on acoustic cavitation of slightly subcritical bubbles · on acoustic cavitation of slightly...

On�

acoustic cavitation of slightly subcritical bubblesAnthony�

HarkinDepartment�

of Mathematics, Boston University, Boston, Massachusetts 02215

Ali NadimDepartment of Aerospace and Mechanical Engineering, Boston University, Boston, Massachusetts 02215

Tasso�

J. KaperDepartment of Mathematics, Boston University, Boston, Massachusetts 02215�Received13 April 1998;accepted15 October1998�

The�

classicalBlake thresholdindicatesthe onsetof quasistaticevolution leadingto cavitationforgas� bubblesin liquids. Whenthemeanpressurein the liquid is reducedto a valuebelowthevaporpressure,� theBlakeanalysisidentifiesa critical radiuswhich separatesquasistaticallystablebubblesfrom

thosewhich would cavitate.In this work, we analyzethe cavitation thresholdfor radiallysymmetric bubbleswhoseradii are slightly lessthan the Blake critical radius,in the presenceoftime-periodic�

acousticpressurefields. A distinguishedlimit equationis derivedthat predictsthethreshold�

for cavitationfor a wide rangeof liquid viscositiesandforcing frequencies.This equationalso� yields frequency-amplituderesponsecurves.Moreover,for fixed liquid viscosity, our studyidentifies

the frequencythat yields the minimal forcing amplitudesufficient to initiate cavitation.Numerical�

simulations of the full Rayleigh–Plessetequation confirm the accuracy of thesepredictions.� Finally, theimplicationsof thesefindingsfor acousticpressurefieldsthatconsistof twofrequencies

will be discussed. © 1999�

American Institute of Physics. � S1070-6631� �

99��

00302-5� �

I.�

INTRODUCTION

The Blake thresholdpressureis the standardmeasureofstatic acousticcavitation.1,2 Bubbles

�forced at pressuresex-

ceeding� the Blake threshold grow quasistaticallywithoutbound.�

This criterion is especiallyimportantfor gasbubblesin

liquids whensurfacetensionis the dominanteffect, suchas� submicronair bubblesin water,wherethenaturaloscilla-tion�

frequenciesarehigh.In contrast,when the acousticpressurefields are not

quasistatic,� bubbles generally evolve in highly nonlinearfashions. 3–6

�To�

begin with, the intrinsic oscillations ofspherically symmetric bubbles in inviscid incompressibleliquids are nonlinear.5

�The phaseportrait of the Rayleigh–

Plesset�

equation7–9

consists� of a large region of bounded,stable statescenteredabout the stable equilibrium radius.The naturaloscillationfrequenciesof thesestatesdependonthe�

initial bubbleradiusand its radial momentum,and thisfamily

of stateslimits on a stateof infinite period,namelyahomoclinicorbit in the phasespace,which actsasa bound-ary� outsideof which lie initial conditionscorrespondingtounstable! bubbles.Time-dependentacousticpressurefieldsthen�

interactnonlinearlywith boththeperiodicorbitsandthehomoclinicorbit. In particular,they canact to breakthe ho-moclinic orbit, permittinginitially stablebubblesto leavethestable region and grow without bound. Theseinteractionshave"

beenstudiedfrom manypointsof view: experimentally,numerically, and analytically via perturbationtheory andtechniques�

from dynamicalsystems.In#

Ref. 7, the transitionbetweenregularandchaoticos-cillations,� aswell astheonsetof rapidradialgrowth,is stud-ied for spherical gas bubbles in time-dependentpressure

fields.$

There,Melnikov theory is appliedto the periodicallyand� quasiperiodicallyforced Rayleigh–Plessetequationforbubbles�

containingan isothermalgas.One of the principalfindingsis that,whentheacousticpressurefield is quasiperi-odic% in time with two or more frequencies,the transitiontochaos� andthe thresholdfor rapid growth occurat lower am-plitudes� of the acousticpressurefield than in the caseofsingle-frequency forcing. Their work was motivatedin turnby�

that in Ref. 10, whereMelnikov theorywasusedto studythe�

time-dependentshapechangesof gas bubblesin time-periodic� axisymmetricstrainfields.

The�

work in Ref. 8 identifiesa rich bifurcation super-structure for radial oscillationsfor bubblesin time-periodicacoustic� pressurefields.Techniquesfrom perturbationtheoryand� dynamicalsystemsareusedto analyzeresonantsubhar-monics,& period-doublingbifurcation sequences,the disap-pearance� of strangeattractors,and transientchaos in theRayleigh–Plessetequationwith small-amplitudeliquid vis-cosity� and isentropicgas. The analysisin Ref. 8 comple-ments& the experimentsof Ref. 11 and the experimentsandnumerical' simulationsof Refs. 12–14. Analyzing subhar-monics, theseworks quantify the impact of increasingtheamplitude� of the acousticpressurefield on the frequency-response( curves.

Other)

works examiningthe thresholdfor acousticcavi-tation�

in time-dependentpressurefields havefocusedon thecase� of a stepchangein pressure.In Ref. 9, the responseofa� gasbubbleto sucha stepchangein pressureis analyzedbynumerical and Melnikov perturbationtechniquesto find acorrelation� betweenthe cavitationpressureandthe viscosityof% the liquid. Oneof theprincipal findingsis that thecavita-tion�

pressurescalesas the one-fifth powerof the liquid vis-

PHYSICSOF FLUIDS VOLUME 11, NUMBER 2 FEBRUARY 1999

2741070-6631/99/11(2)/274/14/$15.00 © 1999 American Institute of Physics

cosity.� A generalmethodto computethe critical conditionsfor an instantaneouspressurestepis also given in Ref. 15.The�

resultsextendnumericalsimulationsof Ref. 16 andex-perimental� findingsof Ref.17,andapplyfor anyvalueof thepolytropic� gasexponent.

Thegoalof thepresentarticle is to applysimilar pertur-bation�

methodsandtechniquesfrom the theoryof nonlineardynamical*

systemsto refine the Blake cavitation thresholdfor isothermalbubbleswhoseradii areslightly smallerthanthe�

critical Blake radiusand whosemotionsare not quasi-static. Specifically,we supposethesebubblesare subjectedto�

time-periodicacousticpressurefieldsand,by reducingtheRayleigh–Plessetequationsto a simpler distinguishedlimitequation,+ we obtain the dynamic cavitation threshold forthese�

subcriticalbubbles.The paperis organizedas follows. In the remainderof

this�

section,thestandardBlakecavitationthresholdis brieflyreviewed.( This also allows us to identify the critical radiuswhich, separatesstableandunstablebubblesthatarein equi-librium. In Sec.II, the distinguishedlimit - or% normal form.equation+ of motion for subcriticalbubbles/ i.e., thosewhoseradii( are slightly smallerthan the critical value0 is

obtained

from

the Rayleigh–Plessetequation.This necessitatesiden-tifying�

thenaturaltimescaleof oscillationof suchsubcriticalbubbles�

which happensto dependuponhow closetheyaretothe�

critical size.We beginSec.III by defininga simplecri-terion�

for determiningwhen cavitation has occurred.Wethen�

analyzethe normal form equationand determinethecavitation� thresholdfor a specificvalueof the acousticforc-ing

frequency 1 at� which the correspondinglinear undampedsystem would resonate2 . This pressurethresholdis thencom-pared� to numericalsimulationsof the full Rayleigh–Plessetequation+ and the good agreementfound betweenthe two isdemonstrated.*

Theself-consistencyof thedistinguishedlimitequation+ is furtherdiscussedin that section.SectionIV gen-eralizes+ the resultsto include arbitrary acousticforcing fre-quencies.� Acousticforcing frequencieswhich facilitate cavi-tation�

using the least forcing pressureare determined.Anunusual! dependenceof the thresholdpressureon forcing fre-quency� is discovered and explained by analyzing the‘‘slowly varying’’ phase-planeof the dynamicalsystem.Atthe�

end of Sec. IV, our choice of a cavitation criterion isdiscussed*

in thesettingof a Melnikov analysis.In Sec.V weextend+ thecavitationresultsto thecaseof anoscillatingsub-critical� bubblethat is driven simultaneouslyat two differentfrequencies.

We recapthe paperin Sec.VI by highlightingthe�

mainresultsanddiscussingtheir applicability.Lastly,weconclude� thepaperwith anappendixwhich qualitativelydis-cusses� therelationof our resultsto somerecentexperimentalfindings.

