Ultrasonics Sonochemistry 27 (2015) 153–164
Contents lists available at ScienceDirect
Ultrasonics Sonochemistry
journal homepage: www.elsevier.com/locate/ultson
The effect of high viscosity on the collapse-like chaotic and regular
periodic oscillations of a harmonically excited gas bubble
} s ⇑, Kálmán Klapcsik
Ferenc Hegedu
Budapest University of Technology and Economics, Faculty of Mechanical Engineering, Department of Hydrodynamic Systems, P.O. Box 91, 1521 Budapest, Hungary
a r t i c l e
i n f o
Article history:
Received 6 December 2014
Received in revised form 20 April 2015
Accepted 12 May 2015
Available online 16 May 2015
Keywords:
Bubble dynamics
Bifurcation structure
Chaos
Keller–Miksis equation
Continuation technique
High viscosity
a b s t r a c t
In the last decade many industrial applications have emerged based on the rapidly developing ultrasonic
technology such as ultrasonic pasteurization, alteration of the viscosity of food systems, and mixing
immiscible liquids. The fundamental physical basis of these applications is the prevailing extreme conditions (high temperature, pressure and even shock waves) during the collapse of acoustically excited bubbles. By applying the sophisticated numerical techniques of modern bifurcation theory, the present study
intends to reveal the regions in the excitation pressure amplitude–ambient temperature parameter plane
where collapse-like motion of an acoustically driven gas bubble in highly viscous glycerine exists. We
report evidence that below a threshold temperature the bubble model, the Keller–Miksis equation,
becomes an overdamped oscillator suppressing collapse-like behaviour. In addition, we have found periodic windows interspersed with chaotic regions indicating the presence of transient chaos, which is
important from application point of view if predictability is required.
Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction
Although the geometry of a single spherical bubble is rather
simple, the physics of the bubble oscillation, however, can be very
complicated. The wall velocity of a bubble can accelerate to extremely high values due to the inertia of the liquid domain, resulting
in a minimum bubble size many orders of magnitude smaller than
the average. This process often referred to as the collapse phase. At
this minimum bubble radius the temperature and pressure can be
as high as 1000 bar and 8000 K, respectively [1]. Due to such extremely high temperatures during the collapse phase, chemical reactions can take place yielding various reaction products. These are
the keen interest of sonochemistry [2–5], or the spectroscopy in
a laser induced cavitation bubble [6,7].
In the last decade ultrasonic technology has began to develop
very rapidly. The main objective of the applications is to enhance
the mass, heat and momentum transfer between the various
phases by taking the advantage of the above mentioned extreme
conditions during bubble collapse. A promising technology in food
preservation, for instance, is the ultrasonic pasteurization. At moderate temperature (50 °C) the membrane of the bacterial organisms weakens enough to become less resistant to cavitational
⇑ Corresponding author. Tel.: +36 1 463 1680; fax: +36 1 463 3091.
} s), [email protected]
E-mail addresses: [email protected] (F. Hegedu
(K. Klapcsik).
http://dx.doi.org/10.1016/j.ultsonch.2015.05.010
1350-4177/Ó 2015 Elsevier B.V. All rights reserved.
damage. With this novel innovation Knorr et al. [8] could successfully reduce the Escherichia coli in liquid whole egg. The alteration
of the viscosity of many food systems such as tomato puree is also
possible with ultrasound since cavitation causes shear stress that
decreases the viscosity of thixotropic fluids. With high enough
energy the alteration becomes permanent by reducing the molecular weight of the substances. Examples for viscosity reduction
were published by Seshadri et al. [9] and Iida et al. [10]. During
the collapse of cavitation bubbles shock waves are generated causing very efficient mixing of two immiscible liquids. Canselier et al.
[11] and Freitas et al. [12] reported the production of fine, highly
stable emulsions. Moreover, the possible occurrence of rectified
diffusion is the basis of novel degassing technologies [13,14].
These applications were the main motivation to investigate the
oscillations of spherical gas bubbles in liquid glycerine, which is
used in many medical, pharmaceutical and personal care preparations. The choice of the substance is also important from point of
view of the available knowledge, since the majority of the papers
are related to water. Some exceptions, for instance, is the paper
of Toegel et al. [15] who found that high viscosity can destabilize
the position of the bubble trapped in an acoustic field; or the study
of Englert et al. [16] revealed that the luminescence pulse duration
in a water–glycerine mixture is increased by a factor of two as the
glycerol concentration increases by 33%. The oscillations of gas
bubbles in other kind of liquid material have also been
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
154
investigated, such as, in hydraulic oil [17], Powell–Eyring fluids
[18] or in polymer solutions [19–22].
The damping effect of an oscillating bubble can be classified
into three physical categories, namely, viscous, acoustic and thermal damping [23]. Due to the very high viscosity of the glycerine,
approximately three orders of magnitude larger than of water,
makes the hunting for collapse-like bubble oscillation difficult.
The very high damping rate tries to decrease the maximum bubble
wall velocity and thus softens the impact of the bubble collapse.
We shall see, however, that with the aid of the modern nonlinear
theory and its rapidly evolving, sophisticated numerical methods,
such as, the pseudo-arc length continuation technique, the determination of the parameter regions of the collapse-like oscillations
becomes only a minor problem. These efficient numerical algorithms, related mostly to the topic of nonlinear dynamics, are
started to spread in the field of bubble dynamics in the recent years
[24,22,25–33], but they have already been applied successfully in
other branches of science [34–40].
As the desired solutions have high bubble wall velocity, the consideration of the liquid compressibility is necessary at least as a
first order approximation. Therefore, the Keller–Miksis equation
[41] was applied during the computations. According to the ultrasonic technology, the most important parameters were the pressure amplitude and frequency of the excitation. Because of the
very strong dependence of the viscosity on the ambient temperature, its influence on the dynamics is also significant, and it was
regarded as a secondary scaling or control parameter.
