Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers

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1 Euopean Jounal of Mechancs B/Fluds Dag foce actng on a bubble n a cloud of compessble sphecal bubbles at lage Reynolds numbes S.L. Gavlyuk a,b,,v.m.teshukov c a Laboatoe de Modélsaton en Mécanque et Themodynamque, Faculté des Scences et Technques de Sant-Jéôme, Case 322, Avenue Escadlle Nomande-Nemen, Maselle cedex 20, Fance b CNRS UMR 6595, IUSTI, 5, ue E. Fem, Maselle cedex 13, Fance c Laventyev Insttute of Hydodynamcs, Laventyev pospect 15, Novosbsk , Russa Receved 2 Septembe 2004; eceved n evsed fom 13 Decembe 2004; accepted 16 Decembe 2004 Avalable onlne 29 Januay 2005 Abstact We have deved the expessons fo the vscous foces actng on a bubble n a cloud of bubbles by usng the appoach by Levch. To obtan the dsspaton functon, an appoxmate expesson fo the velocty potental calculated pevously by the authosuptoodeβ 3 has been used. Hee β = b/d s a small dmensonless paamete, b s the mean bubble adus and d s the mean dstance between bubbles Elseve SAS. All ghts eseved. Keywods: Bubbly fluds; Lagange s fomalsm; Levch s appoach 1. Intoducton Levch [1] calculated the dag foce actng on a bubble movng though a lqud at lage Reynolds and small Webe numbes the last condton guaantees that the bubble shape s sphecal defnng the total vscous dsspaton though the velocty potental of otatonal flow see also Mooe [2] and Batchelo [3]. Wth the use of ths appoach, he obtaned the dag foce F n a tanslatonal moton of a bubble of unchanged adus F = 12πµbU. It dffes fom the dag foce gven by the theoy of vscous potental motons. In the last method, the vscous flow s supposed to be potental, and the vscosty s taken nto account only n the dynamc condton expessng the contnuty of the nomal stess at the gas lqud nteface Mooe [4], see also Joseph and Wang [5]. The coespondng dag foce s then F = 8πµbU. Hee µ s the dynamc vscosty of the flud, U s the bubble velocty, and b s the bubble adus. Mooe [2] ustfed futhe the appoach by Levch by developng of the model of the bounday laye wapped aound bubble. Moeove, ecently Magnaudet and Legende [6] calculated the dag foce on a sphecal compessble bubble by usng the full Nave Stokes equatons. In 1 * Coespondng autho. E-mal addesses: segey.gavlyuk@unv.u-3ms.f S.L. Gavlyuk, teshukov@hydo.nsc.u V.M. Teshukov /$ see font matte 2004 Elseve SAS. All ghts eseved. do: /.euomechflu

2 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds patcula, they found that the Levch fomula 1 s vald not only n the hgh-reynolds lmt, but also fo modeate and even small Reynolds numbes, povded the bubble pulsaton velocty s hgh wth espect to the flow velocty. Both appoaches, the method of dsspaton functon and the method of vscous potental motons, gve the same esult f we calculate the vscous foces on an oscllatng bubble wthout tanslatonal moton the Raylegh Plesset equaton: b d2 b dt db dt 2 = 1 ρ p g p 4µ db ρb dt. 2 Hee ρ s the flud densty, p g b s the pessue n a bubble t s a gven functon of b, and p s the flud pessue at nfnty. The Levch appoach allows to apply the Lagange fomalsm fo constuctng the govenng equatons of moton wth nonpotental foces deved fom the Raylegh dsspaton functon method see, fo example, Goldsten, Poole and Safko [7]. Moe