Blake threshold pressure

To�

facilitate the developmentof subsequentsectionswefirst$

briefly review thederivationof theBlake threshold.63

At4

equilibrium,+ the pressure,p5 B ,6 inside a sphericalbubbleofradiusR is relatedto the pressure,p5 L

7 ,6 of the outsideliquidthrough�

the normalstressbalanceacrossthe surface:

p5 B 8 p5 L 9 2 :R

. ; 1<The�

pressureinside the bubbleconsistsof gaspressureandvapor= pressure,p5 B > p5 g?A@ p5CB ,6 wherethevaporpressurep5ED istaken�

to be constant—p5EF depends*

primarily on the tempera-ture�

of the liquid—andthe pressureof the gasis assumedtobe�

given by the equationof state:

p5 g?HG p5 g? 0I RJ

0K

R

3LNM

,6 O 2Pwith, Q the

�polytropic index of the gas.For isothermalcon-

ditions* RCS

1, whereasfor adiabaticones, T is the ratio ofconstant-pressure� to constant-volumeheat capacities.Atequilibrium,+ the bubblehasradiusR

J0K ,6 the gashaspressure

p5 g? 0K ,6 andthe staticpressureof the liquid is takento be p5 0

K U .Thus, the equilibrium pressureof the gas in the bubble isgiven� by

p5 g? 0IWV p5 0

K XZY p5E[H\ 2 ]RJ

0K .

Upon^

substitutingthis result into _ 2 we, get the followingexpression+ for the pressureof the gasinsidethe bubbleasafunction

of the bubbleradius:

p5 g?Ha p5 0K bZc p5EdHe 2

fhgRJ

0K R

J0K

RJ 3

Lji. k 3lnm

Upon^

combiningEqs. o 1p and� q 3lnr , w6 efi nd

p5 L7ts p5 0

K uwv p5ExHy 2 zRJ

0K R0

KRJ 3

Lj{n|p5E}H~ 2 �

RJ . � 4�n�

Equation � 4� governs� thechangein the radiusof a bubbleinresponse( to quasistaticchangesin the liquid pressurep5 L

7 .More�

precisely,by ‘‘quasistatic’’ we meanthat the liquidpressure� changesslowly anduniformly with inertial andvis-cous� effectsremainingnegligible during expansionor con-traction�

of the bubble.For very small � submicron � bubbles,�

surface tensionis the dominanteffect. Furthermore,typicalacoustic� forcing frequenciesaremuchsmallerthanthe reso-nancefrequenciesof suchtiny bubbles.In this case,thepres-sure in the liquid changesvery slowly and uniformly com-pared� to the naturaltime scaleof the bubble.

For very smallbubbles,thePecletnumberfor heattrans-fer within thebubble—definedasR0

K2�N� /�W�

,6 with � the�

bubblenaturalfrequency� see Sec.II A � and� � the

�thermaldiffusiv-

ity

of the gas—issmall, and,due to the rapidity of thermalconduction� oversuchsmall lengthscales,thebubblemayberegardedasisothermal.We thereforelet �E� 1 for anisother-mal& bubbleanddefine

G�˜ � p5 0

K �Z� p5E�H� 2 �R0K R0

K3L .Then�

Eq. � 4�n� becomes�

p5 L7t� p5E�H� G

�˜R3LA� 2

fh�R

. 5¡n¢

275Phys. Fluids, Vol. 11, No. 2, February 1999 Harkin, Nadim, and Kaper

The�

right-handsideof this equationis plottedin Fig. 1 £ solidcurve� ¤ ,6 which showsa minimum value at a critical radiuslabeledRcrit¥ .

Obviously,)

if the liquid pressureis lowered to a valuebelow�

the correspondingcritical pressurep5 L7

crit¥ ,6 no equilib-rium( radius exists. For valuesof p5 L which, are above thecritical� valuebut belowthevaporpressurep5E¦ ,6 Eq. § 5¡n¨ yields©two�

possiblesolutionsfor the radiusR. Bubbleswhoseradiiare� lessthanthe Blake radius,R

Jcrit¥ ,6 arestableto small dis-

turbances,�

whereasbubbles with RJ«ª

RJ

crit¥ are� unstabletosmall disturbances.

The Blake radius itself can be obtainedby finding theminimum& of the right-handsideof ¬ 5¡n for

RJ«®

0.�

This yieldsthe�

critical Blake radius.

RJ

crit¥°¯ 3l

G�˜

2fh± 1/2

,6 ² 6³n´at� which the correspondingcritical liquid pressureis

p5 Lcritµ·¶ p5E¸H¹ 32l»º 3

L27G�˜

1/2

. ¼ 7½n¾

By�

combining the last two equations,it is also possibletoexpress+ the Blake radiusin the form:

RJ

crit¥°¿ 4 À3lEÁ

p5CÂHÃ p5 LcritµÅÄ ,6relating( the critical bubbleradiusto the critical pressureinthe�

liquid. Bubbles whose radii are smaller than Rcrit¥ are�quasistatically� stable,while biggeronesareunstable.

To�

obtainthestandardBlakepressurewe assumethatp5EÆcan� be ignoredand recall that surfacetensiondominatesinthe�

quasistaticregimewhich amountsto p5 0K ÇZÈ 2

fhÉ/�RJ

0K . Under

these�

approximations,G�˜ Ê 2

fhËRJ

0K2� and� the Blake threshold

pressure� is conventionallydefinedas

p5 BlakeÌ p5 0K ÍZÎ p5 Lcritµ

Ï p5 0K ÐZÑ 0.77

� ÒR0K .

In#

the quasistaticregimewherethe Blake thresholdis valid,p5 Blake is the amplitudeof the low-frequencyacousticpres-sure beyondwhich acousticforcing at higherpressureis sureto�

causecavitation.When the pressurechangesfelt by thebubble�

are no longer quasistatic,a more detailedanalysistaking�

into considerationthe bubbledynamicsand acousticforcing frequencymustbe performedto determinethe cavi-tation�

threshold.This is the typeof analysiswe undertakeinthis�

contribution.

II. THE DISTINGUISHED LIMIT EQUATION

A.Ó

Derivation

To�

makeprogressanalytically,we focusour attentionon‘‘subcritical’’ bubbleswhoseradii are only slightly smallerthan�

the Blake radiusat a given liquid pressurebelow thevapor= pressure.We thusdefinea small parameterÔÖÕ 0 b

�y

×ÖØ 2 1 Ù R0K

Rcrit¥ ,6 Ú 8ÛnÜwhich, measureshow closethe equilibrium bubbleradiusR0

Kis

to the critical valueRJ

crit¥ . The valueof the meanpressurein

theliquid, correspondingto theequilibriumradiusRJ

0K ,6 can

also� be found from Eq. Ý 5¡nÞ to�

be

p5 0K ßZà p5Cáãâ 2 ä

3l

RJ

0K 1 åçæ

2f è 2

� é 3l

êìë 4 í3l

R0K 1 î 1

2 ïãð3l8Ûòñ 2

�AóìôöõW÷3Lùø

. ú 9�nûThe liquid pressurep5 0

K ü and� the critical pressurep5 Lcrit¥ differ*

only% by an ý (þAÿ 2)�

amount.It#

turnsout that thecharacteristictime scalefor thenatu-ral( responseof suchsubcriticalbubblesalsodependson thesmall parameter� . This time scalefor smallamplitudeoscil-lations�

of a sphericalbubble is obtainedby linearizing theisothermal,

unforcedRayleigh–Plessetequation3L

� RR � 3l2

R2� �

p5 0K �� p5�� 2

f� R0K R

J0K

R

3L�

p5�� 2f��R � p5 0

K � ,6�10�

where, the densityof the liquid is given by � and� viscosityhas been neglected.Specifically, we substitute R � R0

K (1þ�x� )�

into � 10� and� keeptermslinear in x� to�

get

x� ¨ � 4 �� RJ

0K3L!

3l"

p5 0K #�$ p5%'&( RJ

0K2 x�*) 0.

� +11,

Solutions�

to - 11. ,6 representingsmall amplitudeoscillationsabout� equilibrium, are thereforex�*/ x� 0

K cos(� 0 t13254 )�

with theangular� frequencygiven by

687 4 9: RJ

0K3L<;

3l=

p5 0K >�? p5@BAC RJ

0K2

1/2

. D 12E

FIG.F

1. Pressurein the liquid, pG LH , versusbubbleradius,R

I, as governedby

Eq.JLK

5M .

276 Phys. Fluids, Vol. 11, No. 2, February 1999 Harkin, Nadim, and Kaper

WeN

now use O to�

definea nondimensionaltime variable: PQSR t1 . We are interestedin analyzingstability for valuesof(þR0K ,6 p5 0

K T )�

near (Rcrit¥ ,6 p5 Lcritµ )�. Hence,upon recalling U 6³WV ,6YX 7½WZ ,6

and� [ 8ÛW\ ,6 we seethat

]_^ 2f�`a R0K3L 2 1 b R

J0K

Rcrit¥1/2

t1dc 2f�egfh R0K3L

1/2

t1 .

