Translational instability of a spherical bubble
in a standing ultrasound wave
Robert Mettin a , Alexander A. Doinikov b
a Drittes
Physikalisches Institut, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1,
37077 Göttingen, Germany
b Institute of Nuclear Problems, Belarus State University, 11 Bobruiskaya Street,
Minsk 220030, Belarus
Abstract
Translational bubble dynamics is much less studied than the dynamics of radial bubble oscillation, while in
many scientific and engineering applications the control of space location of cavitation bubbles is of great practical
importance. This paper aims at the theoretical study of various aspects of the translational motion of a spherical
gas bubble in a high-frequency standing wave. In particular, it is shown that the translational instability that gives
rise to the reciprocal translation of a spherical bubble between the pressure antinode and the pressure node is caused
by the hysteresis in the main resonance of the bubble. Different types of translational trajectories that can occur
in a standing wave are illustrated by numerical simulations. A general classification of the observed translational
trajectories is proposed.
Key words: Cavitation, bubble dynamics, Bjerknes force, translational motion, hysteresis
PACS: 43.25.Yw; 43.35.Ei
1. Introduction
Most of theoretical investigations on bubble dynamics in ultrasound wave fields are devoted to volume and shape oscillations of bubbles. The majority of this work was reviewed by Plesset and Prosperetti [1] and Feng and Leal [2]. The translational
behavior of bubbles is a less studied problem.
A gas bubble in a standing ultrasound wave moves
towards either the pressure antinode or the pressure node. In a relatively weak field, a bubble driven
below resonance (i.e., the driving frequency of the
imposed ultrasound field is below the fundamental resonance frequency of the bubble) moves to
the pressure antinode, while a bubble driven above
resonance moves to the pressure node. The theoretical explanation of this effect is based upon the
well-known formula for the primary Bjerknes force
Preprint submitted to Elsevier
which was derived by Eller [3]. In a high-intensity
field, the translational behavior of bubbles is much
more involved. In particular, bubbles demonstrate
two types of translational instability. The first type
is the well-known erratic “dancing” motion. This
phenomenon was first reported by Gaines [4] and
many others later on [5–9]. Much theoretical work
has been done on the investigation of the dancing
motion [6,8–13], and currently it is generally recognized that this translational instability is caused by
shape oscillations of the bubble which are parametrically excited by the bubble volume oscillation when
the acoustic pressure amplitude exceeds a threshold
value.
The second type of translational instability is unrelated to shape oscillatory modes. This instability
is caused by change in the sign of the primary Bjerknes force at higher acoustic pressures and consists
21 November 2008
in the effect that a spherical bubble driven below
resonance, not undergoing shape distortions, reciprocates between the pressure antinode and the pressure node. This behavior was observed experimentally by Miller [14] and more recently by Khanna et
al. [15] and Kuznetsova et al. [16]. Theoretical investigations were carried out by Watanabe and Kukita
[17] and Doinikov [18]. However, this phenomenon
still remains little-studied, and the physical reason
why such bubbles have no equilibrium position between the antinode and the node, as, for example,
in the case of the counteraction between the Bjerknes force and buoyancy under acoustic levitation, is
not understood so far.
The purpose of this paper is to reveal a physical
mechanism responsible for this second type of translational instability. Another aim is to propose a general classification and to adduce examples of various
translational trajectories that are demonstrated by
bubbles in standing waves.
In Section 2 we describe the theoretical model of
the bubble oscillation and translation. For a larger
parameter space analysis of the primary Bjerknes
force, which is given in Section 3, only the volume
oscillation part of the model is used. The direction
of bubble migration is deduced here from the sign of
the Bjerknes force. The full model considers bubble
translation and its coupling with the radial oscillation. It is used afterwards in Section 4 to verify and
classify the observed translation types of spherical
bubbles in acoustic standing waves.
essary, buoyancy and gravity are added. As a result
of this derivation, the variation of the bubble radius
with time is an input quantity for the translational
equation, while the feedback effect of the translational motion on the radial oscillation is ignored.
This approach was used by Watanabe and Kukita
[17] and Matula [21], and it was also employed in
“particle” models of multibubble fields where the
secondary Bjerknes force is included as well [22,23].
A more correct theoretical approach is to derive
the radial and translational equations simultaneously. This can be done by using the Lagrangian formalism [18,24]. Various refinements of this method
can be found in [25–27]. As a consequence, the radial equation takes the form:
1
3
RR̈ + Ṙ2 = H + G,
2
c
and the translational equation is given by
mb ẍ +
(1)
2π d 3
4π
∂
ρ0 (R u) = − R3 Pex (x, t) + Fdrag .
