EFFECTS OF LIQUID COMPRESSIBILITY ON RADIAL OSCILLATIONS OF GAS BUBBLES IN LIQUIDS

760Availableonlineatwww.sciencedirect.comScienceDirectJoumal of HydrodynamIcsELSEVIER2012,24(5):760-766wwwsciencedirect.comDO1:10.1016S1001-6058(11)60301-6science/journal/1001 6058EFFECTS OF LIQUID COMPRESSIBILITY ON RADIAL OSCILLATIONSOF GAS BUBBLES IN LIQUIDSZHANG Yu-ning, LI Sheng-caiSchool of Engineering, University of Warwick, Coventry, UK CV4 7AL, E-mail: Y Zhang @warwick. ac uk( Received November 10, 2011, Revised July 21, 2012)Abstract: For forced radial oscillations of gas bubbles in liquids, a more rigorous expression of the acoustic damping constant basedon Keller's equation is developed. Comparison with those in published papers is also made. The express on offered in this paper willimprove the predictions of total damping constant in particular for high frequencies and large bubbles, i.e., large oRo/c,(@ is thefrequency of driving sound field, Ro is the equilibrium bubble radius, cr is the sound speed in the liquid). Examples in ultrasoundimaging and acoustical oceanography are demonstratedKey words: gas bubble, damping constant, liquid compressibility, natural frequencyIntroductionone-parameter family of equations of bubble radius toFor forced gas bubble oscillations in liquids with the first order of bubble-wall Mach number, whichliquid compressibility considered, expressions of treats Kellers equation as a special case. For smallacoustic damping constant and natural frequency were oscillations considered in the present paper, expre-derived based on radiation pressure by Chapman and ssions of acoustic damping constant and natural frePlesset! and Prosperetti(2] and on Keller's equation 3 quency will be the same for equation of bubble motionby Prosperetti4). These expressions are currently cited which falls into this one-parameter family (referring toand prevalent in the literature. They are also reviewed Appendix). Here, a more appropriate expression ofrecently by Brenner et al. I5 and Coussios and Roy(ojKellers equation was used! p.466.Here,oscillationsHowever, for high frequencies and large bubbles such of spherical gas bubbles with small amplitude in infias those in biomedical and oceanic applications, a nite liquids are considered. Keller's equation.canmore rigorous expression of acoustic damping con- be written asstant has been derived by the authors basedKeller's equation. The reasons for the improvementsRby using our expressions in those areas are discussedas wellPe (0-p(r d[pe()-P,(o]1. Derivation based on Kellers equationp, crIn this section, express for damping constantsand natural frequency will be dewhereKeller's equation. Prosperetti and Lezzill proposed aBiography: ZHANG Yu-ning(1983-), Male,Pn()=、2o4Ph. D. Candidate中国煤化工Corresponding author: LI Sheng-cai,E-mail: S Li@warwick. ac ukP,()=P0CNMHG761here R is the instantaneous bubble radius and theoverdot denotes its time derivative, c, is the speed of Arus-2uiP, ROMdensity, t is the time, Pin is the pressure at the gasside of the bubble wall, o is the surface tension, Ap, RSM(7)is the viscosity of the liquid, Po is the ambient pre-ssure,e is the non-dimensional amplitude of drivingRosound field, is the frequency of driving sound(8)fieldFor comparison, the framework in Ref[2] forRef[2] for represent the viscous, thermal and acoustic dampingthermal effects will be followed but the equation constants respectively. The natural frequency of thebased on radiation pressure (Pad replaced by harmonic oscillator isKeller's equation. Notations in Ref [2]will also be re-tained the same such as3xP20R=R(+x), Pin=Pin.eg PoP(Ro, t)p,Ro P, R(9)PngR,' PoP(R, /)=-3xP x-4H,i(4) Note that here , 4h and liquid compressibilitymake contributiHere Ro is the equilibrium bubble radius, x is the small. Taking an air bubble oscillating