Although

a partial pressure due to the water vapor is sometimes included in the model,

Pv P is usually much less than (Pv P)c and the result is essentiallythe same as that of a gas bubble with Pv = 0. Numerical investigation ofthe bifurcation structure based on the Rayleigh-Plesset equation is given in

Lauterborn (1976) and discussed by Plesset & Prosperetti (1977).More sophisticated models for gas bubbles in water (these models are to be

discussed shortly) have also been used by Cramer (1980), Lauterborn & Suchla

(1984), Smereka et al (1987), Lauterborn & Parlitz (1988), Holzfuss & Lauter-

born (1989), Parlitz et al (1990), and Lauterborn (1990) to investigate the bubble

response to periodic forcing under various forcing frequencies and amplitudes.

Although the results obtained are based on more sophisticated models, the bi-

furcation structures are typical of generic nonlinear oscillators, and thus are

qualitatively similar to those found for the Rayleigh-Plesset model. It is found

that various resonances, such as period-1, period-2, and period-3, are possible.The resonant creation of periodic orbits in the forcing frequency space obeys

a Farey-tree ordering with respect to the classification of the resonances by

torsion number and period. Multiple steady-states can coexist corresponding

to the same forcing parameters; the final steady-state sensitively depends on

the initial conditions. As the forcing parameter is changed slowly, periodic so-

lutions are created or annihilated through saddle-node bifurcations. Hysteries,

which are associated with saddle-node bifurcations are expected. Furthermore,

period-doubling cascades to chaos are a prominent recurring feature connected

with each resonance. These results, together with the experimental observation

that a bubble cloud under ultrasound follows a subharmonic route to chaos with

a low-dimensional strange attractor (Lauterborn & Koch 1987, Lauterborn &

Holzfuss 1986, Lauterborn et al 1993), provide strong arguments that cavitation

noise from a cloud is caused by chaotic oscillation of the individual bubbles in-

stead of a statistically randomly occurring cavitation of the liquid, as sometimes

proposed.

2.2 Other Models for Spherical Bubble Dynamics

and Experimental WorkIn all of the models we are about to discuss, the partial pressure due to the liquid

vapor is low if it is included in the model at all. Hence Pv P is much less

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210 FENG & LEAL

than ( Pv P)c and bubble dynamics are essentially the same as those of anideal gas bubble.

Theoretical predictions based on the Rayleigh-Plesset model have been com-pared with experimental data for three different gases that have a wide range of

values of the polytropic exponent by Crum (1983). The theoretical predictions

based on the Rayleigh-Plesset equation with a polytropic model were satisfac-

tory, provided the gas bubble was not driven near one of its harmonic resonant

frequencies. However, when the forcing frequency is near one half of the bub-

ble resonance frequency, the experimental data deviate substantially from the

quantitative predictions of the Rayleigh-Plesset model (Crum & Prosperetti

1983, 1984).

The discrepancy between the experimental data and the theoretical predictioncan be attributed to several factors. First of all, the Rayleigh-Plesset equation as-

sumes that the liquid is incompressible. When the bubble oscillation amplitude

is large, the corresponding velocity of the bubble interface can be comparable

to the speed of sound in the liquid, and the compressibility of the liquid can no

longer be ignored. Second, the polytropic relation governing the partial pres-

sure of the gas in the bubble cannot account for the energy loss due to heating

and cooling of the gas, which is known to be an important factor over a wide

range of physical conditions. Third, the spherical shape of the bubble can be

unstable at large oscillation amplitudes.Models that include liquid compressibility and thermal effects in the gas

dynamics of spherical bubbles have been summarized by Prosperetti (1993).

Prosperetti and his coworkers (Prosperetti & Lezzi 1986, Lezzi & Prosperetti

1987, Prosperetti 1987) use singular perturbation analysis to obtain the follow-

ing unique equation for the bubble radius:

R R + 32R2 1

c[R2

...

R + 6R R R + R3] = pB P

+ O

1

c2

,

where P is the pressure at the position occupied by the bubble center in the

absence of the bubble and pB is the pressure on the liquid side of the bubble in-

terface. Note that this equation requires an initial condition for R. An alternative

form which does not require the second order derivative is1+ ( + 1)

Rc

R R + 3

2

1

+ 1

3

Rc

R2

= 1 + ( 1)Rc +

R

c

d

dt pB P

+O 1

c2

where is an arbitrary parameter of order 1. Because the parameter is

arbitrary, the above equation is not truly unique unless one can show that the

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solutions of the equation are in some sense insensitive to the specific choice

of.

Thermal effects for spherical bubbles were considered by Flynn (1975) tobe based on a system of ordinary differential equations derived from the exact

set of conservation principles via the assumption of spherical symmetry. Later,

Prosperetti et al (1988) reduced the exact equations to a nonlinear partial differ-

ential equation for the temperature field and an ordinary differential equation

for the internal pressure. Numerical methods for solution of the reduced equa-

tions are developed by Kamath & Prosperetti (1989). Further simplification is

achieved by assuming that the pressure in the bubble is uniform and that the

gas is perfect gas (Nigmatulin & Khabeev 1974, 1977; Prosperetti 1991).

