Analytical expressions for primary Bjerknes force on inertial cavitation bubbles

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Louisnard, Olivier Analytical expressions for primary Bjerknes

force on inertial cavitation bubbles. (2008) Physical Review E, vol. 78 (n° 3). pp.

036322-1. ISSN 1539-3755

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Analytical expressions for primary Bjerknes force on inertial cavitation bubbles

Olivier Louisnard

*

RAPSODEE research center, UMR EMAC CNRS 2392, Ecole des Mines d’Albi, 81013 Albi, France

共Received 9 April 2008; revised manuscript received 11 July 2008; published 24 September 2008兲

The primary Bjerknes force is responsible for the quick translational motion of radially oscillating bubbles in a sound field. The problem is classical in the case of small-amplitude oscillations, for which an analytical expression of the force can be easily obtained, and predicts attraction of sub-resonant bubbles by pressure antinodes. But for high-amplitude sound fields the bubbles undergo large-amplitude nonlinear oscillations, so that no analytical expression for the force is available in this case. The bubble dynamics is approximated on physical grounds, following the method of Hilgenfeldt et al.关J. Fluid Mech. 365, 171 共1998兲兴, but carefully accounting for surface tension. The analytical expression of the maximum radius of the bubble is recovered, the time of maximum expansion is noticeably refined, and an estimation of the collapse time is found. An analytical expression for the time-varying bubble volume is deduced, and the Bjerknes force is obtained in closed form. The result is valid for any shape of the sound field, including purely standing or purely traveling waves, and is ready to use in a theoretical model of bubble cloud evolution. In addition, the well-known sign inversion of the Bjerknes force for large standing waves is recovered and the inversion threshold in the parameter space is obtained analytically. The results are in good agreement with numerical simulations and allow a quantitative assessment of the effects of physical parameters. It is found that either reduction of the surface tension or increase in the static pressure should produce a widening of the bubble-free region near high-amplitude pressure antinodes.

DOI:10.1103/PhysRevE.78.036322 PACS number共s兲: 47.55.dd, 43.35.⫹d

I. INTRODUCTION

When excited by a sinusoidal sound field, gas bubbles undergo radial oscillations. Most of the practical applications of this phenomenon, known as acoustic cavitation, use high-amplitude sound fields, of typical high-amplitude greater than the static pressure, so that the liquid is under tension for some part of the cycle. In such conditions, whatever the frequency, two distinct dynamic bubble behaviors can be clearly divided by the so-called Blake threshold关1–4兴. In the tension phase,

very small bubbles are retained to grow by surface tension. Conversely, larger ones suffer an explosive expansion fol-lowed by a violent collapse, responsible for chemical关5兴 and

mechanical effects关6,7兴 and sonoluminescence 关8–10兴. The

latter oscillation regime is known as “inertial cavitation.” Bubbles in liquids experience various hydrodynamic forces. The buoyancy force is the most familiar one, and is the pressure force that a sphere of liquid replacing the bubble would experience. This remains true in an accelerating liquid 关11兴, and the generalized buoyancy force experienced by the

bubble is− V ⵱ P where P共r , t兲 is the pressure that would exist at the center of the bubble if it were absent, and V共t兲 is the bubble volume. For a bubble oscillating radially in a sound field, both P共r,t兲 and V共t兲 are oscillatory quantities so that the time average of the product over one cycle is not zero. The bubble experiences therefore a net force known as the “primary Bjerknes force”关12,13兴:

FB= − 具V ⵱ P典. 共1兲

The Bjerknes force can be easily calculated from knowledge of both the shape of the sound field and the bubble dynamics.

A classical result is that, for low-amplitude standing waves, subresonant bubbles are attracted by pressure antinodes, while bubbles larger than resonant size are repelled关14–16兴.

For the case of strong driving pressures, subresonant inertial bubbles can also be attracted by pressure antinodes, which constitutes the basic principle of single bubble SonoLumi-nescence levitation cells关8,17兴. However, it has been shown

by numerical calculations that above a given threshold the primary Bjerknes force on subresonant inertial bubbles un-dergoes a sign change关18兴. This behavior is due to the

reso-nancelike response curve共termed “giant resonance” by Lau-terborn and co-workers 关19兴兲 of the bubble just above the

Blake threshold, which is a physical consequence of the ef-fect of surface tension. Experiments indeed demonstrate that, above a certain driving level, no bubbles are visible in the neighborhood of large pressure antinodes关20兴.

Quantitative agreement between theory and experiment has been found in the case of linear or quasilinear oscilla-tions关15兴. Particle simulations 关20,21兴 were also found to be

in excellent agreement with recent experiments involving in-ertial bubbles 关22兴. While the Bjerknes force can be

calcu-lated analytically for linear bubble oscillations, only numeri-cal results can yet be found for inertial bubbles关18,23兴. An

analytical expression for the latter would first be helpful in particle or continuum models, describing the self-organization of bubbles, in order to get more efficient calcu-lations. Furthermore, analytical results allow a direct assess-ment of the sensitivity of the force to the physical parameters, and the establishment of scaling laws. These two objectives motivated this study.

Owing to the strong nonlinearity of the bubble dynamics equations, inertial cavitation has long been thought intrac-table analytically, up to the seminal papers of Löfstedt et al. 关24兴 and Hilgenfeldt et al. 关1兴 who demonstrated that several

terms of the Rayleigh-Plesset 共RP兲 equation could be ne-*[email protected]

glected during the explosive expansion of the bubble. This theoretical breakthrough allowed scaling laws to be obtained for the maximum radius of the bubble and the time of maxi-mum expansion. In this paper, we closely follow the ap-proach of Hilgenfeldt et al. 关1兴 and refine their analytical

solutions in order to account more precisely for the effect of surface tension. The approximate dynamics found are then used to obtain an analytical expression of the bubble volume. The latter are then conveniently recast in order to obtain the Bjerknes force共1兲 in closed form, in any acoustic field,

in-cluding the two extreme cases of traveling and standing waves. Finally, in the latter case, we seek an approximate expression of the Bjerknes force inversion threshold, evi-dencing the role of surface tension.