in water as annon-dimensional perturbation of instantaneous bubble example, M varies between I and 1.03 if 10 m2radius, P(Ro, t) the non-dimensional pressure devia- Ro 210"m and 10s"20210s"tion from equilibrium pressure, P e the equilibriumForced oscillations of bubbles will enter stapressure in the gas, u the effective thermal viscotionary oscillations once the transient term (i.e thesity, x the polytropic exponent. When the acousticcomplementary function or the solution for the corre-sponding homogenous equation) in the solution appwave is weak (e <1), the amplitude of the bubble roaches to zero For most cases of bubble oscillations,oscillation is small (r <1 ). Substituting above rela- this contribution from the transient term decays quitions into Eq (1)and omitting the 2nd and higher ckly. For example, for a bubble of radius 1 um drivenorders of x (or &)result in an inhomogeneous by an external force at the frequencyequation for a harmonic oscillator i. e,, the oscillating with the non-dimensional amplitude E=0.1, the ratiobubble being studiedof amplitudes between the transient term and the sta-tionary term (i. e,, the particular integral for the inho十2mx+a2x=-a1+Rcos(at +omogeneous equation)reduces down to 0.1 at tM1. 15 Hs. Therefore, for many applications where thetransient term approaches to zero quickly, the oscilla-(5) tion to be analyzed can be simplified as a stationarystudies. Herewherefor the stationary forced oscillations driven by soundM=1+R4(4+H, field of single frequency (a), Lin in Eq(7)and KP, Rop, Roin Eq- (9)can be determined by solving the bubble interior problem following the approach in Ref[2]Though Keller's equation has been used in this paperIn Eq (5), Bo is the total dampto replace the eqbased on the radiatithe natural frequency. The total damping constant as used by Prosperetti", the resultant expressions for(B, )isu and x are the same noticing that expressions of中国煤化工Bot=Bvis + B+Bsolutions ofCNMHmodel with eftective thermal damping, the other basedwhereon solution of bubble interior76uh=ap ro Im(o)2. Comments on published expressions(10)In order to compare with those published expressions, complex analysis is used in this section andx=IaD R2Eqs. 3)and(5)become(11)PiP,(*Po(1+Ee),i+2po i+adr=-ae'eiarMHere p is a function relating to the solution of the(13)bubble interior problem(Eqs. (16)and(17)of Ref [2])lote that the recent study by Zhang and Lil pointed whereout that the approximation of G,=0 where G is anon-dimensional parameter employed in Ref [2] andE'=E1+IaRoreflects the length ratio between the mean free pathand the wavelength in gas should be dropped off ifG, 210. As to the valid regions for the frame workof Prosperetti2, readers are referred to Ref [8]thatHere E1+(oR/ccos(ar+s in Eq ( 5)hascategorized the bubble behavior into three different re- been replaced by a'ele and other equations(particugions according to the ratios of the bubble radius to larly, the natural frequency and dampings )are all thethe wavelengths in gases and liquidssame as those in Section 1generally,A4+)1, therefore,M≈1Firstly, we discuss those published papers basedon Keller's equation If we follow Prosperetti, divi-Thus, Eq (5)reduces toding Eq (13)by(1+ ORo /c,), neglecting all termswith order of c and using x=iox and x+2Bi+ax=-ae 1+ oR cos(@+8)iax, p 7, Eq (13)becomesC2(+1),oyRP, RoHere a and Pot reduce toThis is just the Egs (29)-(30) in Prosperettil4.Theacoustic damping constant (Ro /2c,)shown inPBm1=2(4+A)+R20Eg ( 14)is different from ours, i.e., G3R,/2c,ofP, RoEq (8), by noticing that @*@ for non-resonantThe expression for natural frequency is well cited, in oscillations. This also explains why the prediction ofhich the contribution of compressibility(through acoustic damping constant can be improved by usingM)disappears. However, the contribution of