Nonlinear dynamics of a cavitation bubble pair near a rigid boundary in a standing ultrasonic wave field

Journal Pre-proofs Nonlinear dynamics of a cavitation bubble pair near a rigid boundary in a standing ultrasonic wave field Xiao Huang, Haibao Hu, Shuai Li, A-Man Zhang PII: DOI: Reference:

S1350-4177(19)31447-6 https://doi.org/10.1016/j.ultsonch.2020.104969 ULTSON 104969

To appear in:

Ultrasonics Sonochemistry

Received Date: Accepted Date:

12 September 2019 9 January 2020

Please cite this article as: X. Huang, H. Hu, S. Li, A-M. Zhang, Nonlinear dynamics of a cavitation bubble pair near a rigid boundary in a standing ultrasonic wave field, Ultrasonics Sonochemistry (2020), doi: https://doi.org/10.1016/ j.ultsonch.2020.104969

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© 2020 Published by Elsevier B.V.

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LQi2 i?i i?2 1BMbi2BM bmKKiBQM +QMp2MiBQM Bb mb2/ ?2`2X h?2 BMM2` T`2bbm`2 pb Q7  #m##H2 Bb K/2 mT Q7 i?2 pTQ` T`2bbm`2 pv M/ i?2 MQM@ +QM/2Mb#H2 ;b T`2bbm`2 pn - 7QHHQrBM; .HiQMǶb Hr,  γ V0 pb = pv + pn = pv + pn0 , UjV V r?2`2 V /2MQi2b i?2 #m##H2 pQHmK2- i?2 x2`Q bm#b+`BTi /2MQi2b i?2 pHm2 Q7  [mMiBiv i BMBiBH iBK2 t0 M/ γ = 1.4 BM i?Bb TT2` Bb i?2 `iBQ Q7 bT2+B}+ ?2ib Q7 i?2 ;bX q?2M  KB+`QK2i2`@bBx2/ #m##H2 KQp2b BM i?2 ri2`- i?2 pBb+Qmb 2z2+i Q7 i?2 ri2` Bb H`;2 bBM+2 i?2 _2vMQH/b MmK#2` Bb bKHH- rBi? i?2 Q`/2` O(10)- 7Q` i?2 +b2b +QMbB/2`2/ BM i?Bb TT2`X AM i?2 pBb+Qmb #QmM/`v Hv2` `QmM/ i?2 #m##H2 bm`7+2- i?2 pBb+Qmb T`2bbm`2 +Q``2+iBQM pvc - r?B+? Bb mb2/ iQ +QKT2Mbi2 i?2 iM;2MiBH pBb+Qmb bi`2bb- b r2HH b i?2 MQ`KH pBb+Qmb bi`2bb τn - b?QmH/ #2 iF2M BMiQ +QMbB/2`iBQM 7QHHQrBM; i?2 pBb+Qmb +Q``2+iBQM 7`QK pBb+Qmb TQi2MiBH ~Qr i?2Q`v Uo*oS6V BM CQb2T? M/ qM; (j3)X h?2 ri2` T`2bbm`2 pl QM i?2 #m##H2 bm`7+2 Bb i?2M pl = pb − 2σκ − pvc + τn , τn = 2µ

Ry8

∂ 2ϕ . ∂n2

U9V

r?2`2 σ- κ M/ µ /2MQi2 i?2 bm`7+2 i2MbBQM- i?2 HQ+H K2M +m`pim`2 M/ i?2 /vMKB+ pBb+QbBiv- `2bT2+iBp2HvX h?2 ;Qp2`MBM; 2[miBQM UkV +M #2 bQHp2/ mbBM; :`22MǶb b2+QM/ B/2MiBiv iQ +?B2p2 i?2 #QmM/`v BMi2;`H 2[miBQM ZZ  ∂G (a, b) ∂ϕ (b) + G (a, b) dS, U8V s (a) ϕ (a) = −ϕ (b) ∂n ∂n S r?2`2 a M/ b /2MQi2 i?2 }2H/ M/ bQm`+2 TQBMib- `2bT2+iBp2HvX h?2 bQHB/ M;H2 s (a) 2[mHb 2π B7 i?2 TQBMi a Bb HQ+i2/ QM  bKQQi? bm`7+2 S M/ G (a, b) = 1/|a − b| Bb +HH2/ i?2 e

F2`M2H 7mM+iBQMX AM Q`/2` iQ +QMbB/2` i?2 2z2+i Q7 i?2 `B;B/ #QmM/`v- r2 mb2 Gt iQ `2TH+2 G BM 1[X U8V, Gt (a, b) = G(a, b) + Gim (a, b), UeV r?2`2 Gim (a, b) =

1 . |a − (2Dez − b)|

UdV

>2`2 D /2MQi2b i?2 /BbiM+2 7`QK i?2 bQm`+2 TQBMi b iQ i?2 `B;B/ #QmM/`vX hQ mT/i2 i?2 TQbBiBQM Q7 i?2 B`@ri2` BMi2`7+2- i?2 FBM2KiB+ #QmM/`v +QM/BiBQM Bb /QTi2/ Da = ∇ϕ = v. U3V Dt 6`QK i?2 "2`MQmHHB 2[miBQM Dϕ 1 pl p∞ (t) − |∇ϕ|2 + + gz = , Dt 2 ρ ρ