2. Mathematical model
Because of the possibility of large amplitude oscillations, the
consideration of liquid compressibility is necessary. In this paper,
the well known Keller–Miksis equation [41] is used with minor
modifications [42], in which the retarded time from the original
equation was eliminated. The form of the modified equation is
1
!
!
_
R_
€ þ 1 R 3 R_ 2 ¼
RR
cL
3cL 2
1þ
!
R_
R d ðpL p1 ðtÞÞ
þ
;
cL cL dt
qL
2.1. Parameters and material properties
ð1Þ
where RðtÞ is the time dependent bubble radius; qL , cL are the liquid
density and sound speed, respectively; pL is the pressure at the bubble wall in the liquid domain and p1 ðtÞ is the pressure far away
from the bubble consisting of static and periodic components written as
p1 ðtÞ ¼ P1 þ pA sinðxtÞ;
ð2Þ
where P1 is the ambient pressure, pA and x are the pressure amplitude and angular frequency of the excitation, respectively. The bubble content is a mixture of glycerine vapour and non-condensable
gas; we treat both as ideal gases. This means that the pressure
inside the bubble is the sum of the partial pressures of the vapour
pV and the gas pG . The relationship between the pressures on the
two sides of the bubble wall is described by the mechanical balance
at this interface
pG þ pV ¼ pL þ
R_
2r
þ 4l L ;
R
R
ð3Þ
where r is the surface tension and lL is the liquid dynamic
viscosity.
The vapour pressure inside the bubble is constant but its value
depends on the ambient temperature T 1 ; while the gas content
obeys a simple polytropic relationship
3n
Ro
pG ¼ pgo
R
with a polytropic exponent n = 1.4 (adiabatic behaviour). The reference pressure pgo and radius Ro determine the mass of gas inside the
bubble and therefore the average size of the bubble.
At this point, the discussion of the validity of the applied bubble
model is necessary. It is well-known that the assumption of adiabatic gas behaviour is a severe oversimplification in many cases,
see e.g. [43–45]. However, the comparison between our numerical
and former experimental results, presented in Fig. 1, shows that
the adiabatic behaviour describes the dynamics very well. The
radius-time curve of a laser-induced gas bubble in Fig. 1 is similar
to those in [46]. The ambient properties of the glycerine, T 1 and
P1 , were also similar to those applied throughout the present
study. Although the bubble exhibits free oscillations about its equilibrium radius, the remarkable agreement implies that our present
model provides a good qualitative description of the behaviour of
the harmonically excited bubble as well.
The evaporation/condensation (possibly with non-equilibrium
thermodynamics) can play an important role in the dynamics of
the bubble if the saturation vapour pressure is comparable with
the gas pressure in the bubble interior [47,48]. Due to the very
low amount of glycerine vapour inside the bubble at the applied
temperature range, the ratio of the vapour and gas partial pressures is less than 1:100000, the effect of evaporation and condensation can definitely be neglected.
In spite of the relatively large bubbles observed during the
experiment of [46], in the range of tenth of millimetres, the spherical shape of the bubbles were exceptionally stable. From the differential equation describing the dynamics of the surface waves
of an individual bubble [49], it is obvious that the viscosity has significant effect on the stability. Although the majority of the papers
deal with gas bubbles in water and numerical analyses are absent
for glycerine, its very high viscosity supports the aforementioned
experimental observation. The spherical instability due to the bubble–bubble interaction in clusters [50,51] or the presence of solid
boundary [52,53], liquid surface [54] and positional stability due
to the primary Bjerknes force [55,56] were not modelled.
ð4Þ
} s [27], all the parameters in
By following the concept of Hegedu
system (1)–(4) can be specified with only five quantities. The material properties of a pure substance depend in general on the ambient pressure P1 and temperature T 1 making these two ambient
properties as the main parameters. Specifically, in our case the
material properties depend only on the ambient temperature and
their pressure dependence are neglected. The tabulated values of
the material properties can be found in Appendix A. To describe
the bubble size, one need to prescribe the equilibrium radius RE
of the unexcited bubble (pA ¼ 0) or, equivalently, the mass of gas
inside the bubble, see below. Eventually, the properties of the excitation, namely, the pressure amplitude pA and the frequency x are
also needed.
For a given mass of gas mG and ambient temperature T 1 the
equilibrium radius RE is determined by the static mechanical balance (again pA ¼ 0) at the bubble wall:
0 ¼ pV P1 þ pgo
3n
Ro
2r
:
RE
RE
ð5Þ
As it is noted earlier, the reference quantities pgo and Ro define the
mass of gas
mG ¼
4pgo R3o p
3RT 1
ð6Þ
inside the bubble, where R is the specific gas constant. Therefore,
one can specify the reference properties pgo and Ro , which
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
155
The reference properties are
pref ¼ qL R2E
x 2
2p
ð14Þ
;
lref ¼ cL qL RE ;
A
pref
¼ c L q L RE
pBref ¼ cL qL RE
ð15Þ
x
2p
¼ lref
x
2p
;
x
A 1
¼ pref
:
2
2p
ð2pÞ
ð16Þ
ð17Þ
The dimensionless Mach number is
M¼
Fig. 1. Comparison of the numerically obtained bubble radius vs. time curves (blue
curve) by assuming adiabatic gas behaviour (n = 1.4) with the experimental results
(black curve).
