The dynamic response of a gas bubble entrapped in a cavity on the surface of a submerged solid subject to an acoustic field is investigated in the linear approximation. We derive semi-analytical expressions for the resonance frequency, damping, and interface shape of the bubble. For the liquid phase, we consider two limit cases: potential flow and unsteady Stokes flow. The oscillation frequency and interface shape are found to depend on two dimensionless parameters: the ratio of the gas stiffness to the surface tension stiffness, and the Ohnesorge number, representing the relative importance of viscous forces. We perform a parametric study and show, among others, that an increase in the gas pressure or a decrease in the surface tension leads to an increase in the resonance frequency until an asymptotic value is reached.

1.
D. L.
Miller
, “
Frequency relationships for ultrasonic activation of free microbubbles, encapsulated microbubbles, and gas-filled micropores
,”
J. Acoust. Soc. Am.
104
,
2498
2505
(
1998
).
2.
L. A.
Kuznetsova
,
S.
Khanna
,
N. N.
Amso
,
W. T.
Coakley
, and
A. A.
Doinikov
, “
Cavitation bubble-driven cell and particle behavior in an ultrasound standing wave
,”
J. Acoust. Soc. Am.
117
,
104
112
(
2005
).
3.
P. V.
Zinin
and
J. S.
Allen
, “
Deformation of biological cells in the acoustic field of an oscillating bubble
,”
Phys. Rev. E
79
,
021910
(
2009
).
4.
M.
Postema
and
O. H.
Gilja
, “
Ultrasound-directed drug delivery
,”
Curr. Pharm. Biotechnol.
8
,
335
361
(
2007
).
5.
R. J.
Browning
,
H.
Mulvana
,
M.-X.
Tang
,
J. V.
Hajnal
,
D. J.
Wells
, and
R. J.
Eckersley
, “
Effect of albumin and dextrose concentration on ultrasound and microbubble mediated gene transfection in vivo
,”
Ultrasound Med. Biol.
38
,
1067
1077
(
2012
).
6.
R. H.
Liu
,
J. N.
Yang
,
M. Z.
Pindera
,
M.
Athavale
, and
P.
Grodzinski
, “
Bubble-induced acoustic micromixing
,”
Lab Chip
2
,
151
157
(
2002
).
7.
D. F.
Rivas
,
A.
Prosperetti
,
A. G.
Zijlstra
,
D.
Lohse
, and
H. J. G. E.
Gardeniers
, “
Efficient sonochemistry through microbubbles generated with micromachined surfaces
,”
Angew. Chem., Int. Ed.
49
,
9699
9701
(
2010
).
8.
N.
Bremond
,
M.
Arora
,
C.-D.
Ohl
, and
D.
Lohse
, “
Controlled multibubble surface cavitation
,”
Phys. Rev. Lett.
96
,
224501
(
2006
).
9.
B. M.
Borkent
,
S.
Gekle
,
A.
Prosperetti
, and
D.
Lohse
, “
Nucleation threshold and deactivation mechanisms of nanoscopic cavitation nuclei
,”
Phys. Fluids
21
,
102003
(
2009
).
10.
D. L.
Miller
, “
Experimental investigation of the response of gas-filled micropores to ultrasound
,”
J. Acoust. Soc. Am.
71
,
471
476
(
1982
).
11.
E. A.
Neppiras
,
W. L.
Nyborg
, and
D. L.
Miller
, “
Non-linear behavior and stability of trapped micron-sized cylindrical gas bubbles in an ultrasonic field
,”
Ultrasonics
21
,
109
115
(
1983
).
12.
H.
Rathgen
,
K.
Sugiyama
,
C.-D.
Ohl
,
D.
Lohse
, and
F.
Mugele
, “
Nanometer-resolved collective micromeniscus oscillations through optical diffraction
,”
Phys. Rev. Lett.
99
,
214501
(
2007
).
13.
M. S.
Plesset
and
A.
Prosperetti
, “
Bubble dynamics and cavitation
,”
Ann. Rev. Fluid Mech.
9
,
145
185
(
1977
).
14.
D. L.
Miller
and
W. L.
Nyborg
, “
Theoretical investigation of the response of gas-filled micropores and cavitation nuclei to ultrasound
,”
J. Acoust. Soc. Am.
73
,
1537
1544
(
1983
).
15.
B.
Cuenot
,
J.
Magnaudet
, and
B.
Spennato
, “
The effects of slightly soluble surfactants on the flow around a spherical bubble
,”
J. Fluid Mech.
339
,
25
53
(
1997
).
16.
G. K.
Batchelor
,
An Introduction to Fluid Dynamics
(
Cambridge University Press
,
Cambridge
,
2000
), pp.
353
368
.
17.
L. D.
Landau
and
E. M.
Lifshitz
,
Fluid Mechanics
, 2nd ed. (
Elsevier
,
Amsterdam
,
2004
), p.
525
.
18.
H.
Lamb
,
Hydrodynamics
, 6th ed. (
Dover
,
New York
,
1945
), pp.
46
580
.
19.
A.
Prosperetti
, “
Normal-mode analysis for the oscillations of a viscous liquid drop in an immiscible liquid
,”
J. Mec.
19
,
149
181
(
1980
).
20.
M.
Abramowitz
and
I. A.
Stegun
,
Handbook of Mathematical Functions
(
Dover
,
New York
,
1968
).
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