A fundamental understanding of the effect of a surface on the resonance frequency of bubbles will be useful in the future development of diagnostic medical ultrasound equipment, and specifically in the area of targeted contrast agents for the screening and possible treatment of colon cancer. In this work we turn to the wall effects on the nonlinear resonance frequency response of air bubbles in water, following on from an earlier work which considered linear interactions (E. M. B. Payne, S. Illesinghe, A. Ooi, R. Manasseh, J. Acoust Soc. Am. 118, 2841-2849 (2005)). Numerical results for micron-sized bubbles near a rigid boundary are presented, showing the shift in frequency caused by the presence of the boundary and the presence of other bubbles. Time delay effects are also included, showing a damping of the frequency response. Simulations are limited to the special case where all bubbles are in phase (i.e., the symmetric mode), which refers to the case where all bubbles have the same initial conditions and are subjected to the same excitation pressure field. As a result they have identical time histories. An experimental method for measuring the frequency response of a single bubble attached to a surface is also briefly mentioned

Manasseh, R.

Ooi, A.

Payne, E. M. B.

University of Queensland eSpace

16th Australasian Fluid Mechanics Conference
Crown Plaza, Gold Coast, Australia
2-7 December 2007
Symmetric mode resonance of bubbles near a rigid boundary - the nonlinear case with time
delay effects
E. M. B. Payne1, A. Ooi1 and R. Manasseh2
1Department of Mechanical & Manufacturing Engineering
University of Melbourne, Victoria, 3010 AUSTRALIA
2Thermal & Fluid Dynamics Group CSIRO Manufacturing & Materials Technology,
PO Box 56, Highett, Victoria, 3190 AUSTRALIA
Abstract
A fundamental understanding of the effect of a surface on the
resonance frequency of bubbles will be useful in the future
development of diagnostic medical ultrasound equipment, and
specifically in the area of targeted contrast agents for the screen-
ing and possible treatment of colon cancer. In this work we
turn to the wall effects on the nonlinear resonance frequency
response of air bubbles in water, following on from an earlier
work which considered linear interactions (E. M. B. Payne, S.
Illesinghe, A. Ooi, R. Manasseh, J. Acoust Soc. Am. 118,
2841-2849 (2005)). Numerical results for micron-sized bub-
bles near a rigid boundary are presented, showing the shift in
frequency caused by the presence of the boundary and the pres-
ence of other bubbles. Time delay effects are also included,
showing a damping of the frequency response. Simulations are
limited to the special case where all bubbles are in phase (i.e.,
the symmetric mode), which refers to the case where all bubbles
have the same initial conditions and are subjected to the same
excitation pressure field. As a result they have identical time
histories. An experimental method for measuring the frequency
response of a single bubble attached to a surface is also briefly
mentioned.
Introduction
Nonlinear oscillations of bubbles have been studied extensively
over the years ([1, 2, 3, 4]). Understanding the various mech-
anisms and different types of nonlinear behaviour is important
especially in the field of medical ultrasound diagnostics, where
coated microspheres (known as contrast agents) are injected
into the body for the purpose of highlighting blockages or in-
jured areas.
The current practice in medical contrast imaging is to drive
these contrast agents at ultrasonic frequencies, thereby exciting
them into nonlinear oscillations [5]. The frequency of these os-
cillations is then detected on an ultrasound scanning machine,
which highlights the position of the group of bubbles. In tar-
geted contrast agent imaging, the idea is that the bubbles target a
specific site in the body. A bubble which is attached to its target
site will thus emit a different response from a bubble which is
freely flowing [6]. The difference in frequency response can be
attributed to the presence of the surface, which alters the bubble
dynamics [7, 8]. Modelling the effect of a surface on the fre-
quency response of a system of bubbles is therefore important
in being able to distinguish those bubbles which have attached
from those which are freely flowing.
In this paper, we model a group of bubbles near a rigid surface,
and show how the frequency response shifts due to the presence
of the surface, as well as the presence of other bubbles. We
also consider the effect of time delay, and show how time delay
dampens the maximum bubble radius reached.
Time delay effects for coupled bubble system have been studied
only in recent times, and have been limited to linear oscillations
[9]. We show that time delays can have a significant effect on
the nonlinear frequency response, and, in addition to the other
effects, may help explain discrepencies between theory and ex-
periment.
Theory
The theoretical model used in the present study is the Rayleigh-
Plesset equation with an additional term that represents the
bubble-bubble interaction,
RiR¨i +
3
2
R˙2i −
pmi
ρ
=− pext(t)
ρ
−
N
∑
j=1, j 6=i
1
ri j
d(R2j R˙ j)
dt
, (1)
pmi = pv +
(
P0 +
2σ
Ri0
)(
Ri0
Ri
)3κ
− 2σ
Ri
− 4µ
Ri
R˙i−P0, (2)
where Ri is the time dependent radius of bubble i, pext(t) is the
driving pressure signal, ρ is the density of liquid water, ri j is
the distance between the centers of bubbles i and j, pv is the
vapor pressure, P0 is the atmospheric pressure, σ is the surface
tension, Ri0 is the equilibrium radius of bubble i, and µ is the
viscosity of liquid water. This equation assumes that all the
bubbles oscillate spherically and remain spherical throughout
their cycle. The last term of equation (1) denotes the pressure
of the sounds emitted by the other bubbles measured at the posi-