determine the mass of gas via Eq. (6) and the equilibrium radius by
means of Eq. (5). Alternatively, as it is used in the present paper, one
can specify the equilibrium radius and compute the reference properties pgo and Ro . Now, let us choose Ro to be the equilibrium radius
RE itself, and then the required reference gas pressure pgo to satisfy
Eq. (5) is
pgo
2r
¼
ðpV P 1 Þ:
RE
ð7Þ
RE xy2
:
2pcL
ð18Þ
According to Eqs. (4) and (7), the gas pressure inside the bubble
becomes
pG ¼
2r
ðpV P 1 Þ
RE
3n
1
:
y1
ð19Þ
The pressure outside the bubble at the bubble wall, and the pressure far away from the bubble are
pL ¼ pG þ pV 2r 1 4lL x y2
;
RE y 1
2p y1
ð20Þ
and
p1 ðsÞ ¼ P1 þ pA sinð2psÞ;
ð21Þ
The remaining two parameters are related to the excitation
itself, namely, the pressure amplitude pA and the angular frequency
x, see Eq. (2). As the angular frequency can vary over several
orders of magnitude its normalization with a suitable reference
quantity is reasonable. The linear eigenfrequency of the undamped
system corresponding to the equilibrium radius is
respectively. Observe, that according to Eq. (21) the period of excitation in the dimensionless system is unity (so ¼ 1).
ð8Þ
The usual way to investigate a periodically driven dynamical
system, and seek large amplitude oscillations is to present amplification diagrams or frequency response curves [57–59]. Examples
of such curves are given in Fig. 2 where the maximum dimensionless bubble radius ymax
of the stable periodic solutions are plotted
1
as a function of the relative frequency xR at two different pressure
amplitudes pA and at several ambient temperatures T 1 by keeping
all the other parameters constant.
Each curve was computed by increasing the relative frequency
from 0.01 to 2 with an increment of 0.01. At each frequency five
simulations were carried out by a simple initial value problem
(IVP) solver (Runge–Kutta scheme with fifth order embedded error
estimation) with random initial values to reveal the coexisting
stable solutions (attractors). After convergence the maximum absolute value of each component (ymax
¼ jy1 ðtÞjmax ; ymax
¼ jy2 ðtÞjmax )
1
2
was recorded in every case.
At the smaller pressure amplitude, pA ¼ 0:1 bar, the system
behaves like a linear damped oscillator. There is only one peak in
the amplification diagram near the linear undamped resonant frequency (xR ¼ 1) except at higher temperatures, such as at
T 1 ¼ 70 C, where the second harmonic resonance appears due
to the nonlinearity of the system. The frequency value at which
there is the maximum of the frequency response curve of the linearized equations is the peak frequency
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3nðP1 pV Þ 2ð3n 1Þr
;
xE ¼
þ
qL R2E
qL R3E
see Brennen [1]. In this work the applied relative frequency is
defined as
xR ¼
x
:
xE
ð9Þ
2.2. Dimensionless equation system
By introducing dimensionless variables, namely, the dimensionless bubble radius y1 ¼ R=RE , the dimensionless time s ¼ t=ð2p=xÞ
and the dimensionless bubble wall velocity y2 ¼ y01 (where the 0
stands for the derivative with respect to s) the modified Keller–
Miksis equation can be rewritten as a system of first order dimensionless differential equations:
y01 ¼ y2 ;
N
y02 ¼ ;
D
ð10Þ
ð11Þ
where
N¼
pL p1
y
þ A 2 ðpG ð1 3nÞ p1 ðsÞ þ pV Þ
pref y1
pref y1
p cosð2psÞ
M 3 y22
A B
1
;
3 2 y1
pref
D¼1Mþ
4l L
:
lref y1
ð12Þ
ð13Þ
3. The detection of large amplitude oscillations
3.1. Frequency response curves
xP ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
8l2
x2E 2 L4
qL RE
ð22Þ
of the system [1]. Because of the very high viscosity, the second
term under the root can be dominant, resulting in a complex valued
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
156
Fig. 2. Frequency response curves at pressure amplitudes pA ¼ 0:1 bar (left) and 0:5 bar (right) and at several ambient temperatures T 1 . The red curve corresponds to the
temperature value T 1 ¼ 27:44 C, below which the system behaves like an overdamped oscillator.
frequency. This is the case of an overdamped system, where the
peak in the frequency response curves disappears. As the viscosity
is strongly temperature dependent, the threshold for the overdamped behaviour is temperature dependent, too. This is what
Fig. 2 exactly demonstrates. Under the temperature value of
approximately T 1 ¼ 27:44 C, found by Newton–Raphson method
applied on equation x2P ¼ 0, the peak in the curves is completely
of
invisible and the maximum dimensionless bubble radius ymax
1
the solutions decreases monotonically with increasing frequency.
The amplification curves related to the threshold temperature are
indicated by red curves in Fig. 2. From the application point of view
this result is very crucial as the large amplitude (collapse-like) oscillations probably do not exist at all under the threshold temperature; leading to a very small efficiency of the ultrasonic
technology. At higher pressure amplitudes, such as at pA ¼ 0:5 bar,
the nonlinear effects become more dominant especially at high
temperature values, see the right hand side of Fig. 2. Several harmonic resonances are generated, and a hysteresis appears near
the main resonance, indicating the coexistence of two distinct periodic solutions at the same parameter values. Early references on
these phenomena, including analytical formulae for frequency
response curves, are [60–62].
The aforementioned coexistence is demonstrated via bubble
radius vs. time curves and phase space diagrams in Fig. 3 at relative
frequency xR ¼ 0:8. The solutions corresponding to the upper and
lower branch of the hysteresis at temperature value T 1 ¼ 70 C are
depicted by the black lines. In order to show the effect of the overdamped behaviour, the corresponding periodic solution at
T 1 ¼ 27:44 C is also presented by the red curve. These solutions
are also marked by the blue dots in the right hand side of Fig. 2.
