The translational oscillations of a cylindrical bubble in a bounded volume of a liquid with free deformable interface

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1 Joural of Physcs: Cofrc Srs PAPER OPEN ACCESS Th traslatoal oscllatos of a cyldrcal bubbl a boudd volum of a lqud wth fr dformabl trfac To ct ths artcl: A A Alabuzhv ad M I Kaysa 6 J. Phys.: Cof. Sr Rct ctatos - Th oscllatos of oblat drop udr th fluc of a altratg lctrc fld A A Alabuzhv ad M A Kasha - Th axsymmtrc oscllatos of a cyldrcal bubbl a lqud boudd volum wth fr dformabl trfac A A Alabuzhv ad M I Kaysa - Th dyamcs of hmsphrcal drop udr th fluc of a altratg lctrc fld A A Alabuzhv ad M A Kasha Vw th artcl ol for updats ad hacmts. Ths cott was dowloadd from IP addrss o 7//8 at :53

2 Itratoal Cofrc o Computr Smulato Physcs ad Byod 5 IOP Publshg Joural of Physcs: Cofrc Srs 68 6) 43 do:.88/ /68//43 Th traslatoal oscllatos of a cyldrcal bubbl a boudd volum of a lqud wth fr dformabl trfac A A Alabuzhv, ad M I Kaysa Isttut of Cotuous Mda Mchacs UB RAS, Prm 643, Russa Prm Stat Uvrsty, Prm 6499, Russa E-mal: alabuzhv@mal.ru Abstract. W cosdr th g ad forcd traslato oscllatos of a cyldrcal gas bubbl surroudd by a comprssbl flud wth fr dformabl trfac. Th bubbl has a qulbrum cyldrcal shap ad s boudd axally by two paralll sold surfacs. Dyamcs of cotact ls s tak to accout by a ffctv boudary codto: vlocty of th cotact l s assumd to b proportoal to cotact agl dvato from th qulbrum valu. Th qulbrum cotact agl s rght. Eg frqucy dcrass wth lqud outr fr surfac radus dcrasg ad crass wth th radus-to-hght rato crasg. W foud that th ma traslatoal frqucy of g oscllato vashs at a crta valu of th Hockg paramtr th so-calld wttg paramtr). Th g frqucs of th bubbl ar hghr tha th g frqucs of flud volum.. Itroducto Problms wth th cotact-l dyamcs tak to accout hav b xamd varous formulatos [-4]. I th study of th dyamc cotact agl Hockg codto s wdly usd by vrtu of ts smplcty), ad t s th o mployd a study [5] of th dampg of stadg wavs btw two vrtcal walls. Ths codto assums a lar rlatoshp btw th vlocty of th cotact l moto ad th cotact agl for th cas of rght qulbrum cotact agl) ζ =Λk ζ, ) t whr ζ s th dvato of th trfac from th qulbrum posto, k s th xtral ormal to th sold surfac, Λ s a phomologcal costat th so-calld wttg paramtr or Hockg paramtr) havg th dmso of th vlocty. Th spcal cass of th boudary codto ) ar th rqurmt of th fxd cotact l ζ =, th so-calld pd-d dg codto) ad th costat cotact agl k ζ = ). Artcl [5] also prsts a qualtatv comparso wth th xprmtal studs of th othr authors. Th quato ) was usd th study of oscllatos of th sssl drop ad bubbl [6-8], cyldrcal drop ad bubbl [9,], capllary brdg [] ad sadwchd drop []. Th partcular cas of pd-d dg was usd for capllary brdg [3,4] ad sssl drop [5]; th cas of costat cotact agl was usd for cyldrcal drop [6] ad for sssl drop [7]. Th mor dffcult codto was proposd [8]. Ths codto volvs a ambguous dpdc of th cotact agl o th cotact l vlocty for smplcty, th qulbrum cotact agl s grally assumd to b a rght agl) To whom ay corrspodc should b addrssd. Cott from ths work may b usd udr th trms of th Cratv Commos Attrbuto 3. lcc. Ay furthr dstrbuto of ths work must mata attrbuto to th authors) ad th ttl of th work, joural ctato ad DOI. Publshd udr lcc by IOP Publshg Ltd

