I. FORMULATION. Here, p i is the pressure in the bubble, assumed spatially uniform,

The natual fequency of oscillation of gas bubbles in tubes H. N. Og uz and A. Pospeetti Depatment of Mechanical Engineeing, The Johns Hopkins Univesity, Baltimoe, Mayland 21218 Received 28 July 1997; accepted fo publication 27 Febuay 1998 A numeical study is pesented of the natual fequency of the volume oscillations of gas bubbles in a liquid contained in a finite-length tube, when the bubble is not small with espect to the tube diamete. Tubes igidly teminated at one end, o open at both ends, ae consideed. The open ends may be open to the atmosphee o in contact with a lage mass of liquid. The numeical esults ae compaed with a simple appoximation in which the bubble consists of a cylindical mass of gas filling up the coss section of the tube. It is found that this appoximation is vey good except when the bubble adius is much smalle than that of the tube. An altenative appoximate solution is developed fo this case. The viscous enegy dissipation in the tube is also estimated and found geneally small compaed with the themal damping of the bubble. This wok is motivated by the possibility of using gas bubbles as actuatos in fluid-handling micodevices. 1998 Acoustical Society of Ameica. S0001-49669802606-X PACS numbes: 43.35.Pt HEB INTRODUCTION An extensive liteatue exists on the small-amplitude volume oscillations of gas bubbles in unbounded liquids, nea igid plane boundaies and fee sufaces see, e.g., Stasbeg, 1953; Howkins, 1965; Blue, 1966; Plesset and Pospeetti, 1977; Apfel, 1981; Scott, 1981; Pospeetti et al., 1988; Og uz and Pospeetti, 1990; Pospeetti, 1991. The case of bubbles confined in channels and tubes, howeve, does not seem to have been consideed befoe except in a bief unpublished epot by Devin 1961. Of couse, when the adius of the tube is much lage than the bubble as would be the case, fo example, fo bubbles entained in odinay macoscopic flows the esults fo bubbles in unbounded liquids can be used to a good appoximation. In the situations of concen hee, howeve, the size of the bubble is not small and the effect of the poximity of the bounday vey significant. The situations consideed in this pape ae all axisymmetic and ae sketched in Fig. 1. The bubble is inside a liquid-filled, finite-length, igid-walled tube that may be open at both ends Fig. 1a, a, and b, o igidly teminated at one end and open at the othe Fig. 1c and d. The open ends of the tube may be in contact with the atmosphee Fig. 1b and d, o with a lage mass of the same liquid Fig. 1a, a, and c. Ou inteest in these poblems is motivated by the possibility to use gas bubbles as actuatos in the small fluidhandling systems that advances in silicon manufactuing technology ae endeing possible see, e.g., Fujita and Gabiel, 1991; in et al., 1991; Gavesen et al., 1993. These include bioassay chips, integated mico-dosing systems, miniatuized chemical analysis systems, and othes. The advantage of bubbles in this setting would be the possibility to powe them emotely by ultasonic beams with no need fo diect contact between the actuato and the powe supply. A paticulaly intiguing possibility in this egad may be offeed by the ability of ultasound to popagate though living tissue. While the scale that we envisage is of the ode of one millimete o less and the flow velocities elatively small, so that viscous effects would not be negligible, it seems natual fo a fist analysis of this poblem to stat fom a consideation of the inviscid case, teating viscous effects in an appoximate way see Sec. IV. The attending simplification enables us to focus with geate claity on the inetial aspects of the bubble fluid inteaction, which ae one of the dominant aspects of the system. Second, it will be easie to establish a connection with the available esults fo the unbounded case. Thid, one can envisage situations in which viscosity is indeed negligible, such as an oscillation fequency so lage that the viscous bounday laye is much thinne than the tube. Since, in ode to maximize the effectiveness of the actuato, it is desiable to opeate nea esonance conditions, the natual fequency of the bubble is the most significant quantity to be detemined. This is the objective of this pape. In the futue we shall conside foced