Ultrasonics Sonochemistry 27 (2015) 153–164

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Ultrasonics Sonochemistry

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The effect of high viscosity on the collapse-like chaotic and regularperiodic oscillations of a harmonically excited gas bubble

http://dx.doi.org/10.1016/j.ultsonch.2015.05.0101350-4177/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +36 1 463 1680; fax: +36 1 463 3091.E-mail addresses: hegedusf@hds.bme.hu (F. Heged}us), kklapcsik@hds.bme.hu

(K. Klapcsik).

Ferenc Heged}us ⇑, Kálmán KlapcsikBudapest 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 2014Received in revised form 20 April 2015Accepted 12 May 2015Available online 16 May 2015

Keywords:Bubble dynamicsBifurcation structureChaosKeller–Miksis equationContinuation techniqueHigh viscosity

a b s t r a c t

In the last decade many industrial applications have emerged based on the rapidly developing ultrasonictechnology such as ultrasonic pasteurization, alteration of the viscosity of food systems, and mixingimmiscible liquids. The fundamental physical basis of these applications is the prevailing extreme condi-tions (high temperature, pressure and even shock waves) during the collapse of acoustically excited bub-bles. By applying the sophisticated numerical techniques of modern bifurcation theory, the present studyintends to reveal the regions in the excitation pressure amplitude–ambient temperature parameter planewhere collapse-like motion of an acoustically driven gas bubble in highly viscous glycerine exists. Wereport 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 peri-odic windows interspersed with chaotic regions indicating the presence of transient chaos, which isimportant from application point of view if predictability is required.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction

Although the geometry of a single spherical bubble is rathersimple, the physics of the bubble oscillation, however, can be verycomplicated. The wall velocity of a bubble can accelerate to extre-mely high values due to the inertia of the liquid domain, resultingin a minimum bubble size many orders of magnitude smaller thanthe average. This process often referred to as the collapse phase. Atthis minimum bubble radius the temperature and pressure can beas high as 1000 bar and 8000 K, respectively [1]. Due to such extre-mely high temperatures during the collapse phase, chemical reac-tions can take place yielding various reaction products. These arethe keen interest of sonochemistry [2–5], or the spectroscopy ina laser induced cavitation bubble [6,7].

In the last decade ultrasonic technology has began to developvery rapidly. The main objective of the applications is to enhancethe mass, heat and momentum transfer between the variousphases by taking the advantage of the above mentioned extremeconditions during bubble collapse. A promising technology in foodpreservation, for instance, is the ultrasonic pasteurization. At mod-erate temperature (50 �C) the membrane of the bacterial organ-isms weakens enough to become less resistant to cavitational

damage. With this novel innovation Knorr et al. [8] could success-fully reduce the Escherichia coli in liquid whole egg. The alterationof the viscosity of many food systems such as tomato puree is alsopossible with ultrasound since cavitation causes shear stress thatdecreases the viscosity of thixotropic fluids. With high enoughenergy the alteration becomes permanent by reducing the molec-ular weight of the substances. Examples for viscosity reductionwere published by Seshadri et al. [9] and Iida et al. [10]. Duringthe collapse of cavitation bubbles shock waves are generated caus-ing very efficient mixing of two immiscible liquids. Canselier et al.[11] and Freitas et al. [12] reported the production of fine, highlystable emulsions. Moreover, the possible occurrence of rectifieddiffusion is the basis of novel degassing technologies [13,14].

These applications were the main motivation to investigate theoscillations of spherical gas bubbles in liquid glycerine, which isused in many medical, pharmaceutical and personal care prepara-tions. The choice of the substance is also important from point ofview of the available knowledge, since the majority of the papersare related to water. Some exceptions, for instance, is the paperof Toegel et al. [15] who found that high viscosity can destabilizethe position of the bubble trapped in an acoustic field; or the studyof Englert et al. [16] revealed that the luminescence pulse durationin a water–glycerine mixture is increased by a factor of two as theglycerol concentration increases by 33%. The oscillations of gasbubbles in other kind of liquid material have also been

154 F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164

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 classifiedinto three physical categories, namely, viscous, acoustic and ther-mal 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 bubblewall velocity and thus softens the impact of the bubble collapse.We shall see, however, that with the aid of the modern nonlineartheory and its rapidly evolving, sophisticated numerical methods,such as, the pseudo-arc length continuation technique, the deter-mination of the parameter regions of the collapse-like oscillationsbecomes only a minor problem. These efficient numerical algo-rithms, related mostly to the topic of nonlinear dynamics, arestarted to spread in the field of bubble dynamics in the recent years[24,22,25–33], but they have already been applied successfully inother branches of science [34–40].

As the desired solutions have high bubble wall velocity, the con-sideration of the liquid compressibility is necessary at least as afirst order approximation. Therefore, the Keller–Miksis equation[41] was applied during the computations. According to the ultra-sonic technology, the most important parameters were the pres-sure amplitude and frequency of the excitation. Because of thevery strong dependence of the viscosity on the ambient tempera-ture, its influence on the dynamics is also significant, and it wasregarded as a secondary scaling or control parameter.