3 Itratoal Cofrc o Computr Smulato Physcs ad Byod 5 IOP Publshg Joural of Physcs: Cofrc Srs 68 6) 43 do:.88/ /68//43 ), Λ γ γ c γ > γc ζ =, γ γ c, ) t Λ γ + γc), γ < γc whr γ = ζ z s th cotact agl dvato from th qulbrum valu, z s th coordat ormal to th sold wall ad xtdg dp to th flud. Rsults of physcal xprmts [9,] ar dscrbd wll by formula ) for small dvatos of th cotact agl. Th quato ) was usd th study of oscllatos of th sssl drop [], hmsphrcal bubbl [] o sold plat ad cyldrcal drop [3]. I th prst study, w cosdr th traslatoal oscllatos of cyldrcal bubbl whch surroudd by a lqud wth fr dformabl trfac. W apply codto ) ad cotu th study of th axsymmtrc oscllatos [,4,5]. It s mportat to ot that th xtral lqud has th fr odformabl trfac [,4] ad th rgd wall [5].. Problm formulato Cosdr a gasous bubbl surroudd by a lqud wth a fr xtral surfac s fgur ). Th systm s boudd by two paralll sold surfacs sparatd by a dstac h. Th bubbl ad lqud volum ar assumd to b cyldrcal, wth a radus r ad R qulbrum, rspctvly, mplyg that th vry qulbrum cotact agl s 9. Th systm s actd upo by a xtral vbrato forc wth ampltud A ad oscllato frqucy ω. Th forc drctd paralll to th sold surfacs. h r Fgur. Problm gomtry. Th charactrstc ampltud of oscllatos A s small compard to th qulbrum radus r. W assum that, o th o had, th fudamtal oscllato frqucy ω s larg ough for th vscosty could b gord, ad, o th othr had, th oscllato frqucy s small ough, so that w ca us th comprssblty codtos δ = ν ω << r ad ω r << c δ s th boudary-layr thckss, c s th soud vlocty ad ν s th kmatc vscosty of lqud). Sc th problm s symmtrc, t s covt to troduc cyldrcal coordats r, α, z ). Th azmuthal agl α s rckod from th x axs. Lt th latral surfac of th bubbl ad th xtral surfac of lqud volum b dscrbd by th followg quatos, rspctvly = + ζ α,, r = R + ξ α, z, t. r r z t ), ) 3 Followg [,4], w us ρ ρ σ, r, h, A, Aσ r, A σ ρ ρ as th scals for th tm, lgth, hght, dvato of bubbl surfac ad fr surfac from ts qulbrum posto, prssur, ad vlocty pottal, rspctvly. Hr, σ s th surfac tso ad ρ s th lqud dsty. Thus, th dmsolss boudary valu problm s dtrmd by trmdat stps ca b foud [,4]) ωt p = ϕ + ω rcos α, ϕ=, p = Pr σ ζ P ζ, 3) t ) ) p g = r + + b r r r r α z,

4 Itratoal Cofrc o Computr Smulato Physcs ad Byod 5 IOP Publshg Joural of Physcs: Cofrc Srs 68 6) 43 do:.88/ /68//43 r =: ζ = ϕ p = ζ + ζ + b ζ zz, 4), [ ] t r αα p g r = R : ξt = ϕ r p = ξ + ξαα + b ξzz, 5) z = ± : ϕ z =, 6) r =, z = ± : ζ t = ± ζ z 7) r = R, z = ± : ξt = ± ξz 8) whr p s th lqud prssur, ϕ s pottal of lqud vlocty, p s th gas prssur th bubbl, s polytropc.g., adabatc) xpot, P s dmso gas prssur th bubbl. Th boudaryvalu problm 3) 7) volvs fv paramtrs: th aspct rato, th radus of fr surfac, th wttg paramtr, th frqucy ad ampltud b= r h, R = R r, = Λ ρ ρ σ, 3 ω = ω ρ ρ σ, ε = A r. 3. Natural traslatoal oscllatos of th bubbl I ordr to vstgat th problm, t s covt to bg wth a cosdrato of th atural oscllatos of a cyldrcal bubbl. By th vss of th atural oscllato mods s mat th vss of th fuctos udr a chag of sg of th axal coordat z. I vw of traslatoal symmtry, th soluto of th boudary valu problm 3) 8) wthout xtral forc s wrtt as ) ) ϕ rzt,,, α) = a R ) r + b R m )s r) + ) πz) + = 9) ) ) α Ωt + a R) r + b R)cos r ) π z), = ) α Ωt ζ α, zt, = dz + dz + hs + ) πz) + ccos πz), ) = = ) α Ωt ξ α, z, t = Dz+ Dz + Hs + ) πz) + Ccos πz), ) = = whr Ω s g frqucy, R r) = I + ) πbr ), R r) = K + ) πbr ), R )= r r, R r) = I π br ), ) R r = r, R r) = K π br ), I ad K ar modfd Bssl fuctos of th frst ordr. Substtutg solutos 9) ) to 3) 8), w obta a spctral-ampltud problm whch gvalus ar th valus of th atural oscllato frqucy Ω. From th soluto of ths problm t follows that th gvalus ar foud from th quatos: Ω R) v mods =, ) = π ω Ω R) ω Ω ) ω S8 S5 S7 S6) + 4 = π Ω +Ω S S3 Ω S4 S6 Ω Ω Ω R)) + + =, 3) 4 Ω ) Ω SS 5 4 Ω Ω 6 = π Ω + Ω S S3 Ω 4 = π Ω + Ω S S3 6 Ω Ω R ) ) ) R R R R R) R ) ) ) R R R R R) Ω ) = 4π b, Ω R R) = 4π b, ) ) ) R R R R R ) R ) ) ) R R R R R ) 4bR + S =Ω ) +Ω R ), S3 =Ω ) Ω R ), S4 =, 5 = R b R + ) S S 6 =, R R S b R ). 7 = R + 3 ) R, )