oscillations, damping mechanisms, and nonlinea effects. I. FORMUATION As shown by Stasbeg 1953; see also Og uz and Pospeetti, 1990, it is possible to obtain a elation fo the natual fequency diectly by using the analogy with the capacitance poblem of electostatics. To this end we stat fom the condition expessing the balance of nomal foces at the bubble suface: p i p C. Hee, p i is the pessue in the bubble, assumed spatially unifom, p the pessue in the liquid at the bubble suface, the suface tension coefficient, and C the local cuvatue of the inteface. Upon using the lineaized Benoulli integal to expess p in tems of the velocity potential and the 1 3301 J. Acoust. Soc. Am. 103 (6), June 1998 0001-4966/98/103(6)/3301/8/$10.00 1998 Acoustical Society of Ameica 3301

Fo linea oscillations at a single fequency, any vaiable is popotional to any othe so that we may wite dp i dt dp i dv dv dt, 5 whee V is the instantaneous bubble volume and dp i /dv a possibly complex constant. Futhemoe, dv dt u n S dss n, 6 FIG. 1. The vaious configuations of a bubble in a tube consideed in this pape: a open tube with infinite thickness immesed in an unbounded liquid; a open tube with negligible thickness immesed in an unbounded liquid; b patially filled tube with liquid sufaces exposed to the atmosphee; c igidly teminated tube in the bottom of a lage containe; d patially filled tube closed at one end. static pessue p, the pevious elation becomes p i t p C, whee is the liquid density. The bubble intenal pessue p i can be assumed to emain spatially unifom at all times. Since p is a constant, this equation implies then that will emain essentially unifom ove the bubble suface povided it is unifom e.g., equal to zeo at the initial time, and povided the suface cuvatue is eithe unifom o small. The fome possibility pevails in the case of small bubbles, which tend to emain spheical, while the latte one is encounteed fo lage bubbles fo which the suface tension contibution is negligible. Upon balancing vaiations in cuvatue and vaiations in intenal pessue, it is found that the appopiate scale to judge whethe a bubble is to be consideed small o lage is of the ode of /p which, fo the case of wate at atmospheic pessue, is a few micometes. With the assumption of a unifom intenal pessue, aveaging Eq. 2 ove the bubble suface, we have p i t p C, whee... denotes the suface aveage. In the linea appoximation, to which we confine ouselves, the suface aveage of any fist-ode quantity can consistently be calculated on the unpetubed equilibium suface, athe than on the moving one. As a consequence, time diffeentiation and suface aveaging commute and theefoe, upon diffeentiating once moe with espect to time, we find d2 dt 2 dp i dt d dt C. 2 3 4 whee n is the outwad diected unit nomal to the bubble suface S. With the neglect of gavity, the equilibium configuation of the bubble is necessaily spheical, although the instantaneous shape duing volume oscillations is not necessaily so. Howeve, again in the linea appoximation, it is easy to show that d dt 0 CC n, 7 whee C 0 2/a is the cuvatue of a spheical bubble of adius a. Upon substituting these esults into Eq. 4, and futhe witing i fo d/dt, we find 2 1 S dp i 2 2 1 dv a n. Fo a spheical bubble in an infinite liquid (a 2 /) (da/dt) whee is the distance fom the bubble cente and da/dt the adial velocity and this expession educes to 2 0 1 a S dp i dv 2 a 2, whee 0 denotes the bubble angula fequency in this case. Upon taking the atio with Eq. 8 and intoducing the fequencies f /2, f 0 0 /2, we thus have f f 0 2 a n, 10 which expesses in a compact fom the change in the natual fequency of the bubble due to the pesence of boundaies. The validity of this esult pesupposes of couse that dp i /dv in Eq. 8 has the same value as fo a bubble in an infinite fluid. This assumption may be justified as follows. The ate of change of the intenal pessue with volume is detemined essentially by the themal pocesses in the bubble. It is well known that, to an excellent appoximation, these can be evaluated assuming the bubble suface tempeatue to emain undistubed see, e.g., Kamath et al., 1993, which effectively decouples the themal poblem fom the envionment suounding the bubble. If the length of the tube wee infinite, volume changes of the bubble would only be possible in a compessible fluid. Howeve, if the length of the tube is much smalle than the wavelength of sound in the liquid, we may use the incompessible appoximation so that the velocity potential satisfies aplace s equation 2 0. Fo simplicity we only conside axisymmetic situations. The bounday condition on the suounding solid boundaies is of couse n 0. If the liquid mass in the tube is bounded by a fee suface in con- 8 9 3302 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3302