WeN

note that as i tends�

to zero, the time scalefor bubbleoscillations% j the

�reciprocalof the factor multiplying t1 in the

last�

equationk increases

as lnm 1/2.Havingo

determinedthe properscalingfor the time vari-able� for slightly subcritical bubbles,we can now find thedistinguished*

limit p or% normal formq equation+ for suchbubbles�

in a time-periodicpressurefield. We start with theisothermal,

viscousRayleigh–Plessetequation:3L

r RRJ ¨ s 3

l2f RJ ˙ 2 t 4

�vu RJ ˙RJxw p5 0

K y�z p5{�| 2f�}RJ

0K R

J0K

RJ 3

L

~p5�� 2 �

R � p5 0K �� p5 A sin��3� t1�� . � 13�

The�

amplitudeandfrequencyof theappliedacousticforcingare� given by p5 A and� � ,6 respectively,and � represents( theviscosity= of thefluid. Here,thefar-field pressurein theliquidhasbeentakento be p5 0

K �� p5 A sin(�� t1 ),� with p5 0K � given� by Eq.�

9�W�

. SettingR(þt1 )�W� R0

K (1þ�� x� (þW�

)�), with � the

�samesmall pa-

rameter( introducedabove,we obtainat order � 2 � noting' thatall� of the 8¡ 1¢ and� £8¤3¥§¦ terms

�cancel¨

x� ¨ © 2f�ª

x� ˙ « x�¬ x� 2�¯®

A°

sin�±3² * ³B´ ,6 µ 14¶where,·¹¸ 2 º 2

»½¼¿¾ R0K

1/2

,6 A À p5 AÁ R0K

2f�ÂgÃ 2

� ,6LÄ * ÅÇÆÈ R0K3L

2 ÉgÊ1/2

. Ë 15Ì

In Eq. Í 14Î ,6 eachoverdotrepresentsa derivativewith respectto�ÐÏ

.It is implicit in the abovescalingthat Ñ ,6 A,6 and Ò * are�

nondimensionaland ÓÕÔ 1Ö with, respectto × . To seethat thisis

reasonable,consider an air bubble in water with ØÙ 998�

kg/m3L,6�ÚÜÛ 0.001

�kg/m s, and ÝßÞ 0.0725

�N/m. If we

specify àná 0.1�

and take a modestequilibrium radiusof R0Kâ 2 ã 10ä 6

3m, then å¹æ 0.38.

�Our analysisof ç 14è in subse-

quent� sectionswill concentrateprimarily on valuesof é in

the�

range0 êSëìê 0.4.�

TheparametersA°

and� í * are� relatedtothe�

forcing conditions,and their magnitudescan be madeorder% unity by choosingappropriateforcing parametersp5 A

Áand� î . As an example,if we again chooseR

J0K to�

equal 2microns,then ï * ð (

þ2.35ñ 10ò 7

ós)õô /

�dö ÷. Moreover,settingønù 0.05

�gives ú * û (

þ1.05ü 10ý 6

3s)õþ . Hence, the dimen-

sionless parameterÿ * is ��

1� when, � is

in the megahertzrange,andthis is preciselythe frequencyrangewe areinter-ested+ in exploring.Similarly, with R0

K and� � chosen� asabove,we, find A

°��(þ1.38 10 5

�m& s 2/kg)

�p5 AÁ /� � 2 and� therebywe see

that�

if �� 0.1,�

then p5 A can� becomeon the order of 103L

Pa.�

More datawill bepresentedlater � in Fig. 9� ,6 showingtypicalforcing pressures.

B. Interpretation

In the laboratoryonecancreatea subcriticalbubblebysubjecting the liquid to a low-frequencytransducerwhoseeffect+ is to lower theambientpressurebelowthevaporpres-sure. Thena secondtransducerof high frequency� high rela-tive�

to theslow transducer� will, give rise to the forcing termon% theright-handsideof � 14� . The low-frequencytransducerperiodically� increasesanddecreasesthepressurein theliquid�and� shrinks and expandsthe bubble, which follows this

pressure� field quasistatically� . Whenthe peaknegativepres-sure is reached� and� the bubble has expandedto its maxi-mumsize� ,6 we canimaginethatstateasthenewequilibriumstate, andat that point bring in the effectsof the soundfieldfrom

the secondtransducer.This secondfield canthenpos-sibly make the bubble,which had alreadygrown to somelargesize � but

�still smallerthan the critical radius� ,6 become

unstable.! This would all happenvery fast comparedto thetime�

scaleof the original slow transducer,so the pressurefield$

contributedby the original transducerremainsnearitsmostnegativevaluethroughout.Thestability responseof thebubble�

to thehigh-frequencycomponentof thepressurefieldis

the subjectof the restof this work.

III. ACOUSTIC FORCING THRESHOLDS �� * 1 !The�

value of " * # 1 correspondsto the forcing fre-quency� at which thelinearandundampedcounterpartof $ 14%would, resonate.We thereforechoosethis valueof the forc-ing frequencyas a starting point and perform a detailedanalysis� of the dynamicsinherentin the distinguishedlimitequation+ at this valueof & * . We caution,however,that, aswith, most forced, dampednonlinearoscillators,the largestresonant( responseoccursawayfrom theresonancefrequencyof% the linear oscillator.We use ' * ( 1 mainly asa startingpoint� for theanalysis,andthedynamicsobservedfor a rangeof% other ) * values= is reportedin Sec.IV.

Some�

specialcasesof * 14+ can� bereadilyanalyzedwhen,* - 1. In theabsenceof forcing, i.e., whenA

°�.0,�

thephaseportraits� of / 140 with, 132 0

�are shown in Fig. 2. With no

damping* 4

Fig. 25 a�7698 ,6 the phaseplane has a saddlepoint at:1,0; and� a centerat < 0,0

�>=. The latter representsthe equilib-

rium( radiusof the bubblewhich, when infinitesimally per-turbed,�

resultsin simpleharmonicoscillationsof the bubbleabout� that equilibrium. The saddlepoint at ? 1,0@ representsthe�

effects of the secondnearby root of the equation A 5¡CBwhich, is an unstableequilibrium radius.When dampingisadded� D Fig. 2E b�CF9G ,6 the saddlepoint remainsa saddle,but thecenter� at H 0,0

�>Ibecomes�

a stable spiral, attracting a well-defined*

regionof the phasespacetowardsitself. In the pres-ence+ of weak forcing J small AK but

�with no damping ( LM 0

�), the behaviorof N 14O can� be seenin a Poincare´ section

shown in Fig. 3.

A.Ó

Phase plane criterion for acoustic cavitation

To determinewhena slightly subcriticalbubblebecomesunstable! we choosea simplecriterion baseduponthe phaseportrait� of the distinguishedlimit equationP 14Q . For a givenR,6 there exists a thresholdvalue, AescS , o6 f A such that the

trajectory�

throughthe origin T 0,0�>U

grows� without boundfor


A V AescS ,6 whereas that trajectory stays bounded for AWA°

escS . A stablesubcritical bubble becomesunstableas A°

increases

past A°

escS . Thus there is a stability curve in the(þA,6�X )� -plane separatingthe regionsof this parameterspace

for which the trajectorystartingat the origin in the phase-plane� eitherescapesto infinity or remainsbounded.Numeri-cally,� many suchthresholdY ,6 A° escS pairs� Z represented( by theopen% circlesin Fig. 4[ were, foundwith \ * ] 1. Thedataareseen empirically to be well fitted by a least-squaresstraightline,�

given by A°

escS_^ 1.3563a 0.058.�

Forb

practicalexperimentalpurposesa linear regressioncurve� baseduponour escapecriterion shouldprovidea use-ful cavitationthresholdfor the acousticpressurein the fol-lowing�

dimensionalform:

p5 A c 3.835l d 3/2

Lfe1/2g

h 1/2RJ

0K3/2Lji 0.116

� k 2 lRJ

0K . m 16n

Here,o o

is

given by Eq. p 8ÛCq and� is itself a function of theequilibrium+ radius R

J0K ,6 surfacetension r ,6 and the pressure

differential*

p5 0K sut p5wv .

FIG. 2. Phaseportraitsfor thedistinguishedlimit equationx 14y . In z a{ , A | 0 and }�~ 0. Thefixed point � 0,0� is a center.Thefixed point � 1,0� is a saddle.In�b� �

, A � 0 and �� 0.09.The fixed point � 0,0� is a stablespiral.