exactly, f we consde a bubble of adus bt oscllatng wth the velocty st = db/dt whose cente poston s t and the tanslatonal velocty vt = dt/dt, the equatons of moton ae Vonov and Golovn [8]: d L dt v d L dt s L Φ v, = 1 2 L b = 1 Φ 2 s. Eqs. 3 mply the enegy equaton n the fom d L dt v v + L s s L = Φ. Hee L s the Lagangan of the system, Φ s the Raylegh dsspaton functon. In the case of a massless bubble, the Lagangan s L = πρb3 3 v 2 + 2πρb 3 s 2 ετ. Hee ετ = p g z dz + p τ τ s the ntenal enegy of the gas lqud system, τ = 4πb 3 /3 s the bubble volume. The dsspaton functon Φ can be explctly calculated fo the velocty potental of the flow aound a sngle bubble: ϕ = b2 s x t + b3 2 v 1. x t It s gven by Φ = 2µ R 3 \B D : D dω = µ Γ n ϕ 2 dγ = 4πµb 4s v 2. 4 Hee B s a egon occuped by the bubble, Γ = B s the bubble bounday, D = u/x + u/x T /2 s the ate of defomaton tenso, u = ϕ, n s the nomal vecto to Γ dected to the flud. System 3 can be ewtten n the followng equvalent fom: d 2 dt 3 πρb3 v = 12πµbv, b d2 b dt db dt 4 v 2 = 1 ρ 4µ db pg b p ρb dt. The geneal case of N nteactng gd bubbles was consdeed fst by Golovn [9]. Fo a dlute bubbly mxtue, he deved the equatons of moton as well as the fcton foces by usng Levch s appoach up to ode β 3.Heeβ = b/d s a small paamete whch s the ato of the mean bubble adus b to the mean dstance d between bubbles. Obvously, 4πβ 3 /3sthe volume facton of bubbles n the lmt N. Kok [10,11] and efeences theen, studed analytcally and expementally the dynamcs of a pa of gd bubbles of equal ad. In patcula, he deved the equatons of moton up to any ode of β. Sangan and Ddvana [12] and Smeeka [13] examned the dynamcs of N bubbles numecally. In patcula, they showed that 3 5

3 470 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds the andom state of massless gd bubbles was unstable, and that the bubbles fomed aggegates n plane tansvese to gavty. Ths concluson was confmed analytcally by Van Wngaaden [14]. The themodynamcs of clusteng of a system of gd bubbles has been studed by Yukovetsky and Bady [15]. They poved that suffcently stong bubble velocty oscllatons can pevent clusteng. Russo and Smeeka [16] and Heeo, Lucqun-Deseux and Pethame [17] obtaned a Vlasov type system descbng the moton of massless gd sphees. Hakn, Kape and Nadm [18] obtaned the equatons of moton of two pulsatng and tanslatng sphecal bubbles. They calculated the flow potental up to ode β 4. The fcton foces wee not calculated wth the same pecson: the authos used fo numecal puposes the classcal Levch foces actng on an solated bubble. Teshukov and Gavlyuk [19] studed the system of N compessble bubbles and obtaned a Vlasov type system n the dsspaton-fee lmt. In that wok, they calculated the flow potental up to ode β 3, whch accounts fo pa-wse nteactons between bubbles. They have shown, n patcula, that the bubble oscllatons egulaze the bubble flow dynamcs: n the hydodynamc lmt, the dspeson elaton has eal oots. One can speculate that ths could be esponsble fo the stablty of andom bubble clouds. In that wok, the expessons fo the fcton foces wee not deved. The am of ths atcle s to detemne the dag foces on bubbles n a cloud of N massless compessble bubbles. Ths poblem was consdeed by Vonov and Golovn [8] and Wang and Smeeka [20] who also obtaned appoxmate expessons of these foces. In the pesent wok, the expessons fo the dag foces ae obtaned usng Levch s appoach and an explct fomula fo the velocty potental of the flow wth N compessble bubbles calculated up to ode β 3 n Teshukov and Gavlyuk [19]. 