3
dt
3
∂x
(2)
Here,
γ
3
2σ
R0 − a3
2σ
P0 +
−
−
R0
R 3 − a3
R
#
Ṙ
−4η − P0 − Pex (x, t) ,
(3)
R
1
u2
+
G=
4
ρ0
...
dG
+ 2Ṙ(RR̈ + Ṙ2 ),
H = R2 R + 6RṘR̈ + 2Ṙ3 ≈ R
dt
(4)
u(x, t) = ẋ(t) − vex (x, t),
(5)
π
Fdrag = − CD ρ0 R2 |u|u,
(6)
2
where R(t) is the time-varying radius of the bubble, the overdot denotes the time derivative, c is
the speed of sound in the surrounding liquid, mb =
4πR03 ρg0 /3 is the mass of the bubble, R0 is the equilibrium radius of the bubble, ρg0 is the equilibrium
density of the gas within the bubble, x(t) is the position of the center of the bubble in space (translational trajectory), ρ0 is the equilibrium density of
the surrounding liquid, u is the relative translational
velocity of the bubble with respect to the velocity
of the surrounding liquid, ẋ(t) is the absolute translational velocity of the bubble (with respect to an
inertial frame of reference), vex (x, t) is the velocity
of the surrounding liquid which is generated by the
imposed ultrasound wave at the center of the bubble
as if the bubble were absent, Pex (x, t) is the driving
acoustic pressure at the location of the bubble, Fdrag
2. Theoretical model
There are different approaches to model spherical bubble oscillation and translation. The radial
and the translational equations can be derived
separately. Specifically, for the radial equation the
Rayleigh–Plesset [19] or the Keller–Miksis [20]
model is taken, and the translational equation is
obtained by applying Newton’s second law to a
bubble immersed in a liquid. This means that all
forces acting on the bubble in an acoustically excited liquid are equated to mb ẍ, where mb is the
mass of the gas and vapor inside the bubble, x(t)
is the instantaneous position of the center of the
bubble in space (with respect to an inertial frame of
reference), and the overdot denotes the time derivative. The forces experienced by the bubble normally
include the acoustic radiation (Bjerknes) force, the
added mass force, and the viscous drag force. If nec2
forces for a positionally fixed (non-translating) bubble. This means that equation (1) is solved under
the assumption u ≡ 0, and x is held constant. We
further assume that the bubble is small compared
to the wavelength of the sound field, and thus the
primary Bjerknes force is given by
4π 3
FB = −
R (t)∇Pex (x, t) ,
(9)
3
t
is the viscous drag force on the bubble, P0 is the hydrostatic pressure in the liquid, σ is the surface tension, a is the radius of the bubble’s van der Waals
hard core (a = R0 /8.54 for air), γ is the ratio of specific heats of the gas, and η is the dynamical viscosity of the liquid. The second term on the left-hand
side of (2) is the added mass force and the first term
on the right-hand side of (2) is the primary Bjerknes force. The function H takes into account acoustic radiation losses due to the compressibility of the
surrounding liquid [25]. The drag coefficient CD in
(6) is taken in the form of an empirical law, proposed
by Mei et al. [28,29], that matches the asymptotic
limits of high and low Reynolds numbers:
(
)
−1
8
1
16
−1/2
1+
,
+
1 + 3.315Re
CD =
Re
Re 2
(7)
where Re = 2R|u|ρ0 /η denotes the Reynolds number.
The feedback effect of the translational motion on
the radial oscillation is provided by the term u2 /4 in
(3), which makes equations (1) and (2) really coupled. This term is missing from the equations which
were obtained by Watanabe and Kukita [17]. It is not
difficult to understand the origin of this additional
term. If we calculate the scattered pressure on the
outer surface of the bubble by the time-dependent
Bernoulli equation and then average it over the bubble surface, we obtain:
u2
3
< pscat >R = ρ0 RR̈ + Ṙ2 − ρ0 . (8)
2
4
where < . >t denotes a time average. In the following we suppose a stationary, plane standing wave
which is harmonic in the x-direction and in time,
i.e., Pex (x, t) = Pa cos(kx) cos(2πf t). As the bubble
is assumed to be non-translating, the Bjerknes force
can be split into a spatial component and the time
average,
4π 3
R (t) cos(2πf t)
FB = −∇ [Pa cos(kx)]
3
t
= kPa sin(kx) fB
(10)
with the Bjerknes force coefficient fB , defined by
the term in the averaging brackets. Note that the
sign of this coefficient determines the direction of the
force in the standing wave: a negative fB pushes the
bubble towards higher pressures (the pressure antinode), and a positive value towards lower pressure
(the node). As the bubble oscillation dynamics R(t)
– after transients – depends on the local pressure
amplitude at the bubble’s position x, Pex,a (x) =
Pa cos(kx), and the bubble equilibrium radius R0 ,
we will consider the sign of the primary Bjerknes
force in a parameter plane of Pex,a and R0 .
Similar diagrams have been given before [30,31],
but they were implicitly suggesting that the coefficient fB is unique. However, as is well known from
the study of nonlinear systems in general [32], and
specifically of nonlinear oscillators [33], multiple solutions and attractors can occur. This means that
another important parameter that can determine
the solution is the initial condition. In particular for
the driven bubble it has been shown that saddlenode and period-doubling bifurcations until chaotic
behavior can occur under parameter variation [34],
and thus it is not surprising to observe this phenomenon in broader parameter regions.