2. Mathematical model

Because of the possibility of large amplitude oscillations, theconsideration of liquid compressibility is necessary. In this paper,the well known Keller–Miksis equation [41] is used with minormodifications [42], in which the retarded time from the originalequation was eliminated. The form of the modified equation is

1�_R

cL

!R€Rþ 1�

_R3cL

!32

_R2 ¼ 1þ_R

cLþ R

cL

ddt

!pL � p1ðtÞð Þ

qL;

ð1Þ

where RðtÞ is the time dependent bubble radius; qL, cL are the liquiddensity and sound speed, respectively; pL is the pressure at the bub-ble wall in the liquid domain and p1ðtÞ is the pressure far awayfrom the bubble consisting of static and periodic components writ-ten as

p1ðtÞ ¼ P1 þ pA sinðxtÞ; ð2Þ

where P1 is the ambient pressure, pA and x are the pressure ampli-tude and angular frequency of the excitation, respectively. The bub-ble content is a mixture of glycerine vapour and non-condensablegas; we treat both as ideal gases. This means that the pressureinside the bubble is the sum of the partial pressures of the vapourpV and the gas pG. The relationship between the pressures on thetwo sides of the bubble wall is described by the mechanical balanceat this interface

pG þ pV ¼ pL þ2rRþ 4lL

_RR; ð3Þ

where r is the surface tension and lL is the liquid dynamicviscosity.

The vapour pressure inside the bubble is constant but its valuedepends on the ambient temperature T1; while the gas contentobeys a simple polytropic relationship

pG ¼ pgoRo

R

� �3n

ð4Þ

with a polytropic exponent n = 1.4 (adiabatic behaviour). The refer-ence pressure pgo and radius Ro determine the mass of gas inside thebubble and therefore the average size of the bubble.

At this point, the discussion of the validity of the applied bubblemodel is necessary. It is well-known that the assumption of adia-batic gas behaviour is a severe oversimplification in many cases,see e.g. [43–45]. However, the comparison between our numericaland former experimental results, presented in Fig. 1, shows thatthe adiabatic behaviour describes the dynamics very well. Theradius-time curve of a laser-induced gas bubble in Fig. 1 is similarto those in [46]. The ambient properties of the glycerine, T1 andP1, were also similar to those applied throughout the presentstudy. Although the bubble exhibits free oscillations about its equi-librium radius, the remarkable agreement implies that our presentmodel provides a good qualitative description of the behaviour ofthe harmonically excited bubble as well.

The evaporation/condensation (possibly with non-equilibriumthermodynamics) can play an important role in the dynamics ofthe bubble if the saturation vapour pressure is comparable withthe gas pressure in the bubble interior [47,48]. Due to the verylow amount of glycerine vapour inside the bubble at the appliedtemperature range, the ratio of the vapour and gas partial pres-sures is less than 1:100000, the effect of evaporation and conden-sation can definitely be neglected.

In spite of the relatively large bubbles observed during theexperiment of [46], in the range of tenth of millimetres, the spher-ical shape of the bubbles were exceptionally stable. From the dif-ferential equation describing the dynamics of the surface wavesof an individual bubble [49], it is obvious that the viscosity has sig-nificant effect on the stability. Although the majority of the papersdeal with gas bubbles in water and numerical analyses are absentfor glycerine, its very high viscosity supports the aforementionedexperimental observation. The spherical instability due to the bub-ble–bubble interaction in clusters [50,51] or the presence of solidboundary [52,53], liquid surface [54] and positional stability dueto the primary Bjerknes force [55,56] were not modelled.

2.1. Parameters and material properties

By following the concept of Heged}us [27], all the parameters insystem (1)–(4) can be specified with only five quantities. The mate-rial properties of a pure substance depend in general on the ambi-ent pressure P1 and temperature T1 making these two ambientproperties as the main parameters. Specifically, in our case thematerial properties depend only on the ambient temperature andtheir pressure dependence are neglected. The tabulated values ofthe material properties can be found in Appendix A. To describethe bubble size, one need to prescribe the equilibrium radius RE

of the unexcited bubble (pA ¼ 0) or, equivalently, the mass of gasinside the bubble, see below. Eventually, the properties of the exci-tation, namely, the pressure amplitude pA and the frequency x arealso needed.

For a given mass of gas mG and ambient temperature T1 theequilibrium radius RE is determined by the static mechanical bal-ance (again pA ¼ 0) at the bubble wall:

0 ¼ pV � P1 þ pgoRo

RE

� �3n

� 2rRE

: ð5Þ

As it is noted earlier, the reference quantities pgo and Ro define themass of gas

mG ¼4pgoR3

op3RT1

ð6Þ

inside the bubble, where R is the specific gas constant. Therefore,one can specify the reference properties pgo and Ro, which

Fig. 1. Comparison of the numerically obtained bubble radius vs. time curves (bluecurve) by assuming adiabatic gas behaviour (n = 1.4) with the experimental results(black curve).