UNV

i?2 /vMKB+ #QmM/`v +QM/BiBQM Bb /2`Bp2/ #v bm#biBimiBM; 1[bX URV- UjV M/ U9V BMiQ 1[X UNV,    γ Dϕ 1 1 ∂2ϕ V0 2 = |∇ϕ| − − 2σκ − pvc + 2µ 2 − pv + pn0 Dt 2 ρ V ∂n URyV 1 gz + [p0 + 2pa cos (kz cos θi ) cos (−kx sin θi + 2πf t)] , ρ r?2`2 g /2MQi2b ;`pBiiBQMH ++2H2`iBQMX 6Q` i?2 +b2b +QMbB/2`2/ BM i?Bb TT2`- g Bb y bBM+2 i?2 bBx2 Q7 i?2 #m##H2 Bb p2`v bKHHX jX oHB/iBQM Q7 i?2 MmK2`B+H KQ/2H RRy

jXRX LQM@/BK2MbBQMHBxiBQM h?2 HB[mB/ BM i?2 T`2b2Mi rQ`F Bb ri2`- rBi? i?2 /2MbBiv ρ = 999 kg/m3 - i?2 pBb+QbBiv µ = 1.0016 × 10−3 Pa · sc M/ i?2 ?v/`QbiiB+ T`2bbm`2 r?B+? 2[mHb iQ i?2 iKQbT?2`B+ T`2bbm`2 p0 = 101.3 kPaX h?2 #m##H2 ?b i?2 BMBiBH `/Bmb R0 = 5 µm M/ i?2 pTQ` +QMi`B#mi2b pv = 2339 Pa iQ Bib BMM2` T`2bbm`2X LQM@/BK2MbBQMHBxiBQM Bb +QM/m+i2/ rBi? i?2 BMBiBH #m##H2 `/Bmb R10 b i?2 H2M;i? b+H2- ∆p = p0 − pv b i?2 T`2bbm`2 b+H2 M/ ρ b i?2 /2MbBiv b+H2 (9j)X h?2 /BK2MbBQMH2bb T`K2i2`b `2 /2MQi2/ rBi?  dz Ǵ b bmT2`b+`BTi, r ρ D2 Dbb ∗ D1 ∗ ∗ ∗ , D2 = , Dbb = ,v =v ; URRV D1 = R10 R10 R10 ∆p t t∗ = R10

RR8

s

∆p ∗ , f = f R10 ρ

r

ρ ϕ , ϕ∗ = ∆p R10

r

ρ σ , σ∗ = . ∆p R10 ∆p

URR#V

h?2b2 /BK2MbBQMH2bb pHm2b `2 /QTi2/ i?`Qm;?Qmi i?2 TT2` mMH2bb bT2+B}2/ Qi?2`rBb2X S`K2i`B+ bim/B2b `2 +``B2/ Qmi BM i?2 7QHHQrBM; bm#b2+iBQMbX

d

6B;m`2 k, hBK2 ?BbiQ`v Q7 #m##H2 pQHmK2 +H+mHi2/ mbBM; i?2 o*oS6 KQ/2H M/ i?2 pBb+Qmb _vH2B;?Ĝ SH2bb2i 2[miBQM 7Q`  bBM;H2 #m##H2 Qb+BHHiBM; mM/2`  i`p2HBM; mHi`bQMB+ rp2c i?2 i?2Q`2iB+H `2bmHi rBi?Qmi +QMbB/2`BM; i?2 pBb+Qmb 2z2+ib Bb HbQ ;Bp2M 7Q` +QKT`BbQMX

jXkX *QKT`BbQM rBi? i?2 MHviB+H `2bmHib AM i?Bb bm#b2+iBQM- i?2 MHviB+H _vH2B;?ĜSH2bb2i U_ĜSV 2[miBQM rBi? pBb+Qmb 2z2+ib Bb mb2/ iQ pHB/i2 Qm` MmK2`B+H KQ/2H, ! ˙ σ 1 R 3 ¨ + R˙ 2 = pb − p0 − pw − 4µ − 2 . URkV RR 2 ρ R R

Rky

Rk8

Rjy

Rj8

R9y

>2`2 pw = pa sin(2πf t) /2MQi2b i?2 T`2bbm`2 Q7  i`p2HBM; rp2 BM i?2 HB[mB/ `QmM/ i?2 ¨ `2 i?2 BMbiMiM2Qmb #m##H2 `/Bmb M/ Bib }`bi M/ b2+QM/ iBK2 #m##H2 M/ R- R˙ M/ R /2`BpiBp2b- `2bT2+iBp2HvX h?2 iBK2 ?BbiQ`v Q7 i?2 pQHmK2 Q7  bBM;H2 #m##H2 rBi? BMBiBH `/Bmb R0∗ = 1 BM  7`22 }2H/ Bb T`2b2Mi2/ BM 6B;X kX h?2 rp2 KTHBim/2 M/ 7`2[m2M+v `2 p∗a = 0.5 M/ f ∗ = 0.18 UBX2X 300 kHzV- `2bT2+iBp2HvX b b?QrM BM 6B; k- i?2 "1J `2bmHib #b2/ QM o*oS6 ;`22 r2HH rBi? i?2 pBb+Qmb _ĜS 2[miBQM 7Q` KMv +v+H2b Q7 #m##H2 Qb+BHHiBQM r?2M 7Q+mbBM; QM pQHmK2 iBK2 ?BbiQ`B2bX Ai Bb M2+2bb`v iQ +QMbB/2` i?2 pBb+Qmb 2z2+i Q7 i?2 ri2`- bBM+2  /Bz2`2M+2 TT2`b M/ ;`/mHHv BM+`2b2b #2ir22M pBb+Qmb M/ BMpBb+B/ `2bmHib 7`QK i?2 +QKT`2bbBQM bi;2 Q7 i?2 }`bi Qb+BHHiBQM T2`BQ/X h?2 KtBKmK M/ KBMBKmK Q7 i?2 #m##H2 pQHmK2 #2+QK2 bKHH2` M/ H`;2`- `2bT2+iBp2Hv- /m2 iQ i?2 /KTBM; 2z2+i Q7 i?2 pBb+QbBivX "b2/ QM i?2b2 `2bmHib- i?2 MmK2`B+H KQ/2H BM i?Bb TT2` Bb bmBi#H2 7Q` bBKmHiBM; KB+`QK2i2`@bBx2/ #m##H2 /vMKB+b M/ rBHH #2 mb2/ BM i?2 7QHHQrBM; +H+mHiBQMbX jXjX *QMp2`;2M+2 MHvbBb h?2 K2b? Q7  #m##H2 BM i?Bb rQ`F Bb /Bb+`2iBx2/ #b2/ QM M B+Qb?2/`QM 7QHHQrBM; i?2 rQ`F Q7 qM; (99)X h?2 #bB+ B+Qb?2/`QM Bb `2;`/2/ b G2p2H R K2b?- M/ 7Q`  #m##H2 /Bb+`2iBx2/ BMiQ G2p2H N K2b?- i?2`2 `2 20N 2 2H2K2Mib QM i?2 #m##H2 bm`7+2X q2 i?2`27Q`2 +H+mHi2  bBM;H2 #m##H2 BM biM/BM; rp2 }2H/ M2`  `B;B/ #QmM/`v rBi? /Bz2`2Mi K2b? /2MbBiv BM 6B;X jX h?2 BMBiBH #m##H2 bBx2 M/ rp2 KTHBim/2 `2 i?2 bK2 b BM 6B;X k- i?2 rp2 7`2[m2M+v Bb f ∗ = 0.36 M/ i?2 #m##H2@#QmM/`v /BbiM+2 D∗ = 3X h?2 #m##H2 bm`7+2 Bb /Bb+`2iBx2/ BMiQ G2p2H e- 3- Ry- Rk M/ R9 K2b?- `2bT2+iBp2HvX _2bmHi Q7 bBKmHiBQM i t∗ = 7.23 Bb T`2b2Mi2/ BM HH 7`K2b M/ i?2 G2p2H R9 Bb Tmi QM i?2 `B;?i ?H7 Q7 2+? 7`K2 7Q` +QKT`BbQMX q?2M i?2 K2b? /2MbBiv `2+?2b G2p2H Rk UBX2X k33y 2H2K2MibV- i?2`2 Bb bHB;?iHv /Bz2`2M+2 #2ir22M i?2 b?T2b Q7 i?2 #m##H2- M/ i?2 `2bQHmiBQM Q7 i?2 G2p2H Ry K2b? +M HbQ +?B2p2  biBb7+iQ`v `2bmHiX 3