In the following we will not explore the full richness of possible dynamics of the driven spherical
bubble and its influence on the primary Bjerknes
force in detail. Instead, we will concentrate on a simple parameter variation for a bubble of fixed size,
namely (i) increasing driving pressure and (ii) decreasing driving pressure. This is a generic case for
Comparing (1) and (8), it is easy to see that the
term u2 /4 in the radial equation is a consequence
of the correction to the scattered pressure which is
caused by the translational motion of the bubble. In
a weak field, this correction is negligible compared
to the scattered pressure produced by the radial oscillation. In a strong field, however, its ignoring is
wrong.
In simulations described below, the following
physical parameters are used: P0 = 101.3 kPa,
ρ0 = 1000 kg/m3 , c = 1500 m/s, η = 0.001 Pa·s,
σ = 0.072 N/m, ρg0 = 1.2 kg/m3 , and γ = 1.4.
3. Stationary analysis
To investigate translational stability in an extended parameter region, we first evaluate the bubble pulsations and the resulting primary Bjerknes
3
bubbles translating in standing waves, either towards a pressure antinode, or towards a node. In the
computations it is essential to simulate a continuous
pressure sweep by keeping the previously calculated
bubble state as initial condition for the next parameter step. It will be seen that in particular the region of the (nonlinearly distorted) main resonance
exhibits an extended area of hysteresis, and that this
can result in the absence of positional equilibria, i.e.,
zeros of the primary Bjerknes force.
Figure 1 presents the overviews in the Pex,a − R0
parameter plane for driving frequencies of 20 kHz,
100 kHz, 300 kHz, and 1 MHz. The interesting region
is left of the linear resonance radius Rres , which can
be found from the Minnaert frequency formula,
2σ
1
3γP
+
(2πf )2 =
(3γ
−
1)
, (11)
0
2
ρ0 Rres
Rres
pressure, a bubble is driven back to lower pressures,
and a bubble for decreasing pressure falls on a value
driving it back upwards. In a detailed simulation of
translation, one would expect an oscillating translational motion in this case, and indeed this will be
the outcome (see Fig. 9c).
Similar hysteresis scenarios repeat at nonlinear
resonances of higher order, although they are less
pronounced. A prominent region is formed around
the second harmonic resonance (R0 ≈ Rres /2).
Here, an overlay of further (ultrasubharmonic) resonances, period doubling, and chaotic solutions
result in rather complicated multiattractor dynamics (compare [36]). In Fig. 1, scattered and irregular
values of the Bjerknes force coefficient appear for
such parameters. The example shown in Fig. 3 is
chosen to pass this second harmonic parameter region. For a bubble of R0 = 5.8 µm and otherwise
the same parameter conditions as in Fig. 2, fB and
Rmax are given. The case is now more complicated
as additionally one stable and one unstable position
are present: at about 55 kPa and at 105 kPa the
Bjerknes coefficient crosses zero with positive slope
(stable) and negative slope (unstable), respectively.
This can be seen in the inset. Furthermore, beyond
110 kPa, the upgoing branch shows chaotic bubble
pulsations, visible by the broadened, i.e., multivalued lines of fB and Rmax .
From Figs. 2 and 3, the absolute strength of the
Bjerknes force on both branches, upgoing and downgoing, respectively, can be compared. It turns out
that the motion away from the pressure antinode
should be much faster than the translation towards
it. This leads to asymmetric, sawtooth-like translation curves of bubble position vs. time, and it is confirmed by results in the next Section. It is a consequence of the larger bubble pulsations on the downgoing branch of the hysteresis loop and should be
typical for positional reciprocation of this type.
and which is met by the lower right curve endings for
zero excitation. The main picture is for all frequencies the same: a line running to the upper left, thus
narrowing the bubble size range that is attracted by
the pressure antinode for higher and higher driving.
For medium high acoustic pressures, this border is
“broadened” by bends, due to nonlinear resonances,
and a “scattered” region appears where complicated
bubble oscillations occur. Very pronounced is a hysteresis zone of triangle shape, connected to the distorted main (linear) resonance. The resonance bends
towards lower radii for larger driving because of the
soft spring character of the oscillating bubble [35],
and its hysteresis zone is limited by saddle-node bifurcations [34]. This bifurcation creates (or annihilates) a pair of periodic solutions, one stable (attractor) and one unstable (repellor). As one stable solution already exists before the bifurcation appears,
we find a coexistence of two different stable bubble
oscillations in such an area. It depends on the initial
conditions, i.e., bubble radius and bubble wall velocity at the starting time, which of both possible states
will be finally reached. It is now peculiar to note that
in wide parameter regions the different pulsations
result also in different signs of the Bjerknes force!