2. Velocty potental fo a system of N compessble bubbles n an ncompessble flud In the Levch appoach, the dsspaton functon Φ s calculated by usng the velocty potental fo a system of N bubbles. The coespondng potental has been found n Teshukov [21] and Teshukov and Gavlyuk [19] n explct asymptotc fom whch s coect up to ode β 3. Fo completeness, we shall shotly menton hee esults obtaned n the last atcle. We consde N sphecal gas bubbles movng n an unbounded nvscd ncompessble flud. We assume that the flud flow n the egon between bubbles s otatonal and the velocty vecto vanshes as x. The flud velocty potental ϕt,x s a soluton of the followng bounday-value poblem: ϕ = 0, x Ω = R 3 ϕ B, n = v n + s, Γ 6 ϕ 0, x. Hee v t = x t ae the veloctes of the centes of sphecal bubbles, s = b t ae the veloctes of expanson of the bubbles, x t ae the adus-vectos of the centes, b t ae the bubble ad = 1,...,N, B and ae a ball and a sphee of adus b t wth cente at the pont x t, espectvely, and n s the unt nomal vecto to dected to the flud. Hee the pme denotes the devatve wth espect to tme. Ths poblem descbes the moton of a collecton of compessble bubbles n othewse quescent flud. The unknown potental of the otatonal flow can be wtten as N ϕ = v ψ + s ϕ, =1 whee, n vew of 6, the hamonc functons ψ t, x and ϕ t, x satsfy the condtons ψ ϕ n = δ k n, Γk n = δ k Γk and the gadents vansh at nfnty. Hee δ k ae the Konecke symbols. In what follows, we consde a aefed bubbly flud fo whch β s a small paamete. Asymptotc expessons of ϕ and ψ up to ode β 3 ae gven n Teshukov and Gavlyuk [19]: ϕ = b2 + ϕ, 7 ϕ = b2 n n+1 n b b P n cos θ, 8 n + 1

4 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds ψ = b ψ, 9 Hee ψ = b b 2 b 4 b 2 B x x. 10 = x x, = x x, n = x x, cos θ = x x n, B = I 3n n, P n cos θ ae the Legende polynomals Gadsteyn and Ryzhk [22], I s the unt matx, and a b s the tenso poduct. It s easy to see that ϕ /n Γk = δ k + Oβ 4 by vtue of the estmates ϕ Γk = Oβ 4 and ϕ Γk = Oβ 4, k.the hamonc functon ψ satsfes the condtons ψ 0 + ψ /n Γ = 0, ψ Γk = Oβ 4, ψ /n Γk = δ k n + Oβ 4, and ψ Γk = Oβ 4 fo k. In Teshukov and Gavlyuk [19] thee s a mspnt: the sgn befoe the expesson 10 was absent. Howeve, all futhe calculatons n the above mentoned pape took nto account the ght sgn. Usng the asymptotc expanson n β, we have obtaned an appoxmate epesentaton of the flud velocty potental fo the specfed postons x,adb, tanslatonal veloctes v and dlataton veloctes s of bubbles. Ths enables us to calculate the knetc enegy of the flud: T = ρ N b 2 3 =1 + 2πb 3 b 2 N 3 π v 2 + 4πs 2 + 4πb 2 b =1 2 s n v + πb 3 b 3 v B v b s s + 2πb 3 b 2 s n v. 11 In ths fomula, the fst sum compses the zeo ode tems n β and the second sum ncludes tems of odes β, β 2,andβ 3. The lowe ode appoxmate fomula fo T contanng the tems up to β 2 has also been obtaned by Vonov and Golovn [8]. Fo the case of gd bubbles of equal ad the fomula fo T was obtaned by Golovn [9] see also Russo and Smeeka [16]. The poblem of moton of a flud wth bubbles s Lagangan n the case whee the flow of the flud s completely detemned by the bubble moton Vonov and Golovn [8]. The Eule Lagange equatons fo the genealzed coodnates and coespondng veloctes ae: d L L N = 0, L= T U, U = ετ. 12 dt ẏ y =1 Hee y = x,b T = 1,...,N ae the vectos wth fou components. Now we wll ntoduce the dag foces by usng the Levch appoach. 