F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164 155

determine the mass of gas via Eq. (6) and the equilibrium radius bymeans of Eq. (5). Alternatively, as it is used in the present paper, onecan specify the equilibrium radius and compute the reference prop-erties pgo and Ro. Now, let us choose Ro to be the equilibrium radiusRE itself, and then the required reference gas pressure pgo to satisfyEq. (5) is

pgo ¼2rRE� pV � P1ð Þ: ð7Þ

The remaining two parameters are related to the excitationitself, namely, the pressure amplitude pA and the angular frequencyx, see Eq. (2). As the angular frequency can vary over severalorders of magnitude its normalization with a suitable referencequantity is reasonable. The linear eigenfrequency of the undampedsystem corresponding to the equilibrium radius is

xE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3nðP1 � pV Þ

qLR2E

þ 2ð3n� 1ÞrqLR3

E

s; ð8Þ

see Brennen [1]. In this work the applied relative frequency isdefined as

xR ¼xxE

: ð9Þ

2.2. Dimensionless equation system

By introducing dimensionless variables, namely, the dimension-less bubble radius y1 ¼ R=RE, the dimensionless time s ¼ t=ð2p=xÞand 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 dimen-sionless differential equations:

y01 ¼ y2; ð10Þ

y02 ¼ND; ð11Þ

where

N ¼ pL � p1pref y1

þ y2

p Aref y1

pGð1� 3nÞ � p1ðsÞ þ pVð Þ

� pA cosð2psÞpB

ref

� 1�M3

� �32

y22

y1; ð12Þ

D ¼ 1�M þ 4lL

lref y1: ð13Þ

The reference properties are

pref ¼ qLR2E

x2p

� �2; ð14Þ

lref ¼ cLqLRE; ð15Þ

p Aref ¼ cLqLRE

x2p¼ lref

x2p

; ð16Þ

pBref ¼ cLqLRE

xð2pÞ2

¼ p Aref

12p

: ð17Þ

The dimensionless Mach number is

M ¼ RExy2

2pcL: ð18Þ

According to Eqs. (4) and (7), the gas pressure inside the bubblebecomes

pG ¼2rRE� pV � P1ð Þ

� �1y1

� �3n

: ð19Þ

The pressure outside the bubble at the bubble wall, and the pres-sure far away from the bubble are

pL ¼ pG þ pV �2rRE

1y1� 4lLx

2py2

y1; ð20Þ

and

p1ðsÞ ¼ P1 þ pA sinð2psÞ; ð21Þ

respectively. Observe, that according to Eq. (21) the period of exci-tation in the dimensionless system is unity (so ¼ 1).

3. The detection of large amplitude oscillations

3.1. Frequency response curves

The usual way to investigate a periodically driven dynamicalsystem, and seek large amplitude oscillations is to present amplifi-cation diagrams or frequency response curves [57–59]. Examplesof such curves are given in Fig. 2 where the maximum dimension-less bubble radius ymax

1 of the stable periodic solutions are plottedas a function of the relative frequency xR at two different pressureamplitudes pA and at several ambient temperatures T1 by keepingall the other parameters constant.

Each curve was computed by increasing the relative frequencyfrom 0.01 to 2 with an increment of 0.01. At each frequency fivesimulations were carried out by a simple initial value problem(IVP) solver (Runge–Kutta scheme with fifth order embedded errorestimation) with random initial values to reveal the coexistingstable solutions (attractors). After convergence the maximum abso-lute value of each component (ymax

1 ¼ jy1ðtÞjmax; ymax2 ¼ jy2ðtÞjmax)

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 inthe amplification diagram near the linear undamped resonant fre-quency (xR ¼ 1) except at higher temperatures, such as atT1 ¼ 70 �C, where the second harmonic resonance appears dueto the nonlinearity of the system. The frequency value at whichthere is the maximum of the frequency response curve of the lin-earized equations is the peak frequency

xP ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

E �8l2

L

q2L R4

E

sð22Þ

of the system [1]. Because of the very high viscosity, the secondterm under the root can be dominant, resulting in a complex valued

Fig. 2. Frequency response curves at pressure amplitudes pA ¼ 0:1 bar (left) and 0:5 bar (right) and at several ambient temperatures T1 . The red curve corresponds to thetemperature value T1 ¼ 27:44 �C, below which the system behaves like an overdamped oscillator.

156 F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164

frequency. This is the case of an overdamped system, where thepeak in the frequency response curves disappears. As the viscosityis strongly temperature dependent, the threshold for the over-damped behaviour is temperature dependent, too. This is whatFig. 2 exactly demonstrates. Under the temperature value ofapproximately T1 ¼ 27:44 �C, found by Newton–Raphson methodapplied on equation x2

P ¼ 0, the peak in the curves is completelyinvisible and the maximum dimensionless bubble radius ymax

1 ofthe solutions decreases monotonically with increasing frequency.The amplification curves related to the threshold temperature areindicated by red curves in Fig. 2. From the application point of viewthis result is very crucial as the large amplitude (collapse-like) oscil-lations probably do not exist at all under the threshold tempera-ture; leading to a very small efficiency of the ultrasonictechnology. At higher pressure amplitudes, such as at pA ¼ 0:5 bar,the nonlinear effects become more dominant especially at hightemperature values, see the right hand side of Fig. 2. Several har-monic resonances are generated, and a hysteresis appears nearthe main resonance, indicating the coexistence of two distinct peri-odic solutions at the same parameter values. Early references onthese phenomena, including analytical formulae for frequencyresponse curves, are [60–62].