The most profound distinction between all of the theories described above forspherical bubbles and experimental observations of large amplitude oscillations

is the common occurrence in the experiments of shape oscillations, induced

via a parametric instability. Of course the loss of the spherical symmetry

leads to dramatic changes in the bubble dynamics, and limits the range of real

bubble oscillation intensities where the spherical bubble analyses, including

Rayleigh-Plesset, can be expected to provide a useful description (Vokurka

1990). In fact, experiments intended to verify nonlinear predictions of the

various spherical bubble models have proven extremely elusive because the

onset of shape instability typically occurs before the nonlinear phenomena ofinterest are seen (Holt & Crum 1990, 1992, Leighton et al 1990, Vokurka et al

1992, Vokurka 1993). Recently, however, studies of sonoluminescencea

vast subject which requires a separate review (see Crum 1994 and Putterman

1994)have shown that an upper threshhold of the driving pressure exists

above which a stable radial oscillation of the bubble is re-established (Gaitan

& Crum 1990, Gaitan et al 1992). This allows a direct experimental test of the

various spherical bubble models.

To date, Holt & Crum (1992) have made comparisons with the model of

Prosperetti et al (1993), discovering that the theory overestimates the nonlinear

damping in predictions of the second harmonic response. Other model com-

parisons were made by Lofstedt et al (1993) and Gaitan et al (1992), including

a model due to Keller & Miksis (1980) and the model of Flynn (1975). Al-

though general agreement was found in all cases, it is disappointing that the

measurements have not been precise enough to date to discriminate between

the different models.

Due in part to the lack of experimental verification and to their mathematical

complexity relative to Rayleigh-Plesset theory, none of the models described

above for large amplitude spherical bubble dynamics have enjoyed general

acceptance. In this regard, a recent approximate model by Lofstedt et al (1993)

is worth mentioning due to its simplicity and apparent good agreement with

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212 FENG & LEAL

Figure 3 The radius of an air bubble in water as a function of time for one cycle of the driving

sound field (designated by a broken line). Reproduced from Figure 1 of Lofstedt et al (1993).

experimental results. By including the radiation of acoustic energy from the

bubble to the fluid, their fundamental equation takes the form:

R R

1 2R

c

3

2R2

1 43

Rc

+ P(R, t)

+ Rc

d P(R, t)

dt Pa (0, t)

R

c

d Pa (0, t)

dt P0

= 0,

supplemented by the boundary condition at the fluid-gas interface,

P(R, t)+ 4R

R+ 2

R= Pg(R, t),

and a van der Waals hard core model as the equation of state for the bubble,

Pg (R) =P0R

30

(R3 a3)where 4

3

a3 is the van der Waals excluded volume. Figure 3 shows an example

of the excellent agreement between a theoretical prediction using this model

(the solid curve) and experimental data (the dots) for the bubble radius as a

function of time.

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3. SHAPE OSCILLATIONS OF A GAS OR VAPORBUBBLE WITHOUT RESONANT COUPLINGTO VOLUME OSCILLATIONS

We first examine the free oscillation of a bubble that is at its equilibrium volume,

but is initially nonspherical. When the amplitude of shape oscillations is small,

the bubble surface can be represented as a superposition of spherical, harmonic,

normal modes Ynm (,), where and are the polar and azimuthal angles,

respectively. For an inviscid fluid, the natural frequency of these modes was

shown many years ago by Lamb (1932) to be

2nm = (n + 1)(n 1)(n + 2)R3 (4)

where n and m are the mode numbers. Note that this frequency does not depend

on the azimuthal mode number m. Microgravity experiments by Asaki et al

(1993) have shown very good agreement with this classical formula.

The nonlinear nature of shape oscillations for a bubble in a quiescent fluid is

manifested in the fact that the free oscillation frequencies of the shape modes

are amplitude dependent, as is typical for the generic nonlinear oscillator. This

effect was studied by Tsamopoulos & Brown (1983). By carrying out a pertur-

bation analysis up to cubic order in the magnitude of deformation, they obtainedthe frequency correction for the second, third, and fourth axisymmetric modes

(see corrected results in Tsamopoulos & Brown 1984). The oscillation frequen-

cies of these modes were found to decrease as the amplitude of the oscillation

increased. Since the perturbation analysis is limited to small oscillation am-

plitudes, the dynamics of free oscillation at large amplitude need to be studied

using numerical methods; this is the topic of section 7.

Forced shape oscillations can be excited by various means. For example,

such oscillations can be excited by a modulated acoustic radiation stress (or

pressure). The linear response to time-periodic forcing of a bubble or a drop

was studied theoretically by Marston (1980). Not surprisingly, he found that

the shape modes behave like linear oscillators, with a large response predicted

when the forcing frequency is close to the natural frequencies of the shape

modes. Large amplitude forced shape oscillations will clearly be affected by

nonlinear effects. However, this has not yet been studied.

Another mechanism of inducing time-dependent shape oscillations is to sub-

ject a bubble to a time-dependent velocity gradient. One example of this type

was studied in a series of papers by Kang & Leal (1988, 1989, 1990, 1991), who

considered the dynamics of shape oscillations for an incompressible bubble in

steady, time-periodic, and quasi-periodic axisymmetric straining flows includ-

ing weak viscous effects. The nature of solution trajectories is conveniently

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214 FENG & LEAL

described in terms of a scalar measure of deformation, x , defined as

x = 0

R( , t) P2(cos ) sin d.