3. Dsspaton functon The dag foces ae calculated though the devatves Φ/v, Φ/s of the dsspaton functon 2 ϕ Φ = 2µ D : D dω = 2µ R 3 \ B R 3 \ x 2 : 2 ϕ x 2 dω B whch s a quadatc functon of v and s. Ths fomula can be ewtten n the fom Batchelo [3]: Φ = µ N k=1γ k n ϕ 2 dγ = µ N k=1γ k ϕ 2 dγ. Hee s the dstance fom the bubble cente. We assume that the system of N bubbles s govened by the equatons d L L = 1 Φ, y = x,b T 13 dt ẏ y 2 ẏ

5 472 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds genealzng 12 to the case of non-potental foces. To fnd the dag foces explctly, we need to calculate the devatves of Φ wth espect to v and s. Fo any constant vecto h Φ N h = 2µ v k=1γ k N = 4µ h T ψ x k=1γ k Hence Φ T N = 4µ v k=1γ k In the same manne, Φ N = 4µ s k=1 Γ k ϕ ψ h ϕ N dγ = 4µ n k=1 ψ x ϕdγ. Γ k ψ h ϕdγ ϕdγ. 14 ϕdγ. To calculate 14, 15, we evaluate the values of ψ and ϕ as well as the gadents of these functons on all Γ. In the followng the outwad nomal vecto n to, = 1,...,N, wll be denoted as n. Fom 7 10 we obtan appoxmate expessons of these functons and the devatves coect up to β 3 whch ae necessay to constuct the dsspaton functon. Futhe, we wll use the sgn to say that a gven expesson s coect up to β 3 tems of ode β 4 and hghe ae omtted Calculaton of dag foces In ths secton we calculate the dag foces and gve an expesson of the dsspaton functon n explct fom. Usng the appoxmate fomulae fo ψ and ϕ obtaned n Appendces A and B, we can evaluate the devatves of the dsspaton functon wth espect to v and s dffeng only by a constant multple fom the dag foces. Fst, we shall calculate the dag foces n the equatons govenng the tanslatonal moton of bubbles. The tems up to β 3 n 14 ae gven by Φ T 4µ v 4µ ψ x Γ N ϕ s =1 ψ x + T ψ v dγ x ϕ T ψ s + v dγ. 16 x To evaluate 16 we consde thee ntegals. The fst one s easly evaluated afte substtutng of appoxmate expessons fo the devatves of ϕ and ψ see Appendces A and B: V 1 = 4µ = 6µ b ψ x ϕ ψ T s + v dγ = 4µ x 1 I 3n n s n + 3 I 3n nv dγ 2 2b I + 3n nv dγ = 24µπb v. 17 We used hee the equaltes 1 I n dγ = 0, 4πb 2 n n dγ = Γ Futhe, we substtute appoxmate expessons fo ϕ and ψ nto 15

6 V 2 = 4µ = 2µ + b2 b 2 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds ψ ϕ T ψ s + v dγ x x Γ 3 3 b I 3n n I 3n nb v + s b2 b n 4b Γ b 2 n2n + 1 P n cos θ n 2n + 1 b n P n n + 1 cos θ I n nn dγ. 19 Takng nto account the equaltes I 3n ni 3n n dγ = 8πb 2 I, I 3n ni n n dγ = 8π 3 b2 I and the defnton of the fst Legende polynomals P 0 x = 1, P 1 x = x, P 2 x = 3x2 1, 2 we obtan the expesson 3 b 2 b V 2 = 12πµb B v + 8πµb s n + 12µ s b 2 b 2 n cos θ dγ. To calculate the last ntegal, t s convenent to use the sphecal coodnates, θ, λ wth z-axs dected along n.inthe sphecal coodnates the nomal vecto to s: n =sn θ cos λ, sn θ sn λ, cos θ. Usng ths fomula we fnd n cos θ dγ = 4 3 πb2 n and 2 b 3 b V 2 = 24πµb s n + 12πµb B v. 20 In calculaton of the thd tem we use smla denttes and obtan V 3 = 4µ ψ ϕ T ψ s + v dγ x x Γ = 4µ 3 b 3 B I n n 2s n+ 3 I 3n nv dγ 4 b 2b Γ = 12πµb 3 b 3 B v. Fnally, Φ T V 1 + V 2 + V 3 = 24πµb v + 24πµb b 2 s n v 3 b + 12πµb B v + 12πµb 3 b 3 B v. 21 We poceed now to fnd the vscous foce n the equatons govenng the bubble oscllatons:

7 474 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds Φ N = 4µ s =1Γ ϕ 4µ Γ ϕ Fst, let us calculate the pncpal tem ϕ S 1 = 4µ ϕ s + = 4µ n ϕdγ 4µ ϕ s + ϕ ψ x ϕdγ T v dγ. ψ T v dγ x 2s n + 3 I 3n nv dγ = 32πµb s. 22 b 2b The othogonalty of the Legende polynomals and the fst dentty of 18 mply that fo up to β 3 ϕ ψ T ϕ dγ 0, ϕ v dγ 0. x Hence S 2 = 4µ ϕ s ϕ Now we have to calculate the last tem S 3 = 4µ ϕ Γ = 4µ b2 b Γ ϕ s + T ψ + v dγ 0. x T ψ v dγ x 2n + 1 n b P 3 n n + 1 cos θ I n nn I 3n nv 2s n dγ. 2b b Snce I n nn n = 0, and the polynomal P 2 x s odd, S 3 = 9µb 2 1 b 2 I n nn I 3n nv dγ = 9µb 2 1 b Γ 2 n T I n nv dγ Γ = 24πµb 2 b 2 n v. Hence Φ S 1 + S 2 + S 3 = 32πµb s + 24πµb 2 b s 2 n v. 23 Fnally, t follows fom 21 and 23 that the dag foces ae gven by F v Φ T = 1 2 b 12πµb v + 12πµb s n 2 v b 3 + 6πµb B v + 6πµb 3 b 3 B v 24 and F s = 1 Φ 16πµb s + 12πµb 2 b 2 s 2 n v. 25

8 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds It s woth to note that expesson 25 does not contan Oβ 3 tems: they vansh dentcally. Fomulae 24, 25 allow us to fnd the expesson fo the dsspaton functon Φ. Let us desgnate by U f velocty of flud nduced by the moton of othe bubbles at the cente of th bubble: the ambent U f = x b2 k s k + b3 k 1 k k 2 x v k. k 26 Then the dsspaton functon Φ s gven by Φ = N µπb 16s v U f 2. =1 Ths expesson s exactly the same as that gven by 4 fo a sngle bubble, but n a flud havng the velocty U f at nfnty. Dect calculatons show that up to the tems of ode β 3 t gves us the dag foces 24, 25. A smla fom of the expesson of the dsspaton functon was obtaned by Wang and Smeeka [20] Equatons of moton fo a system of N bubbles The buoyancy foce and the suface tenson can also be added though a smple changng of the Lagangan of the system. It s suffcent to take L defned by 11, 12 n the fom N L = T U 4πσb 2 N 4 3 πρb3 g x, 27 =1 =1 whee σ s the suface tenson coeffcent and g s the gavty. Fnally, the equatons of moton wth Lagangan 27 ae: d T T = dt v x 12πµb v + 12πµb b + 6πµb 2 b s n 3 B v + 6πµb 3 d T T = 4πb 2 dt s b p g b p 2σ ρg x b 4µs 3µ b b 2 n v. b 3 B v 4 3 πρb3 g, 28 Hee the knetc enegy T s gven by 11. System 28 genealzes Eqs. 5 descbng the moton of a sngle bubble. 4. Concluson In ths pape we have pesented a egula asymptotc pocedue fo the calculaton of the dag foces on bubbles n a cloud of N compessble bubbles. By usng the Levch appoach, the vscous foces have been calculated explctly up to ode β 3.The devaton has been based on the asymptotc expanson fo the velocty potental wth the same pecson. Fomulae obtaned ae n ageement wth those found by Wang and Smeeka [20]. A compason wth the lowe ode appoxmaton obtaned by Vonov and Golovn [8] shows that not all tems of ode β 2 ae pesented n the fomulae. Fnally, a geneal system 28 descbng the moton of N bubbles n the pesence of eal effects vscous dsspaton, gavty and suface tenson has been deved. Acknowledgement The eseach was accomplshed when the authos wee vstng the Mathematsches Foschungsnsttut Obewolfach. V.T. was patally suppoted by the Russan Foundaton of Basc Reseach, gant , and by the Russan Mnsty of Educaton, gant E