The coexisting periodic attractors have totally different behaviour. The maximum bubble radius ymax
of the solution at the
1
upper branch of the hysteresis is more than twice of the equilibrium radius of the unexcited system (y1;E ¼ 1) in contrast to the
maximum radius of the solution at the lower branch which is no
more than ymax
¼ 1:4. Consequently, as the bubble starts shrinking
1
due to the inertia of the liquid, the occurring maximum of the bub_
is more than five times greater;
ble wall velocity V max ¼ jRj
max
however, it is still far below the required hundreds or even thousands of m/s, required to produce strong pressure waves. In the following, the paper focuses on the finding of similar, large amplitude,
collapse-like oscillations with even larger bubble wall velocity. The
special solution corresponding to the overdamping threshold temperature, denoted by the red curve, is almost a smooth harmonic
function with a maximum velocity of only V max ¼ 1:68 m=s.
It is worth mentioning that the period of the solutions presented in Fig. 3 is 1, meaning that the solutions return to their
starting point after one cycle of the harmonic forcing. Keep in mind
that the period of the excitation of the dimensionless system is
so ¼ 1 according to the expression (21). Such orbits are usually
called period 1 solutions. We shall see in the next section that a
number of periodic solutions with different periods and chaotic
solutions can coexist even at the same parameter values because
of the strong nonlinearity of the system.
The trajectories of the periodic attractors form closed curves in
the dimensionless phase space (y1 y2 plane), like on the right
hand side of Fig. 3. These, however, can intersect themselves leading to unsuitable representation of a solution in this plane. To overcome this difficulty one can represent only the points of the
so-called Poincaré map obtained by simply sampling the continuous trajectory at time instants s ¼ kso , where k ¼ 0; 1; . . .. If a trajectory of an arbitrarily initiated system returns exactly to its
starting value after N iterations, PN ðyo Þ ¼ yo , then the solution is
a periodic orbit whose period is sp ¼ N so called period N solution.
In our period 1 cases the Poincaré map returns to itself immediately after the first period, that is, yo ¼ Pðyo Þ ¼ P2 ðyo Þ ¼ , see
the black and red dots on the right hand side of Fig. 3.
3.2. Pressure amplitude response diagrams
The results of the previous section revealed that the pressure
amplitude pA and the ambient temperature T 1 have the most significant effect on the amplitude of the oscillation. Simply, the bigger the magnitude of the excitation is, the greater the response of
the system becomes. Moreover, the higher the temperature is, the
smaller the viscosity of the liquid, which leads to weaker damping
rate and more rapid bubble motion. Therefore, pressure amplitude
response curves are more suitable for seeking large amplitude
oscillations. Instead of the relative frequency xR , in this subsection
we use pressure amplitude pA as control parameter. The ambient
temperature is still regarded as a secondary parameter (varied
between 20 °C and 70 °C) while the relative frequency is kept constant at the value of the linear undamped resonant angular frequency xR ¼ 1 (f ¼ x=ð2pÞ ¼ 29:33 kHz). The bubble size is still
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
157
Fig. 3. Examples of periodic solutions marked by the blue dots on the left hand side of Fig. 2. The solutions return to themselves after one period of the excitation so ¼ 1 (left
panel). The trajectories in the phase space are closed curves at which the black and red dots are the corresponding points of the Poincaré map (right panel).
RE ¼ 0:1 mm ¼ 100 lm, which is in the order of the experimentally
observed sizes by [46].
The choice of the xR RE parameter pair can be justified as follows. As the liquid domain is irradiated with high frequency ultrasound, the small, usually micron-sized gas bubbles (nuclei sites)
start to oscillate around their equilibrium radius. As long as the
intensity (pressure amplitude) of the ultrasound is low, it is a relatively smooth, small amplitude oscillation. Such bubbles are
called inactive bubbles. For sufficiently high intensity, the bubbles
become cavitationally active and start to oscillate with large amplitude, dominated by the inertia of the liquid domain. The maximum
radius of a bubble during the oscillation can be several times larger
than its equilibrium size. The value of the irradiation intensity,
which separates the cavitationally active and inactive bubbles, is
known as Blake’s threshold [63,64]. This threshold value is higher
for smaller bubbles, since the effect of the surface tension, which
attempts to contract the bubble as small as possible, is inversely
proportional to the size of the bubble. Conversely, for a given pressure amplitude only bubbles larger than a critical size will become
active.
Another important threshold value, in terms of the pressure
amplitude, corresponds to the growth by rectified diffusion [65].
It is well-known that a bubble at rest would dissolve due to the
effect of surface tension. Contrarily, during the oscillation of a bubble, more gas diffuses into the bubble interior in the expanded
state than diffuses back to the liquid in the collapsed state because
of the large difference in the surface area between the two states.
This phenomenon is called rectified diffusion and its effect
increases with the amplitude of the oscillation, that is, with the
intensity of the ultrasonic irradiation and with the size of the bubble. The magnitudes of the two opposite effects determine the dissolution or growth of an oscillating bubble [66]. Thus, for a given
pressure amplitude, there is another critical size which separates
dissolution from growth.
The numerical investigation of Louisnard and Gomez [67]
revealed that the two thresholds (Blake’s and rectified diffusion)
coincide for small bubbles in water. This means that a cavitationally active bubble always grows by rectified diffusion. Naturally,
the time scale of the rectified diffusion is greater by orders of magnitude than that of the radial oscillation. A natural limitations in
the bubble growth is its spherical instability, see again e.g. [49]
or [68]. As the size of the bubble reaches a limit, it becomes
spherically unstable and disintegrates into daughter bubbles.
These smaller bubbles then again start to grow by rectified diffusion and become cavitationally active. This process is a simple
manifestation of an acoustic cavitation cycle, for details see [69].