For the steady flow problem, at Weber numbers below a critical value, it was

shown that two steady solutions exist for the bubble shape, one stable and

the other unstable, and that solution trajectories in the phase space (x, x ) areseparated into two distinct sets by a homoclinic orbit which passes through the

unstable solution (saddle). Inside in the potential flow limit, the shape oscillates

periodically for all initial conditions, except at the stable node, with a period

that increases as the amplitude increases. Outside the homoclinic orbit, the

shape deformation exhibits unbounded growth and the bubble will break up asa result. With viscous effects included, the stable node becomes an attractor

and the homoclinic orbit breaks into distinct trajectories from the saddle which

are known as the stable and unstable manifolds, respectively. When the flow

strength varies periodically in time, a variety of interesting manifestations of

nonlinearity appear. For example, when the shape mode is resonantly forced,

an O(1/3) shape response can be obtained from an O() forcing. Furthermore,

for a variety of initial conditions, the bubble shape undergoes a chaotic motion

(this occurs for infinitesimal oscillations of the flow in the inviscid limit, and

for oscillations above a critical threshold amplitude in the presence of viscouseffects) which can lead to large deformations and breakup, even though the

mean Weber number is subcritical and the initial degree of deformation would

lead only to stable oscillations of shape in the steady flow. It is believed that

these results are qualitatively typical of the behavior that would occur in any

flow which leads to elongated bubble shapes. Further study is required to verify

this assertion. It may also be noted that Kang and Leals analysis was carried

out prior to the more recent focus on coupling between shape and volume

(or radial) oscillations, and thus assumed that volume oscillations would be

negligible. Although further study is necessary to establish the precise impactof volume changes, the constant volume assumption is clearly not true, at least

near conditions of resonance between a shape mode and thepurely radial volume

oscillations.

4. COUPLING BETWEEN SHAPE AND VOLUMEOSCILLATIONS FOR A GAS OR A VAPOR BUBBLEWITHOUT MEAN DEFORMATION

A major shift in recent theoretical studies of bubble dynamics has been a real-

ization of the importance of coupling (and particularly resonant or near-reson-

ant coupling) between the purely radial (volume) oscillations of the classical

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spherical bubble theories, and one or more of the shape modes that were men-

tioned in the previous section. Although almost all studies have been restricted

to axisymmetric deformations, their results are quite general because of thedegeneracy in the nonaxisymmetric modes that is reflected by independence of

mn [ Equation 4 ] on azimuthal mode number m.

4.1 Energy Transfer Between ModesShape oscillations, in addition to being directly forced as in the previous section,

can also be excited indirectly through the forcing of large amplitude purely

radial volume oscillations. For instance, under acoustic pressure forcing, if

the acoustic wavelength is much larger than the bubble size, the initial bubble

response is spherical oscillations of the radius. However, as the amplitude offorcing increases, the spherical shape of the bubble becomes unstable. This

instability leads to the onset of shape oscillations.

When the spherical volume oscillation becomes unstable, linear stability

theory predicts exponential growth of the shape oscillation amplitude. However,

the shape oscillation amplitude is ultimately controlled by nonlinearity. If the

parameters of the external forcing are fixed, there are two possible outcomes

depending on whether the response of the volume mode to the forcing is affected

by the shape oscillations (Feng & Leal 1994).

If the shape oscillation does not affect the volume response to the forcing (asoccurs when the volume mode is forced away from its resonance frequency)

the shape mode is slaved to the volume mode. Hall & Seminara (1980) have

studied shape oscillations bifurcating from a driven spherical oscillation, when

the shape oscillations are either subharmonic or synchronous with the radial

oscillation. Amplitude equations for the shape mode up to cubic nonlinearity

are obtained for both cases. The amplitude equations of the shape mode are

equivalent to two first-order autonomous ordinary differential equations. Var-

ious bifurcations are shown to occur. However, since the amplitude equations

are two dimensional and autonomous, no chaotic motions are possible.

If the shape oscillation does affect the volume response to the forcing,

the shape mode is no longer slaved to the volume mode. This occurs when

the volume mode is forced near its resonance frequency; the excitation of the

shape mode then causes a change in the resonant frequency thus detuning the

volume mode. In a typical scenario, suppose that the volume mode is forced

near perfect resonance; the parametric instability of the volume mode causes a

shape mode to be excited; the shape mode then causes a change in the natural

frequency of the volume mode so that it is now being forced slightly away from

its resonance; this results in a reduced amplitude response of the volume mode;

consequently, the volume mode is stabilized and the shape oscillation amplitude

thus decays. As the shape amplitude decays, the detuning of the volume mode

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216 FENG & LEAL

is restored to the original near-perfect resonance and the volume oscillation

amplitude again grows and the above process repeats. It thus appears that the

two modes compete with each other.When no external excitation is present, a similar competition between shape

and volume oscillations can occur involving resonant or near-resonant coupling.

Due to the quadratic form of the nonlinearity, the dominant resonance occurs

when the frequency of the volume mode is approximately twice that of the shape

mode, i.e. 0 2n 0. This case has been extensively studied, primarily inthe inviscid limit, by Ffowcs Williams & Guo (1991), Longuet-Higgins (1991),

Mei & Zhou (1991), Yang et al (1993), Feng & Leal (1993, 1994) and Mao

et al (1995). If we specify the bubble shape in the form:

r= 1+ f1 + O(2) (5)with

f1 =

n=0an (t) Pn ()

and

a0(t)

=1

2

0( ) exp(i 0t)

+c.c.

an (t) =1

2n ( ) exp(i n t)+ c.c.