The aforementioned coexistence is demonstrated via bubbleradius vs. time curves and phase space diagrams in Fig. 3 at relativefrequency xR ¼ 0:8. The solutions corresponding to the upper andlower branch of the hysteresis at temperature value T1 ¼ 70 �C aredepicted by the black lines. In order to show the effect of the over-damped behaviour, the corresponding periodic solution atT1 ¼ 27:44 �C is also presented by the red curve. These solutionsare also marked by the blue dots in the right hand side of Fig. 2.

The coexisting periodic attractors have totally different beha-viour. The maximum bubble radius ymax

1 of the solution at theupper branch of the hysteresis is more than twice of the equilib-rium radius of the unexcited system (y1;E ¼ 1) in contrast to themaximum radius of the solution at the lower branch which is nomore than ymax

1 ¼ 1:4. Consequently, as the bubble starts shrinkingdue to the inertia of the liquid, the occurring maximum of the bub-ble wall velocity Vmax ¼ j _Rjmax is more than five times greater;however, it is still far below the required hundreds or even thou-sands of m/s, required to produce strong pressure waves. In the fol-lowing, 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 tem-perature, denoted by the red curve, is almost a smooth harmonicfunction with a maximum velocity of only Vmax ¼ 1:68 m=s.

It is worth mentioning that the period of the solutions pre-sented in Fig. 3 is 1, meaning that the solutions return to theirstarting point after one cycle of the harmonic forcing. Keep in mindthat the period of the excitation of the dimensionless system isso ¼ 1 according to the expression (21). Such orbits are usuallycalled period 1 solutions. We shall see in the next section that anumber of periodic solutions with different periods and chaoticsolutions can coexist even at the same parameter values becauseof the strong nonlinearity of the system.

The trajectories of the periodic attractors form closed curves inthe dimensionless phase space (y1 � y2 plane), like on the righthand side of Fig. 3. These, however, can intersect themselves lead-ing to unsuitable representation of a solution in this plane. To over-come this difficulty one can represent only the points of theso-called Poincaré map obtained by simply sampling the continu-ous trajectory at time instants s ¼ kso, where k ¼ 0;1; . . .. If a tra-jectory of an arbitrarily initiated system returns exactly to itsstarting value after N iterations, PNðyoÞ ¼ yo, then the solution isa periodic orbit whose period is sp ¼ Nso called period N solution.In our period 1 cases the Poincaré map returns to itself immedi-ately after the first period, that is, yo ¼ PðyoÞ ¼ P2ðyoÞ ¼ � � �, seethe 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 pressureamplitude pA and the ambient temperature T1 have the most sig-nificant effect on the amplitude of the oscillation. Simply, the big-ger the magnitude of the excitation is, the greater the response ofthe system becomes. Moreover, the higher the temperature is, thesmaller the viscosity of the liquid, which leads to weaker dampingrate and more rapid bubble motion. Therefore, pressure amplituderesponse curves are more suitable for seeking large amplitudeoscillations. Instead of the relative frequency xR, in this subsectionwe use pressure amplitude pA as control parameter. The ambienttemperature is still regarded as a secondary parameter (variedbetween 20 �C and 70 �C) while the relative frequency is kept con-stant at the value of the linear undamped resonant angular fre-quency xR ¼ 1 (f ¼ x=ð2pÞ ¼ 29:33 kHz). The bubble size is still

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 (leftpanel). 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).

F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164 157

RE ¼ 0:1 mm ¼ 100 lm, which is in the order of the experimentallyobserved sizes by [46].

The choice of the xR � RE parameter pair can be justified as fol-lows. As the liquid domain is irradiated with high frequency ultra-sound, the small, usually micron-sized gas bubbles (nuclei sites)start to oscillate around their equilibrium radius. As long as theintensity (pressure amplitude) of the ultrasound is low, it is a rel-atively smooth, small amplitude oscillation. Such bubbles arecalled inactive bubbles. For sufficiently high intensity, the bubblesbecome cavitationally active and start to oscillate with large ampli-tude, dominated by the inertia of the liquid domain. The maximumradius of a bubble during the oscillation can be several times largerthan its equilibrium size. The value of the irradiation intensity,which separates the cavitationally active and inactive bubbles, isknown as Blake’s threshold [63,64]. This threshold value is higherfor smaller bubbles, since the effect of the surface tension, whichattempts to contract the bubble as small as possible, is inverselyproportional to the size of the bubble. Conversely, for a given pres-sure amplitude only bubbles larger than a critical size will becomeactive.

Another important threshold value, in terms of the pressureamplitude, corresponds to the growth by rectified diffusion [65].It is well-known that a bubble at rest would dissolve due to theeffect of surface tension. Contrarily, during the oscillation of a bub-ble, more gas diffuses into the bubble interior in the expandedstate than diffuses back to the liquid in the collapsed state becauseof the large difference in the surface area between the two states.This phenomenon is called rectified diffusion and its effectincreases with the amplitude of the oscillation, that is, with theintensity of the ultrasonic irradiation and with the size of the bub-ble. The magnitudes of the two opposite effects determine the dis-solution or growth of an oscillating bubble [66]. Thus, for a givenpressure amplitude, there is another critical size which separatesdissolution 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 cavitation-ally active bubble always grows by rectified diffusion. Naturally,the time scale of the rectified diffusion is greater by orders of mag-nitude than that of the radial oscillation. A natural limitations inthe 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 diffu-sion and become cavitationally active. This process is a simplemanifestation of an acoustic cavitation cycle, for details see [69].