II. PRIMARY BJERKNES FORCE A. Acoustic field

We assume that the acoustic field in the liquid is mono-harmonic at angular frequency ␻, and defined in any point r by

P共r,t兲 = P共r兲cos关␻t +␾共r兲兴. 共2兲

This expression may represent a traveling wave, a standing wave, or any combination of both. We also define the pres-sure gradient in general form as

⳵P

⳵_xi共r,t兲 = Gi共r兲cos关␻t +␺i共r兲兴, 共3兲 where the fields Giand␺ican be expressed as functions of P

and␾ once the acoustic field is known. The following two extreme cases deserve special consideration: for a standing wave,␾共r兲=␾0, so that Gi共r兲=⳵P/⳵xi and ␺i共r兲=␾0; for a

traveling wave, P共r兲= P0 and ␾共r兲=−k · r so that Gi共r兲

= kiP0and␺i共r兲=␾共r兲− ␲/2.

B. Bubble model

The radial oscillations of a gas bubble in a liquid under the action of the sound field can be described by the Rayleigh-Plesset equation关1,25–27兴 RR¨ +3 2R ˙2₌1 ␳

冉

pg+ R cl dpg dt − 4␮ R˙ R− 2␴ R − 关p0+ P共t兲兴

冊

, 共4兲 where p0is the hydrostatic pressure, pg共t兲 is the gas pressure,

␳,␮, and clare the density, viscosity, and sound speed of the

liquid, respectively, and ␴ is the surface tension. The ambient radius of the bubble R0is the radius that the gas would have

in the absence of the sound field.

Time is nondimensionalized by the angular frequency ␻, and in order to obtain a formulation consistent with Ref.关1兴,

we set

p0+ P共r,t兲 = p0共1− p cos x兲 , 共5兲 so that

p= P共r兲/p0, 共6兲

x= ␻t +␾共r兲− ␲ . 共7兲

Using x as the time variable, and nondimensionalizing pressure with p0, Eq.共4兲 can be written as

RR_⬙+3 2R⬘ 2₌Rres 2 3

冉

pg*+ R␻ cl dp_g* dx − 4␮␻ p0 R_⬘ R −␣S R0 R + p cos x − 1

冊

, 共8兲

where the primed variables denote d / dx,

Rres= ␻−1共3p0/␳兲1/2 共9兲

is the resonance radius, and

␣S= 2␴/p0R0 共10兲

is the dimensionless Laplace tension.

Several models can be used for the bubble internal pres-sure pg 关10,28–32兴. As will be seen below, we are mainly

interested here in the expansion phase of the bubble, during which the density of the gas in the bubble remains weak, so that the precise choice of the thermal bubble interior’s model is unimportant. However, in order to assess the validity of the approximate expressions developed hereafter, simula-tions will be performed by using the Keller equation关33,34兴.

The bubble interior is modeled by using a thermal diffusion layer following Ref.关32兴, neglecting water evaporation and

condensation through the bubble wall. In the remaining part of the paper, we will consider air bubbles in water 共␴ = 0.072 N m−1_{, p}

0= 101 300 Pa, ␳ = 1000 kg m−3, cl

= 1498 m s−1_{, and␮= 10}−3Pa s兲.

C. The Bjerknes force

The primary Bjerknes force acting on a bubble is defined as

FB= − 具V共t兲 ⵱ P典, 共11兲

where V共t兲 is the instantaneous bubble volume. The average is taken over one acoustic period, so that, using Eq.共3兲,

F_B i= − Gi共r兲 1 T

冕

₀ T V共t兲cos关␻t +␺i共r兲兴dt. 共12兲

Using the dimensionless time x defined by共7兲 and the

peri-odicity of V, the latter expression becomes

F_B i= Gi共r兲 1 2␲

冕

₀ 2␲ V共x兲cos关x−␾共r兲 +␺i共r兲兴dx. 共13兲

The generic problem is therefore to obtain an approximate analytical expression for the integral

I= 1 2␲

冕

₀

2␲

V共x兲cos共x− x0兲dx, 共14兲

valid for any bubble dynamics, and for any value of x0. The

os-cillations关16兴. Here, we focus on the case of inertial

oscil-lations, that is, for any combination of parameters 共p,R0兲

above the Blake threshold. The special cases of standing and traveling waves can be simply recovered by setting, respec-tively, x0= 0 and ␲ / 2.

III. APPROXIMATE EXPRESSIONS A. Bubble radius

The method used to obtain analytical formula for the bubble radius is mainly inspired from the approach of Hilgenfeldt et al.关1兴. For self-consistency, we will recall in

this section the main lines of the method, and, where conve-nient, specify the refinements obtained by our approach.

Figure 1 displays the dimensionless bubble radius 关Fig.

1共a兲兴, bubble volume 关Fig.1共b兲兴, and driving pressure 关Fig.

1共c兲兴 in a typical case of inertial cavitation of 共f =20 kHz,

R0= 3␮m, and p = 1.4兲. With the choice of the dimensionless

time variable Eq.共5兲, x=0 represents the time of maximum

tension of the liquid. We set

x+= arccos

1

p, 共15兲

and we denote by xmthe time of maximum expansion of the

bubble, and by xcthe time of its maximum compression共see

Fig.1兲.