(6)

with

= tthen the complex amplitudes of the volume and resonant shape mode in the

inviscid limit follow the dynamic equations

d0

d= i00 + i H52n

dn

d= i H60n

(7)

where i = 1 , the superscript denotes the complex conjugate, 0 specifiesthe degree of frequency mismatch from exact resonance,

0 2n = 0and

H5 =(4n 1)n

16(n + 1)(2n + 1) and H6 =(4n 1)n

4. (8)

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The competitive exchange of energy between oscillations of the purely radial

and resonant shape mode, Pn (), that is described by these equations was shown

by Feng and Leal (1993) to be both periodic and energy conserving in the sensethat the amplitudes of the two modes satisfy the integral constraint

r20 +H5

H6r2n = constant, (9)

where 0 = r0ei 0 and n = rn ei n . Thus, one mode can only gain amplitudeat the expense of the other, and the relationship in Equation 9 determines the

maximum amplitude of a shape mode that can evolve from a purely radial

oscillation and vice versa. Furthermore, because of the condition in Equation 9,Feng and Leal (1993) noticed that solution trajectories of Equation 7 could be

represented as closed paths on the surface of a sphere.

Figure 4 depicts three sets of solution trajectories for different degrees of

detuning. The precise relationship between r0, rn , 0, n , and the coordinates

on the sphere is very complicated. But the physical meaning of the sphere is the

following. The north pole N corresponds to a purely radial volume oscillation,

whereas the south pole S corresponds to a pure shape oscillation. The latitude

gives the partition of total energy among the two modes. In the figure, the lati-

tude of each orbit changes periodically as we move in the direction of the arrow.Therefore, the energy partition between the two modes changes in a periodic

fashion. Physically, this period corresponds to the period of amplitude modula-

tion, and is much longer than the natural oscillation periods of the two modes.

When the frequency detuning is large, as depicted in Figure 4(a), the solution

trajectories are approximately aligned with the parallels of the spheremeaning

Figure 4 The energy exchange between the volume mode and the shape mode. The north poledenotes the pure volume oscillation. The lattitude denotes the partition of energy between these

two modes. (a) large detuning, (b) no detuning, (c) intermediate detuning (reproduced from Figure

2 of Feng & Leal 1993).

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218 FENG & LEAL

that there is relatively little exchange of energy between the two modesand

there are only two fixed points, one of which is the north pole N. This means that

purely radial oscillations can occur, and the nature of the solution trajectoriesshows that any small perturbation of shape will remain smalli.e. the purely

radial oscillation is stable. The condition of exact resonance is shown in Figure

4(b). In this case, there are three fixed points (one of which is still the north

pole N) and there is a solution trajectory which emanates from N and passes

exactly through the south pole S (which is not a fixed point). We recognize this

trajectory as a homoclinic orbit, and note that N, which was an elliptic fixed

point in Figure 4(a), is now a hyperbolic fixed point. It is clear that purely radial

oscillations are now unstable in the sense that infinitesimal perturbations from

N put one on a solution trajectory where there is nearly complete transfer ofenergy from the radial oscillations to a shape oscillation. The critical condition

for instability thus corresponds to the intermediate degree of detuning where a

saddle-node bifurcation occurs at N, and the elliptic fixed point bifurcates into

a hyperbolic fixed point at N with a homoclinic orbit encircling it. Solution

trajectories for a degree of detuning that is near this critical transition are shown

in Figure 4(c). We shall discuss the stability of spherical oscillations in more

detail in the next subsection, including the condition for instability implied by

the above analysis but modified to include weak viscous effects.

We have already noted that the north pole is a saddle point, and that thehomoclinic orbit which emanates from it under conditions of exact two-to-one

resonance is a great circle which passes directly through the south pole. Hence,

solutions corresponding to initial conditions anywhere on this great circle, in-

cluding a pure shape deformation, end up approaching N asymptotically; this

special branch of solutions thus exhibits one-way energy transferfrom initial

conditions including a nonspherical shape to a purely radial volume oscilla-

tion. Obviously, among all possible initial conditions it is a rare occurrence

to fall precisely on the great circle through N. However, this special case was

studied by a number of the early investigators of this mode-coupling problem

(Longuet-Higgins 1989a, 1989b; Yang et al 1993; Ffowcs Williams & Guo

1991).

4.2 Stability of the Spherical OscillationThere are actually two mechanisms which may cause spherical bubble os-

cillations to become unstable to infinitesimal perturbations of shape. These

are Rayleigh-Taylor instability, and the parametric instability that was already

briefly discussed in the preceding section. The earliest studies of the stability of

an oscillating spherical bubble were reviewed by Plesset & Prosperetti (1977).

It is known that because large radial acceleration occurs only for a small fraction

of a typical oscillation of bubble volume, and because its destabilizing influence

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is counterbalanced by the stretching of the bubble surface, the Rayleigh-Taylor

mechanism rarely applies for oscillating bubbles. The instability of spheri-

cal bubble oscillation is thus dominated by the parametric instability. In thelinearized theory of the dynamics of shape modes for a bubble undergoing peri-

odic oscillations of volume, the existence of a parametric instability is signified

by the fact that the governing equations for the shape modes are Hill equa-

tions, which reduce to Mathieus equations when the amplitude of the radial

oscillation is assumed to be small. Hence, based upon the known behavior of

solutions of Mathieus equation, it is expected that the radial oscillation will

become unstable when the ratio of the frequencies of the radial oscillation to

the frequencies of any shape mode is close to 2, 1, 12

, or other ratios of small

integers.A quantitative criterion for the stability of spherical bubble oscillation de-

pends on many factors. In particular, energy dissipation in the system greatly

affects the stability. Since different bubble models treat the damping differently,

a general criterion for all cases is not available. However, when the amplitude

of the volume mode oscillation and the liquid viscosity are both small so that a

multiple scale analysis can be applied, a quantitative criterion can be obtained

(Feng & Leal 1994). The spherical shape is stable if the oscillation amplitude

f (nondimensionalized by the bubble radius a) satisfies:

f 0),

the frequency of shape oscillations is increased, whereas for prolate shapes

(An < 0) the frequency is decreased.