This is illustrated in Fig. 2, where a representative
case is shown. The different solutions manifest themselves in non-unique values of the maximum bubble
radius, Rmax , which is depicted in the lower diagram
of the figure. The vertical jumps are indications of
the saddle-node bifurcations. The source of translational instability of a bubble with the indicated parameters is now the absence of a zero of fB with finite slope: after crossing the bifurcation for upgoing
4. Full simulation of translational motion
In this section, we propose a general classification of the translational trajectories that are demonstrated by bubbles in the standing wave. Trajectories are modeled by numerically solving the system
of coupled equations that have been introduced in
Section 2. They describe generally the instantaneous
radial and translational motion of a spherical bubble in an ultrasound field, and are applied to the
standing wave case. The aim of this Section is both
4
to corroborate the theory presented in the preceding
Section, and to demonstrate examples of trajectories that are caused by the translational instability
due to the hysteresis effect.
does not move towards the pressure antinode. Instead, it moves to an equilibrium point between the
antinode and the node and remains there, undergoing strong enough radial oscillations. Thus, intermediate bubbles no longer obey Eller’s linear theory.
Examples of the trajectories of intermediate bubbles
are shown in Fig. 6. The horizontal lines mean that
the bubbles reached their equilibrium positions and
are staying there. It can be said that intermediate
bubbles are former small bubbles the Bjerknes force
on which changed the sigh and got directed away
from the pressure antinode because of high acoustic
pressure.
The equilibrium position of an intermediate bubble is dependent on the initial bubble radius, acoustic pressure amplitude, and the driving frequency.
Figure 7 shows the equilibrium position, xE , of an
intermediate bubble as a function of R0 at three values of acoustic pressure for f = 300 kHz and f = 1
MHz. By way of example let us consider the 300 kHz
200 kPa curve. The left end of the curve terminates
at about 2.3 µm, and the right end, at 6.25 µm.
These values indicate the size range of intermediate bubbles for the specified values of frequency and
pressure. Below 2.3 µm, the region of small bubbles
lies. What happens to intermediate bubbles beyond
6.25 µm is described in the following subsection.
Turning back to Fig. 7, one can see that, as the
acoustic pressure amplitude increases, increasingly
smaller bubbles turn to intermediate bubbles. Also,
varying the acoustic pressure, one can change the
position of an intermediate bubble, moving it off or
towards the antinode. The gap in the 200 kPa curve
at 1 MHz means that bubbles in this size range again
become “small” and go to the antinode.
The dependence of the equilibrium position of intermediate bubbles on the driving frequency is illustrated in Fig. 8. It is seen that, by increasing the
driving frequency, one can move increasingly smaller
bubbles off the pressure antinode. However, the upper limit of R0 at which the equilibrium position
curves terminate, decreases as well.
4.1. Translation stable bubbles
By translational stability, we mean that a bubble
stays at a certain point of space or executes insignificant translational oscillations around a certain point
of space. Simulations show that there are three cases
of translational stability, which will be referred to
as “large” bubbles, “small” bubbles, and “intermediate” bubbles. They are described successively below.
4.1.1. “Large” bubbles
A “large” bubble is a bubble driven above resonance, i.e., its fundamental resonance frequency is
lower than the driving frequency of the imposed ultrasound field. In conformity with Eller’s formula
[3], large bubbles move to the pressure node where
their radial oscillation dies down. Figure 4 shows
an example of the translational trajectory of a large
bubble for the driving frequency f = 300 kHz and
the acoustic pressure amplitude Pa = 200 kPa. The
position of the bubble in the ultrasound field is displayed in terms of a normalized distance from the
pressure antinode. The distance is defined as x(t)/λ,
where λ is the wavelength of sound in the surrounding liquid. The position 0 of the ordinate axis corresponds to the pressure antinode and 0.25 to the
pressure node. The initial position of bubbles is set
to be in the immediate vicinity of the pressure antinode. Doing so, we assume that bubbles arise due to
acoustic cavitation in the area of high alternating
pressure. The horizontal line means that the bubble
has reached the node and is staying there.
4.1.2. “Small” bubbles
A “small” bubble is a bubble that moves to the
pressure antinode and settles there, or executes
insignificant translational excursions around the
antinode plane. Small bubbles are driven below
resonance and so, formally, their behavior obeys
Eller’s theory [3]. The trajectory of a small bubble
is exemplified in Fig. 5.
4.2. Translation unstable (“traveling”) bubbles
When R0 exceeds a threshold value, bubbles have
no equilibrium space position. Such bubbles will
be called “traveling” bubbles hereinafter. Traveling bubbles execute translational oscillations in the
space between the antinode and the node. Examples
of the trajectories of traveling bubbles are given in
4.1.3. “Intermediate” bubbles
An ”intermediate” bubble is a bubble that, in
spite of the fact that it is driven below resonance,
5
Fig. 9. According to Fig. 7, for 300 kHz the traveling regime sets in at R0 = 6.25 µm, and for 1 MHz,
at R0 = 2.09 µm. Combined diagrams for intermediate and traveling bubbles at Pa = 200 kPa and
f = 300 kHz and 1 MHz are presented in Fig. 10.