It is shown in Ref. 关1兴 that, during the expansion phase

and most of the collapse phase, the dominant terms in the right-hand side of Rayleigh equation are the driving term

pcos x − 1 and also the surface tension term␣SR0/Rfor

am-bient radii just above the Blake threshold. Following Ref. 关1兴, we neglect the dependence of the surface tension term on

R, and replace␣SR0/Rby␣S/K共p兲, where K共p兲 will be

de-termined later. The approximate Rayleigh equation becomes

RR_⬙+3 2R⬘ 2₌Rres 2 3

冋

pcos x −

冉

1 + ␣S K共p兲

冊

册

. 共16兲

We set, for further use,

A= 1 + ␣S

K共p兲. 共17兲

In addition, noting that

RR_⬙+ R_⬘2_{= 1/2}d 2_共R2_兲

dx2 , 共18兲

the right-hand side of the Rayleigh equation can be written in two different forms:

RR_⬙+3 2R⬘ 2₌1 2 d2共R2兲 dx2 + 1 2R⬘ 2₌3 4 d2共R2兲 dx2 − 1 2RR⬙. Numerical simulations show that R_⬘2Ⰷ RR_⬙on the interval 关−x+, x+兴, while R⬘2Ⰶ RR⬙holds on the interval关x+, xm兴 关1兴.

Additionally, we found that the latter property still holds in fact during almost all the collapse, except in its ultimate phase, where the gas and acoustic terms become significant again. This could be expected since the main part of the collapse is inertially driven and R_⬘becomes significant only when the liquid has acquired enough kinetic energy. We therefore obtain the following equations for the bubble ra-dius, over the interval关−x+, xc兴:

d2共R2兲 dx2 =

冦

4 9Rres 2 _{共p cos x} − A兲 on关− x+,x+兴, 共19兲 2 3Rres 2 _{共p cos x} − A兲 on关x+,xc兴. 共20兲

These equations are the same as those of Hilgenfeldt et al. 关1兴, except that the validity of the second is extended up to

xc. The first equation can be solved with the initial condition

R共−x+兲=␨R0, where ␨⯝1.6 and R⬘共−x+兲⯝R共−x+兲 关1兴. The

second equation is solved by requiring continuity of R共x兲 and

R_⬘共x兲 at x=x+. Integrating both equations twice, we obtain

R − 2_{共x兲 =}4 9Rres 2

冉

1 − p cos x + p共x + x+兲sin x+− A 2共x + x+兲 2

冊

+ ␨2R₀2关1 + 2共x + x+兲兴 共21兲 and R+ 2_{共x兲 =}2 3Rres 2

冋

1 − p cos x + p

冉

x 3+ x+

冊

sin x+ − A 2

冉

x 2_{+ x} + 2₊2 3x+x

冊

册

+ ␨ 2_R 0 2_{关1 + 2共x + x} +兲兴. 共22兲 The point 共xm, Rmax兲 of maximum expansion is obtained

by setting d共R+

2_{兲/dx=0, so that x}

mis given in implicit form

by 0 10 20 0 2000 4000 6000 0 2 4 0 1 2 b c a −x+ 1_{− pcos x} x R R0      R R0      3 x+ xm xc

FIG. 1. 共a兲 Dimensionless bubble radius R/R0;共b兲 dimension-less bubble volume共R/R0兲3;共c兲 dimensionless driving pressure 1 − p cos x. The case considered is a 3 ␮m air bubble in water and

p= 1.4. The times− x+and x+are the two instants of zero crossing of the driving pressure, xmis the time of maximum expansion of the

bubble, and xc the time of maximum compression. The dashed

curve in共a兲 represents the approximate dynamics given by Eqs. 共21兲 and 共22兲. The dashed line in 共b兲 is the final approximation of

psin xm− xm+ 1 3共p sin x+− x+兲− ␣S K共p兲

冉

xm+ 1 3x+

冊

+ 3␨2

冉

R0 Rres

冊

2 = 0, 共23兲

and Rmaxreads

Rmax 2 = R0 2 f共p,xm兲 + Rres 2

冉

g共p,xm兲− 2 3 ␣S K共p兲h共p,xm兲

冊

, 共24兲 where f共p,xm兲 = ␨2关1 + 2共xm+ x+兲兴, 共25兲 g共p,xm兲 = 2 3

冋

1 − p cos xm+ p

冉

xm 3 + x+

冊

sin x+ − 1 2

冉

xm 2 _{+ x} + 2₊2 3x+xm

冊

册

, 共26兲 h共p,xm兲 = 1 2

冉

xm 2 + x+ 2 +2 3x+xm

冊

. 共27兲 In order to obtain xm, the implicit equation共23兲 should be

solved. To avoid this, Hilgenfeldt and co-workers关1兴

devel-oped this equation near ␲ / 2 at first order, neglecting, on the one hand,␣S/K共p兲, and also 共R0/Rres兲2, which is appropriate

for driving the bubble at low frequencies. They obtain

xm₀= p +

1

3共p sin x+− x+兲, 共28兲 which can be further simplified to xm= p, if p is small

enough. Plugging the latter into Eqs.共24兲–共27兲, they obtain

an expression for Rmaxthat depends on R0only through the

␣S term in共24兲. The expression of K共p兲 is then determined

by using the fact, confirmed numerically, that the maximum of the response curve共Rmax/R0兲共R0兲 is obtained for an

am-bient radius R₀Cvery close to the Blake threshold, ⳵ ⳵_R0

冉

Rmax共p,R0兲 R0

冊

= 0 for R0= R0 c₌4

冑

3 9 ␴ p0 1 p − 1. 共29兲

This scheme yields a good approximation for Rmax, which

was the main objective of Hilgenfeldt and co-workers 关1兴,

but the approximation共28兲 of xm yields a rather large error

共see dotted line in Fig.2兲.1 Since the value of the integral 共14兲 is found to be very sensitive to the precise location of

xm, we seek a better approximation.