The other two consequences of the change in bubble volume and shape are:

(1) the two-to-one resonance (0 = 2n ) that exists for the case of a spherical

bubble is modified primarily because the natural frequencies for volume andshape oscillations are shifted as described by Equations (20a, 20b); (2) the

mean deformation of shape also produces a one-to-one (0 = n ) resonantinteraction between shape and volume modes, described in the form

d0

d= i H1n + i

0 +

2+ 3

4n+ 3 1

83n

A0

0

dn

d= i H30 + i H4n ,

(22)

where

H1 =(n 3)An

2(2n + 1)n, H3 =

(n + 1)(n 3)An2n

.

Equation 22 shows both a frequency shift due to shape and volume deformation,

and coupling terms. As with the two-to-one resonance described earlier, there

is an exchange of energy between the shape and volume mode, with a constant

total energy (inviscid flow), expressed via the constraint

r2

0 +H1

H3r2

n =constant. (23)

Unlike the two-to-one resonance, however, neither the purely radial oscillation

nor the pure shape oscillation is a fixed point, meaning that even a purely radial

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230 FENG & LEAL

initial perturbation will lead to a continuous exchange of energy. There are two

(and only two) fixed points on the solution trajectories of Equation 22 for any

value of. These fixed points represent conditions in which oscillations of bothshape and volume occur on the short time scales of the natural frequencies, but

there is no exchange of energy between the modes (i.e. 0, n are constants).

More realistic, and therefore interesting, are the less detailed studies that

have been carried out for gas bubble dynamics in a flow. The shape oscillations

of a bubble in a uniform flow were studied by Feng (1992), using a domain

perturbation technique. The external flow, in this case, breaks the degeneracy

of nonaxisymmetric modes that was found by Lamb (1932). Feng found that

all modal frequencies decrease quadratically with increase of the translational

velocity (i.e. Weber number). This decrease in frequency appears surprisingat first because the mean shape becomes increasingly oblate with increase of

W (cf Section 3), but it appears that the increase of volume in this case (cf

Equations 16 and 17b) is sufficient to counteract the shape effect on frequency

so that the first term in H4 (Equation 21) dominates the second.

Shape oscillations of a bubble in a uniaxial straining flow have been studied

by Kang & Leal (1988, 1989, 1990, 1991) and by Yang et al (1993). The uni-

axial straining flow stretches the bubble into a prolate shape and the frequency

of the P2 mode shape oscillation is thus decreased in accord with Equation

20b. The decrease in oscillation frequency suggests, in both of the above cases,the possibility of estimating a critical mean flow condition for breakup, cor-

responding to n = 0. Extrapolation of the result for n to finite amplitudeis, however, especially problematical in the uniform translation problem where

the decrease in n is due to a delicate balance between two terms of oppositesign.

Much more important than these changes in the frequency of existing resonant

modes is that the steady uniaxial flow introduces a new mode of coupling

between volume and shape oscillations, which is stronger than the resonant

coupling in the absence of flow. Specifically, when 0 = 2, resonant couplingbetween the volume mode and the P2 mode is found to occur on a time scale,

1/2t, that is much shorter than the t time scale associated with either the two-

to-one or one-to-one resonance described above. This suggests that viscous

damping will play a less important role in inhibiting the growth of resonant

modes in the presence of the flow than it does for a quiescent fluid. The

governing amplitude equations are

d0

d T = 1

6 2

d2

d T= 5

20

(24)

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where T = 1/2t. The solution, incorporating an initial perturbation of shape,

n (0) =i A

n(n+

1)

n

and the steady deformation to O(W), Equations 16 and 17c, is

0(T) = i

A20 +

3

5A22

1/20

cos

5

12

1/2T+

(25)

and

2(T) = i3

53

A20 + A221/2

2sin

5

12

1/2T+

(26)

where

= tan1

(15)1/2A2

5A0

.

We infer that enhanced mode coupling would also occur for other types of

mean flow, though the coupling with the P2 mode in this case is presumably a

consequence of the specific form of the flow considered. We may also infer by

comparison with the results of Feng (1992) for a uniform external flow, that the

interaction with the flow to produce enhanced mode coupling requires a mean

velocity gradient. Beyond these qualitative observations, we cannot say much

about flow-enhanced mode coupling except that it is likely to be important in

many applications.

The mechanism for this flow-enhanced coupling is simply that a bubble

which oscillates radially is subjected to flow of different strengths as it changes

radius, so that the natural tendency of the flow to deform the bubble produces a

time-dependent shape. When the volume oscillation is at the natural frequency

of the shape mode that is generated, a resonant coupling occurs. Although

this idea is extremely simple, one puzzling aspect is that the uniaxial flow

produces a steady state shape that contains both P2 and P4 at the leading order

of approximation, but the above mechanism produces no P4 oscillation and no

resonant interactions on the 1/2t time scale when 0 = 4.