UV G2p2H e

U#V G2p2H 3

U+V G2p2H Ry

U/V G2p2H Rk

6B;m`2 j, *QKT`BbQM Q7 i?2 T`Q}H2 Q7  bBM;H2 #m##H2 M2` i?2 `B;B/ #QmM/`v #27Q`2 D2i T2M2i`iBQM rBi? /Bz2`2Mi K2b? /2MbBivX

6B;m`2 9, hBK2 ?BbiQ`B2b Q7 i?2 pQHmK2 /2pBiBQM `iBQ δV ∗ #2ir22M i?2 #m##H2 rBi? G2p2H N K2b? M/ G2p2H R9 K2b?X h?2 pQHmK2 Q7 i?2 #m##H2 rBi? G2p2H R9 K2b? Bb HbQ T`2b2Mi2/ 7Q` +QKT`BbQMX

h?2 [mMiBiiBp2 MHvbBb Bb ;Bp2M BM 6B;X 9- r?2`2 /2pBiBQM `iBQ Q7  #m##H2 7Q` i?2 pQHmK2 δV ∗ #2ir22M G2p2H N K2b? M/ G2p2H R9 K2b? Bb ;Bp2M b, δV ∗ =

R98

R8y

R88

∗ ∗ (t∗ ) (t∗ ) − VL14 VLN . ∗ VL14 (t∗ )

URjV

h?2 2``Q` ++mKmHi2b rBi? i?2 BM+`2bBM; iBK2 M/ #2+QK2 Q#pBQmb r?2M  #m##H2 b?`BMFb iQ Bib KBMBKmK bBx2X h?Bb bBimiBQM Bb HH2pBi2/ 7Q` +b2 +H+mHi2/ rBi? }M2` K2b?X 6Q` G2p2H Ry K2b?- i?2 KtBKmK pQHmK2 /2pBiBQM `iBQ δV ∗ Bb H2bb i?M 9W BM irQ +v+H2b Q7 #m##H2 Qb+BHHiBQM M/ Bi Bb 7m`i?2` /2+`2b2/ iQ #Qmi RX8W 7Q` i?2 G2p2H Rk K2b?X q2 }M/ i?i i?2 G2p2H Ry K2b? UrBi? kyyy 2H2K2Mib 7Q` 2+? #m##H2V +M T`Q/m+2 biBb7+iQ`v ++m`+v- rBi? i?2 2{+B2M+v HbQ #2BM; +QMbB/2`2/X >2M+2- i?Bb K2b? `2bQHmiBQM Bb /QTi2/ BM HH +b2b BM i?Bb TT2`X 9X _2bmHib M/ /Bb+mbbBQMb 9XRX 1z2+i Q7 i?2 rp2 /B`2+iBQM h?2 2z2+i Q7 i?2 rp2 /B`2+iBQM QM i?2 #m##H2 #2?pBQ` Bb }`biHv /Bb+mbb2/ BM i?Bb bm#b2+iBQMX h?2 `2bmHib 7Q`  #m##H2 TB` BM  biM/BM; rp2 UBX2X- θi = 0 BM 6B;X RV M/  i`p2HBM; rp2 UBX2X- θi = π/2V }2H/ `2 +QKT`2/X h?2 #m##H2b ?p2 i?2 bK2 BMBiBH `/Bmb ∗ ∗ R10 = R20 = 1 M/ i?2 bK2 /BbiM+2 iQ i?2 #QmM/`v D1∗ = D2∗ = 3c i?2 BMBiBH ?Q`BxQMiH ∗ BMi2`@#m##H2 /BbiM+2 Bb Dbb = 6 M/ i?2 rp2 KTHBim/2 M/ 7`2[m2M+v `2 p∗a = 0.5 M/ f ∗ = 0.36- `2bT2+iBp2HvX N

UV

U#V

6B;m`2 8, 1pQHmiBQM Q7  #m##H2 TB` mM/2` mHi`bQMB+ rp2 2t+BiiBQM BM /Bz2`2Mi /B`2+iBQMb, UV i`p2HBM; ∗ ∗ ∗ rp2 θi = π/2- U#V biM/BM; rp2 θi = 0X R10 = R20 = 1- D1∗ = D2∗ = 3- Dbb = 6- p∗a = 0.5 M/ f ∗ = 0.36X