We therefore revert to the original equations 共23兲–共27兲.

The main difficulty lies in the presence of the ␣S term in

共23兲, which makes xmrigorously a function of both p and R0.

Thus Rmax depends on R0 not only through ␣S but also

through xm in the expressions for f, g, and h. The condition

共29兲 therefore becomes more complex, and should be solved

simultaneously with共23兲. We initially followed this complex

process, but finally found that a better approximation of xm

could be obtained by using a simple trick. First, as was done in Ref.关35兴, we neglect the␣Sterm in共23兲 and develop the

latter near ␲ / 2, but up to second order:

xm₁= ␲ 2 − 1 p+ 1 p

再

1 + 2p

冋

xm0− ␲ 2 + 3␨ 2

冉

R0 Rres

冊

2

册

冎

1/2 . 共30兲 For low-frequency driving, R0Ⰶ Rres, and xm₁ depends only

slightly on R0. We then plug 共30兲 into Eqs. 共24兲–共27兲 and

express the condition共29兲, neglecting ⳵xm/ ⳵R0, to obtain

K1共p兲 = xm₁ 2 + x+ 2 +2 3x+xm1 g共p,xm₁兲 9 4

冑

3共p− 1兲 . 共31兲 We now expand 共23兲 again near ␲/2, but keeping the ␣S

term, in which we set K = K1共p兲, to obtain

xm₂= ␲ 2 − A1 p + 1 p

再

A1 2 + 2p

冋

xm₀− A1 ␲ 2 +共1− A1兲 x+ 3 + 3␨2

冉

R0 Rres

冊

2

册

冎

1/2 , 共32兲 where A1= 1 + ␣S K1共p兲 . 共33兲 Setting A₁= 1 in x_m

2, that is, neglecting the effect of surface

tension, the result of Ref.关35兴, Eq. 共30兲, is recovered.

Figure 2 represents the variations of xm for a bubble of

ambient radius R0= 1␮m 关Fig. 2共a兲兴 and R0= 3␮m 关Fig.

1

Curiously, it can be checked that, although xm= p is a

simplifica-tion of共28兲, it still yields better results in the whole range

consid-ered in Fig.2. This is why we did not represent共28兲 on the latter.

1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 b a p xc xm p

FIG. 2. Variation of xm and xc with p. Thick solid line, xm

calculated from numerical solutions of Eq.共4兲; thin solid line, xm₂

from Eq.共32兲; dash-dotted line, xm₁from Eq.共30兲 共Ref. 关35兴兲;

dot-ted line, xm= p 共Ref. 关1兴兲; thick dashed line, xc calculated from

numerical solutions of Eq.共4兲; thin dashed line, xcfrom Eq.共43兲.

The results are calculated for a bubble of ambient radius共a兲 R0= 1 and共b兲 3 ␮m.

2共b兲兴 in water. The thick solid lines are the exact values obtained numerically, and the thin solid lines represent xm₂.

The agreement is seen to be excellent, although a noticeable difference can be seen for R0= 1␮m, which originates from

the oversimplification of accounting for surface tension by the simple term␣S/K共p兲 in Eq. 共16兲. Also shown is the

ap-proximation xm₁ 共dash-dotted line兲, which does not take

sur-face tension into account. This clearly introduces a notice-able error on xm, reasonably corrected by Eq.共32兲. Finally,

the approximation xm= p proposed in Ref. 关1兴 is displayed

共dotted line兲.

The approximation of Rmaxcan then easily be made by

plugging an approximation of xm into Eq. 共24兲. This was

done in Ref.关35兴 using xm₁, and an excellent agreement was

found. The gain brought by using xm₂ instead of xm₁in 共24兲

remains unimportant, and for brevity, we do not present the comparison between the analytical and numerical expres-sions of Rmaxhere.

B. Bubble volume

Approximations of the bubble volume could readily be obtained from the approximations 共21兲 and 共22兲, of the

bubble radius. However, such expressions do not yield ana-lytical expressions of the integral 共14兲 in closed form, and

further approximations are therefore required. First, we con-sider frequencies low enough to have RresⰇ R0, so that the ␨

term can be safely neglected in Eqs.共21兲 and 共22兲.

1. Approximate expression on [−x+, x+]

Numerical simulations demonstrate that R₋is almost lin-ear between 0 and x+关Fig.1共a兲兴, which suggests that Eq. 共21兲

is almost a perfect square in this interval. We then develop the cosine term in 共21兲 near x=0 up to second order, and

write the result as

R − 2_{共x兲 =}4 9Rres 2

冋

p − A 2

冉

x+ psin x+− Ax+ p − A

冊

2 + px+sin x+ − A 2x+ 2 + 1 − p −共p sin x+− Ax+兲 2 2共p− A兲

册

. 共34兲

For R₋ to be linear in x, the part of the expression in the square bracket out of the large parentheses must be negli-gible, so that R₋can be simplified as

R₋共x兲 =2 3Rres

冑

p − A 2

冉

x − Ax+− p sin x+ p − A

冊

. 共35兲

The expression for the bubble volume on关−x+, x+兴 therefore

reads V₋共x兲 =

冉

2 3Rres

冑

p − A 2

冊

3 共x− x1兲3, 共36兲 where x₁=Ax+− p sin x+ p − A , 共37兲

which allows us to calculate integral共14兲 in closed form.