FIG. 3. Poincare´ sectionshowingthe unstablemanifold of the saddlefixedpoint� � 0.999769375,� 0.024007197� for A

��0.048 and �� 0. Asymptoti-

cally, the saddlepoint is locatedat a distance� (A) from � 1,0� , specifically�1 � A� 2�/10��

(A� 4 ), ¡ A�

/2�£¢

(13/200)A� 3¤¦¥�§

(A� 5¨) © . Invariant tori areshown

inside a portion of the unstablemanifold. The centerpoint has moved alarge distancefrom ª 0,0« , in this casedue to the 1:1 resonancewhen ¬ * 1. We notefor comparisonthatwith nonresonantvaluesof ® * , Poincare´sectionsshow that the centeronly movesan ¯ (A

�) distance.For example,

when ° * ± 0.6, 0.7, and0.85, the centersare locatedapproximatelyat thepoints� ² 0.009,0.045³ , ´ 0.009,0.066µ and ¶ 0.029,0.163· , respectively.

FIG. 4. Escapeparametersfor the trajectoryof the origin ¸ 0,0¹ . For valuesof A�

, º abovethe regressionline, the trajectoryof the origin growswithoutbound.�

Below this line the trajectoryof the origin remainsbounded.Leastsquaresfit: A » 1.356¼�½ 0.058.


B. Period doubling in the distinguished limit equation

It#

sohappensthat thestability curvefor the trajectoryofthe�

origin canalsobe interpretedin termsof theperioddou-bling�

route to chaosfor the escapeoscillator ¾ 14¿ . In otherwords,, thevalueof AescS happensto bevery nearthe limitingvalue= at which the oscillationsbecomechaotic, just beforegetting� unbounded.For a fixed value of À3Á 0

�and a small

enough+ A,6 thetrajectoryof theorigin will settleupona stablelimit�

cycle in the phaseplane.As A°

is

increasedgradually,the�

period of this stable limit cycle undergoesa doublingcascade� asshownin Fig. 5 for a fixed valueof ÂÄÃ 0.35.

�The

period� doubling sequencewill continue as A is increaseduntil! the trajectoryof the origin eventuallybecomeschaotic,but�

still remainsbounded.Finally, at a thresholdvalueof A°

the�

trajectoryof theorigin will escapeto infinity. This is thevalue= of A that

�is given by the opencircles on the stability

diagram* Å

Fig.b

4Æ . A typical bifurcation diagramfor the es-cape� oscillator Ç 14È with, É3Ê 0

�is shownin Fig. 6, in whichË3Ì

0.375.�

C.Í

Robustness of the simple cavitation criterion

In this subsectionwe justify defininga cavitationcrite-rion basedupon the fate of a single initial condition. In allsimulations with ÎÄÏ 0,

�thereis a largeregionof initial con-

ditions*

whosefate Ð escaping+ or stayingboundedÑ is

thesameas� that of the origin Ò Fig. 7Ó . In fact, when the trajectorythrough�

the origin staysbounded,it is clear from the simu-lations�

that the origin lies in the basin of attraction of abounded,�

attractingperiodic orbit, and points in a large re-gion� aroundit all lie in the basinof attractionof the sameorbit.% The trajectoriesthroughall pointsin thatbasinremainbounded.�

Then, as the forcing amplitude is increased,theattractor� is observed to undergo a sequenceof period-doubling*

bifurcations,and this sequenceculminatesat theforcing

magnitudewhen the origin and other initial condi-

tions�

in a large region about it escape,becausethere is nolongera boundedattractingorbit in whosebasinof attractionthey�

lie.

D. Comparison with full Rayleigh–PlessetsimulationsÔ

Thestability thresholdpredictedby Eq. Õ 16Ö can� becom-pared� to dataobtainedfrom simulatingtheRayleigh–Plessetequation+ × 13Ø directly.

*For small valuesof Ù ,6 Fig. 8 shows

the�

resultinggoodagreement.The following is a brief descriptionof how the simula-

tions�

werecarriedout. The materialparametersusedto pro-duce*

Fig. 8 were ÚwÛ 998�

kg/m3L,6ÝÜßÞ 0.001

�kg/m s, and àá 0.0725

�N/m. Four valuesof â were, chosen,ãåä 0.01,

�0.05,

0.1,�

and0.2. For eachfixed valueof æ and� for a selectedsetof% valuesof ç ranging( from 0 to 0.4, the parametersR

J0K ,6

FIG. 5. Perioddoubling route to chaosin the distinguishedlimit equationè14é . For ê�ë 0.35, the limit cycles undergoperiod doubling as A

ìisí

in-creased.

FIG. 6. Bifurcation diagram( î�ï 0.375). Plotted is xð ˙ versusA. For eachfixed valueof A

ì, the origin ñ 0,0ò is

íintegratednumericallyandthe valueof

xð ˙ is plottedevery óõô¦ö 2 ÷ .

FIG.F

7. Boundedtrajectoriesfor the distinguishedlimit equationwith øù 0.2,A ú 0.3, and û * ü 1.0. The dark regionis the setof initial conditionswhosetrajectoriesremainbounded;it is the basinof attractionof the peri-odic orbit that existsin the period-doublinghierarchyfor this valueof A.


RJ

crit¥ , (6 p5wýÝþ p5 0K ÿ )�, and � were, calculatedsuccessivelyusing

the�

formulas R0K�� 2 � 2

�/(�� 2��

),�

Rcrit¥ � R0K /(1��

/2)�

, p5�� p5 0K �� (2

þ��/�R0K )�� 1 � (1

þ/3)/(1 �� /2)

� 2� �

,6 and !#"%$ (2þ�&(')/�

(þ )

R0K3L )�+* 1/2. , Note

�thatthis successionof computationsis done

for eachchosenvalueof - in eachof the four plots..Havingo

obtainedthedimensionalparametersrequiredforthe�

simulationof the full Rayleigh–Plessetequationscorre-sponding to a given /10 ,632+4 pair,� we useda bisectionprocedureto�

determine AescSRP,6 the threshold value of A separatingbounded�

and unboundedbubble trajectories.The bisectionprocedure� wasinitiatedby choosinga valueof A

°close� to the

linear�

regressionline. For this choiceof A°

,6 the dimensionalpressure,� p5 A ,6 was calculatedusing the middle equationin5156 . Then,the initial conditionR(0)

þ87R0K and� R(0)

þ:90�

wasintegratedforward in time using an implicit

; 18 fourth-orderRunge<

–Kutta scheme.The adaptive, implicit schemeweused! offers an accurateand stable meansto integratethegoverning� equations.The time stepsarelargein thoseinter-vals= in which thebubbleradiusdoesnot changerapidly, andthey�

are extremelyshort for the intervalswhere R or% R islarge = see, for example,figure4.7on p. 309of Ref. 6> . If thebubble�

radiusremainedboundedduring the simulation,thenthe�

valueof A°

was, increasedslightly, a new p5 A was, calcu-lated,anda newsimulationwasbegun.If, on theotherhand,the�

bubbleradiusbecameunboundedduring the simulation,then�

the valueof A°

was, slightly decreasedanda new simu-lation�

wasinitiated.Continuingwith this bisectionof A°

,6 the

threshold�

value, A°

escSRP,6 where the bubble first becomesun-stable wasdetermined.

The�

dimensionalcounterpart(p5 AÁ versus= R

J0K )� to Fig. 8 is

shown in Fig. 9 along with the dimensionalstability curvegiven� by Eq. ? 16@ . Note that for a given parameterA ,6 therelationship( which definesthe dimensionlessdamping pa-rameterB ,6 i.e., R0

KDC 2 E 2�/(��F 2��GH�I

)�, canbe thoughtof asde-

fining the bubbleradiusR0K . That is, for a given liquid vis-

cosity� J and� with all other physical parametersbeingconstant,� K can� only changeby varying the equilibrium ra-dius*

R0K . As such,the dimensionlessA-versus-L curves� can

be�

put in terms of the dimensionalp5 AÁ versus= R

J0K curves,�

drawn*

in Fig. 9.To�

show the way in which the bubble radius actuallybecomes�

unboundedin the full Rayleigh–Plessetsimula-tions,�

Fig. 10 providesthe radiusversustime plots for threetypical�

simulationswith the samevalue of MON 0.3,�

wheretime�

is nondimensional.In this case,A°

escSRPPRQ

0.51.�

Thetop twocurves� are obtainedfor valuesof A

°of% 0.3 and 0.5, respec-

tively.�

They showstableoscillationsalthougha perioddou-bling�

canbe seento haveoccurredin going from oneto theother.% The bottomfigure correspondsto A S 0.53

�andshows

that�

the bubble radius is becomingunbounded.The corre-sponding dimensionalparametersfor the Rayleigh–Plessetsimulations aregiven in the figure caption.