The dashed lines show the far and near limits of the
translational trajectories of traveling bubbles. For
300 kHz, the dashed curves merge with the pressure
node at R0 = 12 µm. From this value, the region of
large bubbles begins. For 1 MHz, the dashed curves
merge at R0 = 3.59 µm, then the region of intermediate bubbles follows, and at R0 = 3.74 µm the
region of large bubbles begins.
The existence and the behavior of traveling bubbles are a result of the translational instability described in Section 3. This is also confirmed by Fig. 11
which shows a sharp jump in the amplitude of the
bubble radial oscillation at the moment of the transition from the intermediate to the traveling regime.
that at 110 kPa the bubble goes to the antinode if
it starts at x(0)/λ < 0.052. If, however, the bubble
starts at x(0)/λ > 0.052, it goes to an equilibrium
point with xE /λ ≈ 0.168. At 130 kPa, Fig. 14b,
the boundary between the two types of trajectories
passes at x(0)/λ = 0.1.
Other examples for the start position dependence
are presented in Fig. 15. Figure 15a shows a situation opposite to that shown in Fig. 14. A 4.5µm-radius bubble starting near the antinode at 110
kPa has an equilibrium position situated at a distance from the antinode. If, however, the same bubble starts closer to the node, it has the equilibrium
position at the very antinode. At 200 kPa, Fig. 15b,
the same bubble has two equilibrium positions off
the antinode.
Following the terminology of chaos physics [37],
one can say that in the cases considered the bubble has two coexisting attractors of the type “fixed
point” in the translation state space, the basins of
which are divided at a certain value of the initial
position of the bubble x(0).
4.3. Cumulative translation diagram
Figure 12 shows the type of translational behavior of a bubble depending on the equilibrium radius
and the acoustic pressure amplitude at 300 kHz and
1 MHz. One can see that at f = 300 kHz there is
an area denoted as “overnode bubbles”. It demonstrates that at higher pressures, exceeding about 245
kPa, some of intermediate bubbles go through the
pressure node and settle between the node and the
next antinode, not between the node and the antinode from which they started. Such bubbles will be
called “overnode” bubbles. An example of the trajectory of an overnode bubble is presented in Fig. 13.
Figure 12 reveals that small bubbles can become
traveling bubbles skipping the intermediate regime.
It is also interesting to note that at 1 MHz, for higher
pressures, traveling bubbles are a dominant group.
5. Conclusions
This paper shows that the translational behavior
of bubbles in standing ultrasound waves is rather
involved. At higher acoustic pressures, in addition
to the equilibrium positions at the pressure antinode and the pressure node which are know from the
linear theory for the primary Bjerknes force, spatial equilibria between the antinode and the node
occur. As a result, three groups of bubbles can be
distinguished: “large” bubbles going to the node,
“small” bubbles going to the antinode, and “intermediate” bubbles having equilibrium positions between the antinode and the node. Moreover, there
are translation unstable bubbles, which were named
here “traveling” bubbles. These latter have no equilibrium space position and have to execute translational reciprocating oscillations between the antinode and the node. This study discloses the physical
mechanism responsible for the behavior of traveling bubbles. It has been shown that the occurrence
of traveling bubbles is caused by a specific translational instability that results from the hysteresis in
the main bubble resonance. This study also revealed
that the translational behavior of bubbles possesses
features similar to those recognized for the nonlinear
volume bubble oscillation, such as bifurcations. It
was shown that in certain parameter regions a bub-
4.4. Effect of start position
All the results presented above were obtained assuming that bubbles start in the immediate vicinity
of the pressure antinode. In many cases, the final position of a bubble in space is independent of its initial position. However, there are parameter regions
where this is not the case. For instance, at 300 kHz,
in the range of acoustic pressure from 104 kPa to 133
kPa, a 5.8-µm-radius bubble can reach two equilibrium positions depending on its start position. This
situation is illustrated in Fig. 14. Figure 14a shows
6
ble can have two equilibrium space positions which
are realized depending on the start position of the
bubble in space.
From the given theory, there are no limitations
on driving frequencies and the size of bubbles for
which the translational phenomena described in
this paper can occur. Practically, one should expect
that parameter change and further physical effects
can have a crucial influence. We have checked that
our findings are robust against a change of the polytropic exponent to isothermal bubble oscillations,
and against a slow bubble growth/dissolution by
rectified diffusion. Nevertheless, spherical surface
instability might limit the observability of traveling
bubbles to certain parameter ranges. Preliminary
calculations of parametric instabilities confirmed
the expectation that the described phenomena will
be observed more readily for higher frequencies
and micron-sized bubbles. Under such conditions
the bubbles are more resistant to surface instabilities and can retain the spherical form even at high
enough acoustic pressures. This fact contributes to
the importance of the findings, since MHz frequencies and micron-sized bubbles just form a region
of currently growing interest of various engineering
and medical ultrasound applications.