In water, the shape instability threshold is much smaller thanthe linear resonant size at a given frequency, approximately inthe orders of few micrometres. In glycerine, however, the thresholdof the shape instability is much higher than in water (tenth of mil-limetres), observed experimentally in [46]. This can lead to bubblesizes close to or even larger than the frequency-dependent reso-nance size. From the linear theory of rectified diffusion [70], it isknown that diffusionally stable bubbles exist above the resonancesize. Therefore, it is possible to drive the bubble at (or at least near)its resonance frequency for a few hundred acoustic cycles duringthe 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 sub-section. The pressure amplitude pA was increased from 0.01 bar to5 bar with 0.01 bar increments, and five randomly generated initialconditions were tried at each value of the control parameter toreveal coexistent attractors. After convergence, as many pointsfrom the Poincaré plane as the period of the solution were recorded(in case of chaos the number of the points were 512). An exampleof such a diagram at 40 �C is presented on the left hand side ofFig. 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 equilib-rium point of the unexcited system (pA ¼ 0; y1;E ¼ 1), undergoes aperiod doubling sequence resulting in the appearance of chaoticoscillation approximately at pA ¼ 3:69 bar. Then, windows of chao-tic and regular periodic solutions show up successively, and eachperiodic solution again ends in a period doubling cascade. A peri-odic window appears through a saddle–node bifurcation generat-ing a pair of stable and unstable periodic orbits. Immediatelyafter the bifurcation point, the stable periodic orbit replaces theoriginal chaotic attractor, which becomes unstable (transientchaos) [71]. The periods of the regular solutions are marked by ara-bic numbers in Fig. 4 left. An experimentally observedperiod-doubling route to chaos can be found in Refs. [72,73].

The Poincaré representation of a bifurcation structure of a sys-tem is a very efficient tool in general. For our purpose, however, it

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 numberas a function of the pressure amplitude (right).

158 F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164

is not completely satisfactory since it gives no information aboutthe maximum bubble wall velocity, which is a good indicator forthe strength of the collapse [70]. Therefore, after convergence themaxima of the absolute value of y2 were also recorded in each per-iod of the excitation. Now, on the right panel of Fig. 4 the samebifurcation structure can be seen as on the left hand side, butinstead of Pðy1Þ, the maximum of the bubble wall velocitiesjy2j

max rescaled to the Mach number Mmax by means of (18) isdemonstrated as a function of the control parameter pA. It is clearthat with increasing pressure amplitude the maximum Mach num-ber also increases, and at pA ¼ 5 bar it reaches Mmax ¼ 0:1, which isapproximately 190 m=s.

In order to reveal the effect of the temperature on the bifurca-tion structure of the system and on the achievable maximumMach number, several numerical computations, similar to the onespresented in Fig. 4, were performed between T1 ¼ 20 �C andT1 ¼ 70 �C with DT1 ¼ 5 �C increments. According to the resultsshown in Fig. 5, the strong influence of the temperature throughthe alteration of the viscosity is evident. With increasing tempera-ture the maximum bubble wall velocity increases, too. Moreover,the bifurcation structures become more complicated; however,above T1 ¼ 35 �C they are similar in the sense that chaotic andperiodic windows appear successively. The relevant periodic solu-tions found are again marked by arabic numbers.

The most important feature of the bifurcation structure is thatevery dominant periodic solution appears approximately at a cer-tain level of the maximum Mach number Mmax regardless of thevalue of the temperature, see the horizontal lines in Fig. 5C–F.For instance, it is a very good approximation that above the firstperiod doubling bifurcation point of the period 1 solution, the max-imum Mach number is more than 0.02. After the emergence of theperiod 3 solution via a saddle–node bifurcation, Mmax is definitelygreater than 0.06. Finally, the period 2 orbit that evolves from asaddle–node bifurcation and dominates the dynamics at high pres-sure amplitudes implies Mmax > 0:25. It is clear that the differenttypes of regular solutions, enclosed by bifurcation points, corre-spond to different achievable maximum Mach numbers.Fortunately, for finding periodic solutions, very efficient andsophisticated numerical techniques exist, including detection ofunstable solutions and bifurcation points. These methods shallopen the way to determine the boundaries of the different typeof orbits and thus the achievable maximum Mach numbers notonly as a function of a single control parameter, but in the whole

pA � T1 parameter space itself. Such computations are the topicof the next subsections.

3.3. Exploration of the periodic solutions by continuation technique

The idea to compute periodic solutions efficiently is to apply aboundary value problem (BVP) solver with periodic boundary con-ditions, and obtain the desired orbits directly. More precisely:

_y ¼ f ðy; sÞ; ð23Þ

with boundary conditions

yð0Þ ¼ yðspÞ; 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 vari-ous efficient numerical techniques to solve such BVPs, for instance,shooting, finite differences or orthogonal collocation method, whichare all insensitive to the stability of the periodic orbit. Once such asolution is computed, its evolution with respect to a control param-eter can be traced. This curve is called bifurcation curve, and thebifurcation points, where the change of the stability type takesplace, can also be detected. It was shown in Fig. 5 that these pointsplay an important role in separating different levels in the achiev-able maximum Mach number Mmax in the parameter space. Oneof the most popular parameter continuation method is the pseudoarc-length continuation technique as it is capable of followingcurves containing turning points (folds). A thorough discussion ofthe aforementioned numerical methods can be found in Ref. [74].