Rey

Re8

Rdy

Rd8

R3y

R38

RNy

AM 6B;X 8- i?2 #m##H2 T`Q}H2b `2 `2+Q`/2/ i i?2 #2;BMMBM; Q7 i?2 bBKmHiBQM- M/ i 2+? iBK2 r?2M i?2 ri2` D2ib 7Q`Kb /m`BM; i?2 +QKT`2bbBQM bi;2 Q7 i?2 #m##H2X AM i?2b2 irQ +b2b- Bi Bb i?2 p`vBM; rp2 T`2bbm`2 i?i KF2b i?2 #m##H2 Qb+BHHi2X h?2 BMi2`@#m##H2 M/ #m##H2@`B;B/@rHH BMi2`+iBQMb `2bmHi BM i?2 #m##H2b #2+QKBM; +HQb2` iQ 2+? Qi?2`- M/ KQpBM; iQr`/b i?2 rHH i i?2 bK2 iBK2- /m2 iQ i?2 "D2`FM2b 7Q`+2b (98)X h?2 `B;B/ #QmM/`v ii`+ib i?2 #m##H2b M/ BM/m+2b i?2 ri2` D2ib TQBMiBM; iQ i?2 #QmM/`v- r?B+? KF2b i?2 i`p2HBM; rp2 +b2 (6B;X 8UV) [mBi2 /Bz2`2Mi 7`QK i?2 +b2 Q7 #m##H2b BM  7`22 }2H/ bm#D2+i iQ i?2 mHi`bQmM/ rBi? i?2 bK2 KTHBim/2 (j9)X h?2 #QmM/`v HbQ 2M?M+2b i?2 #m##H2 +QHHTb2 bQ i?i D2i T2M2i`iBQM Q++m`b rBi?BM Dmbi  72r Qb+BHHiBQM +v+H2b- i t∗ = 7.79Xq?2M  HB[mB/ D2i BKT+ib QM i?2 Qi?2` bB/2 Q7  #m##H2- T?vbB+HHv- i?2 p2HQ+Biv BM ~Qr }2H/ #2+QK2b KmHiB@pHm2/c M/ MmK2`B+HHv- i?2 +QH2b+2M+2 M/ `2KQpH Q7 K2b? /m`BM; T2M2i`iBQM ?b iQ #2 +`27mHHv ?M/H2/ BM Q`/2` iQ KBMiBM i?2 MmK2`B+H bi#BHBivX AM i?Bb rQ`F- i?2 pQ`i2t `BM; KQ/2H Bb /QTi2/ iQ bQHp2 i?2 #Qp2 bii2/ T?vbB+H T`Q#H2K (9e- 9d)c M/ bQK2 MmK2`B+H i2+?MB[m2b `2 mb2/ BM Q`/2` iQ KBMiBM i?2 bi#BHBiv /m`BM; +H+mHiBQM rBi? i?2 j. "1J KQ/2H (93- 9N)X q?2M +QKT`BM; i?2 `2bmHib 7Q` i?2 i`p2HBM; M/ biM/BM; rp2b- r2 }M/ i?i i?2`2 Bb Hrvb  T?b2 /Bz2`2M+2 BM i?2 7Q`K2` bBimiBQM- BX2X- i?2 #m##H2 r?B+? 7+2b i?2 rp2 7`QMi 2`HB2` mM/2`;Q2b  Hi2` D2i T2M2i`iBQM- M/ i?2 Qi?2` QM2 2pQHp2b BMiQ  iQ`QB/H #m##H2 Km+? 2`HB2`- b b?QrM BM 6B;X 8UVX h?Bb Bb #2+mb2 i?2 i`p2HHBM; rp2 BM/m+2b T`2bbm`2 ;`/B2Mi BM i?2 ~Qr }2H/ BM x /B`2+iBQM M/ +mb2b i?2 T?b2 /Bz2`2M+2 Q7 i?2 #m##H2Ƕb 2pQHmiBQMX h?2 H27i #m##H2 i?mb b?`BMFb 7bi2` M/ +mb2b ?B;?2` K#B2Mi T`2bbm`2 Q7 i?2 `B;?i #m##H2X 6BMHHv- i?2 `B;?i #m##H2 r?B+? BKK2`b2/ BM i?2 ?B;?2` T`2bbm`2 +QHHTb2b 2`HB2`X 6Q` i?2 biM/BM; rp2 +b2- i?2 #m##H2b 2pQHp2 i i?2 bK2 T+2 M/ i?2 ri2` D2i Bb MQi bi`QM; 2MQm;? iQ T2M2i`i2 i?2 Qi?2` bB/2 Q7 i?2 #m##H2 rHH #27Q`2 `2@2tTMbBQM Q7 i?2 #m##H2 (b22 6B;X 8U#V)X AM 6B;X e- i?2 H27i M/ `B;?i #m##H2b BM 2+? +b2 `2 /2MQi2/ b dz"RǴ M/ dz"kǴ`2bT2+iBp2HvX b +M #2 b22M 7`QK i?2 pQHmK2 ?BbiQ`v +m`p2b- i?2 #m##H2b BM  biM/BM; rp2 }2H/ Qb+BHHi2 7Q` KMv +v+H2b- r?BHbi i?2 Qb+BHHiBQM BMi2MbBiB2b U`2~2+i2/ #v i?2 p`BiBQM BM #m##H2 pQHmK2V Q7 i?2 H27i M/ `B;?i #m##H2b bm#D2+i iQ i?2 i`p2HBM; rp2 `2- `2bT2+iBp2HvH2bb i?M M/ ;`2i2` i?M i?2 BMi2MbBiv BM i?2 biM/BM; rp2 +b2X h?2`27Q`2- i?2 `B;?i #m##H2 BM i?2 i`p2HBM; rp2 T`2b2Mib bi`QM;2` +QHHTb2 rBi?  ?B;?2` D2i p2HQ+Biv- b +M #2 b22M 7`QK 6B;bX eU+V M/ eU/VX h?2 /QrMr`/b KQp2K2Mi Q7 i?2 #m##H2b KBMHv Q++m`b i i?2 +QHHTb2 T?b2 BM 2+? Qb+BHHiBQM +v+H2 (b22 6B;X eU#V)X h?2 biM/BM; rp2 BM/m+2b KQ`2 2pB/2Mi i`MbHiBQM Q7 i?2 #m##H2b iQr`/b i?2 #QmM/`v /m2 iQ i?2 HQM;2` iBK2 Q7 Qb+BHHiBQMX Ry