2. Approximate expression on [x+, xc]

Using Eqs. 共23兲 and 共24兲, it can be easily checked that,

setting y = x − xm, the expression共22兲 for R+can be recast as

R₊2= R_max2 +2 3Rres

2 _{L共y兲, 共38兲}

where

L共y兲 = 2p cos xmsin2

y

2− A

y2

2 + p sin xm共sin y− y兲 . 共39兲 The bubble volume on关x+, xc兴 becomes therefore

V+= Rmax 3

冋

_{1 +}2 3

冉

Rres Rmax

冊

2 L共y兲

册

3/2 , 共40兲

which unfortunately does not yield an explicit integration of 共14兲. Further progress can be made by noting that, from Eq.

共24兲, Rresand Rmaxare of the same order of magnitude, and

that, from 共39兲, L共y兲=O共y2兲 near y =0. Equation 共40兲 can

therefore be approximated by V+= Rmax 3

冋

_{1 +}

冉

Rres Rmax

冊

2 L共y兲 + O共y4兲

册

. 共41兲 Thus, to the same order of approximation, L共y兲 can be re-placed by any equivalent expression up to order 4 in y, and the choice must be directed by the ability of V₊cos共x− x₀兲 to be integrable in closed form. We therefore choose to set

y2_{/2 = 2sin}2_{共y /2兲+O共y}4_{兲 and sin y}

− y = −共1/6兲 sin3y+ O共y5兲 in Eq.共39兲 to finally obtain

V+共x兲 = Rmax 3 _{+ R} maxRres 2

冉

_{2共p cos x} m− A兲sin2 y 2 − 1 6psin xmsin 3_y

冊

_{+ O共y}4_{兲, 共42兲}

which can now yield an explicit expression for integral共14兲.

It can further be noted that if the sin3yterm in the large parentheses is neglected and we set sin2共y /2兲⯝ y2/_{4, V+ is} found to be zero for

yc= xc− xm= Rmax Rres

冉

2 A − p cos xm

冊

1/2 , 共43兲

which constitutes a simple approximation for the collapse time. The comparison between this expression and the exact instant of minimum radius is visible in Fig.2共dashed lines兲.

Here again, an excellent agreement is found, but it deterio-rates toward small bubble radii.

IV. BJERKNES FORCE A. Analytical expression

With the expressions for the bubble volume共36兲 and 共42兲

at hand, the integral 共14兲 can be calculated in analytical

form, keeping the contribution of the integrand only in the intervals关0,x+兴 and 关x+, xc兴, since V can be neglected in the

other regions关see Fig. 1共b兲兴. The integral is thus the sum of the two contributions

I= I₋+ I+, 共44兲 where I −=

冕

0 x₊ V −共x兲cos共x− x0兲dx, I+=

冕

x₊ x_c V+共x兲cos共x− x0兲dx.

Using the approximate expressions 共36兲 and 共42兲 of the

bubble volume, integration yields

I₋= 8 27Rres 3

冉

p − A 2

冊

3/2 关⌬x共⌬x2 − 6兲sin共x+− x0兲 + 3共⌬x2 − 2兲cos共x+− x0兲 + x1共6− x1 2_{兲sin x} 0+ 3共2− x1 2_{兲cos x} 0兴, 共45兲 with ⌬x = x+− x1.

The contribution I+reads

I+= Rmax 3 _关sin共x

c− x0兲−sin 共x+− x0兲兴

+ RmaxRres 2

冉

1

4共p cos xm− A兲关 f2共yc兲− f2共y+兲兴 −

1

192psin xm关f3共yc兲− f3共y+兲兴

冊

, 共46兲 where

f2共y兲 = 4 sin共y− y0兲−sin 共2y − y0兲− 2y cos y0,

f3共y兲 = 2 cos共2y + y0兲 + cos共4y− y0兲

+ 12y sin y0− 6 cos共2y − y0兲

and

y0= x0− xm, y+= x+− xm.

The value of I from共44兲–共46兲 is displayed in Fig.3共thick

lines兲 for R0= 1␮m 关Fig.3共a兲兴, 3␮m 关Fig.3共b兲兴, and 6␮m

关Fig.3共c兲兴, in the case of a standing wave 共x0= 0兲, for

driv-ings ranging from the Blake threshold to p = 2.5. In order to get a clear picture, I is drawn in logarithmic scale, the solid part of the curves representing a positive sign and the dashed part a negative sign. The thin lines are the results obtained by solving共4兲 and calculating 共14兲 numerically, for f =20 kHz.

It is seen that excellent agreement is obtained, except for

R0= 1␮m关Fig.3共a兲兴. In particular, the point of inversion of

the Bjerknes force is shifted toward large drivings. This fea-ture originates from the errors induced in the values of xm, xc

共see Fig.2兲, and Rmaxfor small ambient radii, by replacing

the surface tension in the RP equation by␣S/K共p兲 in 共16兲. It

should be noticed that even the small errors visible on the curves of Fig.2共a兲yield large differences in the estimation of

I. This is to be expected since the phase between V and cos共x− x0兲 crucially influences the value of integral I.

B. Bjerknes force inversion threshold in standing waves

We consider the case of a standing wave x0= 0, and look

for an approximate locus in the parameter space where the Bjerknes force changes sign. Summing Eqs.共45兲 and 共46兲, it

is seen that integral共14兲 is zero for

a3X3+ a1X+ a0= 0, 共47兲

where

X=Rmax

Rres

,

and the coefficients ai depend on x+, which is just

arccos共1/ p兲; x1, which from 共37兲 depends on p and ␣S; xc,

which from共43兲 depends on p, x_m, X, and␣S; and xm, which

from 共32兲 depends only on p and ␣S, for R0/RresⰆ 1.