tact with the atmosphee, as in Fig. 1b and d, 0 is the appopiate bounday condition thee. Futhemoe, unless the bubble is vey close to this suface, thee will be little eo in assuming it to be plane. If, on the othe hand, the tube is pat of an extended infinite mass of liquid, as in Fig. 1a, a, and c, is equied to vanish at infinity. As noted befoe, povided the bubble is eithe elatively small o elatively lage, one may assume that emains unifom ove the bubble suface. Since the poblem fo is linea and, aside fom the bounday condition on the bubble suface, it is homogeneous, it follows that /n will also be popotional to the aveage value of evaluated on the bubble suface. It is theefoe sufficient to calculate the suface aveages appeaing in Eq. 10 by solving aplace s equation subject to the bounday condition 1 on the undistubed spheical bubble. The solution of the potential poblem fomulated befoe can eadily be obtained by using the bounday integal method see, e.g., Pozikidis, 1992. Ou vesion of this method is aleady well documented in the liteatue, to which the eade is efeed see, e.g., Og uz and Pospeetti, 1990. The deivation of the numeical esults shown below, the accuacy of which was veified by the standad convegence and gid-independence tests, did not equie a vey high degee of discetization. The line epesenting the bubble suface in the meidian plane was appoximated by cubic splines with 10 nodal points. Up to 70 points wee used on the tube s wall, depending on its length. In situations whee the solid bounday extends to infinity Fig. 1a and c, the integation ove its suface must be stopped at some lage distance fom the axis of symmety. Ten tube adii poved sufficient fo convegence. II. APPROXIMATIONS The numeical calculation of the natual fequency accoding to the method descibed befoe is a matte of some complexity and it is useful theefoe to obtain appoximate expessions. We conside sepaately the case of lage and small bubbles. A. age bubbles When the adius a of the bubble is not small compaed with the adius R of the tube, an obvious appoximation to the situation envisaged hee is that of a one-dimensional slice of gas filling the entie coss section of the tube and with a thickness h adjusted to give the same volume as the eal bubble: Ah 4 3a 3, 11 whee A is the tube s coss sectional aea. Since the consideations that follow ae applicable to tubes of geneal coss section, we do not specialize the fomulae to cicula tubes in this subsection. et the bubble cente be at a distance 1 fom one end of the tube and 2 1 fom the othe. Fo geate accuacy, these geometical paametes can be adjusted to eflect moe closely the physical situation. In the fist place, in ode to peseve the total volume, an amount l 1 2 h 2 a3 12 3 A must be subtacted fom both 1 and 2. If the tube s ends ae in contact with the atmosphee Fig. 1b and d, this is the only adjustment to the lengths of the liquid columns. If an end is immesed in an unbounded liquid Fig. 1a, a, and c, howeve, thee is an added mass effect that can be accounted fo by augmenting the geometical length by an amount. Fo the situation of Fig. 1a one can simply estimate this end coection by noting that, fom the point of view of the fluid outside the tube, the effect of the liquid enteing and exiting the tube opening is simila to the pulsations of a half-bubble with diamete equal to the hydaulic diamete D h of the tube. The hydaulic diamete is fou times the atio of the coss-sectional aea A to the peimete P of the tube. Since such a bubble in an unbounded liquid would have an added, o vitual, mass 4(D h /2) 3, the added mass fo the half-bubble is 2(D h /2) 3, which can be accounted fo by extending the tube by an amount chosen so that A contains an equal mass of liquid. The esult is 16 A2 P 3, 13 and equals 2R fo a cicula tube. This esult can also be deived in an altenative, moe igoous way Og uz and Zeng, 1995, 1997. The same pocedue applied to the thinwalled tube of Fig. 1a is inaccuate, howeve, as shown by evine and Schwinge 1948. In this case, fo a cicula tube, one finds 1.22R. On the basis of these aguments we define equivalent lengths of the liquid columns on the two sides of the bubble by i e i l, i1,2. 