UV

U#V

U+V

U/V

6B;m`2 e, hBK2 ?BbiQ`B2b Q7 UV i?2 pQHmK2b Q7 i?2 #m##H2b- U#V i?2 +2Mi2` TQbBiBQMb Q7 i?2 #m##H2b- U+V i?2 D2i p2HQ+BiB2b Q7 i?2 }`bi +QHHTb2 M/ U/V i?2 D2i p2HQ+BiB2b Q7 i?2 b2+QM/ +QHHTb2- 7Q` i?2 +b2b BM 6B;X 8X

RR

9XkX 1z2+i Q7 i?2 rp2 7`2[m2M+v h?2 rp2 7`2[m2M+v Bb MQi?2` BKTQ`iMi 7+iQ` i?i +M BM~m2M+2 i?2 #m##H2 +?`+@ i2`BbiB+b- 2bT2+BHHv i?2 #m##H2Ƕb +QHHTbBM; #2?pBQ`X q?2M  #m##H2 /Q2b MQi Qb+BHHi2 iQQ bi`QM;Hv- Bib Mim`H Qb+BHHiBM; 7`2[m2M+v fn +M #2 2tT`2bb2/ b (e- jk) s 1 3∆p 4γ fn = + . UR9V 2πR10 ρ ρR10 RN8

kyy

ky8

kRy

kR8

6Q` i?2 #m##H2b rBi? M BMBiBH bBx2 Q7 5 µm BM i?Bb TT2`- Bib Mim`H 7`2[m2M+v Bb 718 kHzBX2X- i?2 /BK2MbBQMH2bb pHm2 fn∗ = 0.36- ++Q`/BM; iQ i?2 #Qp2 2[miBQMX AM i?Bb bm#b2+iBQMbiM/BM; rp2b rBi? /Bz2`2Mi 7`2[m2M+B2b f ∗ = 0.18- f ∗ = 0.36 M/ f ∗ = 0.72 `2 +QMbB/2`2/ iQ +i QM i?2 #m##H2 TB`X h?2 `2bmHib `2 b?QrM BM 6B;X dX b i?2 rp2 7`2[m2M+v BM+`2b2b- i?2 #m##H2Ƕb pQHmK2 Qb+BHHiBQM T2`BQ/ #2+QK2b b?Q`i2`r?B+? +M #2 b22M 7`QK 6B;X dUVX h?2 rp2 7`2[m2M+v i?i Bb HQr2` i?M i?2 Mim`H 7`2@ [m2M+v BM/m+2b Km+? bi`QM;2` #m##H2Ƕb pQHmK2 Qb+BHHiBQM M/ i?2 #m##H2 +QHHTb2 Q++m`b /m`BM; Bib }`bi +v+H2 Q7 Qb+BHHiBQMX h?2`27Q`2- 7Q` f ∗ = 0.18- i?2 KtBKmK D2i bT22/ 2t@ +22/b Ry UKQ`2 i?M 100 m/sV- b +M #2 b22M BM 6B;X dU#VX M/ 6B;X dU+V b?Qrb i?i i?2 T`2bbm`2 M2` i?2 #QmM/`v i i?2 TQBMi (0, 0, 0.1) BM+`2b2b `TB/Hv iQ R9 UR3 iBK2b Q7 i?2 rp2 KTHBim/2V BM i?2 #m##H2Ƕb +QKT`2bbBQM bi;2X h?2b2 T?2MQK2M `2 /m2 iQ i?2 HQr2` rp2 7`2[m2M+v f ∗ = 0.18 BM/m+2b bi`QM;2` #m##H2 Qb+BHHiBQM- bBM+2 Bib HQM;2` rp2 i`Qm;? KF2b i?2 #m##H2 2tTM/ iQ  H`;2` pQHmK2 M/ i?2 HQM;2` rp2 +`2bi Tmb?2b i?2 #m##H2 BMiQ  bKHH2` pQHmK2 M/ }MHHv +mb2b  7bi +QHHTb2X q?2M i?2 rp2 7`2[m2M+v 2[mHb i?2 Mim`H 7`2[m2M+v Q7 i?2 #m##H2 UBX2X f ∗ = 0.36V`2bQMM+2 TT2`b 7`QK i?2 #2;BMMBM; M/ Hbib 7Q` i?2 }`bi irQ Qb+BHHiBQM +v+H2b- `2bmHiBM; BM  bi`QM;2` TmHbiBQM- ri2` D2i M/ KB;`iBQM (6B;bX dUV- dU#V M/ dU/V)X >Qr2p2`- i?2 bi`QM; #m##H2 Qb+BHHiBQM BM/B+i2b i?i i?2 ?B;?2`@Q`/2` 7`2[m2M+B2b ;`/mHHv bi`i iQ /QKBMi2 Bib KQiBQM M/ i?2 `2bQMMi 7`2[m2M+v i?mb b?B7ibc //BiBQMHHv- i?2 pBb+Qmb 2z2+i ++mKmHi2b M/ ii2Mmi2b i?2 #m##H2 TmHbiBQMX q?2M i?2 rp2 7`2[m2M+v Bb ?B;?2` i?M i?2 Mim`H 7`2[m2M+v Q7 i?2 #m##H2- i?2 #m##H2 TB` Qb+BHHi2b r2FHv M2` Bib 2[mBHB#`BmK bii2 M/ KQp2b iQr`/b i?2 #QmM/`v bHQrHvr?B+? +QMi`B#mi2b HBiiH2 iQ i?2 ~Qr }2H/ p2HQ+Biv M/ i?2 T`2bbm`2 M2` i?2 #QmM/`vX h?2 E2HpBM BKTmHb2 ;Bp2b  ;2M2`H /2b+`BTiBQM Q7 i?2 #m##H2 KQiBQM- BM+Hm/BM; i?2 /B`2+iBQMb Q7 i?2 D2i /2p2HQTK2Mi M/ KB;`iBQM Q7 i?2 #m##H2 (8y)X Ai Bb ;Bp2M #v i?2 bm`7+2 BMi2;`H Q7 i?2 TQi2MiBH ϕ∗ Qp2` i?2 #m##H2 bm`7+2- b ZZ ∗ IK = ϕ∗ n dS. UR8V S