FIG.F

8. Simulations of the fullRayleighT

–Plesset equation for fourdifferent valuesof U . EachopencirclerepresentsV an (A

W, X ) pair at which the

bubbleY

first goes unstable.Superim-posedZ is the linear regressionline ob-tained[

from the simplecriterion basedupon\ the distinguishedlimit equation.In]_^

a – a db , c�d 0.01,0.05,0.1,and0.2,respectively.


E. Consistency of the distinguished limit equation

In#

this subsection,we argue,ae posteriori,6 that it is self-consistent� to use the escapeoscillator as the distinguishedlimit equationf 14g for thefull Rayleigh–Plesseth 13i ,6 i.e.,weshow that the higher-orderterms encounteredduring thechange� of variablesfrom R

Jto�

x� may& be neglectedin a con-sistent fashion.

Recall that, in the derivationof the escapeoscillator,allof% the jlk 1m and� npo1qsr terms

�droppedout, and the ordinary

differential*

equationt 14u was, obtainedby equatingthetermsof%wv (

þyx 2�)�. The remainingtermsareof z (

þy{ 3L)�

andhigher.Tobe�

precise,at | (þy} 3L)�, we find on the left-handside:

xx� ¨ ~ 2 � xx� ˙ � 3�2 x� ˙ 2�,6

and� on the right-handside:

� 3�4x�� 2x� 2

�D�203� x� 3L.

Moreover,�

we note that, for i;+�

4,�

all termsof � (þy� i)�

on theleft-hand�

sideareof theform x� i � 2�x� ˙ ,6 while all termsof � (

þy� i)�

on% the right-handsidearepolynomialsin x� .WeN

alreadyknow that, for trajectoriesof the escapeos-cillator� thatremainbounded,thex� and� x� ˙ variables= stay �l� 1� .Hence,all of the higher-ordertermsremainhigherorderforthese�

trajectories.Next, for trajectoriesthat eventuallyes-cape� � i.e.,

thosewhosex� coordinate� exceedssomelargecut-

off% at somefinite time� ,6 we know that x� and� x� ˙ are� boundeduntil! that time andafterwardstheygrow without bound.Thefact that thenR also� growswithout boundfor thesetrajecto-ries( � due

*to the changeof variablesthat definesx�� is

consis-

tent�

with thedynamicsof the full Rayleigh-Plessetequation.Potentialtroublecouldarisewith trajectoriesfor which x�

becomes�

negative and large in magnitude, e.g., whenx�� 1/ � the

�coefficientof x� ¨ vanishes.= � This

�correspondsto

small RJ

.� A4

glanceat the Poincare´ map& for the escapeoscil-lator reveals, however, that trajectories which have

FIG.F

9. Simulations of the fullRayleigh–Plesset equation for fourdifferent valuesof � . Eachopencirclerepresents� a (p� A

� ,R�

0� ) pair at which the

bubble

first goes unstable.Superim-posed¡ is the thresholdcurve ¢ 16£ ob-tained¤

from the simplecriterion basedupon¥ the distinguishedlimit equation.In ¦ a§ – ¨ d© , ª�« 0.01,0.05,0.1,and0.2,respectively.

FIG. 10. Radiusversustime plots ( ¬� 0.1, ®¯ 0.3, A ° 0.3, 0.5, 0.53± . Di-mensional parameters:R

�0�³² 3.0 ´ m, R

�critµp¶ 3.2 · m, p��¸³¹ p� 0

� º¼» 29.7kPa, ½¾ 0.7MHz. p� A�À¿ 141.6Pa. Middle: p� A

�ÀÁ 236.0Pa. Bottom: p� A�ÀÂ 250.2Pa.


x��Ã�Ä 1/ Å atÆ sometime Ç canÈ neverhavex� (þyÉ

)�

and x� ˙ (þyÊ

) o�

fËlÌ1Í simultaneously,Î for any Ï . Hence,thesetrajectoriesare

notÐ in the regimeof interest,neitherfor theescapeoscillatornorÐ for the full Rayleigh–Plessetequation.This completestheÑ

argumentthat it is self-consistentto usetheescapeoscil-lator for this study.

IV.Ò

PRESSURE THRESHOLDS FOR GENERAL Ó *

A.Ó

Stability curves for various Ô *

UntilÕ

now, we haveonly examinedthe caseÖ * × 1 intheÑ

distinguishedlimit equation Ø 14Ù . In this sectionwe ex-amineÆ thedependenceof thestability thresholdon theacous-ticÑ

frequency, Ú ,Û for subcritical bubbles.Specifically, forfrequenciesÜ * between

Ý0.1 and1.1, we performednumeri-

calÈ simulationsof the distinguishedlimit equation Þ 14ß toÑ

determineà

many ( á ,Û Aâ escS )�

pairs. Thesepairs are plotted inFig. 11, andthe datapointsat eachdimensionlessfrequencyã

* areÆ connectedby straight lines ä in contrastto the leastsquaresÎ fitting donein Sec.III A å .

Asæ

in Sec. III A, good agreementbetweenthe distin-guishedç limit equation threshold and the full Rayleigh–Plessetequationis observedfor variousvaluesof è * ; thiscanÈ beenseenin Fig. 12 which comparesthe two resultsatfouré

different valuesof ê * givenç by 0.6, 0.7, 0.8, and0.9,for a fixed valueof ëíì 0.05.

îVariousï

featuresobservedin Fig. 11, suchastheflatten-ingð

of thesecurvesas ñ * decreases,à

are explainedin thenext subsection.

B. Minimum forcing threshold

Supposeò

that we wish to determine the driving fre-quency,ó for a givenbubblewith anequilibriumradiusR0

ô andÆaÆ critical radiusR

õcrit¥ ,Û so that the acousticforcing amplitude

FIG. 11. Stability thresholdcurvesfor the origin trajectoryof the distin-guishedlimit equationö 14÷ for manydifferent valuesof ø * . The valuesofù

* on the right-handborderlabel the different thresholdcurves.

FIG.ú

12. Simulations of the fullRayleighû

–Plesset equation for fourdifferent valuesof ü * . In eachplotý�þ 0.05. Each open circle representsan A

ÿ, � pair¡ at which the bubblefirst

goes unstable. Superimposedis thethreshold�

curveobtainedfrom the dis-tinguished�

limit equation.In � a� – � d� ,�* � 0.6, 0.7, 0.8, and 0.9, respec-

tively.�


necessaryÐ to make the bubble unstableis minimized. ThiscanÈ be done by choosingin Fig. 11, the value of � * for

éwhich the correspondingthresholdcurve is below all theothers for a given � . The result of sucha procedureis pro-vided� in Fig. 13 as follows. Figure 13 aÆ�� provides� the fre-quencyó of harmonicforcing at a given valueof the dampingparameter� � for which the requiredamplitudeof theacousticpressure� field to create cavitation is the smallest.Figure13� b�� showsÎ thedimensionlessminimum pressureamplitudeAescS correspondingÈ to thevalueof � * just

�presented.In Fig.

13� aÆ�� ,Û for � between�

0 and 0.225, the frequencycurve isnearlyÐ a straightline andwe fit a linearregressionline to thedataà

in that interval: � * �� 1.12�! 0.90"

for 0 #%$&# 0.225."

Correspondingly,'

in Fig. 13( b��) ,Û we see that the minimumpressure� curveis alsonearlystraightfor thesameintervalof*

values.� The leastsquaresline fitting the datain Fig. 13+ b��,isð

A-/.

1.030!1 0.02"

for 0 2%3!2 0.225."

When4 5!6

0.225,"

thereis a discontinuityin thefrequencycurve,È as seenin Fig. 137 aÆ98 . At the samevalue of : ,Û thepressure� curve levelsoff to A

-/;0.25."

This canbe explainedby�

a brief analysisof the normal form equation.The keyobservation will be that, in the escapeoscillator with con-stantÎ forcing < i.e.,constantright-handside= ,Û thereis a saddle-nodeÐ bifurcationwhen the magnitudeof the forcing is 1

4> . In

order to carry out this brief analysis,we considerthe cases?!@0"

and A&B 0"

separately,beginningwith C!D 0."