14 If the system is egaded as an oscillato, its equivalent mass is see Eq. 33 in Sec. IV: M eq A 1 e 1 1 1 2 e while the sping constant is, 15 KA dp 2 i dv. The natual fequency of the system is theefoe A dp i dv. a 2 K M eq 1 1 e 1 2 e 16 17 With f a a /2, upon taking the atio of this expession to the natual fequency of a bubble of equal adius in an unbounded liquid neglecting suface tension effects, wefind f a f 0 2 4a A 1 e 1 1 e. 18 2 If one of the ends is igidly teminated, we assume that the liquid on that side does not patake of the motion in this one-dimensional appoximation. This limit is contained in 3303 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3303

the pevious fomulae by taking the coesponding effective length to be infinite. It will be seen in the next section that this is a good appoximation. B. Small bubbles The model just descibed is evidently a poo appoximation when the bubble is small compaed with the tube adius. We now tun to this case consideing explicitly a tube open at the two ends. The adjustment 14 to the length of the liquid columns pemits one to adapt the esults to the othe cases depicted in Fig. 1. It is paticulaly convenient to use the following special epesentation of the velocity potential: n0 a n1 A n n1 P n cos B n I n1 0 n sin nz C 0 z D 0 z n1 sinh n z/r D n sinh J n /R 0 n C n sinh n z/r sinh n /R R. 19 Hee the P n s ae egende polynomials, and and ae pola coodinates centeed at the bubble cente; and z ae cylindical coodinates with the two fee sufaces of the liquid at z0 and z. The modified and odinay Bessel functions of ode 0 ae denoted by I 0 and J 0, the n s ae the zeos of J 1, and the coefficients A n, B n, C n, D n ae to be detemined fom the bounday conditions. This paticula fom fo is constucted in such a way that the fist summation descibes the flow nea the bubble, the second one in the tube away fom the bubble, the next two tems the bulk tanslation of the liquid away fom the bubble, and the last summation the end effects. The poof that Eq. 19 gives an accuate epesentation of the potential follows fom the fact that, as will be seen shotly, all the coefficients ae uniquely detemined and all the bounday conditions ae satisfied. On the tube wall the velocity must vanish, which equies n0 n1 A n a n1 n1 P n cos B n n I 1 nr sin nz 0. 20 At the lowe and uppe fee sufaces of the tube, z0 and z, the condition of vanishing pessue petubation simply equies 0, i.e., a n1 A n n0 n1 P n cos D n J n1 0 n D R 0 0, n0 21 a n1 A n n1 P n cos C n J n1 0 n C R 0 0. 22 A consideation of these bounday conditions funishes a ationale fo the epesentation 19 of the potential. Indeed, fom these thee homogeneous bounday conditions one can conceptually think of expessing B n, C n, and D n in tems of the A n, which ae in tun detemined by the pessue condition on the bubble suface. If the bubble adius is small, the magnitude of the highe-ode tems of the egende polynomial expansion is apidly deceasing and theefoe we tuncate this infinite sum to just the fist tem which, in the cylindical coodinate system used to expess the othe tems, is a A 0 P a 0A 0 2 zd 2 1/2, 23 whee d is the position of the bubble cente. Using wellknown othogonality popeties we then have fom Eqs. 20 to 22: B n A 0 2 n D 0 2 R A 0 R 2 0 1 I 1 nr/ 0 D n 2 A 0 R 2 J 2 0 n 0 C 0 2 R A 0 R 2 0 C n 2 A 0 R 2 J 2 0 n 0 2 d 2 d, R R sin nz R 2 zd 2 3/2 dz, 2 d J 0 2 n Rd, 2 d 2 d, R 2 d J 0 2 n Rd. 24 25 26 27 28 In pinciple, A 0 should now be detemined by imposing a condition on at the bubble suface. As is evident fom the pevious elations, howeve, all the coefficients ae popotional to A 0 and it will be ecalled fom Eq. 10 that we ae only inteested in the atio (1/)/n that is obviously independent of A 0. It is theefoe unnecessay to impose the last bounday condition explicitly and A 0 can simply be taken as 1. Of couse, it is not necessay to tuncate the spheical hamonic