In the present study, the AUTO continuation and bifurcationanalysis software was used, see the manual of Doedel et al. [75].AUTO discretizes boundary value problems (including periodicsolutions) of ordinary differential equations (ODEs) by the methodof orthogonal collocation using piecewise polynomials with 2–7collocation points per mesh interval [76]. The mesh automaticallyadapts to the solution to equidistribute the local discretizationerror [77]. During our computations the relative error was 10�10.AUTO can handle only autonomous systems (free of explicit timedependence), thus system (10)–(21) has to be extended with twoadditional decoupled ODEs defined as

y03 ¼ y3 þ 2py4 � y3ðy23 þ y2

4Þ; ð25Þy04 ¼ �2py3 þ y4 � y4ðy2

3 þ y24Þ; ð26Þ

Fig. 5. Pressure amplitude response diagrams at different ambient temperatures T1 .

F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164 159

where the periodic solutions of the variables y3 and y4 are exactlycosð2psÞ and � sinð2psÞ, respectively. Thus, the fourth and secondterms in the right hand side of Eqs. (12) and (21) can be replaced by

� pAy3

p Aref

; ð27Þ

and

�pAy4; ð28Þ

respectively. This description has the disadvantage that AUTO canhandle a periodic solution only as a whole object regardless of itsperiod. Therefore, the illustrative representation in the Poincarémap has been lost, and one can obtain only the maximum valuesof the dimensionless variables (ymax

1 ; ymax2 ) of each kind of solution.

It is worth mentioning that the same bifurcation analysis software(AUTO) was used by Fyrillas and Szeri [66] for the rectified diffusionproblem.

In Fig. 6 the capabilities of the BVP solver is demonstrated bycomparing some of its results (coloured curves) with the corre-sponding bifurcation structure of the IVP solver (black dots) atT1 ¼ 50 �C. The family of the period 1 solution is presented bythe 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 atpA ¼ 2 bar denoted by the red cross. The solid and dashed partsof 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 branchswitching 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 aredisconnected. In fact, from the larger scale representation in Fig. 7,it is evident that the two parts are actually connected via multipleturning points called saddle–node (FL) bifurcation marked by theblue dots. This case demonstrates one of the several strength ofthe BVP solver, namely, distinct stable solutions can be foundsimultaneously if they are connected through unstable solutions.

A similar analysis can be made for the period 3 orbits (greencurves in Fig. 6). From the large scale representation displayed inFig. 8, it turns out that this solution forms a closed curve in thepressure amplitude response diagram. This behaviour of the solu-tions with odd periods was also found by Heged}us et al. [26].

As mentioned earlier, each domain of existence of stable peri-odic solutions, either period 2 or period 3 (solid branches inFig. 6), corresponds to a given range of maximum Mach number.Although in Fig. 6 the maximum of the dimensionless bubbleradius ymax

1 is plotted against the control parameter pA due to some

Fig. 6. Comparison of the results computed by the BVP solver (coloured curves) and the IVP solver (black dots) at T1 ¼ 50 �C. The solid and dashed lines are the stable andunstable solutions, respectively. The detected period doubling bifurcation points are denoted by the crosses, while the saddle–node (fold) bifurcations are represented by thedots at the turning points. The red, blue and green curves are period 1, 2 and 3 solutions, respectively.

Fig. 7. Large scale representation of the period 1 and the period 2 solutionscomputed by the BVP solver, see also the red and blue curves in Fig. 6. The blackrectangle shows the diagram limits of Fig. 6.

Fig. 8. Large scale representation of the period 3 solution computed by the BVPsolver, see also the green curve in Fig. 6.

160 F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164

specifics of AUTO, the maximum Mach number can be estimatedaccording to the results of Fig. 5 and the corresponding discussion.These achievable Mach numbers are also highlighted in Fig. 6.

3.4. Phase diagram

The results of Fig. 6 have shown that the detected bifurcationpoints (coloured crosses and dots) are the boundaries of the exis-tence of the different type of stable periodic orbits. The appearanceof such attractors define an approximated threshold in terms of thecontrol parameter pA for the accessible maximum Mach numberMmax. These threshold values, however, vary with the ambienttemperature T1, cf. Fig. 5C–F. Naturally, the higher the tempera-ture, the lower the threshold value of the pressure amplitude fora given value of the maximum Mach number. For instance, the per-iod doubling bifurcation related to Mmax > 0:02 takes place atpA ¼ 2:43 bar at T1 ¼ 45 �C, while at T1 ¼ 70 �C the pressureamplitude is only pA ¼ 1:75 bar, see Fig. 5. This means that therequired energy of the sound radiation for a given Mmax decreases,allowing lower operational costs. On the other hand, increasing thetemperature also requires a certain amount of energy transfer intothe liquid domain acting oppositely on the costs. Moreover, toohigh temperature can cause unwanted chemical reactions ormolecular degradation in the glycerine. In order to apply optimaloperational parameters (pressure amplitude pA–temperature T1pair), it is essentially important to reveal the evolution of thethreshold values of pA as a function of the ambient temperature,that is, tracking the path of the detected bifurcation points in thepA � T1 plane.