kky

kk8

6`QK 6B;X dU2V- r?2M 7Q+mbBM; QM i?2 z ∗ +QKTQM2Mi Q7 i?2 E2HpBM BKTmHb2- r2 }M/ i?i- 7Q` i?2 HQr rp2 7`2[m2M+v- i?2 #m##H2 TB` mM/2`;Q2b pBQH2Mi Qb+BHHiBQM M/ `2KBMb ii`+i2/ iQ i?2 `B;B/ #QmM/`vX h?Bb i2M/2M+v #2+QK2b r2F2` r?2M i?2 ri2` D2i #2;BMb iQ 7Q`KX 6Q` i?2 rp2 7`2[m2M+v 2[mH iQ i?2 Mim`H 7`2[m2M+v- i?2 #m##H2 TB` KQp2b `TB/Hv iQr`/b i?2 #QmM/`v r?BH2 i?2v Qb+BHHi2 pBQH2MiHv- M/ bHQrb /QrM r?2M i?2 Qb+BHHiBQM #2+QK2b KQ/2`i2X 6Q` i?2 ?B;? rp2 7`2[m2M+v- i?2 #m##H2 TB` Bb `2HiBp2Hv bi#H2 M/ QMHv KQp2b p2`v bHB;?iHv iQr`/b i?2 #QmM/`vX 9XjX 1z2+i Q7 i?2 BMi2`@#m##H2 /BbiM+2 .m`BM; i?2 BM@T?b2 BMi2`+iBQM Q7 i?2 #m##H2 TB`- i?2 ri2` D2i Q7 2+? #m##H2 i2M/ iQ b?QQi iQr`/b i?2 Qi?2` QM2 i i?2 +QHHTb2 bi;2 (b22 6B;X 8U#V)X h?Bb BMi2`@#m##H2 KQiBQM Rk

UV

U#V

U+V

U/V

U2V 6B;m`2 d, hBK2 ?BbiQ`B2b Q7 #m##H2 TB`b bm#D2+i iQ biM/BM; mHi`bQmM/ rp2b rBi? /Bz2`2Mi 7`2[m2M+B2b, UV i?2 #m##H2 pQHmK2- U#V i?2 D2i p2HQ+Biv- U+V i?2 T`2bbm`2 M2` i?2 #QmM/`v i i?2 TQBMi (0, 0, 0.1)- U/V i?2 +2Mi2` TQbBiBQM BM i?2 z ∗ /B`2+iBQM- M/ U2V i?2 E2HpBM BKTmHb2 BM i?2 z ∗ /B`2+iBQMX >2`2 θi = 0 M/ i?2 Qi?2` T`K2i2`b `2 i?2 bK2 b BM 6B;X 8X

Rj

UV Dbb∗ = 6

U#V Dbb∗ = 10

U+V Dbb∗ = 20

U/V Dbb∗ = 40

U2V Dbb∗ = 80

6B;m`2 3, S`Q}H2 Q7 "R Ui?2 H27i #m##H2V i i?2 +QHHTb2 bi;2 BM i?2 biM/BM; mHi`bQmM/ }2H/X h?2 M;H2 θjet #2ir22M i?2 −z ∗ tBb M/ i?2 ri2` D2i Bb UV 45◦ - U#V 25◦ - U+V 9◦ - U/V 3◦ M/ U2V 0◦ - 7Q` θi = 0 M/ Qi?2` T`K2i2`b `2 i?2 bK2 b BM 6B;X 8X

kjy

kj8

∗ #2+QK2b r2F2` b i?2 /BbiM+2 Dbb BM+`2b2bX AM i?Bb bm#b2+iBQM- #m##H2 TB`b /Bbi`B#mi2/ ∗ ?Q`BxQMiHHv rBi? /Bz2`2Mi /BbiM+2b Dbb - bm#D2+i2/ iQ  biM/BM; rp2- `2 BMp2biB;i2/X Pi?2` T`K2i2`b `2 b2i bK2 b BM 6B;X 8X AM 6B;X 3- T`Q}H2b Q7 "R Ui?2 H27i #m##H2V ∗ `2 T`2b2Mi2/X b i?2 /BbiM+2 Dbb BM+`2b2b 7`QK e iQ 3y- i?2 M;H2 #2ir22M i?2 ri2` ∗ ∗ D2i M/ i?2 −z tBb ;`/mHHv /2+`2b2bX 6Q` Dbb = 6 i?2 bi`2M;i?b Q7 i?2 BMi2`@#m##H2 BMi2`+iBQM M/ #m##H2@#QmM/`v BMi2`+iBQM `2 2[mH iQ 2+? Qi?2`- bQ i?2 M;H2 Q7 i?2 D2i Bb 45◦ X h?Bb BMi2`@#m##H2 BMi2`+iBQM `2/m+2b iQ M2;HB;B#H2 b i?2 /BbiM+2 BM+`2b2b iQ 3yX JQ`2Qp2`- i?2 #m##H2 pQHmK2 i D2i T2M2i`iBQM #2+QK2b bKHH2` b i?2 /BbiM+2 BM+`2b2bX h?2 BMi2`@#m##H2 BMi2`+iBQM /2pBi2b i?2 ri2` D2i M/ /2+`2b2b i?2 bi`2M;i? Q7 Bib BKT+i QM i?2 `B;B/ #QmM/`vX ZmMiBiiBp2 `2bmHib `2 b?QrM BM 6B;X NX h?2 #m##H2b 2pQHp2 bvKK2i`B+HHv rBi? `2bT2+i iQ i?2 THM2 y ∗ Oz ∗ BM i?2 biM/BM; rp2 }2H/- bQ QMHv `2bmHib 7Q` i?2 H27i #m##H2 ∗ "R `2 T`2b2Mi2/ 7Q` #`2pBivX q2 2tKBM2/ /Bz2`2Mi #m##H2 /BbiM+2b Dbb p`vBM; 7`QK ∗ e iQ eyy M/ 7QmM/ i?i 7Q` KQbi T`K2i2`b- r?2M Dbb > 80- i?2 BMi2`+iBQM #2ir22M #m##H2b Bb iQQ r2F iQ #2 /2i2+i2/X 6`QK i?2 iBK2 ?BbiQ`B2b Q7 #m##H2 pQHmK2- Bi Bb 7QmM/ i?i i?2 bi`QM;2` #m##H2 BMi2`+iBQM BKT2/2b i?2 #m##H2 Qb+BHHiBQMX h?mb- i?2 #m##H2 i ∗ Dbb = 6 Qb+BHHi2b `2HiBp2Hv r2FHv M/ D2i T2M2i`iBQM /Q2b MQi Q++m` /m`BM; i?2 BMBiBH +v+H2bX 6Q` H`;2` #m##H2 /BbiM+2b- i?2 #m##H2 Kv `2+? ;`2i2` KtBKmK M/ bKHH2` KBMBKmK pQHmK2b- M/ i?2 #m##H2 T2`BQ/ Bb HbQ b?Q`i2`X q?2M i?2 BMi2`+iBQM #2+QK2b r2F2`- i?2 bi`QM;2` #m##H2 TmHbiBQM `2bmHib BM D2i T2M2i`iBQM i i?2 2M/ Q7 i?2 b2+QM/ Qb+BHHiBQM +v+H2- b22 6B;X NUVX h?2 KtBKmK ri2` D2i p2HQ+Biv Bb KQ`2 b2MbBiBp2 iQ i?2 BMi2`@#m##H2 /BbiM+2- b +M #2 b22M 7`QK 6B;X NU#VX h?2 KtBKmK D2i p2HQ+Biv 7Q` r2F ∗ ∗ BMi2`+iBQM UDbb > 300V Bb HKQbi j iBK2b Q7 i?i BM i?2 bi`QM; BMi2`+iBQM +b2 UDbb = 6VX  }iiBM; +m`p2 ?b #22M 7QmM/ 7Q` i?Bb rp2 T`K2i2`−0.58 ∗ ∗ vmax (Dbb ) = −18.55Dbb + 10.11, ∗