ForE F!G

0,"

thePoincare´ mapH of thenormalform equationI14J has

Kan asymptoticallystablefixed point L aÆ sinkM ,Û which

correspondsÈ to anattractingperiodicorbit for thefull normalform equation.Now, duringeachperiodof theexternalforc-ing,ð

the location of this periodic orbit in the (x� ,Û x� ˙ )�-plane

changes.È In fact, for thesmallvaluesof N * we areinterestedin here( O * P 0.3

"approximatelyQ ,Û thechangein locationoc-

cursÈ slowly, and one can write down a perturbationexpan-sionÎ for its position in powersof the small parameterR * .TheS

coefficientsat eachorderarefunctionsof the slow timezTVUXW * Y . To leadingorder, i.e., at Z\[ 1] ,Û the attractingperi-

odic orbit is locatedat the point x� (þzT ),� 0 ,Û wherex� (

þzT )� is the

smallerÎ root of x�_^ x� 2`ba

A sin(Î zT ),� namely,

x�_c zTedef 12 g 1

2 h 1 i 4A sinÎkj zT�l .Therefore,oneseesdirectly thatA m 1

4 is a critical value.Inn

particular,if oneconsidersany fixed valueof A-/o 1

4> ,Û then

theÑ

attractingperiodic orbit exists for all time, and the tra-jectory�

of our initial condition p 0,0"rq

will be always be at-tractedÑ

to it. s Notet

that the viscosity u&v 0.225"

is largeenoughw soorbitsareattractedto thestableperiodicorbit at afasteré

ratethantherateat which theperiodicorbit’s positionmovesin the(x� ,Û x� ˙ )

�-planedueto theslow modulation.x How-

ever,w for any fixed value of A y 14,Û the function giving x� (

þzT )�

becomes�

complex after the slow time zT reachesz a criticalvalue� zT * (

þA-

)�, whereA

-sin(Î zT * )

�|{ 14> ,Û andwherewe write zT * (

þA-

)�

sinceÎ zT * dependsà

on A. Moreover,x� (þzT )� remainscomplexin

theÑ

interval zT * (þA),�/}�~

zT * (þA)�

duringà

which A sin(Î zT )�|� 14.

Viewed�

in terms of the slowly varying phaseportrait, theslowlyÎ movingsink mergeswith theslowly movingsaddleinaÆ saddle-nodebifurcation when zT reacheszT * (

þA)�, and they

disappearà

togetherfor zT * (þA)��

zT�� zT * (þA)�. Hence,theat-

tractingÑ

periodic orbit no longer exists when zT reacheszzT * (þA)�, andthe trajectorythat startedat � 0,0

"r�—andthat was

spiralingÎ in towardtheslowly movingattractingperiodicor-bit�

while zT was lessthan zT * (þA-

)�—escapes,becausethereis

noÐ longerany attractorto which it is drawn.For thesakeof completenessin presentingthis analysis,

we notethat whenA � 14,Û thenzT * (

þA)�e��

/2;�

hence,it is pre-ciselyÈ nearthis lowestvalueof A

-,Û namelyA

-/� 14> ,Û thatwe find

theÑ

thresholdfor the acousticforcing amplitude,andthe es-capeÈ happensnearthe slow time zT�� /2.

�Moreover,for val-

ues� of A � 14, 0Û�� zT * (

þA)�e��

/2,�

andso the escapehappensatanÆ earlier time.

Numerically,t

the minimal frequency� * appearsÆ to be�* � 0,

"where � * � 0.01

"is the lowest value for which we

conductedÈ simulations.Moreover,this alsoexplainswhy, as

FIG.�

13. In � a� , the * that¡

minimizesA¢

esc£ is¤

plottedversus¥ . In ¦ b§9¨ , the minimum valueof A¢

esc£ correspondingto the valueof © * in¤«ª

a¬ is¤

plottedversus. The opencircles representRayleigh–Plessetcalculationsof the minimum thresholdwith ®e¯ 0.05. Note that A

¢_° 14>_± the¡

flat portion of the curve in ² b§9³µ´representsthe Blake quasistaticthresholdfor the distinguishedlimit equation.


we see from Fig. 11 already, the curves are flat with A¶ 0.25"

for · * ¸ 0.3."

This is therangeof small valuesof ¹ *foré

which the aboveanalysisapplies.Next,t

having analyzedthe regime in which º!» 0,"

weturnÑ

briefly to thecase¼!½ 0."

For small values,¾ * ¿ 0.3,"

thecurvesÈ in Fig. 11 remainflat nearA À 0.25

"all the way down

toÑÂÁ!Ã

0."

Thefull normalform equationwith Ä!Å 0"

is a slowlymodulatedH Hamiltonian system. One can again use theslowlyÎ varying phaseplanesas a guide to the analysis Æ al-ÆthoughÑ

the periodic orbit is only neutrally stable when ÇÈ 0"

andno longerattractingasaboveÉ ,Û andthe saddle-nodebifurcation�

in the leading order problem at A sin(Î zT * )ÊÌË 1

4 istheÑ

main phenomenonresponsiblefor the observationthattheÑ

thresholdforcing amplitudeis near0.25. Í We4

also notethatÑ

for a detailedanalysisof the trappedorbits, one needsadiabaticÆ separatrix-crossingtheory Î seeÎ Ref. 19, for ex-ampleÆ Ï ,Û but we shall not needthat here.Ð

Finally,E

and most importantly, simulationsof the fullRayleighÑ

–Plessetequationconfirm all the quantitativefea-turesÑ

of this analysisof the normalform equation.The opencirclesÈ in Fig. 13 representthe numericallyobservedthresh-old forcing amplitudes,andthesecircleslie very closeto thecurvesÈ obtainedas predictionsfrom the normal form equa-tion.Ñ

We attribute this similarity to the fact that the phaseportrait� of the isothermalRayleigh–PlessetequationhasthesameÎ structure—stableand unstableequilibria, separatrixbounding�

the stableoscillations,and a saddle-nodebifurca-tionÑ

when the forcing amplitudeexceedsthe threshold—astheÑ

normal form equationÒ seeÎ Ref. 7Ó .

C.Í

The dimensional form of the minimum forcingthresholdÔ

RecallÑ

that, for Õ between�

0 and 0.225, we fit linearregressionz lines to portionsof Figs.13Ö aÆ�× andÆ 13Ø b�eÙ . Specifi-cally,È for Fig. 13Ú aÆ�Û we found that, for a particularchoiceofÜ,Û the frequencywhich yields the smallestvalueof AescÝ canÈ

be�

expressedas Þ * ß�à 1.12á&â 0.90"

for 0 ã%ä&ã 0.225."

Andforé

Fig. 13å b��æ ,Û we found that the stability boundaryfor theminimum forcing is given by

A ç 1.03è!é 0.02"

for 0 ê%ë!ê 0.225,"

0.25"

for 0.225ì%í!ì 0.4." î 17ï

Usingð

the definitionsof ñ ,Û A,Û and ò * asÆ given by ó 15ô ,Û the‘‘optimal’’ acousticfrequencyto causecavitationof a sub-criticalÈ bubbleis given in dimensionalform by

õ�ö�÷2.24 øù R

ú0û2 ü 1.27 ý 1/2þ 1/2

ÿ 1/2Rú

0û3/2�

foré

0 � 2 � 2`

�� Rú

0û

1/2�0.225."

Correspondingly,'

theminimumacousticpressurethresholdis

p A��

2.91 3/2��

1/2�� 1/2R

ú0û3/2�� 0.04

" � 2 �R0û

for 0 � 2�� 2

�� R0û

1/2�0.225,"

� 2`!

2�

Rú

0û for

é0.225" 2

��# 2`

$�%'& Rú

0û

1/2(0.4."

D. A lower bound for Aesc) via* Melnikov analysis

The distinguishedlimit equation + 14, canÈ be written astheÑ

perturbedsystem

x - f . x/�021 g354 x,Û7698 ,Ûwhere x:<; (

=x> ,Û y? ),Ê

f@(=x> ,Û y? )Ê�A

(=y? ,Û x> 2 B x> )

Ê, and g3 (= x> ,Û y? ,Û7C )ÊD 0,

"A sin(ÎFE * G )ÊIH 2J y? with A K2L A,Û'M�N2O P . When Q R 0

"the

systemÎ hasa centerat S 0,0"UT

andÆ a saddlepoint at V 1,0W . ThehomoclinicK

orbit to the unperturbedsaddle is given byX0û (=ZY )ÊZ[ x> (

=Z\),Ê

y? (=Z]

)Ê

where x> (=Z^

)ÊZ_a`

(= 1

2)Ê�b

(= 3�2)Êtanh2

`(=7c

/2)�

and

y? (=Zd

)Ê�e

(= 3�2f )Ê tanh(g /2)

�sech2(

=7h/2)�

. Following Ref. 20, theMelni-kovi

function takesthe form

M j!k 0û!l�m n5o

of p 0ûrqts9u ∧g3 v 0

ûZwtx9y ,Û7z|{2} 0û d~��

� 3�2� A-¯ �<�

�sinÎU�� * �!�|�2� 0

û!�7� tanhÑ �

2� sechÎ 2 �

2� d~��

� 9�2�2� �5�

�tanhÑ 2

` �2� sechÎ 4

� �2� d~��

.