5 Itratoal Cofrc o Computr Smulato Physcs ad Byod 5 IOP Publshg Joural of Physcs: Cofrc Srs 68 6) 43 do:.88/ /68//43 Odd mods 4 Ω + =, 4) π + ) Ω ) Ω Ω = 4 Ω R ) =, 5) + ) Ω Ω R ) Ω = π R ) R R ) R R ) R ) Ω = + b, R ) R R ) R R ) R ) ) ) π + ) π b R ) R R ) R R ) R )) Ω =. R) R ) ) ) R R R R R ) ) ) Hr, Ω ), Ω R ), Ω, Ω R ar th atural oscllato frqucs of th frly movg cotact l. Th complx algbrac quatos ) 5) hav complx solutos, ths lads to oscllato dampg du to th dsspato o th cotact l. RΩ ) ImΩ ) 5 5,,,, a) b) Fgur. Frqucy a) ad dampg rato b) of atural oscllatos vs wttg paramtr for Ω R = 5, P = 5 ). b = l, b = l, lqud sold l, bubbl dashd l. 6 RΩ ) 8 ImΩ ) ,,,, a) b) Fgur 3. Frqucy a) ad dampg rato b) of atural oscllatos vs wttg paramtr for Ω R = 5, P = 5, b = ). lqud sold l, bubbl dashd l. Equatos ) 5) wr solvd umrcally wth th usag of th two-dmsoal scat mthod. Fgurs ad 3 show th ral part of R Ω ) oscllato frqucy) ad magary part 4

6 Itratoal Cofrc o Computr Smulato Physcs ad Byod 5 IOP Publshg Joural of Physcs: Cofrc Srs 68 6) 43 do:.88/ /68//43 Im Ω ) dampg rato) of th complx atural frqucy Ω for th oscllato v mods Ω.., k = s th wavumbr) ad Ω. Hr ad blow for th frqucy dcs, coscutv umbrg of th wavumbr s usd: v valus of k corrspod to th v mods th soluto of Eqs. ),3)) ad odd valus of k corrspod to th odd mods th soluto of Eqs. 4),5)). Th zro traslatoal mod k = dscrbs th dsplacmt of th lqud volum as a whol. I th cosdrd cas wth th cotact l dyamcs tak to accout) th dsplacmt s largr th ctral part of th colum tha ar th ds. Elastc forcs caus th lqud volum to tak th orgal shap, rsultg th rtur moto of ts ctr of mass. As th capllary paramtr crass, th shft btw th ctr ad prphry of th fr surfac dcrass. For a crta valu of th capllary paramtr, th dffrc of th valu of th shft dsappars ad th zro g frqucy vashs fgur a). For larg valus of, th dampg rato taks two valus fgur b). From fgur 3 t follows that as crass, th frqucy dcrass mootocally, th dampg rato has a maxmum for a ft valu of th capllary paramtr ad tds to zro as ad. Not, that th g frqucs of th bubbl ar hghr tha th g frqucs of th flud volum. It s th rsult of th fluc of th surfac tso forc o xtral trfac. Fgur 4 shows th ral part of R Ω ) oscllato frqucy) ad magary part Im Ω ) dampg rato) of th complx atural frqucy Ω for th oscllato odd mod Ω. From fgurs ad 4 t follows that th atural oscllato frqucy crass as th paramtr b crass.., as th bubbl qulbrum radus crass or as th bubbl hght dcrass). 4 RΩ ) 3 ImΩ ) 5 5,,,, a) b) Fgur 4. Frqucy a) ad dampg rato b) of atural oscllatos vs wttg paramtr for Ω R = 5, P = 5 ). b = l, b = l, lqud sold l, bubbl dashd l. 4. Forcd traslato oscllatos of a bubbl Hr w cosdr th problm of forcd oscllatos of a cyldrcal bubbl. Th flds of vlocty pottal ad surfacs dflctos ar dfd by ω ϕ r,,, z t α) = R a ) + )cos ) π ) cos α) t R r br r z, 6) = ) ωt ζ α, zt, = R zd+ c cos πz) cos α), 7) = ξ α,, ) = R + cos π ) cos α) Substtutg solutos 6) 8) to 3) 8), w obta xprssos for th ukow ampltuds a, b, c, C, d ad D. Ths xprssos ar ot gv du to thr cumbrsom forms. zt zd C z ωt. 8) = 5