In water, the shape instability threshold is much smaller than
the linear resonant size at a given frequency, approximately in
the orders of few micrometres. In glycerine, however, the threshold
of the shape instability is much higher than in water (tenth of millimetres), observed experimentally in [46]. This can lead to bubble
sizes close to or even larger than the frequency-dependent resonance size. From the linear theory of rectified diffusion [70], it is
known that diffusionally stable bubbles exist above the resonance
size. Therefore, it is possible to drive the bubble at (or at least near)
its resonance frequency for a few hundred acoustic cycles during
the growth phase. This is the reason for the choice of the xR RE
parameter pair.
The pressure amplitude response curves were computed in a
similar way as the frequency response curves of the previous subsection. The pressure amplitude pA was increased from 0.01 bar to
5 bar with 0.01 bar increments, and five randomly generated initial
conditions were tried at each value of the control parameter to
reveal coexistent attractors. After convergence, as many points
from the Poincaré plane as the period of the solution were recorded
(in case of chaos the number of the points were 512). An example
of such a diagram at 40 °C is presented on the left hand side of
Fig. 4, in which only the y1 part of the Poincaré plane were plotted.
The system is very feature rich from the dynamical point of view.
The period 1 solution, which has been bifurcated from the equilibrium point of the unexcited system (pA ¼ 0; y1;E ¼ 1), undergoes a
period doubling sequence resulting in the appearance of chaotic
oscillation approximately at pA ¼ 3:69 bar. Then, windows of chaotic and regular periodic solutions show up successively, and each
periodic solution again ends in a period doubling cascade. A periodic window appears through a saddle–node bifurcation generating a pair of stable and unstable periodic orbits. Immediately
after the bifurcation point, the stable periodic orbit replaces the
original chaotic attractor, which becomes unstable (transient
chaos) [71]. The periods of the regular solutions are marked by arabic numbers in Fig. 4 left. An experimentally observed
period-doubling route to chaos can be found in Refs. [72,73].
The Poincaré representation of a bifurcation structure of a system is a very efficient tool in general. For our purpose, however, it
158
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
Fig. 4. Example of a pressure amplitude response diagram. The first component of the Poincaré section as a function of the pressure amplitude (left). Maximum Mach number
as a function of the pressure amplitude (right).
is not completely satisfactory since it gives no information about
the maximum bubble wall velocity, which is a good indicator for
the strength of the collapse [70]. Therefore, after convergence the
maxima of the absolute value of y2 were also recorded in each period of the excitation. Now, on the right panel of Fig. 4 the same
bifurcation structure can be seen as on the left hand side, but
instead of Pðy1 Þ, the maximum of the bubble wall velocities
jy2 jmax rescaled to the Mach number M max by means of (18) is
demonstrated as a function of the control parameter pA . It is clear
that with increasing pressure amplitude the maximum Mach number also increases, and at pA ¼ 5 bar it reaches M max ¼ 0:1, which is
approximately 190 m=s.
In order to reveal the effect of the temperature on the bifurcation structure of the system and on the achievable maximum
Mach number, several numerical computations, similar to the ones
presented in Fig. 4, were performed between T 1 ¼ 20 C and
T 1 ¼ 70 C with DT 1 ¼ 5 C increments. According to the results
shown in Fig. 5, the strong influence of the temperature through
the alteration of the viscosity is evident. With increasing temperature the maximum bubble wall velocity increases, too. Moreover,
the bifurcation structures become more complicated; however,
above T 1 ¼ 35 C they are similar in the sense that chaotic and
periodic windows appear successively. The relevant periodic solutions found are again marked by arabic numbers.
The most important feature of the bifurcation structure is that
every dominant periodic solution appears approximately at a certain level of the maximum Mach number M max regardless of the
value of the temperature, see the horizontal lines in Fig. 5C–F.
For instance, it is a very good approximation that above the first
period doubling bifurcation point of the period 1 solution, the maximum Mach number is more than 0.02. After the emergence of the
period 3 solution via a saddle–node bifurcation, M max is definitely
greater than 0.06. Finally, the period 2 orbit that evolves from a
saddle–node bifurcation and dominates the dynamics at high pressure amplitudes implies Mmax > 0:25. It is clear that the different
types of regular solutions, enclosed by bifurcation points, correspond to different achievable maximum Mach numbers.
Fortunately, for finding periodic solutions, very efficient and
sophisticated numerical techniques exist, including detection of
unstable solutions and bifurcation points. These methods shall
open the way to determine the boundaries of the different type
of orbits and thus the achievable maximum Mach numbers not
only as a function of a single control parameter, but in the whole
pA T 1 parameter space itself. Such computations are the topic
of the next subsections.
3.3. Exploration of the periodic solutions by continuation technique
The idea to compute periodic solutions efficiently is to apply a
boundary value problem (BVP) solver with periodic boundary conditions, and obtain the desired orbits directly. More precisely:
y_ ¼ f ðy; sÞ;
ð23Þ
with boundary conditions
yð0Þ ¼ yðsp Þ;
ð24Þ
where y ¼ ½y1 ; y2 T ; f is defined by Eqs. (10) and (11), sp ¼ Nso is
again the period of the solution with periodicity N. There are various efficient numerical techniques to solve such BVPs, for instance,
shooting, finite differences or orthogonal collocation method, which
are all insensitive to the stability of the periodic orbit. Once such a
solution is computed, its evolution with respect to a control parameter can be traced. This curve is called bifurcation curve, and the
bifurcation points, where the change of the stability type takes
place, can also be detected. It was shown in Fig. 5 that these points
play an important role in separating different levels in the achievable maximum Mach number Mmax in the parameter space. One
of the most popular parameter continuation method is the pseudo
arc-length continuation technique as it is capable of following
curves containing turning points (folds). A thorough discussion of
the aforementioned numerical methods can be found in Ref. [74].
In the present study, the AUTO continuation and bifurcation
analysis software was used, see the manual of Doedel et al. [75].