The AUTO bifurcation analysis programme enables to obtain therequired two-parameter (codim 2) bifurcation curves easily bychoosing the ambient temperature T1 as a secondary controlparameter. The results of the computations that correspond to allthe detected bifurcation points presented in Fig. 6 are shown inFig. 9. The dashed and solid curves are the period doubling (PD,crosses in Fig. 6) and fold (FL, dots in Fig. 6) bifurcations, respec-tively. The piecewise smooth nature of the curves is an artifact ofthe linear interpolation between the tabulated values of the mate-rial properties. The vertical thick red line is the threshold temper-ature T1 ¼ 27:44 �C related to the linear overdamped system.

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 foldbifurcations, respectively. The vertical thick red line is the threshold temperature T1 ¼ 27:44 �C, under which the linear system is overdamped. The boundaries of theachievable maximum Mach numbers Mmax are denoted by the rectangular based arrows. The period 1, 2 and 3 domains are illustrated by the light blue, light brown andyellow regions, respectively. The periodicities are also marked by arabic numbers.

F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164 161

The two black dashed curves are the period doubling bifurca-tions of the period 1 and the period 2 solutions shown by the redand blue crosses in Fig. 6, respectively. Because of the hyperbolictype behaviour of the curves, the pressure amplitude pA requiredto reach these PD bifurcations increases extremely fast as the ambi-ent temperature approaches to the threshold temperatureT1 ¼ 27:44 �C. Choosing control parameters above the lower blackcurve guarantees only Mmax > 0:02 (approximately Vmax > 40 m=s).The domains of the period 1 and 2 solutions, defined by the twocurves, are highlighted by the light blue and light brown regions,respectively. Above the upper black curve, the solutions becomechaotic via period doubling cascades.

The next definite step to increase the maximum Mach numberis to reach the period 3 solution, which has more complicatedstructure than the period doubling sequence discussed above. Astable period 3 solution, for instance, lies between the green solidand the green dashed lines, both are turning back at approximatelyT1 ¼ 40 �C. Inside this scythe-shaped domain a hysteresis appears(saddle–node bifurcation pair) at T1 ¼ 52:5 �C via a cusp bifurca-tion towards the decreasing temperature values, which resultsthe birth of another stable period 3 domain. This secondary period3 attractor is enclosed by the solid and dashed brown curves. Theupper and lower branch of the solid brown curve, which are initi-ated from the cusp point, turn back at T1 ¼ 44:6 �C andT1 ¼ 37:7 �C, respectively, causing a three arm shaped stable per-iod 3 domain. By reaching either the solid green line or the solidbrown line by increasing the pressure amplitude pA and/or theambient temperature T1 the maximum Mach number shall behigher than Mmax > 0:06 (Vmax > 110 m=s). The found stable period3 domains are denoted by the yellow areas.

The appearance of the period 2 solution in the high pressureamplitude region, see the blue dot in Fig. 6, ensures greater maxi-mum Mach number than Mmax ¼ 0:25 (Vmax > 470 m=s). The corre-sponding two-parameter bifurcation curve of this saddle–nodepoint is the blue solid line in Fig. 9. Again, due to the hyperbolictype nature, the required pressure amplitude can be decreasedonly in case of high enough ambient temperature. The dashed blueline is related to the first period doubling bifurcation after the

saddle–node point. This can only be seen in the large scale repre-sentation in Fig. 7. Between the two blue curves, there is a rela-tively large period 2 domain represented by a light brown area.After the dashed blue curve, the solutions again become chaoticthrough a period doubling cascade.

Fig. 9 is a good summary of the bifurcation structure in the pres-sure amplitude pA – ambient temperature T1 parameter plane,which helps identify the different levels of the strength of bubblecollapse in terms of the maximum Mach number Mmax. The maxi-mum Mach number is a useful indicator in many applications, e.g.for the generation of shock waves in ultrasonic cleaning [78], or incase of micromixing and microstreaming which are quite impor-tant in sonochemistry for competitive reactions [79]. In the litera-ture, however, there are many other, well-defined quantitiesserving to measure the collapse intensity. High pressure and tem-perature inside the bubble are also required to start chemical reac-tions, and they play a role especially when the controllingmechanism is pyrolysis [80,81]. If the production of free radicalsis the keen interest, such as in wastewater treatment [82], themaximum bubble radius Rmax seems to be the most prominentindicator [83,84]. These various quantities are actually scalestogether, for instance, the larger the maximum bubble radiusRmax the higher the maximum Mach number Mmax, see the recentinvestigation of Varga and Paál [28]. Therefore, our results pre-sented in Fig. 9 and the conclusions drawn are useful in generalfor the above mentioned applications.