Thefirst integralcanbedonewith a residuecalculation,andtheÑ

secondintegralis evaluatedin a straight-forwardmanner,resultingz in

M �!� 0û!��a� 6

�¡ £¢�¤* ¥ 2 cosÈ§¦�¨ * © 0

û!ªsinhÎ «¬¯® * ° A ± 12

5²´³ .

The Melnikov function has simple zeros when AµA-¯

h.tan.¶ ,Û where

Ah.tan.· 2�

sinh9¹»º * ¼5²¡½¿¾�À

* Á 2` Â . Ã 18Ä

HenceÅ

the stable and unstablemanifolds of the perturbedsaddleÎ point intersecttransverselyfor all sufficiently smallÆ Ç 0

"when A

-¯ È A-¯

h.tan.É .20 The

Sresulting chaotic dynamicsis

evidentw in Fig. 5, for example.Since homoclinic tangencymust occur beforethe trajectory throughthe origin can es-cape,È A

-¯h.tan.mayH beviewedasa precursorto A

-escÝ . Figure14

demonstratesà

that, for small enough Ê ,Û Eq. Ë 18Ì provides� alower boundfor the stability curvesseenin Fig. 11.

TheS

reasonwhy Melnikov analysisyieldsa lower boundforé

the cavitationthresholdrelatesto how deeplythe stableandÆ unstablemanifoldsof thesaddlefixed point of thePoin-careÈ mapfor Eq. Í 14Î penetrate� into the regionboundedbytheÑ

separatrixin the A-

,Û�Ï�Ð 0"

case.For sufficiently small val-ues� of Ñ ,Û long segmentsof the perturbedlocal stableandunstable� manifoldswill stay Ò (

=ZÓ)Ê

closeto the unperturbedhomoclinicK

orbit. However,as Ô growsç Õ andÆ onegetsout of


theÑ

regimein which the asymptoticMelnikov theorystrictlyappliesÆ Ö ,Û theselocal manifolds will penetratemore deeplyinto theregionboundedby theseparatrixin theA,Û�×�Ø 0

"case.

Inn

fact, thereis a sizablegapin theparameterspacebetweentheÑ

homoclinictangencyvaluesandtheescapevaluescorre-spondingÎ to our cavitationcriteria, i.e., when the trajectorythroughÑ

the origin grows without bound.Thereis a similargapç whenother initial conditionsarechosen.

TheS

Melnikov function was also calculatedin Ref. 21.There,a detailedanalysisof escapefrom a cubicpotentialisdescribedà

andthe fractalbasinboundariesandoccurrenceofhomoclinicK

tangenciesaregiven.We alsorefer the readertoRef.Ñ

22 in which a closelyrelatedsecond-order,dampedand

drivenà

oscillatorwith quadraticnonlinearityis studiedusingboth�

homoclinic Melnikov theory, as was done here, andsubharmonicÎ Melnikov theory.The existenceof periodicor-bits�

is demonstratedthere,andperioddoublingbifurcationsof theseperiodic orbits are examined.Their equationarisesfrom the study of traveling waves in a forced, dampedKorteweg–de Vries Ù KdV Ú equation.wV. PRESSURE FIELDS WITH TWO FASTFREQUENCIES

Inn

this section,we considerwhat happensto the cavita-tionÑ

thresholdif two fast frequencycomponentsarepresentin theacousticpressurefield, andtheslow transducer,whichlowersÛ

the ambientpressureandwhoseeffect is quasistatic,isð

also still present.In Fig. 15, we show the results fromsimulationsÎ with quasiperiodicpressurefields. Thesewereobtained from simulationsof Ü 14Ý with the forcing replacedby�

(A/2)�

sin(Î�Þ1* ß )ÊIà sin(ÎFá

2`* â )Ê ,Û anda wide rangeof values

for ã 1* andÆ ä2`* . For a fixed valueof å�æ 0.25,

"thecavitation

surfaceÎ shown in the figure was plotted by computingthetriplesÑ

( ç 1* ,Ûéè 2`* ,Û Aescê ).

ÊWe4

notethat in Fig. 15, the intersectionof thecavitationsurfaceÎ and the vertical planegiven by ë 1* ìîí 2* representszcavitationÈ thresholds for acoustic forcing of the formA sin(Î�ï * ð )Êòñ i.e., a single fast frequencycomponentand aquasistaticó componentó . Furthermore,we seethat the globalminimum of the cavitation surfacelies along the line ô 1*õîö

2`* . Hence,for A/2

�asour particularchoiceof quasiperi-

odic forcing coefficient, the addition of a secondfast fre-quencyó componentin the pressurefield doesnot lower thecavitationÈ thresholdbeyondthat of the singlefast frequencycase.È

VI. DISCUSSION

A÷

distinguishedlimit equationhasbeenderivedwhich issuitableÎ for usein determiningcavitationeventsof slightlysubcriticalÎ bubbles.This ‘‘normal form’’ equationallows us

FIG. 14. Comparisonsof thestability curvesof Fig. 11 with theMelnikov analysisfor two valuesof ø * . Thedottedline is thestability curveobtainedfromtheù

distinguishedlimit equation.The solid straightline is Eq. ú 18û . In ü aý and þ bÿ�� , � * � 0.9 and1.1, respectively.

FIG.�

15. Stability thresholdsurfacefor the trajectory through the originobtainedby integratingthe distinguishedlimit equationwith quasiperiodicforcing. This was obtainedfrom simulationsof � 14� with the forcing re-placed� by (A

�/2)�

sin(� 1* ) � sin(� 2�* � )� , andfor 0 �� 1* , � 2

�* � 1.3. The valueof � wasfixed at �� 0.25.The pointsbelow the surfacecorrespondto pa-rameter� values for which the trajectory of the origin remains boundedwhereasthosepoints abovethe surfaceare parameterswhich lead to anescapetrajectoryfor the origin. Qualitativelyandquantitativelysimilar re-sultswereobtainedfor �� 0.15and �! 0.35.


toÑ

studycavitationthresholdsfor a rangeof acousticforcingfrequencies.For " * # 1, we find an explicit expressionfortheÑ

cavitationthresholdvia linear regression,sincethesimu-lationÛ

datarevealan approximatelinear dependenceof thenondimensionalthresholdamplitude,A,Û on the nondimen-sionalÎ liquid viscosity, $ . When convertedto dimensionalform,é

this linearexpressiontranslatesinto a nonlineardepen-dence,à

cf. Eq. % 16& ,Û on the materialparameters.In all of oursimulations,Î the acousticthresholdamplitudecoincideswiththeÑ

amplitudesat which thecascadesof period-doublingsub-harmonicsK

terminate.Particularattentionhasalsobeenpaid to calculatingthe

frequency,' * ,Û at which a givensubcriticalbubblewill mosteasilyw cavitate.Expression( 17) for

éthe correspondingmini-

mumH threshold amplitude A-

growsç linearly in * foré +

,0.225"

until the critical amplitudeA - 14 is reached,andthe

thresholdÑ

amplitudestaysconstantat A . 14 for larger / . For

theseÑ

largervaluesof 021 0.225,"

the ‘‘optimal’’ frequencyisessentiallyw zero, as we showedby doing a slowly varyingphase� portraitanalysisandexploitingthefact thatthenormalform equationundergoesa saddle-nodebifurcation at A 3 1

4

inð

which theentireregionof boundedstableorbits vanishes.TheS

transition observedat 465 0.225"

marks a boundarybe-tweenÑ

the regime 7986: 0.225" ;

in which the minimum thresh-old amplitudeoccursfor dynamiccavitationandthe regime<0.225" =?>A@

0.4"CB

inð

which theminimumoccursfor quasistaticcavitation,È via the Blake mechanism.The full Rayleigh–Plessetequationundergoesa similar bifurcation at forcingamplitudesÆ very nearA

-ED 14> foré

sufficiently small F . Overall,theÑ

resultsfrom the normal form equationare in excellentagreementÆ with thoseof the full Rayleigh–Plessetequation,andÆ this may be attributed to the high level of similaritybetween�

the phase-spacestructuresof both equations.Inn

view of the findingsin Ref. 7, we may draw an addi-tionalÑ

conclusionfrom the presentwork. In a certainsense,we have extendedthe finding of lowered transition ampli-tudesÑ

reportedin Ref. 7 to the limiting caseof one low fre-quencyó and one fast frequency. We find that if a low-frequencytransducerpreparesa bubble to becomeslightlysubcritical,Î thenthe presenceof a high-frequencytransducercanÈ lower the cavitation thresholdof the bubblebelow theBlakeG

threshold.OurH

resultson theoptimumforcing frequencyandmini-mum pressurethresholdto cavitatea subcriticalbubblemayalsoÆ be useful in fine-tuning experimentalwork on single-bubble�

sonoluminescenceI SBSLJ K

. In SBSL,23–25 aÆ singlebubble�

is acousticallyforcedto undergorepeatedcavitation/collapseÈ cycles,in eachof which a short-livedflashof lightisð

produced.While the processthroughwhich a collapsingbubble�

emits light is very complexand involvesmanynon-linearphenomena,thepossibilityof bettercontrolovercavi-tationÑ

and collapse, e.g., through the use of multiple-frequencyé

forcing,canperhapsbeinvestigatedusingthetypeof analysispresentedin this paper.