UReV

r?B+? T`QpB/2b  T`2/B+iBQM Q7 i?2 ri2` D2i p2HQ+Biv 7Q`  ;Bp2M rp2 }2H/ M/ BMBiBH #m##H2 TB` +QM};m`iBQMX *QK#BMBM; 6B;bX 3 M/ NU#V- i?2 BMi2MbBiv Q7 i?2 ri2` D2i BM i?2 /B`2+iBQM T2`T2M/B+mH` iQ i?2 `B;B/ #QmM/`v +M #2 /2i2`KBM2/ 7`QK ∗ Ijet = vmax cos θjet .

k9y

URdV

h?Bb [mMiBiv Ijet `2T`2b2Mib i?2 BMi2MbBiv Q7 i?2 ri2` D2i i?i BKT+ib QMiQ i?2 #QmM/`v∗ M/ i?Bb BKT+i #2+QK2b bi`QM;2` b Dbb BM+`2b2bX h?2 #m##H2 KQp2b iQ i?2 +2Mi2` THM2 y ∗ Oz ∗ M/ i?2 `B;B/ #QmM/`v 2pB/2MiHv- i i?2 +QHHTb2 T?b2 BM 2+? Qb+BHHiBM; +v+H2- b +M #2 b22M 7`QK 6B;bX NU+V M/ NU/VX b i?2 R9

UV

U#V

U+V

U/V

U2V

U7V

∗ BM  6B;m`2 N, S`K2i2`b Q7 "R Ui?2 H27i #m##H2V BM i?2 #m##H2 TB` 7Q` /Bz2`2Mi #m##H2 /BbiM+2b Dbb biM/BM; mHi`bQmM/ rp2 }2H/, iBK2 ?BbiQ`B2b Q7 UV i?2 #m##H2 pQHmK2- U#V i?2 KtBKmK D2i p2HQ+BivU+V i?2 i`MbHiBQM Q7 i?2 #m##H2 +2Mi2` BM i?2 x∗ /B`2+iBQM M/ U/V BM i?2 −z ∗ /B`2+iBQM- U2V i?2 E2HpBM BKTmHb2 BM i?2 x∗ /B`2+iBQM M/ U7V BM i?2 −z ∗ /B`2+iBQM- 7Q` θi = 0 M/ Qi?2` T`K2i2`b `2 i?2 bK2 b BM 6B;X 8X

R8

UV

U#V

6B;m`2 Ry, *QKT`BbQM Q7 i?2 T`2bbm`2 M2` i?2 rHH i (0, 0, 0.1) M/ D2i p2HQ+Biv Q7 "R 7Q` /Bz2`2Mi /Bbi`B#miBQM 7Q`Kb U?Q`BxQMiH M/ p2`iB+HV Q7 i?2 #m##H2 TB` rBi? i?2 bBM;H2 #m##H2 +b2X S`K2i2`b `2 ∗ ∗ ∗ ∗ = 6 7Q` = 0 7Q` i?2 p2`iB+H /Bbi`B#miBQMc M/ D1∗ = D2∗ = 3- Dbb = 1c D1∗ = 3- D2∗ = 9 M/ Dbb = R20 R10 i?2 ?Q`BxQMiH /Bbi`B#miBQMX h?2 Qi?2` T`K2i2`b Q7 i?2 mHi`bQMB+ rp2 `2 i?2 bK2 b BM 6B;X 8X