6. TRANSLATIONAL MOTIONDANCING

BUBBLE PROBLEMWhen a spherical bubble is placed in an acoustic standing wave, the acoustic

pressure gradient will exert a force that pushes the bubble either toward or away

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232 FENG & LEAL

from the pressure node depending on the bubble size in comparison with a bub-

ble whose resonance frequency is the same as the acoustic field. Rath (1979)

and Watanabe & Kukita (1993) have considered the coupling of the translationaland radial motions of a spherical bubble via this simple pressure gradient mech-

anism. However, though very complex motions can occur, including chaotic

oscillation of volume and position, these analyses ignore the shape oscillations

that occur when the amplitude of the radial motion has reached a critical value.

When the shape oscillations set in, the bubble may also start to drift errati-

callythe so-called dancing bubble problem first observed by Strasberg &

Benjamin (1958) and Eller & Crum (1970). It was understood then that the mo-

tions observed were caused by the parametric excitation of shape oscillations.

However, mechanisms to explain the erratic nature of the bubble motion haveonly been proposed very recently. Utilizing a mechanism for self-propulsion of

a nearly spherical body in an infinite perfect fluid, discovered 20 years earlier by

Saffman (1967), Benjamin & Ellis (1990) derived a formula for the drift velocity

of an oscillating bubble as a consequence of second-order interactions between

two neighboring time-dependent shape modes. Mei & Zhou (1991) utilized

this formula to suggest possible connections between the chaotic modal inter-

action of the volume mode, and two neighboring shape modes and the erratic

drift. Zardi & Seminara (1995) considered the case when the two neighboring

shape modes are in one-to-two resonance and both modes are parametricallyforced by a volume mode which is not in resonance with either of the two

shape modes. They found that chaos could occur for an acoustic excitation of

sufficient amplitude.

A precise prediction of the erratic drift of the bubble requires consideration

of the coupling of the translational motion as well as the volume and shape

coupling. As we have mentioned earlier, a uniform flow has an effect on the

shape oscillations. Indeed, a recent analysis of Feng & Leal (1995) suggests

that the coupling between translational motion and the two neighboring shape

modes is an essential element of the bubble dancing mechanism. A typical

scenario is presented in Figure 8. Here we plot the amplitudes of the two

neighboring axisymmetric shape oscillation modes (an and am , m = n+1) andthe translational velocity of the bubble (w1) as functions of time, all driven by a

periodic oscillation of the bubble volume. The forcing parameters are such that

the shape mode m is unstable and the shape mode n is stable. Corresponding to

small initial perturbations in both shape modes, the nth mode initially decays

and the mth mode grows as predicted by the linear stability theory. However,

the instability of the mth mode ultimately leads to an instability of the nth mode

as a result of the nonlinear coupling. The interaction of these two modes results

in a translational motion of the bubble. This translational motion appears to

detune the resonance and thus damp out the shape oscillations as shown by

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Figure 8 The dynamics of two shape modes and the translational mode as functions of time.

Reproduced from Figure 6 of Feng & Leal (1995).

the subsequent decay of amplitude ofan and am . As a result, the translational

velocity decreases and eventually becomes too small to keep the unstable shape

mode damped. Subsequently, the mth mode becomes unstable and the above

process repeats again and again. Owing to the interactions of these three modes,

the translational motion as well as the shape oscillations can become chaotic.

7. NUMERICAL METHODS AND APPLICATIONS IN THESTUDY OF NONLINEAR BUBBLE DYNAMICS

The theoretical studies outlined in preceding sections of this review are gen-

erally referred to as analytical, though both the Rayleigh-Plesset equation for

spherical bubbles, and the coupled amplitude equations for slightly nonspher-

ical bubbles are nonlinear and thus must be numerically integrated for all but

the simplest of problems. Here, we consider numerical methods, applied to

study large amplitude changes in bubble volume and shape, where we must

solve the full free-boundary problem of bubble motion and the coupled flow of

the surrounding liquid.

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234 FENG & LEAL

We have tried, in previous sections, to emphasize that very significant qual-

itative changes have occurred in our understanding of bubble dynamics as the

tools and insights of nonlinear dynamical systems theory have been applied tothe analysis of resonant and near-resonant coupling between small amplitude

volume and shape oscillations. However, it is important to recognize the limi-

tations of this work. Because the amplitudes of shape and volume changes are

assumed to be small, significant coupling between modes only occurs when the

natural frequencies, say 0 and n , are very near to exact resonance, and then

the coupling is generally between P0 and only a single resonant shape mode,

Pn . Furthermore, for the same reasons, the time scale for significant coupling

is very long, usually O(1), where is a measure of the shape perturbationamplitude. Thus, in the presence of damping mechanisms, such as viscous dis-sipation, it would often be the case that the modes would simply damp to very

small amplitude before any significant degree of resonant coupling has actually

taken place. As a consequence, the theoretical work was actually restricted in

every case to either the inviscid fluid limit, or to very small viscous effects.

We should thus view the analytical results as providing important qualitative

insight into the kinds of phenomena that can occur, but recognize the need

for either laboratory or numerical experiments to understand how they may be

manifested when the changes in volume or shape are either moderate or large.