AUTO discretizes boundary value problems (including periodic
solutions) of ordinary differential equations (ODEs) by the method
of orthogonal collocation using piecewise polynomials with 2–7
collocation points per mesh interval [76]. The mesh automatically
adapts to the solution to equidistribute the local discretization
error [77]. During our computations the relative error was 1010 .
AUTO can handle only autonomous systems (free of explicit time
dependence), thus system (10)–(21) has to be extended with two
additional decoupled ODEs defined as
y03 ¼ y3 þ 2py4 y3 ðy23 þ y24 Þ;
ð25Þ
y04 ¼ 2py3 þ y4 y4 ðy23 þ y24 Þ;
ð26Þ
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
159
Fig. 5. Pressure amplitude response diagrams at different ambient temperatures T 1 .
where the periodic solutions of the variables y3 and y4 are exactly
cosð2psÞ and sinð2psÞ, respectively. Thus, the fourth and second
terms in the right hand side of Eqs. (12) and (21) can be replaced by
pA y3
;
A
pref
ð27Þ
and
pA y4 ;
ð28Þ
respectively. This description has the disadvantage that AUTO can
handle a periodic solution only as a whole object regardless of its
period. Therefore, the illustrative representation in the Poincaré
map has been lost, and one can obtain only the maximum values
of the dimensionless variables (ymax
; ymax
) of each kind of solution.
1
2
It is worth mentioning that the same bifurcation analysis software
(AUTO) was used by Fyrillas and Szeri [66] for the rectified diffusion
problem.
In Fig. 6 the capabilities of the BVP solver is demonstrated by
comparing some of its results (coloured curves) with the corresponding bifurcation structure of the IVP solver (black dots) at
T 1 ¼ 50 C. The family of the period 1 solution is presented by
the red curve initiated from the IVP solution at pA ¼ 0:01 bar
(almost from the equilibrium state of the unexcited system).
Along the curve a period doubling (PD) bifurcation takes place at
pA ¼ 2 bar denoted by the red cross. The solid and dashed parts
of the curve are the stable and unstable solutions, respectively.
Initiating the BVP solver from the detected bifurcation point,
the period 2 curve can be traced by applying a suitable branch
switching algorithm, see the blue curve in Fig. 6. At the first sight,
it seems that the period 2 segments (blue solid lines) in Fig. 6 are
disconnected. In fact, from the larger scale representation in Fig. 7,
it is evident that the two parts are actually connected via multiple
turning points called saddle–node (FL) bifurcation marked by the
blue dots. This case demonstrates one of the several strength of
the BVP solver, namely, distinct stable solutions can be found
simultaneously if they are connected through unstable solutions.
A similar analysis can be made for the period 3 orbits (green
curves in Fig. 6). From the large scale representation displayed in
Fig. 8, it turns out that this solution forms a closed curve in the
pressure amplitude response diagram. This behaviour of the solu} s et al. [26].
tions with odd periods was also found by Hegedu
As mentioned earlier, each domain of existence of stable periodic solutions, either period 2 or period 3 (solid branches in
Fig. 6), corresponds to a given range of maximum Mach number.
Although in Fig. 6 the maximum of the dimensionless bubble
radius ymax
is plotted against the control parameter pA due to some
1
160
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
Fig. 6. Comparison of the results computed by the BVP solver (coloured curves) and the IVP solver (black dots) at T 1 ¼ 50 C. The solid and dashed lines are the stable and
unstable solutions, respectively. The detected period doubling bifurcation points are denoted by the crosses, while the saddle–node (fold) bifurcations are represented by the
dots at the turning points. The red, blue and green curves are period 1, 2 and 3 solutions, respectively.
specifics of AUTO, the maximum Mach number can be estimated
according to the results of Fig. 5 and the corresponding discussion.
These achievable Mach numbers are also highlighted in Fig. 6.
3.4. Phase diagram
Fig. 7. Large scale representation of the period 1 and the period 2 solutions
computed by the BVP solver, see also the red and blue curves in Fig. 6. The black
rectangle shows the diagram limits of Fig. 6.
Fig. 8. Large scale representation of the period 3 solution computed by the BVP
solver, see also the green curve in Fig. 6.
The results of Fig. 6 have shown that the detected bifurcation
points (coloured crosses and dots) are the boundaries of the existence of the different type of stable periodic orbits. The appearance
of such attractors define an approximated threshold in terms of the
control parameter pA for the accessible maximum Mach number
M max . These threshold values, however, vary with the ambient
temperature T 1 , cf. Fig. 5C–F. Naturally, the higher the temperature, the lower the threshold value of the pressure amplitude for
a given value of the maximum Mach number. For instance, the period doubling bifurcation related to M max > 0:02 takes place at
pA ¼ 2:43 bar at T 1 ¼ 45 C, while at T 1 ¼ 70 C the pressure
amplitude is only pA ¼ 1:75 bar, see Fig. 5. This means that the
required energy of the sound radiation for a given M max decreases,
allowing lower operational costs. On the other hand, increasing the
temperature also requires a certain amount of energy transfer into
the liquid domain acting oppositely on the costs. Moreover, too
high temperature can cause unwanted chemical reactions or
molecular degradation in the glycerine. In order to apply optimal
operational parameters (pressure amplitude pA –temperature T 1
pair), it is essentially important to reveal the evolution of the
threshold values of pA as a function of the ambient temperature,
that is, tracking the path of the detected bifurcation points in the
pA T 1 plane.