9 476 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds Appendx A. Appoxmate fomulae fo ψ and the devatves Fst, we begn wth ψ.wehavefom9,10: ψ Γ 1 b 3 x x 2 = b Γ 2 n, ψ x 1 b 3 I 3 x x x x Γ 2 2 ψ x 3 2b I 3n n, = 1 I 3n n, 2 ψ Γ b3 2 2 n 1 b B x x 1 b 3 3 b 4 3 B x x = b3 Γ 2 2 n 3 b B n, ψ x Γ 1 b B 1 b 3 b B + 3 b 3 b 3 x x x x B 2 = 3 b 3 B I n n, Γ 4 ψ 3 b 3 B I 3n n. x Γ 4b Appendx B. Appoxmate fomulae fo ϕ and the devatves Snce Thefomulaefoϕ followng fom 7 8 ae: ϕ Γ b2 = b, Γ ϕ Γ b2 3 x x = n, Γ ϕ 2 n, Γ b ϕ Γ b2 = b2 n b n=0 n=0 b n P n cos θ b2 2n + 1 n b P n cos θ. n + 1 n b n+1 b n P n cos θ n + 1 Γ cos θ = x x n, cos θ Γ = 1 I x x x x 2 n = Γ 1 I n nn, b we have b 2 n b b ϕ Γ = b2 b b n n n 1 b n n + n n + 1 n+1 P n cos θ bn+1 n+2 P n cos θ x x Γ

10 S.L. Gavlyuk, V.M. Teshukov / Euopean Jounal of Mechancs B/Fluds b2 n n b + n n+1 b P n b n + 1 cos θ 1 I x x x x 2 n = Γ b2 b Hee means the devatve of P n wth espect to cos θ. Fnally, ϕ = b2 n b Γ b 2 n2n + 1 P n cos θ n + b2 b 2 2n + 1 n b P n n + 1 cos θ I n nn. 2n + 1 n b P n n + 1 cos θ I n nn. Refeences [1] V.G. Levch, Physcochemcal Hydodynamcs, Pentce-Hall, [2] D.W. Mooe, The bounday laye on a sphecal gas bubble, J. Flud Mech [3] G.K. Batchelo, An Intoducton to Flud Dynamcs, Cambdge Unvesty Pess, [4] D.W. Mooe, The se of a gas bubble n a vscous lqud, J. Flud Mech [5] D.D. Joseph, J. Wang, The dsspaton appoxmaton and vscous potental flow, J. Flud Mech [6] J. Magnaudet, D. Legende, The vscous dag foce on a sphecal bubble wth tme-dependent adus, Phys. Fluds [7] H. Goldsten, C. Poole, J. Safko, Classcal Mechancs, Addson-Wesley, [8] O.V. Vonov, A.M. Golovn, Lagange equatons fo a system of bubbles wth vaable ad n a small vscosty flud, Izv. Akad. Nauk SSSR Mekh. Zhdk. Gaza n Russan. [9] A.M. Golovn, Lagange equatons fo a system of bubbles n a small vscosty flud, Zh. Pkl. Mekh. Tekhnt. Fzk n Russan. [10] J.B.W. Kok, Dynamcs of a pa of gas bubbles movng though lqud. Pat I. Theoy, Eu. J. Mech. B Fluds [11] J.B.W. Kok, Dynamcs of a pa of gas bubbles movng though lqud. Pat II. Expement, Eu. J. Mech. B Fluds [12] A.S. Sangan, A.K. Ddwana, Dynamc smulatons of flows of bubbly lquds at lage Reynolds numbes, J. Flud Mech [13] P. Smeeka, On the moton of bubbles n a peodc box, J. Flud Mech [14] L. Van Wngaaden, The mean se velocty of pawse-nteactng bubbles, J. Flud Mech [15] Y. Yukovetsky, J. Bady, Statstcal mechancs of bubbly lquds, Phys. Fluds [16] G. Russo, P. Smeeka, Knetc theoy fo bubbly flow I: Collsonless case and II: Flud dynamc lmt, SIAM J. Appl. Math [17] H. Heeo, B. Lucqun-Deseux, B. Pethame, On the moton of dspesed balls n a potental flow: a knetc descpton of the added mass effect, SIAM J. Appl. Math [18] A. Hakn, T.J. Kape, A. Nadm, Coupled pulsaton and tanslaton of two gas bubbles n a lqud, J. Flud Mech [19] V.M. Teshukov, S.L. Gavlyuk, Knetc model fo the moton of compessble bubbles n a pefect flud, Eu. J. Mech. B Fluds [20] N. Wang, P. Smeeka, Effectve equatons fo sound and vod wave popagaton n bubbly fluds, SIAM J. Appl. Math [21] V.M. Teshukov, Knetc model of bubbly flow, J. Appl. Mech. Techn. Phys [22] I.I. Gadshteyn, I.M. Ryzhk, Tables of Integals, Sees, and Poducts, Academc Pess, New Yok, 1980.