From a qualitative point of view, we know only that finite-amplitude inter-actions should occur between a much broader spectrum of deformation modes,

and that coupling between these modes and volume oscillation will occur on

short time scales comparable to the inverse of the natural frequencies of the

modes. As one can imagine, the difficulty of solving the bubble dynamics prob-

lem numerically increases as the degree of deformation increases. The popular

method of formulating the bubble problem using boundary integral methods

can cope with complex shapes provided there is sufficient spatial resolution

to accurately calculate the interface curvature, but this method requires that

flow in the surrounding fluid be characterized by a potential. The solution of

the full fluid mechanics problem described by the Navier-Stokes equations has

been addressed with adaptive grid and interface tracking methods, using both

finite-difference and finite-element formulations, but these techniques currently

suffer from some significant limitations as outlined below.

7.1 Boundary Integral MethodsThe numerical simulation of an oscillating bubble is complicated by the fact that

the discretization must follow the evolving free surface within the calculation

domain. It is well known that boundary integral methods are particularly well

suited to this class of problems as they involve discretization of the boundaries

only. However, the transformation via Greens functions from the equations

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of motion in differential form to an equivalent form involving only boundary

integrals is possible only in the creeping and potential flow limits. Since many

of the most commonly encountered bubble dynamics configurations involveair or water vapor bubbles in water, where the Reynolds number is actually

quite large, and also deal with a surfactant-free interface so that boundary-

layer effects are relatively weak, the limitation to potential flows is often not

a severe restriction. Although so-called hybrid boundary integral methods

for solving the full Navier-Stokes equations have been developed and used for

problems with fixed geometry, the problem of accounting for the time dependent

geometry dominates the computational effort in a free-boundary problem and

any advantage of boundary integral methods over other techniques disappears

if viscous effects are included. More general information on boundary integralformulations for potential flow problems can be found in the review by Canot

& Achard (1991) and a comparison paper by Boulton-Stone (1993a, 1993b).

Boundary integral methods have been particularly favored for investigation of

bubble formation and oscillation near bounding walls or free surfaces (Boulton-

Stone 1993a, Oguz & Prosperetti 1990a, 1990b). Application of the boundary

integral method to study single bubble dynamics in an infinite fluid has been

recently undertaken by McDougald & Leal (1994, 1996) and Chahine (1994).

The restriction of boundary integral methods to potential flow problems pre-

cludes an exact accounting of the role viscous effects play in the dynamics ofoscillating bubbles. However, it is possible to include weak viscous effects in

the boundary integral formulation if it is assumed that these effects are limited

to a thin region near the interface so that the bulk of the fluid remains irrota-

tional. Lundgren & Mansour (1988) performed this analysis to include weak

viscous effects in a boundary integral simulation for an oscillating drop. One

result that they report is a surprisingly large effect of small viscosities when

energy is transfered via resonant interaction to high mode number shape oscil-

lations. Longuet-Higgins (1992) has suggested that a similar mechanism may

be responsible for nonlinear damping of bubble oscillations, but the Lundgren-

Mansour approach has yet to be tried for gas bubbles.

7.2 Finite-Difference MethodsConceptually the ideal numerical simulation of the bubble dynamics problem

would solve the unapproximated Navier-Stokes equations, with appropriate

boundary conditions, for the entire domain. A key difficulty in accomplish-

ing this task arises from the changing geometry of the problem in time, as the

bubble oscillates in volume and shape. The location of the interface must be

accurately known in the computational domain so that boundary conditions

may be applied. A finite difference method for steady free-boundary problems

was developed specifically for fluid mechanics applications by Ryskin & Leal

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236 FENG & LEAL

(1983, 1984a, 1984b,1984c) using boundary-fitted orthogonal coordinates to

map the physical domain to a square computational grid. The numerical gen-

eration of the boundary-fitted coordinates assures that the free surface of thebubble coincides with a coordinate line thus eliminating the difficulties asso-

ciated with tracking the evolving interface. Ryskin and Leals method was

generalized to time dependent bubble deformation without volume change by

Kang & Leal (1987, 1989). This method of simulation was found to exhibit

some artificial dissipation that gave rise to a small phase lag in the bubble re-

sponse to externally forced changes of shape, but still captured most of the

interesting dynamical features of the oscillating bubble. When applied to free

oscillation of a compressible nonspherical bubble, on the other hand, the nu-

merical dissipation is found to play a more important role. As a consequence,all comprehensive studies of time-dependent motions of compressible bubbles

have so far been carried out in the inviscid limit using the boundary integral

method.

7.3 Representative ResultsWe present in this section a few examples of numerical simulations which

identify new effects in the dynamics of a nonspherical bubble in an unbounded

fluid. In doing so we omit a number of numerical studies of oscillating spherical

bubbles, deformation of rising bubbles, and the collapse of bubbles near solidor free boundaries.

We begin by considering the dynamics of a gas bubble in a quiescent fluid,

which is the subject of an ongoing investigation by McDougald & Leal (1994,

1996). One focus of this study is a comparison of the acoustic radiation strength

for a nonspherical bubble with predictions for a spherical bubble from the

Rayleigh-Plesset theory. Another is an exploration of the mechanisms of bubble

break-up involving interaction between volume and shape oscillations. To date,

we have considered two types of problems: (1) free oscillations in a quiescent

fluid from an initial perturbation of shape or volume; and (2) the effect of

shape deformation on the acoustic output from a bubble that is subjected to an

initial pressure impulse. Many of the results confirm qualitative expectations:

a reduction with increased amplitude of the time scale for modal interactions;

an increase of the number of excited modes; shifts in resonance conditions

due to mean shape deformation; breakup due to an initial change in bubble

volume caused by coupling with shape deformation; and a general decrease in

the acoustic output relative to that expected via Rayleigh-Plesset theory.

