Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.
REFERENCES
1.
G. N.
Kuznetsov
and I. E.
Shchekin
, “Interaction of pulsating bubbles in a viscous liquid
,” Sov. Phys. Acoust.
18
, 466
–469
(1973
).2.
E. A.
Zabolotskaya
, “Interaction of gas bubbles in a sound field
,” Sov. Phys. Acoust.
30
, 365
–368
(1984
).3.
A.
Harkin
, T. J.
Kaper
, and A.
Nadim
, “Coupled pulsation and translation of two gas bubbles in a liquid
,” J. Fluid Mech.
445
, 377
–411
(2001
).4.
A. A.
Doinikov
, “Translational motion of two interacting bubbles in a strong acoustic field
,” Phys. Rev. E
64
, 026301
–1
026301
–6
(2001
).5.
O. V.
Voinov
and A. M.
Golovin
, “Lagrange equations for a system of bubbles of varying radii in a liquid of small viscosity
,” Fluid Dyn.
5
, 458
–464
(1970
).6.
Yu. A.
Ilinskii
and E. A.
Zabolotskaya
, “Cooperative radiation and scattering of acoustic waves by bubbles in liquid
,” J. Acoust. Soc. Am.
92
, 2837
–2841
(1992
).7.
N.
Wang
and P.
Smereka
, “Effective equations for sound and void wave propagation in bubbly fluids
,” SIAM J. Appl. Math.
63
, 1849
–1888
(2003
).8.
A. A.
Doinikov
, “Mathematical model for collective bubble dynamics in strong ultrasound fields
,” J. Acoust. Soc. Am.
116
, 821
–827
(2004
).9.
E. A.
Zabolotskaya
, Yu. A.
Ilinskii
, G. D.
Meegan
, and M. F.
Hamilton
, “Bubble interactions in clouds produced during shock wave lithotripsy
,” Proceedings of the 2004 IEEE International UFFC Joint 50th Anniversary Conference
, Montreal, Canada
, pp. 890
–893
. The first two terms on the right-hand side of Eq. (10) should be divided by a factor of 6. The results in Fig. 5 were obtained using the corrected equation.10.
M. F.
Hamilton
, Yu. A.
Ilinskii
, G. D.
Meegan
, and E. A.
Zabolotskaya
, “Interaction of bubbles in a cluster near a rigid surface
,” ARLO
6
, 207
–213
(2005
).11.
Yu. A.
Ilinskii
, M. F.
Hamilton
, E. A.
Zabolotskaya
, and G. D.
Meegan
, “Influence of compressibility on bubble interaction
,” in Innovations in Nonlinear Acoustics
, Proceedings of the 17th International Symposium on Nonlinear Acoustics
, edited by A. A.
Atchley
, V. W.
Sparrow
, and R. M.
Keolian
(AIP
, New York
, 2006
), pp. 303
–310
.12.
Yu. A.
Pishchalnikov
, O. A.
Sapozhnikov
, M. R.
Bailey
, J. C.
Williams
Jr., R. O.
Cleveland
, T.
Colonius
, L. A.
Crum
, A. P.
Evan
, and J. A.
McAteer
, “Cavitation bubble cluster activity in the breakage of kidney stones by lithotripter shockwaves
,” J. Endourol.
17
, 435
–446
(2003
).13.
G.
Russo
and P.
Smereka
, “Kinetic theory for bubbly flow. I. Collisionless case
,” SIAM J. Appl. Math.
56
, 327
–357
(1996
).14.
V. M.
Teshukov
and S. L.
Gavrilyuk
, “Kinetic model for the motion of compressible bubbles in a perfect fluid
,” Eur. J. Mech. B/Fluids
21
, 469
–491
(2002
).15.
16.
17.
In Eqs. (2.1) and (2.2) of
Harkin
et al. (Ref. 3) the positions of the fourth-order terms and should be reversed. As a result, changes are required in the fourth-order terms of their Eqs. (3.3) and (3.5) for the radial motion. Sign changes are also required in the third-order terms of their Eqs. (3.5), (3.8), and (3.10) for the translational motion. However, subsequent calculations in the paper are based on equations in which terms above second order are discarded, and therefore the indicated changes are of no consequence to the conclusions in the paper.18.
L. D.
Landau
and E. M.
Lifshitz
, The Classical Theory of Fields
, 4th English ed. (Butterworth-Heinemann
, Oxford
, 2000
), Sec. 42.19.
L. D.
Landau
and E. M.
Lifshitz
, Mechanics
, 3rd ed. (Butterworth-Heinemann
, Oxford
, 1993
), Secs. 2 and 40.20.
A. J.
Reddy
and A. J.
Szeri
, “Coupled dynamics of translation and collapse of acoustically driven microbubbles
,” J. Acoust. Soc. Am.
112
, 1346
–1351
(2002
).21.
J.
Magnaudet
and D.
Legendre
, “The viscous drag force on a spherical bubble with a time-dependent radius
,” Phys. Fluids
10
, 550
–554
(1998
).22.
S. L.
Gavrilyuk
and V. M.
Teshukov
, “Drag force acting on a bubble in a cloud of compressible spherical bubbles at large Reynolds numbers
,” Eur. J. Mech. B/Fluids
24
, 468
–477
(2005
).© 2007 Acoustical Society of America.
2007
Acoustical Society of America
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