The AUTO bifurcation analysis programme enables to obtain the
required two-parameter (codim 2) bifurcation curves easily by
choosing the ambient temperature T 1 as a secondary control
parameter. The results of the computations that correspond to all
the detected bifurcation points presented in Fig. 6 are shown in
Fig. 9. The dashed and solid curves are the period doubling (PD,
crosses in Fig. 6) and fold (FL, dots in Fig. 6) bifurcations, respectively. The piecewise smooth nature of the curves is an artifact of
the linear interpolation between the tabulated values of the material properties. The vertical thick red line is the threshold temperature T 1 ¼ 27:44 C related to the linear overdamped system.
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
161
Fig. 9. Two-parameter bifurcation curves corresponding to the detected bifurcation points presented in Fig. 6. The dashed and solid curves are the period doubling and fold
bifurcations, respectively. The vertical thick red line is the threshold temperature T 1 ¼ 27:44 C, under which the linear system is overdamped. The boundaries of the
achievable maximum Mach numbers M max are denoted by the rectangular based arrows. The period 1, 2 and 3 domains are illustrated by the light blue, light brown and
yellow regions, respectively. The periodicities are also marked by arabic numbers.
The two black dashed curves are the period doubling bifurcations of the period 1 and the period 2 solutions shown by the red
and blue crosses in Fig. 6, respectively. Because of the hyperbolic
type behaviour of the curves, the pressure amplitude pA required
to reach these PD bifurcations increases extremely fast as the ambient temperature approaches to the threshold temperature
T 1 ¼ 27:44 C. Choosing control parameters above the lower black
curve guarantees only Mmax > 0:02 (approximately V max > 40 m=s).
The domains of the period 1 and 2 solutions, defined by the two
curves, are highlighted by the light blue and light brown regions,
respectively. Above the upper black curve, the solutions become
chaotic via period doubling cascades.
The next definite step to increase the maximum Mach number
is to reach the period 3 solution, which has more complicated
structure than the period doubling sequence discussed above. A
stable period 3 solution, for instance, lies between the green solid
and the green dashed lines, both are turning back at approximately
T 1 ¼ 40 C. Inside this scythe-shaped domain a hysteresis appears
(saddle–node bifurcation pair) at T 1 ¼ 52:5 C via a cusp bifurcation towards the decreasing temperature values, which results
the birth of another stable period 3 domain. This secondary period
3 attractor is enclosed by the solid and dashed brown curves. The
upper and lower branch of the solid brown curve, which are initiated from the cusp point, turn back at T 1 ¼ 44:6 C and
T 1 ¼ 37:7 C, respectively, causing a three arm shaped stable period 3 domain. By reaching either the solid green line or the solid
brown line by increasing the pressure amplitude pA and/or the
ambient temperature T 1 the maximum Mach number shall be
higher than M max > 0:06 (V max > 110 m=s). The found stable period
3 domains are denoted by the yellow areas.
The appearance of the period 2 solution in the high pressure
amplitude region, see the blue dot in Fig. 6, ensures greater maximum Mach number than M max ¼ 0:25 (V max > 470 m=s). The corresponding two-parameter bifurcation curve of this saddle–node
point is the blue solid line in Fig. 9. Again, due to the hyperbolic
type nature, the required pressure amplitude can be decreased
only in case of high enough ambient temperature. The dashed blue
line is related to the first period doubling bifurcation after the
saddle–node point. This can only be seen in the large scale representation in Fig. 7. Between the two blue curves, there is a relatively large period 2 domain represented by a light brown area.
After the dashed blue curve, the solutions again become chaotic
through a period doubling cascade.
Fig. 9 is a good summary of the bifurcation structure in the pressure amplitude pA – ambient temperature T 1 parameter plane,
which helps identify the different levels of the strength of bubble
collapse in terms of the maximum Mach number M max . The maximum Mach number is a useful indicator in many applications, e.g.
for the generation of shock waves in ultrasonic cleaning [78], or in
case of micromixing and microstreaming which are quite important in sonochemistry for competitive reactions [79]. In the literature, however, there are many other, well-defined quantities
serving to measure the collapse intensity. High pressure and temperature inside the bubble are also required to start chemical reactions, and they play a role especially when the controlling
mechanism is pyrolysis [80,81]. If the production of free radicals
is the keen interest, such as in wastewater treatment [82], the
maximum bubble radius Rmax seems to be the most prominent
indicator [83,84]. These various quantities are actually scales
together, for instance, the larger the maximum bubble radius
Rmax the higher the maximum Mach number M max , see the recent
investigation of Varga and Paál [28]. Therefore, our results presented in Fig. 9 and the conclusions drawn are useful in general
for the above mentioned applications.
It is important to emphasize that the bifurcation curves in Fig. 9
are not exactly iso-curves of the Mach number but they are good
lower estimates. Although it is not a precise description, it has
the advantage that high Mach number oscillations can be detected
by monitoring the subharmonic component of the acoustic emission spectra of the bubble/bubble cluster. This idea is supported
by the study of Mettin [50], who reported a strong connection
between the appearance of subharmoinc emission and cavitation
erosion. The values of the maximum Mach numbers, presented in
Fig. 9, may differ for other bubble sizes RE or frequencies xR . The
bifurcation structure, however, should be qualitatively similar in
the T 1 PA parameter plane as the critical temperature
162
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
T 1 ¼ 27:44 C (red vertical line) dominates the behaviour of the
system. That is, high subharmoinc component can always indicate
an increase in the strength of the collapse without the explicit
knowledge of the maximum Mach number. It is worth noting that
high periodicity, e.g. period N, can decrease the emitted
energy/acoustic cycle, since strong collapse occurs only once at
every N acoustic periods [85]. This behaviour is also seen, for
instance, in Fig. 5F where the maximum Mach numbers of the subsequent acoustic cycles of the period 2 solution, existing above
pA ¼ 3 bar, differ by at least two orders of magnitude.