7 Itratoal Cofrc o Computr Smulato Physcs ad Byod 5 IOP Publshg Joural of Physcs: Cofrc Srs 68 6) 43 do:.88/ /68//43 5. Coclusos Eg frqucy dcrass wth th dcras of th lqud outr fr surfac radus ad cras wth th bubbl radus-to-hght rato crasg. W foud that th ma traslatoal frqucy of th atural oscllatos vashs at a crta valu of th Hockg paramtr. Th g frqucs of th bubbl ar hghr tha th g frqucs of th flud volum. It s th rsult of th fluc of th surfac tso forc o xtral trfac. Also ot that th frqucs of th traslatoal mods ar dpdt of th gas prssur sd th bubbl. Wll-markd rsoac ffcts ar foud th study of th forcd oscllatos. Thus, o ca choos th bubbl radus-to-hght rato such that th charactrstc frqucy of ay mod s qual to zro, ordr to ultmatly, to dtrm th wttg paramtr. Thr ar ot th «at-rsoat» frqucs udr th traslatoal vbratos,.. th vbrato frqucs at whch th ampltud of cotact l dvato s zro [7,]. Ackowldgmts Ths work was supportd by th Russa Foudato for Basc Rsarch projct r-ural). 6. Rfrcs [] D Gs P G 985 Rv. Mod. Phys [] Voov O V 976 Flud Dy 74 7 [3] Bo D, Eggrs J, Idku J, Mur J ad Rolly E 9 Rv. Mod. Phys [4] Zhag L ad Thss D B 3 J. Flud Mch [5] Hockg L M 987 J. Flud Mch [6] Lyubmov D V, Lyubmova T P ad Shklyav S V 4 Flud Dyamcs [7] Lyubmov D V, Lyubmova T P ad Shklyav S V 6 Phys. Fluds 8 [8] Shklyav S ad Straub A V 8 Phys. Fluds 5 [9] Alabuzhv A A ad Lyubmov D V 7 J. Appl. Mch. Tch. Phys [] Alabuzhv A A 4 Computatoal Cotuum Mchacs Russa) [] Borkar A ad Tsamopoulus J 99 Phys. Fluds A [] Alabuzhv A A ad Lyubmov D V J. Appl. Mch. Tch. Phys [3] Tsamopoulos J, Ch T ad Borkar A 99 J. Flud Mch [4] Dm V A 8 Flud Dy [5] Kartavh N N ad Shklyav S V 7 Bullt of Prm Uvrsty. Srs: Physcs o 6) 3 8 Russa) [6] Alabuzhv A A ad Lyubmov D V 5 Flud Dy [7] Ivatsov A O Bullt of Prm Uvrsty. Srs: Physcs o 3) 6 3 Russa) [8] Hockg L M 987 J. Flud Mch [9] Abltt R 93 Phlos. Mag [] Dussa V E B 979 Au. Rv. Flud Mch. 37 [] Fayzrakhmaova I S ad Straub A V 9 J. Flud Mch. 74 [] Fayzrakhmaova I S, Straub A V ad Shklyav S J. Flud Mch. 3 5 [3] Alabuzhv A A Bullt of Prm Uvrsty. Srs: Physcs o 9) 35-4 Russa) [4] Alabuzhv A A ad Kaysa M I 5 Bullt of Prm Uvrsty. Srs: Physcs o 9) 35-4 Russa) [5] Alabuzhv A A ad Kaysa M I 5 Bullt of Prm Uvrsty. Srs: Physcs o 3) Russa) 6