compre-our approach for high frequencies and large bubblesssibility to the total damping constant b does not because for those cases the decreasing O is furtherdisappear completely since the B term still re- deviating from the increasing @. Indeed, the appmainsFurthermore,if c, -o0(corresponding to R/ natural frequency can be determined directly from theCr and Ro/c-0),i.e,, for incompressible cases, coefficients of the harmonic oscillator based on theEq ( 1)reduces to the Rayleigh-Plesset equation. And linearization of Keller's equation. The term (Eq (12)reduces toiaR/c,) can still remain on the right hand side ofEqs. 5)and(13), without using the relations of xx(+)BR2/i+a2 x=-aE cos(ar)iax and x=iax. This is exactly the approach employed for the derivations of damping constants andnatural frequency in Section 1. Furthermore, only foidentical to the cases where the Rayleigh-Plesset equaoscillationstion replaces the radiation pressure in Ref[2]. Natura- term disappeH中国煤化工se, the transientthat x=ioxlly, the acoustic damping disappears since liquid com-CNMHG. Neverthelesspressibility is not considered in the Rayleigh-Plesset to be strictly speaking, these two relations should notequationbe used for the purpose of determining the expressionsof damping constants and natural frequency. Other- Substituting it into Eq(18)leads towise, they would have changed the coefficients of thisinhomogeneous second-order equation that representthe damping constants and natural frequency of theRharmonic oscillator defined by this equation(Eq (5)or4(4+th)Eq (13)). For demonstrating examples, readers are re-ORoRferred to the following parts of this paper. ConsequeCntly, the resultant expressions defined by Eq (14) willdeviate from the true damping constants and naturalfrequencyOx=-ae(19Now, we turn our attention to those expressionsbased on radiation pressure, e. g, in Ref [2]If all terms with order of c are neglected in Eq (19),this will yield the acoustic damping constant as expre-Pad=, R,R I+IOR)-1ssed by Eq (27)in Ref[I]. For completeness, all terms(15)with order of c are kept in the following derivationsFor more original studies relating to the use of radiaFollowing Prosperetti2, if the first term intion pressure, readers are referred to the list of refere- Eq (19) is further treated by using the relations ofnces given in Ref [2]. Then one can obtain as in i=i@x and x-1@x, i.eRef[12],RPP(1+e“)-PmR,P十OROCRoSubstituting Eqs. (4)and(15)into Eq (16) yields+分)b/+hRRP, RoRaee(17)In order to demonstrate how the expressions in those Equation(19) will again become another inhomogeliteratures were reached, the relations of x=i@x and neous 2nd order equation with different coefficientsi=iax are to be employed, which will lead to varia-ble expressions for"natural frequency"and"dampingDRoIf both sides of E(17) divided by (+ioR./ i+4( t Hn2+LGP RoCi), it becomes1+/@e+R4({1+)Ii+a0x=-aeeia(18如/+Ie, R22,a2|x=-aeThe first term in Eq (18)then is rearranged using xThis is just the equation used for determining theexpressions of damping and natural frequency inaRoabV凵中国煤化工 how the use of1+(RORthe equation, Icsunthng lu various ululuient expressionsCtCfor the natural frequency and damping constants of theharmonic oscillator as appeared in the literaturesbe considered with the same properties as in Ref [2]ifas the contradiction between those two groups ofnot specified. The value of k, and De in Prospere-lished studies represented by Eqs. (19)and (20)ttil2, p27 have typographical errors and should bespectively. It also explains why there is a difference 0.59 J/msK and 0. 2912 ms(5.9x 104 erg/cm's-'Kbetween Prosperetti's 4) and oursand 0. 