Fur-thermore, looking at Eq.共24兲, for R0Ⰶ Rres, X = Rmax/Rrescan

be written as X=

冉

g共p,xm兲− 2 3 ␣S K共p兲h共p,xm兲

冊

1/2 , 共48兲

and from Eqs. 共25兲–共27兲 and 共30兲–共33兲, still under the

as-sumption R0Ⰶ Rres, the terms g and h in the above equation

depend on R0 only through␣S. We conclude that, provided

that R0Ⰶ Rres, Eq.共47兲 becomes frequency independent, and

can in fact be written in implicit form as

I共␣S, p兲 = 0. 共49兲

This equation can easily be solved for␣S共p兲, in order to find

the approximate frequency-independent threshold for

inver-1 1.5 2 2.5 10−16 10−15 10−14 10−13 10−12 1 1.5 2 2.5 10−16 10−15 10−14 10−13 10−12 1 1.5 2 2.5 10−16 10−15 10−14 10−13 10−12 a c b p p p I (m 3) I (m 3) I (m 3)

FIG. 3. Variation of I with p. Thick lines, value of I predicted by approximation 共44兲–共46兲, for x0= 0 共standing wave兲, for an air bubble in water, of radius R0= 1共a兲, 3 共b兲, and 6 ␮m 共c兲. The solid parts of the curves correspond to I ⬎ 0 and the dashed parts to

sion of the Bjerknes force. The solution is presented in the inset of Fig.4. Below the curve, I ⬎ 0, so that the Bjerknes force attracts the bubble toward pressure antinodes, while it becomes repulsive above.

From␣S= 2␴ /共p0R0兲, the inversion threshold can also be

plotted in the 共R0, p兲 plane in the case of water at ambient

pressure 共␴=0.072 N m−1_{, p}

0= 101 300 Pa兲. The result is

displayed in Fig. 4 共thick solid line兲 and compared to the

exact inversion thresholds calculated from numerical simula-tion for three driving frequencies 20 共dash-dotted line兲, 40 共dashed line兲, and 80 kHz 共thin solid line兲. The labels on the two latter curves represent the value of R0/Rres. It is seen that

the above procedure yields a good estimation of the inver-sion threshold, up to R0/Rres= 0.1, above which it starts to

diverge from the exact value. The reason for this disagree-ment comes from the neglected R₀/_Rres term in all expres-sions, and also from the fact that, for increasing frequency, the bubble rebounds become more important, so that the bubble dynamics for x ⬎ xc also contributes to expression

共14兲. In addition, a cascade of period-doubling bifurcations

and chaos关19,36,37兴 appear in some cases 共and are

respon-sible for the noisy oscillations on the 80 kHz curve兲, so that the correct averaging of the Bjerknes force in such cases should be carried out over more than a single acoustic pe-riod. We did not pursue this issue further, since analytical predictions for these bifurcations are beyond the scope of the present paper.

Marginally, it can be seen that the inversion threshold in the共␣S, p兲 plane is almost linear, so that the following linear

fit共represented by a dashed line in the inset of Fig.4兲 can be

proposed for practical applications:

p= 0.269␣S+ 1.62. 共50兲

These results suggest that the inversion threshold is indepen-dent of frequency, and of the properties of the gas and liquid other than surface tension, as long as R0/RresⰆ 1. This

aston-ishing result originates from the fact that the Bjerknes force mainly depends on the expansion phase of the bubble, which, within the approximations leading to Eq.共16兲, is merely

gov-erned by the driving pressure amplitude and surface tension. The reasonably good agreement found in Figs.2–4partially supports this analysis.

In order to further investigate this issue, we first recalcu-lated the three inversion thresholds of Fig. 4 共f =20, 40,

80 kHz兲, replacing the thermal model of Ref. 关32兴 by an

isothermal behavior for the bubble interior. Figure5displays the results obtained共thick solid lines兲 and recalls the thresh-olds calculated in Fig.4共thin solid lines兲. It can be seen that

the thresholds slightly diverge for increasing R0, but remain

almost indistinguishable for R0/Rres⬍ 0.15. We also repeated

the calculations with the thermal model of Ref.关32兴, but for

argon bubbles共not shown兲, and found a negligible deviation from the air curves. We therefore conclude that the detailed bubble interior has a very weak influence on the expansion phase, at least for low enough values of R0/Rres, so that Eq.

共50兲 indeed constitutes a gas-independent law, within its

range of validity.

Another issue is the sensitivity of the results to the liquid viscosity. The latter has been neglected in the analytical ap-proach, when approximating the RP equation共8兲 by Eq. 共16兲.

The good agreement found in Fig.4between analytical and numerical results, calculated for water at ambient tempera-ture共␮= 10−3Pa s兲, suggests that, for such low values,

vis-cosity indeed plays a minor role during the bubble expan-sion. One should, however, check whether this is still the case for larger viscosities. We therefore repeated the numeri-cal numeri-calculation of the inversion threshold for viscosities 10 and 20 times larger than that of water共Fig. 6, thick dashed line and thick dash-dotted line兲. It is clearly seen that the threshold increases noticeably with viscosity. Conversely, we

0 2 4 6 8 10 1.5 1.7 1.9 2.1 0.1 0.15 0.1 0.15 0.2 0 1 2 3 4 5 1.5 2 2.5 3 R0(µm) p I <0 I <0 I >0 αS I >0 p

FIG. 4. Threshold of Bjerknes force inversion in the 共R0, p兲 plane, for a bubble in water in ambient conditions 共␴ = 0.072 N m−1_{, p}