expansion in Eq. 19 at the fist tem. In pinciple, one can etain any numbe of tems in the sums. Upon taking scala poducts, one is then educed to a linea system fo the coefficients. As discussed in the next section, we have found that the tuncation used hee is sufficient fo the pesent puposes of appoximation. Solution 19, howeve, is in pinciple exact and epesents a valid altenative to the bounday integal calculation, at least fo situations of the type shown in Fig. 1b and d. It is inteesting to note that this pocedue can be extended to deal with bubbles off-axis, and tubes of noncicula coss section, moe simply than the bounday integal method. Devin 1961 calculated the natual fequency fo the situation of Fig. 1a in tems of the potential and kinetic enegies of the system. The fome is simply expessed in tems of the elation between the pessue and volume of the bubble, which he assumed to be adiabatic. To estimate the kinetic enegy, he used the solution fo a point souce in an 3304 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3304

FIG. 2. The natual fequency of a bubble of adius a in a tube of adius R and length as a function of the axial distance of the bubble cente fom the tube bottom fo /R10, a/r0.5. The dotted lines show the esult given by the appoximate fomula 18 and the symbols show the bounday integal esults; case of Fig. 1a; case of Fig. 1a; case of Fig. 1b. infinite tube up to a distance of 1.108 R fom the bubble cente at which point the potential along the axis vanishes combined with that of solid-body motion of the liquid in the emainde of the tube. His agument fo chosing the paticula value 1.108 is that, in this way, the decease in the tube potential fom the suface of the bubble... is exactly equal to the decease in the fee field potential fom the suface of an identical bubble to a point at infinity. His final esult is f D f 0 2 1 2a 1 1 2 1.108. 29 R R Hee the bubble is assumed to be located at the midpoint of the tube and 1.22R. It is evident fom the manne of its deivation that the esult is only applicable povided the bubble adius is much smalle than that of the tube, and that the tem in backets is geate than 1. FIG. 3. The natual fequency of a bubble of adius a in a tube of adius R and length as a function of the axial distance of the bubble cente fom the tube bottom fo /R10, a/r0.5. The dotted lines show the esult given by the appoximate fomula 18 and the symbols show the bounday integal esults; case of Fig. 1c; case of Fig. 1d. Any one of the situations shown in Fig. 1 is chaacteized by fou dimensional lengths: the bubble adius a, the tube adius R, the tube length, and the distance of the bubble cente fom the lowe end of the tube as sketched in Fig. 1, z. In the case of Fig. 1a, the tube thickness would also appea, but we take it as negligibly small in the following. One can thus fom thee dimensionless atios that fully chaacteize each case. The pesentation of a sufficient numbe of esults to cove the entie paamete space is impactical. Thus we limit ouselves to a few examples which also seve to illustate the excellent pefomance of the appoximations descibed in the pevious section. It may be noted that, by symmety, a bubble placed at the tube s midpoint in the situations of Fig. 1a and b is equivalent to a halfbubble esting on the igid bottom of Fig. 1c and d fo a tube of half the length. Figues 2 to 5 show a few epesentative esults. In all these figues the open symbols ae the numeically exact esults obtained with the bounday integal method, the dotted lines ae the lage-bubble appoximation of Sec. II A, and the solid lines the small-bubble appoximations of Sec. II B. Figues 2 and 3 give the atio f / f 0 as a function of the position of the bubble cente along the tube fo the five situations depicted in Fig. 1. Hee the tube s adius is twice that of the bubble. Figues 4 and 5 ae gaphs of f / f 0 as a function of a/r fo /R10, again fo fou of the situations of Fig. 1. Hee the bubble cente is at the midpoint of the tube axis. The small-bubble appoximation of Sec. II B solid lines has been evaluated etaining only B 1, C 0, C 1, D 0, and D 1. The fist obvious featue shown by these figues is that the effect of the tube can be lage. Fo example, fom Fig. 3, we see that a bubble in a tube closed at one end Fig. 1c and d has a 50% eduction in the natual fequency when the tube adius is twice the bubble adius and the depth of III. RESUTS FIG. 4. The natual fequency of a bubble centeed at the midpoint of the axis of a tube of adius R and length as a function of the nomalized bubble adius a/r fo /R10. The dotted lines show the esult given by the appoximate fomula 18, the solid lines those given by the smallbubble appoximation, and the symbols show the bounday integal esults; case of Fig. 1a; case of Fig. 1c. 3305 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3305