[7] Strasberg M, Benjamin TB. Excitation of oscillations in
the shape of pulsating gas bubbles; experimental work
(abstract). J Acoust Soc Am 1958;30(7):697.
[8] Benjamin TB. Surface effects in non-spherical motions
of small cavities. In: Davies R, editor. Cavitation in real
liquids. Elsevier; 1964. p. 164-180.
[9] Eller AI, Crum LA. Instability of the motion of a
pulsating bubble in a sound field. J Acoust Soc Am
1970;47(3):762-7.
[10] Benjamin
TB,
Ellis
AT.
Self-propulsion of asymmetrically vibrating bubbles. J
Fluid Mech 1990;212:65-80.
[11] Mei CC, Zhou X. Parametric resonance of a spherical
bubble. J Fluid Mech 1991;229:29-50.
[12] Feng ZC, Leal LG. Translational instability of a bubble
undergoing shape oscillations. Phys Fluids 1995;7:132536.
[13] Doinikov A. Translational motion of a bubble
undergoing shape oscillations. J Fluid Mech 2004;501:124.
[14] Miller DL. Stable arrays of resonant bubbles in a 1MHz standing-wave acoustic field. J Acoust Soc Am
1977;62(1):12-19.
[15] Khanna S, Amso NN, Paynter SJ, Coakley WT.
Contrast agent bubble and erythrocyte behavior in a 1.5
MHz standing ultrasound wave. Ultrasound Med Biol
2003;29:1463-70.
[16] Kuznetsova LA, Khanna S, Amso NN, Coakley TW,
Doinikov AA. Cavitation bubble-driven cell and particle
behavior in an ultrasound standing wave. J Acoust Soc
Am 2005;117(1):104-12.
Acknowledgements
The authors like to thank F. Holsteyns and A.
Lippert for stimulating discussions. R.M. thanks
the members of the cavitation group at Drittes
Physikalisches Institut, Göttingen University, for
continuous support and fruitful joined theoretical
and experimental work.
[17] Watanabe T, Kukita Y. Translational and radial
motions of a bubble in an acoustic standing wave field.
Phys Fluids A 1993;5(11):2682-88.
References
[20] Keller JB, Miksis M. Bubble oscillations of large
amplitude. J Acoust Soc Am 1980;68(2):628-33.
[1] Plesset MS, Prosperetti A. Bubble dynamics and
cavitation. Ann Rev Fluid Mech 1977;9:145-185.
[21] Matula TJ. Bubble levitation and translation under
single-bubble sonoluminescence conditions. J Acoust
Soc Am 2003;114(2):775-81.
[18] Doinikov AA. Translational motion of a spherical bubble
in an acoustic standing wave of high intensity. Phys
Fluids 2002;14(4):1420-25.
[19] Leighton TG. The Acoustic Bubble. London: Academic
Press; 1994.
[2] Feng ZC, Leal LG. Nonlinear bubble dynamics. Ann
Rev Fluid Mech 1997;29:201-43.
[22] Mettin R, Luther S, Ohl C-D, Lauterborn W. Acoustic
cavitation structures and simulations by a particle
model. Ultrason Sonochem 1999;6:25-29.
[3] Eller A. Force on a bubble in a standing acoustic wave.
J Acoust Soc Am 1968;43(1):170-1.
[23] Appel J, Koch P, Mettin R, Krefting D, Lauterborn W.
Stereoscopic high-speed recording of bubble filaments.
Ultrason Sonochem 2004;11:39-42.
[4] Gaines N. Magnetostriction oscillator producing intense
audible sound and some effects obtained. Physics
1932;3:209-29.
[24] Luther S, Mettin R, Lauterborn W. Modeling acoustic
cavitation by a Lagrangian approach. In: Lauterborn
W, Kurz T, editors. Nonlinear acoustics at the turn of
the millennium - Proceedings of the 15th international
symposium on nonlinear acoustics. AIP Conference
Proceedings; 2000. Vol. 524. p. 351-54.
[5] Kornfeld M, Suvorov L. On the destructive action of
cavitation. J Appl Phys 1944;15:495-506.
[6] Benjamin TB, Strasberg M. Excitation of oscillations
in the shape of pulsating gas bubbles; theoretical work
(abstract). J Acoust Soc Am 1958;30(7):697.
7
[25] Doinikov AA. Equations of coupled radial and
translational motions of a bubble in a weakly
compressible liquid. Phys Fluids 2005;17(12):128101.
[26] Doinikov AA. Lagrangian formalism in bubble
dynamics. Far East J Appl Math 2006;25(2):159-77.
[27] Doinikov AA, Dayton PA. Spatio-temporal dynamics of
an encapsulated gas bubble in an ultrasound field. J
Acoust Soc Am 2006;120(2):661-69.
[28] Mei R, Klausner JF, Lawrence CJ. A note on the history
force on a spherical bubble at finite Reynolds number.
Phys Fluids 1994;6:418-420.