tion of bubble i, and represents bubble-bubble coupling through
sound. Equation (1) also assumes that the spherical bubbles are
in an unbounded domain.
A rigid wall
To include the effect of a rigid wall, we use the mirror image
theory, and model the presence of the wall by assuming a mirror
image of the real bubble system [7]. Assume that the distance
between the center of bubble i and the wall is Di. For simplicity,
we take all Di to have the same value equal to D, and so equation
(1) is rewritten into
RiR¨i +
3
2
R˙2i −
pmi
ρ
= − pext(t)
ρ
− 1
2D
d(R2i R˙i)
dt
−
N
∑
j=1, j 6=i
(
1
ri j
+
1
si j
) d(R2j R˙ j)
dt
, (3)
where si j = (r2i j + 4D
2)1/2 is the distance between the centers
of bubble i and the mirror image of bubble j. The second term
and the last term on the right-hand side, respectively, denote the
amplitudes of the sounds emitted by the mirror image of bubble
i and by bubble j and its mirror image.
Time delay
Time delay is modelled by assuming a finite sound propaga-
tion between bubble i and bubble j. No time delay is modelled
1336
between image bubbles (i.e., between bubble i and its image),
nor is time delay modelled between bubble i and the image of
bubble j. Thus, with these assumptions, equation (3) becomes
RiR¨i +
3
2
R˙2i −
pmi
ρ
= − pext(t)
ρ
− 1
2D
d(R2i R˙i)
dt
−
N
∑
j=1, j 6=i
(
1
ri j
d
dt
(
R2j(t− τ)R˙ j(t− τ)
)
+
1
si j
d
dt
(
R2j R˙ j
))
, (4)
where τ = ri jc represents the finite time delay between bubble i
and bubble j, and c is the speed of sound in liquid water.
For simplicity, we assume that all bubbles oscillate in phase
(Ri = R j = R), have the same equilibrium radius (Ri0 = R j0 =
R0), and have the same distance between them (ri j = r). Equa-
tion (4) thus simplifies to a single nonlinear ordinary differential
equation
RR¨+
3
2
R˙2− pm
ρ
= − pext
ρ
− 1
2D
(
2RR˙2 +R2R¨
)
− (Nbub−1)
(
1
r
(
2R(t− τ)R˙2(t− τ)
+ R2(t− τ)R¨(t− τ)
)
+
1
s
(
2RR˙2 +R2R¨
))
, (5)
where d(R2R˙)/dt = 2RR˙2 +R2R¨, and Nbub denotes the number
of bubbles in the system. Equation (5) is only valid for Nbub ≤ 3,
and represents the special case where all bubbles have the same
time history and are in phase.
To clarify what we mean by “in phase”, consider two bubbles
(bubble 1 and bubble 2), separated by some distance r in an un-
bounded domain. The bubbles have the same equilibrium radii
and are subjected to the same external excitation pressure field.
The external excitation pressure drives both bubbles into oscil-
lation, resulting in an additional pressure field radiating from
each individual bubble – at the same instant in time. Assuming
a time delay of τ seconds, then bubble 1’s pressure field will take
τ seconds to reach bubble 2, and bubble 2’s pressure field will
take τ seconds to reach bubble 1 (i.e., each bubble’s pressure
field takes the same time to reach its respective neighbour). Be-
cause the bubbles are the same size and are driven by the same
external pressure field, each one will respond in the same way
to their respective neighbour’s delayed pressure field. Thus, the
bubbles will have identical time histories and are considered to
be “in phase”. The same reasoning can be extended to arrange-
ments with higher numbers of bubbles, where the bubbles are
spaced equally from each other.
Numerical Method
A simple Euler time-stepping scheme was used to solve equa-
tion (5). A time delay history buffer was needed to store the
history of R, R˙ and R¨, and used to calculate R(t− τ), R˙(t− τ),
and R¨(t−τ). In all the simulations the bubble system was forced
by an oscillating pressure field pext(t) = PA sin(2pi fextt), where
PA and fext are, respectively, the amplitude and frequency of the
external pressure field.
Table 1 lists the physical parameter values that were used in
simulations.
In all of the simulations, the system was initialised with R(t =
0) = R0, R˙(t = 0) = 0, and R¨(t = 0) = 0. Figure 1 shows a typ-
ical time series for a simulation with Nbub = 1 and D = ∞ (i.e.,
Symbol Value Units
ρ 1000 kg/m3
µ 0.001 kg/ms
σ 0.072 N/m
κ 1
c 1480 m/s
P0 0.1013 MPa
PA 0.01 MPa
Pv 2330 Pa
Table 1: Physical parameters used in simulations.
a single bubble in an unbounded domain). One can see that the
radius R(t) goes through an initial transient phase before set-
tling down to a steady-state oscillation. If the applied pressure
amplitude PA is too large then R(t) undergoes large fluctuations
and does not settle down to a periodic behaviour. In such cases,
the Rayleigh-Plesset equation needs modification [10, 11]. To
avoid such situations PA was set to 0.01 MPa.
0 50 100 150 200
t/T
0.95
1
1.05
R
(t)
/R
0
Figure 1: Time series response for a single bubble (Nbub = 1) in
an unbounded domain (D = ∞). R0 = 100 µm, PA = 0.01 MPa,
fext/ f0 = 0.3.
Figure 2 shows a typical frequency response curve for a bubble
of R0 = 100 µm without time delay. This represents a plot of
(Rmax − R0)/R0 versus fext/ f0, where Rmax is the maximum
radius of the bubble during its steady-state oscillation, and f0 =
1
2piR0
√
3κP0
ρ is the linear natural frequency for a single bubble
in an unbounded domain. The main resonance peak occurs in
the region around fext/ f0 = 1, while harmonic resonances occur
to the left and subharmonic resonances to the right of the main
resonance peak. Such behaviour is in general agreement with
the single-bubble calculations of Lauterborn [1].
To ensure that any effects from the initial transients can be ig-
nored, data samples for the calculation of Rmax were only taken
from 150 < (t/T )< 200, where
T = max(T0,Text),
T0 = 1/ f0,
Text = 1/ fext . (6)
For all simulations a time step increment of 1× 10−9 seconds
was sufficient to ensure stability of the solution. Further de-
creasing the time step increment (to 1×10−10 seconds) yielded
identical results and thus such accuracy was not required.
1337
1
fext/f0
0
0.5
1
(R
m
ax
 