FIG. 5. The natual fequency of a bubble centeed at the midpoint of the axis of a tube of adius R and length as a function of the nomalized bubble adius a/r fo /R10. The dotted lines show the esult given by the appoximate fomula 18, the solid lines those given by the smallbubble appoximation, and the symbols show the bounday integal esults; case of Fig. 1b; case of Fig. 1d. submegence below the tube mouth is of the ode of twice the tube adius. Anothe obvious emak suggested by the numeical esults is the supising degee to which the appoximations of the pevious section ae able to epoduce the exact esults. In paticula, the adjustments to the liquid column length descibed in Sec. II A ae seen to wok vey well. The lagebubble appoximation beaks down aound a/r0.2, while the small bubble model woks elatively well at least up to a/r0.5. It is theefoe found that thee is a domain in which both appoximations ae easonably accuate. We have examined the effect of etaining moe tems in the summations of epesentation 19 of the velocity potential. The effect of adding two tems to each of the sums is FIG. 6. Natual fequency f in Hz solid line and total damping paamete b in s 1 Eq. 40, dashed line as a function of bubble adius in a tube of adius 1 mm and length 10 mm fo an ai bubble in wate. The bubble is positioned at the bottom of the tube. The dotted line is the themal contibution to the damping. This figue efes to case d of Fig. 1 but, with the adjustment to the tube length descibed in Sec. II A, the esults can be adapted to the case of Fig. 1c as well. FIG. 7. Natual fequency f in Hz solid line and total damping paamete b in s 1 Eq. 40, dashed line as a function of bubble adius in a tube of adius 0.1 mm and length 1 mm fo an ai bubble in wate. The bubble is positioned at the bottom of the tube. The dotted line is the themal contibution to the damping. This figue efes to case d of Fig. 1 but, with the adjustment to the tube length descibed in Sec. II A, the esults can be adapted to the case of Fig. 1c as well. small, and any moe tems give diffeences that ae indistinguishable in a gaph such as those of Figs. 4 to 7. In addition to the theoetical development leading to Eq. 29 quoted befoe, Devin s epot contains a few data taken in an expeimental setup simila to that of ou Fig. 1a. The bubbles wee geneated by a needle placed at the midpoint of the axis of vetical bass cylindes with a diamete of 30 mm, a wall thickness of 3.2 mm, and a length of 120 o 240 mm. In ode to investigate the effect of static pessue, two depths of submegence of the tube below the suface of a lage wate tank wee used, 5 and 15 ft. A hydophone placed at a distance of 0.1 m ecoded the sound emitted by the bubbles pinching off the needle and a few gaphs of the acoustic powe spectal density ae shown in the epot. By digitizing these figues, we have ead off the position of the maximum of these specta which, in view of the small damping, give a good estimate of the natual fequency. Values of the bubble adius ae not given but, in his gaphs, Devin shows the natual fequency of the bubble geneated by the same method in an unbounded liquid fom which the adius can be deduced accoding to the esults of Pospeetti 1991. Table I shows all of Devin s data togethe with the esult given by the fist fou tems of the seies solution of Sec. II B and two estimates obtained fom Devin s epot. The fist one is found fom his appoximate fomula 29, while the second one is the theoetical value ead fom his gaph. These two numbes should agee but, fo the fist case, we find a 2% discepancy the oigin of which is not clea. This data point also exhibits a geate diffeence with the theoy, about 5%. Fo the second and thid data points ageement with theoy is within about 3% and 1%, espectively, and seems to be slightly bette fo the pesent theoy than fo Devin s although, on the basis of the infomation povided, it is not possible to estimate accuately the eo in his data. 3306 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3306