[29] Magnaudet J, Eames I. The motion of high-Reynoldsnumber bubbles in inhomogeneous flows. Ann Rev Fluid
Mech 2000;32:659-708.
[30] Lauterborn W, Kurz T, Mettin R, Ohl CD.
Experimental and theoretical bubble dynamics. In:
Prigogine I, Rice SA, editors. Advances in Chemical
Physics. Vol. 110. New York: John Wiley and Sons; 1999.
p. 295-380.
[31] Mettin R. From a single bubble to bubble structures in
acoustic cavitation. In: Kurz T, Parlitz U, Kaatze U,
editors. Oscillations, Waves and Interactions. Göttingen:
Universitätsverlag Göttingen; 2007. p. 171-198.
[32] Guckenheimer J, Holmes PJ. Nonlinear Oscillations,
Dynamical Systems and Bifurcations of Vector Fields.
New York: Springer; 1983.
[33] Thompson JMT, Stewart HB. Nonlinear Dynamics and
Chaos. New York: John Wiley and Sons; 1986.
[34] Parlitz U, Englisch V, Scheffczyk C, Lauterborn W.
Bifurcation structure of bubble oscillators. J. Acoust.
Soc. Am. 1990;88(2):1061-77.
[35] Lauterborn W. Numerical investigation of nonlinear
oscillations of gas bubbles in liquids. J. Acoust. Soc.
Am. 1976;59(2):283-93.
[36] Lauterborn W, Mettin R. Nonlinear Bubble Dynamics
- Response Curves and More. In: Crum LA, Mason
TJ, Reisse JL, Suslick KS, editors. Sonochemistry
and Sonoluminescence. Dordrecht: Kluwer Academic
Publishers; 1999. p. 63-72.
[37] Lauterborn W, Parlitz U. Methods of chaos physics
and their application to acoustics. J Acoust Soc Am
1988;84(6):1975-93.
8
250
200
up
down
up
down
200
Pex,a [kPa]
Pex,a [kPa]
150
100
150
100
50
50
20 kHz
100 kHz
0
0
0
20
40
60
80
100
R0 [µm]
120
140
160
0
180
300
5
10
15
20
R0 [µm]
25
30
35
400
up
down
up
down
350
250
300
Pex,a [kPa]
Pex,a [kPa]
200
150
100
250
200
150
100
1 MHz
50
50
300 kHz
0
0
0
2
4
6
R0 [µm]
8
10
12
0
0.5
1
1.5
2
R0 [µm]
2.5
3
3.5
4
Fig. 1. Locations of zero Bjerknes coefficient, fB = 0, for the acoustic frequencies 20 kHz, 100 kHz, 300 kHz, and 1 MHz.
Zero crossings for increasing (“up”) and for decreasing (“down”) driving pressure are indicated by solid and dashed lines,
respectively. The vertical lines in the 300 kHz plot at 8 and 5.8 µm mark the parameter scans of Figs. 2 and 3.
9
600
1200
R0 = 8 µm
500
10
400
0
300
-10
1000
fB [µm3]
fB [µm3]
800
600
0
200
400
100
200
0
-100
0
up
down
50
75
Pex,a [kPa]
100
60
80
100 120
R0 = 5.8 µm
0
25
40
up
down
-200
-200
0
20
125
25
50
150
75
Pex,a [kPa]
100
125
150
25
25
R0 = 8 µm
20
Rmax [µm]
up
down
15
10
R(t) [µm]
Rmax [µm]
R(t) [µm]
20
20
5
20
15
10
5
0
50
5
0
0
1
2
periods
3
4
10
R0 = 5.8 µm
1 per.
2
3
4
75
Pex,a [kPa]
100
125
up
down
0
0
25
10
5
0
0
15
15
0
150
25
50
75
Pex,a [kPa]
100
125
150
Fig. 3. Hysteresis in fB (top) and Rmax (bottom) for variable
Pex,a and f = 300 kHz, R0 = 5.8 µm. Dynamics resulting
from upgoing (downgoing) driving pressure is given by solid
(dashed) lines. The top inset magnifies part of the graph,
showing two driving pressures with zero Bjerknes force. The
bottom inset shows the bubble radius R(t) vs. time in driving
periods for the coexisting solutions at Pex,a = 125 kPa, one
oscillation being chaotic.
Fig. 2. Hysteresis in the Bjerknes force coefficient fB (top)
and the maximum radius Rmax (bottom) for variable driving
pressure Pex,a and f = 300 kHz, R0 = 8 µm. Dynamics
resulting from upgoing (downgoing) driving pressure is given
by solid (dashed) lines. The inset shows the bubble radius
R(t) vs. time in driving periods for the coexisting solutions
at Pex,a = 50 kPa.
10
Pressure node
0.25
x(t) / l
0.2
f = 300 kHz
Pa = 200 kPa
R0 = 12 mm
0.15
0.1
0.05
Pressure antinode
200
400
600
800
1000
Acoustic cycles
Fig. 4. Translational trajectory of a large bubble in a plane
standing wave.
x(t) / l
0.00001
f = 300 kHz
Pa = 200 kPa
R0 = 1.5 mm
0.0
500
1000
1500
2000
Acoustic cycles
Fig. 5. Translational trajectory of a small bubble in a plane
standing wave.