-
 
R
0) 
/ R
0
Figure 2: Frequency response for a single bubble (Nbub = 1,
R0 = 100 µm) in an unbounded domain (D=∞, r =∞), without
time delay.
Results
This section presents numerical results showing the effect of a
rigid surface, nearby bubbles, and time delay on the frequency
response of a system of bubbles.
The effect of a surface
Figure 3 shows a clear shift in the main resonance frequency
towards lower frequencies as the distance to the boundary de-
creases. The same is true for the harmonic and subharmonic
resonance peaks. As shown in previous work (for linear the-
ory [7]), the effect of the wall is to increase the mass loading
on the system, and thereby lower the system’s resonance fre-
quency. As the distance to the boundary decreases, the mass
loading effect increases; further decreasing the resonance fre-
quency. Another effect of the surface is that as the distance to
the boundary decreases, Rmax tends to decrease.
1
fext/f0
0
0.5
1
(R
m
ax
 
-
 
R
0) 
/ R
0
Figure 3: Frequency response for a single bubble (R0 = 100 µm)
without time delay, with varying separation distance to a rigid
boundary. From left to right: D= 3R0, D= 5R0,
D = 10R0, D = 50R0.
The effect of nearby bubbles
Figure 4 shows the effect of the number of bubbles on the fre-
quency response for up to Nbub = 3. As the number of bubbles
in the system is increased, the resonance peaks shift towards
lower frequencies. This can be explained by the fact that nearby
bubbles add to the effective mass of the system, and thus lower
the resonance frequency. This is in agreement with the linear
theory in [7]. Furthermore, Rmax decreases as Nbub increases.
1
fext/f0
0
0.5
1
(R
m
ax
 