TABE I. Compaison between Devin s data, the pesent seies solution of Sec. II B, and Devin s theoy. The tube was bass with a adius of 15 mm. a mm cm Exp. Pesent theoy Devin, Eq. 29 Devin, gaph 1.51 24 0.67 0.636 0.638 0.653 1.52 12 0.74 0.762 0.777 0.779 1.50 12 0.76 0.764 0.780 0.777 IV. DAMPING RATE In eality, the oscillations executed by the bubbles shown in Fig. 1 ae, of couse, damped. In the pevious developments we have disegaded dissipative effects which, as is well known, affect the natual fequency only to second ode. The decay ate is howeve a fist-ode effect, that we now conside. A bubble oscillating in an unbounded liquid loses enegy by themal conduction acoss the gas liquid inteface, acoustic adiation, and the action of viscous stesses at the inteface. In wate, acoustic losses only dominate fo bubble adii lage than seveal millimetes, while viscous losses ae significant only fo bubbles smalle than about 10 m. The dominant enegy loss fo intemediate values of the adius is of themal oigin, and this can be assumed to happen also in the cases of Fig. 1. Indeed, in the undelying pocess, the significant aspects ae the gas volume expansion and contaction and the fact that the bubble suface emains essentially at the undistubed liquid tempeatue duing the oscillations due to the lage themal capacity of the liquid. Both cicumstances occu also in the situations of pesent concen, as aleady noted in connection with Eq. 10. In the case of a bubble in a tube, howeve, a new enegy loss mechanism is pesent, namely viscous dissipation due to liquid flow along the suface of the tube suounding the bubble. An estimate of the ate of damping due to this effect can be found on the basis of the simple one-dimensional model of Sec. II A as follows. Conside the bubble as occupying a slice of the tube extending between z 1 (t) and z 2 (t). If m 1 and m 2 ae the effective masses of the two liquid columns, and 1, 2 the damping ates due to viscous dissipation, the equations of motion of the two intefaces ae m 1 z 12 1 ż 1 Kz 1 z 2 0, m 2 z 22 2 ż 2 Kz 1 z 2 0, 30 31 with the sping constant K given by Eq. 16. The equation fo the complex fequencies of oscillation of this system is eadily witten down and is m 1 2 K2i 1 K K m 2 2 32 K2i 2 0. Upon setting ib t, up to tems of the fist ode in i, one eadily finds 2 K 1 m 1 1 m 2, which is the same as Eq. 17, and 33 b t 1 m m 1 m 2 2 m 1 1 m 2 1 m 2. 34 In the spiit of Sec. II A, the masses appeaing hee ae given by m j j e A, j1,2. To estimate the damping paametes i we poceed appoximately as follows disegading the index i fo the moment. The enegy dissipated duing one cycle by each oscillato is E d 2 0 2/ ż 2 dt, 35 which funishes an estimate of if the othe two quantities can be evaluated. Since, to leading ode, z oscillates sinusoidally with a fequency and velocity amplitude V, the integal has the value (/)V 2. The enegy loss can be estimated by integating the dissipation function ove the volume occupied by the fluid. With the appoximation of peiodic, paallel flow, we have E d 0 2/ dt A da u 2, 36 whee is the liquid viscosity, the integal is ove the coss section of the tube, and u is the axial velocity. Since, in fully developed paallel flow, the poblem fo u is linea, we have uv and theefoe 2 2/ dt 0 A da V u 2. 37 The velocity field equied hee is eadily calculated fom the Navie Stokes equations see, e.g., eal, 1992, but the answe is in tems of Bessel functions with complex agument and it does not appea possible to obtain closed-fom expessions fo this integal at abitay fequency. Appoximations fo /R and /R ae, howeve, eadily found. In the fist case we have 4m R 2, 38 which can also be obtained fom the Poiseuille flow solution, while, in the latte one, m 2R 2 39 which, in the spiit of a bounday laye appoximation, can also be obtained fom the known fom of the velocity field ove an infinite oscillating flat plate. This latte esult is theefoe valid fo tubes of abitay coss section. 3307 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3307