R0 = 3.5 mm
0.12
3.2 mm
x(t) / l
0.10
0.08
3.0 mm
0.06
2.8 mm
0.04
R0 = 2.5 mm
0.02
500
1000
1500
2000
2500
Acoustic cycles
Fig. 6. Translational trajectories of intermediate bubbles for
f = 300 kHz and Pa = 200 kPa.
11
Pressure node
0.25
Pressure node
0.25
f = 1 MHz
f = 300 kHz
0.20
xE / l
xE / l
0.20
0.15
225 kPa
0.10
0.15
300 kPa
200 kPa
0.10
250 kPa
0.05
Pressure antinode
Pressure antinode
2
3
200 kPa
0.05
175 kPa
200 kPa
4
5
6
0.6
0.8
1
1.2
1.4
R0 [mm]
R0 [mm]
Fig. 7. Equilibrium position of intermediate bubbles at different acoustic pressures.
12
1.6
1.8
2
Pressure node
0.25
Pa = 200 kPa
xE / l
0.20
750 kHz
500 kHz
300 kHz
0.15
0.10
0.05
Pressure antinode
2
3
4
5
6
R0 [mm]
Fig. 8. Equilibrium position of intermediate bubbles at different driving frequencies.
13
(b)
0.25
0.2
0.2
x(t) / l
x(t) / l
(a)
0.25
0.15
0.1
0.15
0.1
R0 = 6.3 mm
R0 = 7.0 mm
0.05
0.05
500
1000
1500
500
2000
Acoustic cycles
1500
2000
(d)
0.25
0.25
0.2
0.2
x(t) / l
x(t) / l
(c)
0.15
R0 = 8.0 mm
0.1
0.15
R0 = 9.0 mm
0.1
0.05
0.05
500
1000
1500
2000
500
Acoustic cycles
1000
1500
2000
Acoustic cycles
(e)
(f)
0.25
0.25
0.2
0.2
x(t) / l
x(t) / l
1000
Acoustic cycles
0.15
R0 = 10.0 mm
0.1
0.15
R0 = 10.5 mm
0.1
0.05
0.05
500
1000
1500
2000
500
Acoustic cycles
1000
1500
Acoustic cycles
Fig. 9. Translational trajectories of traveling bubbles for f = 300 kHz and Pa = 200 kPa.
14
2000
Pressure node
0.25
0.25
Pressure node
0.20
0.15
xE / l
xE / l
0.20
f = 300 kHz
Pa = 200 kPa
0.10
0.15
f = 1 MHz
Pa = 200 kPa
0.10
0.05
0.05
Pressure antinode
0
Pressure antinode
2
4
6
8
10
1.5
12
R0 [mm]
2
2.5
3
3.5
R0 [mm]
Fig. 10. Equilibrium position of intermediate bubbles (curved solid lines) and the far and near limits of the translational
trajectories of traveling bubbles (dashed lines). The two horizontal solid lines in the 1 MHz plot display the equilibrium position
of small bubbles.
15
25
Rmax [mm]
20
f = 300 kHz
Pa = 200 kPa
15
10
5
Intermediate
bubbles
4
Traveling bubbles
6
8
10
12
R0 [mm]
Fig. 11. The maximum radius of a bubble versus its equilibrium radius: Transition from the intermediate to the traveling regime.
Fig. 12. Cumulative translation diagrams for f = 300 kHz
and 1 MHz.
16
0.3
x(t) / l
0.25
Pressure node
0.2
R0 = 4.0 mm
0.1
Pressure antinode
200
400
600
800
1000
Acoustic cycles
Fig. 13. Example of the trajectory of an overnode bubble at
f = 300 kHz and Pa = 275 kPa.
17
0.25
(a)
0.25
0.20
x(t) / l
x(t) / l
0.20
(b)
0.15
f = 300 kHz
Pa = 110 kPa
R0 = 5.8 mm
0.10
0.05
0.15
f = 300 kHz
Pa = 130 kPa
R0 = 5.8 mm
0.10
0.05
500
1000
1500
2000
2500
3000
500
1000
1500
2000
Acoustic cycles
Acoustic cycles
Fig. 14. Example of a bubble with two equilibrium space positions depending on the start position of the bubble.
(a)
(b)
0.25
0.25
f = 300 kHz
Pa = 110 kPa
R0 = 4.5 mm
0.15
0.20
x(t) / l
x(t) / l
0.20
0.15
0.10
0.10
0.05
0.05
500
1000
1500
2000
2500
3000
f = 300 kHz
Pa = 200 kPa
R0 = 4.5 mm
500
Acoustic cycles
1000
1500
Acoustic cycles
Fig. 15. Examples of the start position dependence of bubble trajectories.
18
2000