0= 101 300 Pa, ␮ = 10−3Pa s兲. The region I⬎0 corresponds to attraction by the pressure antinode, and I ⬍ 0 to re-pulsion. The thin lines are calculated from numerical simulations of the RP equation. Thin solid line, f = 80 kHz; dashed line, f = 40 kHz; dash-dotted line, f = 20 kHz. The labels on the curves indicate the ratio R₀/_Rres 共triangles, f =80 kHz; filled circles, f = 40 kHz兲. Thick solid line, universal threshold calculated from ap-proximate dynamics by solving 共49兲. Thick dashed line, Blake

threshold. The inset represents the solution of 共49兲 in the 共␣S, p兲

plane. 0 2 4 6 8 10 1.5 1.7 1.9 2.1 R0(µm) p I <0 I >0

FIG. 5. Same as Fig.4calculated with an isothermal model. The thin solid lines are the numerical curves of Fig. 4 共f =20, 40,

80 kHz兲. The thick solid lines are calculated in the same conditions, except that the gas behavior is considered isothermal. The thin dashed line is the analytical threshold calculated from Eq.共49兲.

also checked that the result was unaffected by decreasing the viscosity below that of water, by computing the threshold for

␮= 0.1␮water 共thick solid line兲. This indicates that viscous

friction plays a non-negligible role in the bubble expansion for viscosities above some critical value. As already men-tioned in Ref.关1兴, increasing viscosity decreases R_max, and we also checked that it decreases xm too, so that, strictly

speaking, the Bjerknes force and its inversion threshold are viscosity dependent. Following our results, this influence is negligible for viscosities near to or lower than that of water, but for slightly larger values, the viscous term should be kept in the RP equation.

V. DISCUSSION

Important conclusions can be drawn from these results. Figure4shows that the inversion thresholds for all frequen-cies共thin lines兲 asymptotically merge with the Blake thresh-old共thick dashed line兲 for small bubble radii, in reasonable agreement with the analytical approximation 共thick solid line兲. Thus, as the driving pressure reaches, say 1.8 bar, the range of ambient radii of inertial bubbles attracted toward the antinode is suddenly reduced, with an upper limit lower than 2␮m. This explains why a well-defined bubble-free region can be observed around the pressure antinode for high-amplitude standing waves 关21兴. The range of attracted

bubbles is, however, not void, which suggests that the zone around the antinode could still be filled with inertial bubbles, of ambient radii very close to the Blake threshold, but too small to be visible. As noticed in Ref. 关18兴, in a

high-amplitude standing wave, the Bjerknes force acts as a sorter of inertial bubbles, leaving the smallest ones approaching or even reaching the pressure antinodes. The advantage of the present analysis is that it yields, through Eq. 共49兲, or its

simpler form 共50兲, an explicit classification of the bubble

size as a function of the local acoustic pressure, parametrized by the ratio ␴ / p₀.

As the increasingly small bubbles approach the pressure antinode, they may coalesce or quickly grow by rectified diffusion关35兴. As their size increases, they may again enter

the repulsion zone in the共R0, p兲 plane and move back again.

This picture is still complicated by the potential appearance of surface instabilities. Thus, the apparently void region ob-served around large pressure antinodes may be in fact the locus of the complex evolution of very small bubbles, of sizes close to the Blake threshold.

Finally, it is seen from the inset of Fig. 4that decreasing

␣S lowers the driving at which the pressure antinode

be-comes repulsive. The dimensionless parameter␣Scan be

var-ied experimentally by modifying the surface tension ␴共for example, by adding ionic salts or surfactants兲, or by chang-ing the static pressure p0. The present results suggest that, for

identical bubble ambient radii, the Bjerknes force would be-come repulsive for lower drivings, when either ␴ is de-creased or p0 is increased. This should have an observable

effect on the size of the bubble-free region around the pres-sure antinode. However, it should be noted that surface ten-sion also plays a crucial role for bubble surface instabilities 关10,38,39兴, and also for rectified diffusion 关35兴, through the

same dimensionless parameter␣S. Thus, a change in␣Smay

also directly influence these two processes, with probable consequences on the bubble cloud behavior. The present re-sult just demonstrates that surface tension can influence the shape of the bubble cloud through its direct effect on the bubble dynamics, and on the primary Bjerknes force.

Figure 6 also indicates that the size of the bubble-free region around the pressure antinode will decrease noticeably when the viscosity is increased slightly above that of water. As mentioned in Ref. 关1兴, this may be easily achieved

ex-perimentally by adding glycerin to water. Here again, such a macroscopic effect is mediated by the sensitivity of the bubble dynamics to the physical properties. To account ana-lytically for this dependence on viscosity, the viscous term should be kept in the Rayleigh equation, which renders the approximation scheme more involved. A generalization of our analytical results to this case may be addressed in a fu-ture study.

Finally, it is highly probable that the same effect of sur-face tension could be observed on the secondary Bjerknes force, as suggested by numerical simulations关40兴. The

ex-tension of the present analytical method to the latter effect is difficult, first because the expression of the secondary Bjerknes force also involves the bubble velocities, which are much more sensitive to approximations than the bubble ra-dius itself, and second because the dynamic equations of the two bubbles must be coupled by a radiation term.

0 2 4 6 8 10 1.5 1.7 1.9 2.1 p R0(µm)

FIG. 6. Same as Fig. 4, but for f = 20 kHz, and for different liquid viscosities. Thin solid line, water共same as dash-dotted line of Fig. 4兲; thick solid line, ␮=0.1␮water; thick dashed line, ␮ = 10␮water; thick dash-dotted line, ␮ = 20␮water. The thin dashed line is the analytical threshold calculated from Eq.共49兲.

关1兴 S. Hilgenfeldt, M. P. Brenner, S. Grossman, and D. Lohse, J. Fluid Mech. 365, 171共1998兲.