2912 cm/s)respectively. D. in ProspereL2, p 9 was defined using constant volume which is3. Comparisonscorrect. However, the value of D in ProspereThe expression of acoustic damping constant derived by us is different from those by Chapman and ttil2, p-2) was using constant pressure, which should belessee and Prosperetti 24). For natural frequency, Dg/r(r is specific heat ratio). In figures, "Pro-our expression is almost identical to theirs except forefers to Eq (20) based on radiation pre-Prosperettil2).Therefore, a comparison with Ref[2] ssure and"Present"refers to Eqs. (6)-(9)based onshould be essential. The slight difference of acoustic Keller's equation deri- ved by us. To focus on liquiddamping between Eq ( 19)and Eq (20), i.e,(aR/ compressibility, natural frequency, acoustic and total)2, is trivial and excluded from discussions. The damping constants are comparedassumption of spherical bubble (i. e, uniform pressureoutside bubble when Ro /2<0.1, where A is wave-length in liquids)limits the value of aRo/c (i.e.2TR, /A, )up to 0.628, referring to Ref[8]. Therefore,those bubbles within this range are to be considered inProsperetti"Bthe following discussionsa10°Present BProsperettB10°R。l(c)=10°s1ol(a)o=104sooo0oogoPresent BPresent阝sperettIFig 1 Comparison of acoustic and total damping constants(B.10and B respectively)between Prosperetti's2) and oursProsperett-B(present)Figures I and 2 show the comparisons of theR。/macoustic and the total damping constants and the natu-1(b)a=10°sral frequenc中国煤化工s,10sand10′ respIn this section, values predicted by Prosperettil2) that the acdHCNMH Gns emon stratebased on radiation pressure will be compared with are quite different from those in Ref[2]. For the totalours. Forced oscillations of air bubbles in water will damping constant and the natural frequency, our pre-dictions are much smaller than those by Prosperetti o B+B. o. Boin particular for large @Ro/c that is the regiono. B0where the ultrasound induced bubbles are currentlwidely used in oceanic and biomedical applicationsFor gas bubble oscillating in liquids, in order to limitwhere a, is the acoustical scattering cross sectione error of prediction within a reasonable range, ourTherefore, the prediction of acoustic damping will sigapproach should be employed for predicting total nificantly affect the values of o. and adamping constants and natural frequency if ORo/IS5. ConclusionFor forced gas bubble oscillations in liquids, amore rigorous expression of acoustic damping hasbeen developed based on Keller's equation. The difference between the published literatures and ours ismainly owing to the use of the relations x=iax andx=i@x. This correction to the published papers isessential when high frequencies and large bubbles (i.elarge aR/,)are involved such as those acousticbubbles in biomedical and oceanic applicationsAcknowledgementsRaThis work was supported by the UK EPSrCFig 2 Comparison of natural frequency between Prosperetti s(21 WIMrC(Grant No. RESCM 9219) the EpSRCand ours. The labeled values are @(s)WIMRC Ph. D. studentship ( Grant No. RESCM4. Application examplesAcoustic damping and natural frequency are funReferencesdamental issues for gas bubbles oscillating in liquidsThe findings presented in this paper are thus essential[1] CHAPMAN R. B. PLESSET M. S. Thermal effects inthe free oscillations of gas bubbles[J]. Journal of Basicfor many circumstances. Only a few of them are brie-Engineering, 1971, 94: 142-145fly mentioned here as examples. It should be emphasi- [2] PROSPERETTI A. Thermal effects and damping me-zed that the present findings are only valid for thechanisms in the forced radial oscillations of gas bubblesoscillations of spherical gas bubbles. For non-spheriin liquids[U]. Journal of the Acoustical Society ofcal bubble dynamics, more sophisticated model(e.gAmerica,1977,6l(1):17-27the direct numerical simulation developed byKELLER J B, MIKSIS M. Bubble oscillations of largeamplitude[J]. Journal of the Acoustical Society ofreferred