From the work of Yasui et al. [85], it is also revealed that in case
of moderately high viscosity (in the order of 200 mPa s) the maximum Mach number can be higher that in low viscosity liquids,
such as in pure water. For even higher viscosity, which is our case,
the maximum bubble wall velocity again decreases very rapidly
with the viscosity. This observation raises the issue of the application of water–glycerine mixture with optimised concentration.
Moreover, the numerical study of Yasui et al. [85] reported that
incorporating the thermal processes into the bubble model the
resulting collapse strength can be stronger. In this sense, our
numerical results (without thermal effects) serve as a good lower
estimate for the bubble wall velocities.
The scan for the stable bubble oscillation presented in Fig. 5
implies that there is a global limit in the maximum Mach number
M max , see especially subplot F. The long-term behaviour of M max
presented in a linear scale in Fig. 10, however, reveals that M max
increases almost linearly with the pressure amplitude pA .
Although the maximum value of pA ¼ 15 bar is difficult to achieve
during an experiment, Fig. 10 is a perfect demonstration of how
difficult it is to achieve extremely high bubble wall velocities. For
instance, to achieve M max ¼ 1 (supersonic speed) one needs to
increase the pressure amplitude above pA ¼ 10 bar. Observe, that
the ambient temperature is already as high as T 1 ¼ 70 C. At such
high bubble wall velocities (Mach number near or higher than 1),
our model ceases to be valid as the Keller–Miksis equation is only
a first order approximation in terms of the Mach number [86].
Moreover, according to the derivation of Yasui [47] based on the
fundamental theory of fluid dynamics, the bubble wall velocity R_
at the collapse never exceeds the sound velocity of the liquid at
the bubble wall cL . The author continuously monitored this condition during the computer simulations, and the bubble wall velocity
R_ was replaced by cL if it exceeds cL . During our simulations this
4. Summary
Excited spherical gas/vapour bubbles can exhibit collapse-like
oscillations, causing extreme conditions (high temperature, pressure and shock waves) in the collapse phase. These conditions are
exploited by the rapidly developing ultrasonic technology, which
has several industrial applications. This was the main motivation
to study a spherical bubble placed into highly viscous glycerine driven by a purely harmonic signal. The bubble model was the modified Keller–Miksis equation, a second order nonlinear ordinary
differential equation, which takes into account the liquid compressibility. High viscosity causes a massive, temperature-dependent
damping rate, which can be a limiting factor on the applicability
of the ultrasonic technology on such liquids. The findings of the present paper can help increase the efficiency of the applications and
decrease the costs of the operation.
The investigation of the frequency response curves revealed
that due to the high viscosity, the system behaves like an overdamped linear oscillator below the threshold ambient temperature
of 27:44 C. This very high damping rate weakens the strength of
the collapse by decreasing the peak bubble wall velocity.
Therefore, increasing the ambient temperature above this threshold value is highly recommended.
The results of the pressure amplitude response curves at different ambient temperatures and at constant excitation frequency
demonstrated that the appearing bifurcation points of the simple
periodic solutions can be used as a good estimate of the maximum
bubble wall velocity. This observation enabled us to determine the
boundaries of the achievable maximum bubble wall velocity in the
excitation pressure amplitude–ambient temperature parameter
space. The resulting parametric map can aid the applications to
operate the technology in an efficient way.
Acknowledgement
The research described in this paper was supported by the
Hungarian Scientific Research Fund – OTKA, from the Grant No.
K81621.
Appendix A. Material properties
A.1. KDB equation for vapour pressure
condition was not tested.
The vapour pressure of the glycerine were calculated by means
of the KDB correlation equation ([87]):
ln pV ¼ A ln T 1 þ
B
þ C þ DT 21 ;
T1
ðA:1Þ
where pV is in kPa, T 1 is in K and the coefficients are
A ¼ 2:125867 101 ;
4
ðA:2Þ
B ¼ 1:672626 10 ;
ðA:3Þ
C ¼ þ1:655099 102 ;
ðA:4Þ
D ¼ þ1:100480 105 :
ðA:5Þ
A.2. Tabulated values of the material properties
In the following, the tabulated values of the material properties
of the glycerine1 can be found as a function of the ambient temperature T 1 . For other ambient temperatures, the corresponding values
were calculated with linear interpolation (see Tables A.1–A.4).
Fig. 10. Long-term behaviour of the maximum Mach number Mmax as a function of
the pressure amplitude pA .
1
The tabulated values of the material properties were taken from the results of The
Dow Chemical Company (1995–2014); web page: http://www.dow.com/.
} s, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164
F. Hegedu
Table A.1
Tabulated values of the glycerine density qL as a function of the ambient temperature
T1.
T 1 ð CÞ
qL ðkg=m3 Þ
0
1272.7
10
1267.0
15
1264.4
20
1261.3
30
1255.1
40
1249.0
54
1239.7
T 1 ð CÞ
75.5
1225.6
99.5
1209.7
110
1201.8
120
1194.5
130
1187.2
140
1179.5
160
1164.4
qL ðkg=m3 Þ
Table A.2
Tabulated values of the glycerine viscosity
temperature T 1 .
lL as a function of the ambient
T 1 ð CÞ
lL ðPa sÞ
0
12.07
10
3.9
20
1.41
30
0.612
40
0.284
50
0.142
60
0.0813
T 1 ð CÞ
70
0.0506
80
0.0319
90
0.0213
100
0.0148
110
0.0105
120
0.00780
130
0.00599
lL ðPa sÞ
Table A.3
Tabulated values of the glycerine sound speed cL as a function of the ambient
temperature T 1 .
T 1 ð CÞ
cL ðm=sÞ
10
1941.5
20
1923
30
1905
Table A.4
Tabulated values of the glycerine surface tension
temperature T 1 .
T 1 ð CÞ
r ðN=mÞ
20
0.0634
40
1886.5
50
1869.5
r as a function of the ambient
90
0.0586
150
0.0519
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