k98

k8y

k88

key

ke8

kdy

#m##H2b `2 b2i 7`i?2` rv 7`QK 2+? Qi?2`- i?2 i`MbHiBQM BM x∗ /B`2+iBQM ;`/mHHv 7/2b rv b 2tT2+i2/- ?Qr2p2`- i?2 KQp2K2Mi iQr`/b i?2 −z ∗ /B`2+iBQM #2+QK2b bi`QM;2` M/ 7bi2`X h?Bb `2bmHi +M #2 2tTHBM2/ #v +QMbB/2`BM; i?i i?2 #m##H2 Qb+BHHiBQM Bb }`biHv BM/m+2/ #v i?2 mHi`bQmM/- /m2 iQ i?2 T`BK`v "D2`FM2b 7Q`+2- M/ i?2M i?2 b2+QM/`v "D2`FM2b 7Q`+2 /QKBMi2b i?2 bm#b2[m2Mi #m##H2 #2?pBQ`- `2bmHiBM; BM i?2 #m##H2 TB` ii`+iBM; 2+? Qi?2` M/ KB;`iBM; iQr`/b i?2 #QmM/`v (8R)X 6B;m`2b NU2V M/ NU7V /2b+`B#2 i?2 E2HpBM BKTmHb2X 6Q` Bib +QKTQM2Mi BM i?2 −z ∗ ∗ /B`2+iBQM IKz - i?2 M2;iBp2 pHm2 K2Mb i?i i?2 ;2M2`H KQp2K2Mi Q7 i?2 #m##H2 Bb BM ∗ i?2 −z /B`2+iBQMX q?2M +QK#BM; i?2b2 irQ };m`2b rBi? i?2 pQHmK2 /B;`K- 6B;X NUVr2 }M/ i?i i?2 #bQHmi2 pHm2 Q7 i?2 E2HpBM BKTmHb2 i2M/b iQ /2+`2b2 r?2M i?2 #m##H2 2tTM/b M/ BM+`2b2 r?2M i?2 #m##H2 +QHHTb2bX >2M+2- i?2 E2HpBM BKTmHb2 `2p2Hb i?2 pBQH2M+2 Q7 i?2 #m##H2 Qb+BHHiBQM, i?2 H`;2` pHm2b K2M i?i i?2 #m##H2 Bb ;QBM; i?`Qm;?  ;`2i2` 2tTMbBQM M/ +QKT`2bbBQMX 9X9X 1z2+i Q7 i?2 #m##H2 /Bbi`B#miBQM AM T`2pBQmb bm#b2+iBQMb- Bi rb b?QrM i?i i?2 BM@T?b2 BMi2`+iBQM Q7  ?Q`BxQMiH #m##H2 TB` `2i`/b i?2 #m##H2 +QHHTb2 M/ D2i p2HQ+Biv- +QKT`2/ rBi? i?2 bBM;H2 #m##H2 +b2X AM i?Bb T`i- i?2 /Bbi`B#miBQM Q7 i?2 #m##H2 TB` Bb +QKT`2/ M/ BMp2biB;i2/ BM Q`/2` iQ }M/ B7 i?2`2 Bb Mv FBM/ Q7 +QM};m`iBQM i?i Kv 2M?M+2 i?2 #m##H2 +QHHTb2X ∗ ∗ 6B`biHv-  p2`iB+H #m##H2 TB` rBi? i?2 bK2 BMBiBH #m##H2 bBx2 R10 = R20 = 1 Bb ∗ ∗ +H+mHi2/X h?2 BMi2`@#m##H2 /BbiM+2 Bb D2 − D1 = 6 M/ i?2 #m##H2 "R Bb b2i +HQb2 iQ i?2 #QmM/`v rBi? D1∗ = 3X 6`QK 6B;X RyUV- i?2 #m##H2@BM/m+2/ }2H/ T`2bbm`2 M2` i?2 rHH 7Q` p2`iB+H /Bbi`B#miBQM +b2 Bb Km+? bi`QM;2` i?M i?i Q7 i?2 ?Q`BxQMiH +b2X h?Bb Bb #2+mb2 i?2`2 Bb T`2bbm`2 ;`/B2Mi BM i?2 p2`iB+H /B`2+iBQM i?`Qm;? i?2 ~Qr }2H/- r?B+? Kv BM/m+2 i?2 Qmi@Q7@T?b2 #m##H2 /vMKB+bX h?mb- i?2 Qb+BHHiBQM Q7 "R Kv #2 bi`2M;i?2M2/ #v i?2 mTT2` #m##H2 M/ T`Q/m+2b ?B;?2` T`2bbm`2X >Qr2p2`- i?2 T`2bbm`2 Bb biBHH r2F2` i?M i?2 T`2bbm`2 7`QK i?2 bBM;H2 #m##H2 +b2X M/ b b?QrM BM 6B;X RyU#V- i?2 KtBKmK ri2` D2i bT22/ Bb Km+? bKHH2` i?M i?i Q7 i?2 bBM;H2 #m##H2 +b2X 6m`i?2`KQ`2- i?2 D2i bT22/ bi`ib iQ /2b+2M/ 7i2` Bi `2+?2b i?2 T2FX h?Bb Bb /m2 iQ i?2 #m##H2 Bb 2tTM/BM; /m`BM; i?2 D2i 7Q`KiBQM M/ i?2 D2i rBHH MQi T2M2i`i2 i?2 Qi?2` bB/2 Q7 i?2 #m##H2 7Q` i?Bb bBimiBQMX 6`QK i?2 +b2 #Qp2- r2 }M/ i?i i?2 #m##H2 +QHHTb2 +M #2 2M?M+2/ #v mbBM; i?2 T`2bbm`2 ;`/B2MiX M/ HbQ- B7 i?2 mTT2` #m##H2 ?b  HQM;2` +QKT`2bbBQM bi;2- i?2 #m##H2 Re

UV t∗ = 0.03

U#V t∗ = 6.67

U+V t∗ = 8.00

U/V t∗ = 9.34

U2V t∗ = 9.58

U7V t∗ = 9.66

6B;m`2 RR, h?2 T`2bbm`2 BM i?2 ~Qr }2H/ `QmM/ i?2 #m##H2 "R U+HQb2 iQ i?2 `B;B/ #QmM/`vV i /Bz2`2Mi ∗ ∗ ∗ = 0 M/ θi = 0X h?2 Qi?2` T`K2i2`b Q7 i?2 mHi`bQMB+ = 1.5- D1∗ = 3- D2∗ = 9- Dbb = 1- R20 iBK2bX R10 rp2 `2 i?2 bK2 b BM 6B;X 8X kd8

k3y

k38

kNy

kN8

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Cover letter

1. A modified BEM model considering the viscous effect is used to study the micrometresized bubble pair in ultrasonic waves. 2. Nonlinear bubble deformation results in the shift of the bubbles’ first order frequency. 3. The excitation frequency which is lower than the natural frequency induces the stronger collapse of the bubbles. 4. The vertical distribution of the bubble pair may result in better surface cleaning effect.

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