ACKNOWLEDGMENTS

We4

are grateful to ProfessorS. Madanshettyfor manyhelpfulK

discussions.Theauthorswould alsolike to thankthe

refereesfor their comments.This researchwas madepos-sibleÎ by GroupInfrastructureGrantNo. DMS-9631755fromtheÑ

National ScienceFoundation.AH gratefully acknowl-edgesw financial supportfrom the National ScienceFounda-tionÑ

via this grant.TK gratefullyacknowledgessupportfromtheÑ

Alfred P. SloanFoundationin the form of a SloanRe-searchÎ Fellowship.

APPENDIX:L

COAXING EXPERIMENTS IN ACOUSTICMICROCAVITATION

ToS

illustrate one application of our results, we nowbriefly�

considerthe experimentalfindingsof Ref. 26 on so-calledÈ ‘‘coaxing’’ of acousticmicrocavitation.In theseex-periments,� smooth submicrometersphereswere added tocleanÈ water and were found to facilitate the nucleationofcavitationÈ eventsM i.e.,

ðreducethe cavitationthresholdN when

aÆ high-frequencytransducerO originally aimedasa detectorPwas turnedon at a relatively low-pressureamplitude.Spe-cifically,È the main cavitation transducerwas operatingat afrequencyé

of 0.75MHz, while the activedetectorhada fre-quencyó of 30 MHz. In a typical experiment,with 0.984 QspheresÎ addedto cleanwater,the cavitationthresholdin theabsenceÆ of the activedetectorwasfound to be about15 barpeak� negative.Whenthe activedetectorwasturnedon, pro-ducingà

a minimumpressureof only 0.5 barpeaknegativebyitself,ð

it causedthe cavitation thresholdof the main trans-ducerà

to be reducedfrom 15 to 7 bar peak negative.Thepolystyrene� latex sphereswere observedunder scanningelectronw microscopesandtheir surfacewasdeterminedto besmoothÎ to about 50 nm. It was thus thought that any gaspockets� which were trappedon their surfacedue to incom-plete� wetting andwhich servedasnucleationsitesfor cavi-tationÑ

weresmallerin size than this length.In Ref. 26, it isconjecturedÈ that the extremelyhigh fluid accelerationscre-atedÆ by the high-frequencyactivedetector,coupledwith thedensityà

mismatchbetweenthe gas and the liquid, causedtheseÑ

gaspocketsto accumulateon thesurfaceof thespheresandÆ form much larger ‘‘gas caps’’ R on the orderof the par-ticleÑ

sizeS ,Û which thencavitatedat the lower threshold.Herewe attemptto providean alternativeexplanationfor the ob-servedÎ loweringof thethresholdin thepresenceof theactivedetector.à

To effect our estimates,we shall usethe samephysicalparameters� as earlier: TVU 0.001

"kg/m s,ÎXWZY 0.998

"kg/m3

[,Û

andÆ \^] 0.0725"

N/m. We also ignore the vapor pressureoftheÑ

liquid at room temperature. Note also that1 bar_ 105

`N/mt 2

`andÆ that the transducerfrequenciesf

acitedÈ

aboveÆ arerelatedto the radianfrequenciesb used� earlierbyced2�gf

fa

.Leth

us begin by estimatinga typical size for the nucle-ationÆ siteswhich cavitateat pi LcritjlkXm 15bar in the absenceof the activedetector.Upon usingBlake’s classicalestimateof pi L

ncritjloXp 0.77

" q/�Rr

0s ,Û the equilibrium radiusof the trapped

airÆ pocketsis estimatedto be Rr

0s6t 3.7�vu

10w 8 m oH r3 7n m.This sizeis consistentwith the observationthat the surfacesof the sphereswere smoothto within 50 nm. We note thatsuchÎ a small cavitationnucleuscannotexist within the ho-mogeneousH liquid itself since it would dissolve away ex-


tremelyÑ

fast due to its overpressureresulting from surfacetension.Ñ

However,when trappedin a creviceor within theroughnessz on solid surfaces,it canbe stabilizedagainstdis-solutionÎ with the aid of the meniscusshapewhich separatesit from the liquid. The naturalfrequencyof a 37 nm bubblexif it weresphericaly foundfrom Eq. z 12{ would be385MHz

which is very large comparedto forcing frequencyof thecavitationÈ transducerwhich is 0.75MHz. Therefore,consis-tentÑ

with Blake’s classicalcriterion, the pressurechangesintheÑ

liquid would appearquasistaticto thebubbleandat suchaÆ small size, surfacetensiondoesdominatethe bubbledy-namics.The Blake critical radiusRcritj which correspondstothisÑ

equilibrium radiusR0s of 37 nm canbe calculatedto be

Rr

critjl| 64}

nm.Leth

us now supposethat the cavitationtransduceris op-eratingw at pi L ~X� 7

�bar peaknegativeas in the experiments

with the activetransduceralsoturnedon. Using Eq. � 5²�� ,Û thefinal�

expandedradiusof thebubblewhenthe liquid pressureisð

quasistaticallyreducedto � 7�

bar is found to be 4.2�10� 8 m or 42 nm. In other words, a bubble of original

radius37 nm at a liquid pressureof 1 bar,growsto a maxi-mumH size of 42 nm when the liquid pressureis reducedto� 7�

bar. Its critical radiusis still 64 nm, reachedif the liquidpressure� wereto be reducedfurther to � 15 bar.

At this point,sincethepressurechangesin theliquid duetoÑ

the 0.75 MHz cavitationtransducerare occurringslowlycomparedÈ both with the naturaltimescaleof the bubbleandtheÑ

30 MHz detector,let us take the meanpressurein theliquid to be the pi 0

s ��X� 7�

bar, andimaginethe bubblesizeatthisÑ

pressureto be its new equilibrium radius R0s6� 42nm,

with the critical radius still given by Rr

critjl� 64}

nm. Thisbubble�

is now assumedto be forced by the 30 MHz trans-ducerà

at anacousticpressureamplitudeof 1.5 bar � i.e., � 0.5"

bar�

peaknegative� . Using thesevalues,the perturbationpa-rameterz � is

ðcalculatedfrom equation� 8�� to

Ñbe �� 0.69.

"This

parameter� is too big for the resultsof the asymptotictheorytoÑ

provide meaningfulquantitativeagreement;nevertheless,we proceedwith the discussionto see if we can at leastobtain the right order of magnitudefor the pressurethresh-old.

With4

the given physicalparameters,andusingthe forc-ingð

pressureof pi A�2� 1.5barand �� 2

�g�¡ 30��¢

10£ 6¤

sÎ�¥ 1,Û theparameters� ¦ ,Û A

-,Û and § * areÆ calculatedfrom Eq. ¨ 15© to

Ñbeª2«

0.98,"

A ¬ 0.09"

and * ® 0.16."

UponexaminingFig. 13,attheÑ

relatively largedampingparameter±° 0.98" ²

beyond�

therangez originally considered³ the

Ñminimum forcing threshold

would appear to correspondto the constantvalue of A-

´ 0.25."

Herewe alsonotethat this sameforcing thresholdisalsoÆ observedwith a range of small µ * ,Û including ¶ *· 0.16" ¸

seeÎ Fig. 11¹ . In otherwords,the predictedthresholdpressure� for the active detector to causecavitation is Aº 0.25"

which correspondsroughly to pi A » 4 bar, whereasintheÑ

experimentsthe thresholdwas seento be A-E¼

0.09"

orpi A ½ 1.5bar. Despitethe lack of quantitativeagreement,thetheoreticalÑ

predictions and the experimentsdo show thesameÎ trends.Namely,in theabsenceof the30 MHz detector,theÑ

pressurein the liquid had to be reducedto ¾ 15 bar forcavitationÈ to occur. With the high-frequency transducerturnedÑ

on, however,cavitationoccurredat a minimum pres-

sureÎ of ¿ 7�ÁÀ

1.5ÂXÃ 8.5Ä

bar in the experimentsand at Å 7�

Æ 4ÇÁÈXÉ

11barbasedon thetheory. Ê We4

areaddingthenega-tiveÑ

pressurecontributionfrom the two transducersto arriveatÆ the final minimum pressureË . Thus, the presenceof thesecondÎ high-frequencytransducerdoesreducethe pressurethresholdÑ

for cavitationin both cases.

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