-
 
R
0) 
/ R
0
Figure 4: Frequency response (from right to left) for
Nbub = 1, Nbub = 2, Nbub = 3, near a rigid bound-
ary, without time delay (R0 = 100 µm, D= 1.2 mm, r = 2 mm).
The effect of time delay
The effect of time delay on the frequency response for Nbub = 2
and Nbub = 3 are shown in figures 5 and 6 respectively. In both
cases there is no obvious shift in the frequency response as a
result of time delay, although there is a significant reduction
in Rmax around the main resonance peak. This is also true for
the harmonics and subharmonics of the system. In particular,
in figure 6 for Nbub = 3, the subharmonic around fext/ f0 = 1.8
appears to be completely damped out.
1
fext/f0
0
0.5
1
(R
m
ax
 
-
 
R
0) 
/ R
0
Figure 5: Frequency response for Nbub = 2 near a wall (R0 =
100 µm, D = 1.2 mm, r = 2 mm). without time delay,
with time delay.
Previous investigations of time delay effects [9] show that time
delay alters the damping rather than the resonance frequency of
the system. To confirm that this holds for all parameter values,
a more in-depth investigation than presented here is needed.
Experiments
Holt and Crum [3] give experimental results for the resonance
curves of an oscillating, acoustically levitated air bubble in
water. Their results clearly show peaks in experimental data
around the second harmonic resonance, coinciding with their
numerical results. However, there seems to be no equivalent
study for the case where the bubble is oscillating near a surface,
or near other bubbles. The aim of future experiments, therefore,
1338
1
fext/f0
0
0.5
1
(R
m
ax
 
-
 
R
0) 
/ R
0
Figure 6: Frequency response for Nbub = 3 near a wall (R0 =
100 µm, D = 1.2 mm, r = 2 mm). without time delay,
with time delay.
is to measure the resonance response of a bubble near and at-
tached to a rigid surface. This can be done by measuring the
maximum radius of the bubble during its oscillation cycle (a
stroboscopic technique is planned), while sweeping the excita-
tion frequency over the expected resonance range, and identi-
fying the peaks in the frequency response. Allowing for time
delays (which were neglected in the study of Holt and Crum),
may lead to better agreement with experimental data, as was
the case for the study of the pressure distribution around a ris-
ing chain of bubbles [9].
Conclusions
We have shown that, as for linear theory, the effect of a surface
on the nonlinear frequency response is to cause a distinct shift
towards lower frequencies. As the surface is brought closer to
the bubbles, the frequency response shifts further towards lower
frequencies because of the increased mass loading on the sys-
tem. We have also shown that as the number of bubbles in-
creases, the frequency response shifts towards lower frequen-
cies. The magnitude of the resonance peaks also reduced as a
result of the above factors.
For the parameters considered in this paper, the inclusion of
time delay significantly dampened the harmonics and subhar-
monics of the frequency response, with simulations showing a
significant reduction in the height of the peaks. This may help
to explain future experimental results, and ultimately lead to the
fine-tuning of diagnostic tools.
In terms of all the physically achievable combinations of driv-
ing pressure, bubble size, distance to the surface, and distance
between bubbles, this work has only just touched the surface,
and because of the nonlinearity of the response, different trends
may result for other combinations. The assumptions involved in
modelling time delay also need to be thoroughly explored be-
fore any general conlusions can be made regarding the effect of
time delay on the resonance response.
References
[1] Lauterborn, W., Numerical Investigation of nonlinear os-
cillations of gas bubbles in liquids, J. Acoust. Soc. Am.,
59, 1976, 283–293.
[2] Leighton, T.G., Wilkinspn, M., Walton, A.J. and Field,
J.E., Studies of non-linear bubble oscillations in a simu-
lated acoustic field, Eur. J. Phys., 11, 1990, 352–358.
[3] Holt, R.G. and Crum, L.A., Acoustically forced oscil-
lations of air bubbles in water: Experimental results, J.
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[4] Feng, Z.C. and Leal, L.G., Nonlinear Bubble Dynamics,
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[5] Lindner, J.R., Microbubbles in medical imaging: cur-
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[6] Zhao, S., Ferrara, K.W. and Dayton, P.A., Asymmetric os-
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[9] Doinikov, A.A., Manasseh, R., Ooi, A., Time delays in
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Symmetric mode resonance of bubbles near a rigid boundary - the nonlinear case with time delay effects

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Symmetric mode resonance of bubbles near a rigid boundary - the nonlinear case with time delay effects

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