In the case of bubbles subjected to an initial pressure pulse, we also have

observed a geometric amplification of the amplitude of shape modes as the

bubble collapses which is not present in the small amplitude analysis, and leads

to a very strong reduction of the amplitude of volume oscillations (and thus in

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the intensity of sound produced). In Figure 9(a), the response of the bubble

for a typical case is represented by the amplitudes of Legendre modes that

describe the bubble shape as it evolves in time. The geometric amplification ofthe P2, P4, and P6 shape modes as the volume decreases is clearly seen (even

though the volume changes are relatively modest for this particular case) and

the corresponding effect on the bubble volume versus time is compared with

predictions of the Rayleigh-Plesset theory for the same initial pressure pulse in

Figure 9(b). When the initial shape is very simple, as in this case, there is a delay

in the onset of a noticeable change in the bubble volume relative to Rayleigh-

Plesset predictions, because the major effect only occurs when the higher order

shape modes become significant, and this requires a finite induction time, as

the higher order modes only become active through direct interaction betweenthe P2 mode and the purely radial P0 mode. However, the shifting of energy

away from the volume mode eventually makes the bubble a much less efficient

source of sound than predicted by the Rayleigh-Plesset equation.

An obvious question is whether an alternative to the Rayleigh-Plesset theory

can be developed for cases like these, involving a small number of dynami-

cal equations, to model the response (or predict the sound) for nonspherical

gas bubbles where the Rayleigh-Plesset equation does not work. This is an

important topic for future study as it is clearly unacceptable for practical ap-

plications to end up having to solve a full 2D or 3D flow problem numericallyto predict sound production for each case where the Rayleigh-Plesset theory

is not adequate. All we can say, at the moment, is that the small amplitude

equations clearly do not provide such a model, as they involve coupling with

only a single shape mode and do not capture the geometric amplification effect.

In addition to bubble motions produced by a time-dependent mean pressure,

there have been a few numerical studies of the effects of time-dependent ve-

locity gradients. In particular, Kang & Leal (1987, 1988, 1989) studied the

deformation of a constant volume bubble due to both uniaxial and biaxial ex-

tensional flow using a finite-difference method. Only two of their many results

are of direct relevance to the topics of this review. First, they found that the

frequency of the shape mode, in this case the P2 mode, decreased as the bub-

ble became an elongated ellipsoid and eventually approached zero at a Weber

number, W 2.8, corresponding to the limit point of the stable steady-statesolutions, W = Wssc . A second result of general relevance to this review is thefact that time-periodic perturbations in the strength of the mean flow can lead

to chaotic oscillations in the bubble shape, even though the bubble volume was

held constant.

The only other finite amplitude study of flow effects that we are aware of is the

work by Chahine (1994), who examined the interaction between a collapsing

bubble and a line vortex. This study clearly shows that volume oscillations are

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238 FENG & LEAL

a

b

Figure 9 A bubble deformed by an anisotropic pressure field so that its equilibrium shape is an

oblate speroid, experiences an impulsive pressure decreaseat time=0. The frequency of thevolumemode is nearly equal to the frequency of the P2 shape mode so that 1 : 1 resonanceinteraction occurs.(a) The amplitudes of the Legendre modes comprising the bubble shape as a function of time. (b)

The volume of the bubble from the simulation is compared to the Rayleigh-Plesset prediction.

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a

b

Figure 10 Comparison between the contours of an elongated bubble during its collapse in the

absence and in the presence of swirl. Initial elongation ratio of 3; p/pi = 3.27. (a) No swirl.(b) = 0.56. Re /Rmax = 3. From Figure 12 of Chahine (1994).

strongly coupled with changes in bubble shape in the presence of a rotating

external flow. In Figure 10, the influence of swirl on the bubble dynamics

and the mechanism of breakup can be seen. When the rotation of the vortex

is included, the portions of the bubble surface away from the axis experience

higher pressures than portions near the axis and move much faster as a result.

This leads to an elongation of the bubble along the vorticity axis and eventually

causes the formation of a waist as the bubble fissions into two tear-shaped

bubbles, Figure 10(b). This is in contrast to the case without rotational motion,

Figure 10(a), where the bubble collapses via formation of strong jets at the

extreme ends of the bubble along the vorticity axis. While this result neglects

the influence of the bubble on the motion in the fluid, the author also presents

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240 FENG & LEAL

an outline of how interaction between the bubble and the flow in the external

fluid may be accounted for, and presents some preliminary results.

8. CONCLUSIONS

In our view, major changes have occurred in understanding the dynamics of

spherical and nonspherical gas bubbles over the past 20 years. Nevertheless,

studies of large amplitude effects are still at an early stage, and this is one of two

main directions where continued research is clearly needed. A second major

problem for the acoustics engineer is to develop an alternative to Rayleigh-

Plesset theory for nonspherical bubbles.

ACKNOWLEDGMENTS

Section 7 was written in collaboration with N McDougald. The preparation

of this paper was supported by grants to LGL from the Microgravity Science

Program of NASA and the Ocean Technology Program of the Office of Naval

Research.

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http://www.annurev.org.

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