to Chen et al.710) and Zhang and lm o loyeTryggvason and coworkers)should be employedAmerica,1980,68(2):628-633For some recent progress in this subject, reare [4] PROSPERETTI A Bubble phenomena in sound fieldsPart onea Typical cases in medical applications(eg-, con- [5] BRENNER M. P, HILGENFELDT Sand LOHSE Dagent in diagnostic ultrasound imaging, gene the-Single-bubble sonoluminescence[J]. Reviews ofModem Physics,2002,74(2):425-484[6] COUSSIOS CC, ROY R A Applications of acousticsand cavitation to non-invasive therapy and drug delicorresponding to Ro/c, ranging between 0.00628very[]. Annual Review of Fluid Mechanics, 2008, 40395-420and 0.628. In this region, the difference between ours [7 PROSPERETTI A, LEZZI A Bubble dynamics in aand published studies is noticeably large. Currentlycompressible liquid. Part 1. First-order theory[J]. Jour.oblems with even higher frequencies(up to 30 MHznal of Fluid Mechanics, 1986, 168: 457-478or above), e.g., ultrasonic wave propagation in dilute [8] ZHANG Y, LI S.C. Notes on radial oscillations of gasbubbly mixture, are of interest. For more examples,bubbles in liquids: Thermal effects[J]. Journal of thereaders are referred to our earlier paper as well asAcoustical Society of America, 2010, 128(5): EL306-EL3other recent publicati[9 TRYGGVASON G, SCARDOVELLI R and ZALESKIThe other example is the absorption and extisimulations nfas-liquid multi-nction cross sectio)of forced bubblephase中国煤化工rcdge Universityoscillations, which were determined by Medwin and [10] CHENCNMHG Zhi-gang et alDirect numerical simuiation of bubble-cluster s dyna766mic characteristics[J]. Journal of Hydrodynamics2008,206:689-695R(+x)x-(λ+1)RI1] ZHANG Yu-ning and LI Sheng-cai Direct numericalsimulation of collective bubble behavior[J]. Journal ofHydrodynamics, 2010, 22(5 Suppl. ) 827-831[12] SZABO T. L. Diagnostic ultrasound imaging: Insideout[M]. London: Elsevier Academic Press, 2004, 4551-(3+1)2R2=3R22(A3)[13] ANDO K, COLONIUS T and BRENNEN C E Im-lute bubbly liquids[]. Journal of the Acoustical 1+(-1R|Pa2(R,1)-p()=-1+(1-)Society of America, 2009, 126(3): EL69-EL74[14] MIRI A. K, MITRI F. G. Acoustic radiation force on aspherical contrast agent shell near a vessel porous wall-theory[J]. Ultrasound Medicine and Biology, 2011, 37P2σR_204A4+An0I-311[15] MEDWIN H, CLAY C S Fundamentals of acousti-cal oceanography [M]. San Diego, USA: AcademicPress,1998,302-304RP(+ec-")=n|1+(1-4)AppendixIn this appendix, the influences of one-parameterfamily equation on the expressions of damping and(-3x)++x-4(1+1)x-natural frequency for the radial oscillations of gasRobubbles in liquids are discussed. Prosperetti andLezzi proposed the following one-parameter familyequation(general Keller-Miksis equation) if written inlcc+|3(3x+1)(20)20x2+RoRoterms of pressure201-(4+1)=RR+=1-(3A+1)4(1+)x={1(3k)PRo1+(-x)212n(R)-2(3K(3X+1)x-4(+h)x-ed[Pa(R,)-P2()20120R厂R+4(+Ax}+where a is an arbitrary parameter which is ofsmaller order of 1/ Ma Ma is the bubble wall22Mach number)', p. 66 and Eq(4.3), Pe, (R, t) and P, (t)(1-A(-3x)PRo)Rogiven by Eqs. (2)and(3)respectively. If 2=0Eq (Al)reduces to Keller's equation (i.e, Eq (1))The procedure of derivations of damping4(u, +u)i-poeeconstants and natural frequency for linear gas bubbleoscillations in liquids based on Eq (Al)are exactly the From Eqs. (A2)-(A4), it is clearlythat termssame as those based on Keller's equation. Here, we involving n in Eq (Al)are all of the second orwill only keeps terms up to second order of e (or higher order of e(or x). Therefore, for linearx). Substituting R=R(l+x) into terms related oscillations (up to the Ist order of x), all thewith A in Eq (Al), we obtainequations falling in this one-parameter familyequation of bubble motion will giveexpressions for damping constantsnatural1-(+1)RR-1-(+1)~R(+x)=frequency as中国煤化工CCNMHG