The peceding aguments povide an appoximation to the viscous damping in the tube b t. As aleady mentioned, the gas liquid heat exchange gives ise to anothe dissipation mechanism. If the coesponding damping ate is much less than, the two damping mechanisms ae additive and the total damping of the oscillations is theefoe bb t b b, 40 whee b b is the bubble damping constant that has been exhaustively studied in the liteatue see, e.g., Pospeetti et al., 1988; Pospeetti, 1991. To illustate the magnitude of the effect, we conside two paticula cases in Figs. 6 and 7. These figues efe to the situation of Fig. 1d fo an ai bubble in wate at 1 atm but, with the adjustment to the tube length descibed in Sec. II A, the esults ae also epesentative of case 1c. In Fig. 6 the tube has a length of 10 mm and a adius of 1 mm, while in Fig. 7 the coesponding values ae 1 and 0.1 mm, espectively. The bubble is positioned at the bottom of the tube in the sense of the appoximate conceptual model of Sec. II A; stictly speaking, theefoe, the adius shown is an equivalent spheical adius. The hoizontal axis shows the bubble adius, the solid line the natual fequency, the dotted line the themal damping, and the dashed line the total damping. The viscous contibution is just the diffeence between the two lines, and is theefoe seen to be small in both cases. Just as in the case of a bubble in an unbounded fluid, we thus see that themal damping is the dominant mechanism of enegy dissipation. ACKNOWEDGMENTS The authos ae gateful to D. Muay Stasbeg fo calling thei attention to Devin s wok on this poblem. Thanks ae also due to X. M. Chen fo he help with the calculations of Sec. IV. This study has been suppoted by the Ai Foce Office of Scientific Reseach unde Gant No. F49620-96-1-0386. Apfel, R. E. 1981. Acoustic Cavitation, in Methods of Expeimental Physics Vol. 19 Ultasonics, edited by P. D. Edmonds Academic, New Yok, pp. 355 411. Blue, J. E. 1966. Resonance of a bubble on an infinite igid bounday, J. Acoust. Soc. Am. 41, 369 372. Devin, C. 1961. Resonant fequencies of pulsating ai bubbles geneated in shot, open-ended pipes, Technical Repot 1522, David Taylo Model Basin, Hydomechanics aboatoy. Fujita, H., and Gabiel, K. J. 1991. New oppotunities fo mico actuatos, in Tansduces 91 IEEE, New Yok, pp. 14 20. Gavesen, P., Banebjeg, J., and Jensen, O. S. 1993. Micofluidics A eview, J. Micomech. Micoeng. 3, 168 182. Howkins, S. D. 1965. Measuements of the esonant fequency of a bubble nea a igid bounday, J. Acoust. Soc. Am. 37, 504 508. Kamath, V., Pospeetti, A., and Egolfopoulos, F. 1993. A theoetical study of sonoluminescence, J. Acoust. Soc. Am. 93, 248 260. in,., Pisano, A. P., and ee, A. P. 1991. Micobubble poweed actuato, in Tansduces 91 IEEE, New Yok, pp. 1041 1044. eal,. Gay 1992. amina Flow and Convective Tanspot Pocesses, Scaling Pinciples and Asymptotic Analysis Buttewoth-Heinemann, Boston. evine, H., and Schwinge, J. 1948. On the adiation of sound fom an unflanged cicula pipe, Phys. Rev. 73, 383 405. Og uz, H. N., and Zeng, J. 1995. Bounday integal simulations of bubble gowth and detachment in a tube, in Bounday Elements XVII Computational Mechanics Publications, Madison, WI, pp. 645 652. Og uz, H. N., and Zeng, J. 1997. Axisymmetic and thee-dimensional bounday integal simulations of bubble gowth fom an undewate oifice, Engineeing Analysis with Bounday Elements 19, 313 330. Og uz, H. N., and Pospeetti, A. 1990. Bubble entainment by the impact of dops on liquid sufaces, J. Fluid Mech. 219, 143 179. Plesset, M. S., and Pospeetti, A. 1977. Bubble dynamics and cavitation, Annu. Rev. Fluid Mech. 9, 145 185. Pozikidis, C. 1992. Bounday Integal and Singulaity Methods fo ineaized Viscous Flow Cambidge U.P., Cambidge. Pospeetti, A., Cum,. A., and Commande, K. W. 1988. Nonlinea bubble dynamics, J. Acoust. Soc. Am. 83, 502 514. Pospeetti, A. 1991. The themal behaviou of oscillating gas bubbles, J. Fluid Mech. 222, 587 616. Scott, J. F. 1981. Singula petubation theoy applied to the collective oscillation of gas bubbles in a liquid, J. Fluid Mech. 113, 487 511. Stasbeg, M. 1953. The pulsation fequency of nonspheical gas bubbles in liquids, J. Acoust. Soc. Am. 25, 536 537. 3308 J. Acoust. Soc. Am., Vol. 103, No. 6, June 1998 H. N. Og uz and A. Pospeetti: Oscillation of gas bubbles in tubes 3308