关2兴 F. G. Blake, Harvard University Acoustics Research Labora-tory Technical Memorandum, 1949共unpublished兲.

关3兴 E. A. Neppiras, Phys. Rep. 61, 159 共1980兲.

关4兴 I. Akhatov, N. Gumerov, C. D. Ohl, U. Parlitz, and W. Laut-erborn, Phys. Rev. Lett. 78, 227共1997兲.

关5兴 K. S. Suslick, W. B. McNamara, and Y. Didenko, in

Sonochemistry and Sonoluminescence, edited by L. A. Crum, T. J. Mason, J. L. Reisse, and K. S. Suslick, NATO Advanced Studies Institute, Ser. C: Physics 共Kluwer Academic, Dor-drecht, 1999兲, pp. 191–204.

关6兴 W. Lauterborn, T. Kurz, R. Mettin, and C. D. Ohl, Adv. Chem. Phys. 110, 295共1999兲.

关7兴 D. Krefting, R. Mettin, and W. Lauterborn, Ultrason. Sonochem. 11, 119共2004兲.

关8兴 D. F. Gaitan, L. A. Crum, C. C. Church, and R. A. Roy, J. Acoust. Soc. Am. 91, 3166共1992兲.

关9兴 S. J. Putterman and K. R. Weninger, Annu. Rev. Fluid Mech. 32, 445共2000兲.

关10兴 M. P. Brenner, S. Hilgenfeldt, and D. Lohse, Rev. Mod. Phys. 74, 425共2002兲.

关11兴 J. Magnaudet, in Proceedings of ASME Fluids Engineering

Division Summer Meeting 1997.

关12兴 V. F. K. Bjerknes, Fields of Force 共Columbia University Press, New York, 1906兲.

关13兴 F. G. Blake, J. Acoust. Soc. Am. 21, 551 共1949兲.

关14兴 D. E. Goldman and G. R. Ringo, J. Acoust. Soc. Am. 21, 270 共1949兲.

关15兴 L. A. Crum and A. I. Eller, J. Acoust. Soc. Am. 48, 181 共1970兲.

关16兴 T. G. Leighton, A. J. Walton, and M. J. W. Pickworth, Eur. J. Phys. 11, 47共1990兲.

关17兴 B. P. Barber, R. A. Hiller, R. Löfstedt, S. J. Putterman, and K. R. Weninger, Phys. Rep. 281, 65共1997兲.

关18兴 I. Akhatov, R. Mettin, C. D. Ohl, U. Parlitz, and W. Lauter-born, Phys. Rev. E 55, 3747共1997兲.

关19兴 W. Lauterborn and R. Mettin, in Sonochemistry and

Sonolumi-nescence共Ref. 关5兴兲, pp. 63–72.

关20兴 R. Mettin, S. Luther, C. D. Ohl, and W. Lauterborn, Ultrason. Sonochem. 6, 25共1999兲.

关21兴 U. Parlitz, R. Mettin, S. Luther, I. Akhatov, M. Voss, and W. Lauterborn, Philos. Trans. R. Soc. London, Ser. A 357, 313 共1999兲.

关22兴 P. Koch, D. Krefting, R. Mettin, and W. Lauterborn, in

Pro-ceedings of the IEEE International Ultrasonics Symposium, Honolulu, 2003, edited by D. E. Yuhas and S. C. Schneider共IEEE, Piscataway, NY, 2003兲, pp. 1475–1478. 关23兴 R. Mettin, in Oscillations, Waves and Interactions, edited by T.

Kurz, U. Parlitz, and U. Kaatze共Universitätsverlag Göttingen, Göttingen, 2007兲, pp. 171–198.

关24兴 R. Löfstedt, B. P. Barber, and S. J. Putterman, Phys. Fluids A 5, 2911共1993兲.

关25兴 L. Rayleigh, Philos. Mag. 34, 94 共1917兲.

关26兴 E. A. Neppiras and B. E. Noltingk, Proc. Phys. Soc. London, Sect. B 63, 1032共1951兲.

关27兴 H. Lin, B. D. Storey, and A. J. Szeri, J. Fluid Mech. 452, 145 共2002兲.

关28兴 A. Prosperetti, J. Fluid Mech. 222, 587 共1991兲.

关29兴 A. Prosperetti and Y. Hao, Philos. Trans. R. Soc. London, Ser. A 357, 203共1999兲.

关30兴 A. Prosperetti, in Sonochemistry and Sonoluminescence 共Ref. 关5兴兲, pp. 39–62.

关31兴 B. D. Storey and A. Szeri, Proc. R. Soc. London, Ser. A 457, 1685共2001兲.

关32兴 R. Toegel, B. Gompf, R. Pecha, and D. Lohse, Phys. Rev. Lett. 85, 3165共2000兲.

关33兴 J. B. Keller and I. I. Kolodner, J. Appl. Phys. 27, 1152 共1956兲. 关34兴 A. Prosperetti and A. Lezzi, J. Fluid Mech. 168, 457 共1986兲. 关35兴 O. Louisnard and F. Gomez, Phys. Rev. E 67, 036610 共2003兲. 关36兴 W. Lauterborn and E. Cramer, Phys. Rev. Lett. 47, 1445

共1981兲.

关37兴 U. Parlitz, V. English, C. Scheffczyk, and W. Lauterborn, J. Acoust. Soc. Am. 88, 1061共1990兲.

关38兴 M. S. Plesset, J. Appl. Phys. 25, 96 共1954兲.

关39兴 Y. Hao and A. Prosperetti, Phys. Fluids 11, 1309 共1999兲. 关40兴 R. Mettin, I. Akhatov, U. Parlitz, C. D. Ohl, and W.