It is important to emphasize that the bifurcation curves in Fig. 9are not exactly iso-curves of the Mach number but they are goodlower estimates. Although it is not a precise description, it hasthe advantage that high Mach number oscillations can be detectedby monitoring the subharmonic component of the acoustic emis-sion spectra of the bubble/bubble cluster. This idea is supportedby the study of Mettin [50], who reported a strong connectionbetween the appearance of subharmoinc emission and cavitationerosion. The values of the maximum Mach numbers, presented inFig. 9, may differ for other bubble sizes RE or frequencies xR. Thebifurcation structure, however, should be qualitatively similar inthe T1 � PA parameter plane as the critical temperature

162 F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164

T1 ¼ 27:44 �C (red vertical line) dominates the behaviour of thesystem. That is, high subharmoinc component can always indicatean increase in the strength of the collapse without the explicitknowledge of the maximum Mach number. It is worth noting thathigh periodicity, e.g. period N, can decrease the emittedenergy/acoustic cycle, since strong collapse occurs only once atevery N acoustic periods [85]. This behaviour is also seen, forinstance, in Fig. 5F where the maximum Mach numbers of the sub-sequent acoustic cycles of the period 2 solution, existing abovepA 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 caseof moderately high viscosity (in the order of 200 mPa s) the maxi-mum 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 rapidlywith the viscosity. This observation raises the issue of the applica-tion of water–glycerine mixture with optimised concentration.Moreover, the numerical study of Yasui et al. [85] reported thatincorporating the thermal processes into the bubble model theresulting collapse strength can be stronger. In this sense, ournumerical results (without thermal effects) serve as a good lowerestimate for the bubble wall velocities.

The scan for the stable bubble oscillation presented in Fig. 5implies that there is a global limit in the maximum Mach numberMmax, see especially subplot F. The long-term behaviour of Mmax

presented in a linear scale in Fig. 10, however, reveals that Mmax

increases almost linearly with the pressure amplitude pA.Although the maximum value of pA ¼ 15 bar is difficult to achieveduring an experiment, Fig. 10 is a perfect demonstration of howdifficult it is to achieve extremely high bubble wall velocities. Forinstance, to achieve Mmax ¼ 1 (supersonic speed) one needs toincrease the pressure amplitude above pA ¼ 10 bar. Observe, thatthe ambient temperature is already as high as T1 ¼ 70 �C. At suchhigh bubble wall velocities (Mach number near or higher than 1),our model ceases to be valid as the Keller–Miksis equation is onlya first order approximation in terms of the Mach number [86].Moreover, according to the derivation of Yasui [47] based on thefundamental theory of fluid dynamics, the bubble wall velocity _Rat the collapse never exceeds the sound velocity of the liquid atthe bubble wall cL. The author continuously monitored this condi-tion during the computer simulations, and the bubble wall velocity_R was replaced by cL if it exceeds cL. During our simulations thiscondition was not tested.

Fig. 10. Long-term behaviour of the maximum Mach number Mmax as a function ofthe pressure amplitude pA .

4. Summary

Excited spherical gas/vapour bubbles can exhibit collapse-likeoscillations, causing extreme conditions (high temperature, pres-sure and shock waves) in the collapse phase. These conditions areexploited by the rapidly developing ultrasonic technology, whichhas several industrial applications. This was the main motivationto study a spherical bubble placed into highly viscous glycerine dri-ven by a purely harmonic signal. The bubble model was the modi-fied Keller–Miksis equation, a second order nonlinear ordinarydifferential equation, which takes into account the liquid compress-ibility. High viscosity causes a massive, temperature-dependentdamping rate, which can be a limiting factor on the applicabilityof the ultrasonic technology on such liquids. The findings of the pre-sent paper can help increase the efficiency of the applications anddecrease the costs of the operation.

The investigation of the frequency response curves revealedthat due to the high viscosity, the system behaves like an over-damped linear oscillator below the threshold ambient temperatureof 27:44 �C. This very high damping rate weakens the strength ofthe collapse by decreasing the peak bubble wall velocity.Therefore, increasing the ambient temperature above this thresh-old value is highly recommended.

The results of the pressure amplitude response curves at differ-ent ambient temperatures and at constant excitation frequencydemonstrated that the appearing bifurcation points of the simpleperiodic solutions can be used as a good estimate of the maximumbubble wall velocity. This observation enabled us to determine theboundaries of the achievable maximum bubble wall velocity in theexcitation pressure amplitude–ambient temperature parameterspace. The resulting parametric map can aid the applications tooperate the technology in an efficient way.

Acknowledgement

The research described in this paper was supported by theHungarian Scientific Research Fund – OTKA, from the Grant No.K81621.

Appendix A. Material properties

A.1. KDB equation for vapour pressure

The vapour pressure of the glycerine were calculated by meansof the KDB correlation equation ([87]):

ln pV ¼ A ln T1 þB

T1þ C þ DT2

1; ðA:1Þ

where pV is in kPa, T1 is in K and the coefficients are

A ¼ �2:125867 � 101; ðA:2ÞB ¼ �1:672626 � 104; ðA:3ÞC ¼ þ1:655099 � 102; ðA:4ÞD ¼ þ1:100480 � 10�5: ðA:5Þ

A.2. Tabulated values of the material properties

In the following, the tabulated values of the material propertiesof the glycerine1 can be found as a function of the ambient temper-ature T1. For other ambient temperatures, the corresponding valueswere calculated with linear interpolation (see Tables A.1–A.4).

1 The tabulated values of the material properties were taken from the results of TheDow Chemical Company (1995–2014); web page: http://www.dow.com/.

Table A.3Tabulated values of the glycerine sound speed cL as a function of the ambienttemperature T1.

T1 ð�CÞ 10 20 30 40 50cL ðm=sÞ 1941.5 1923 1905 1886.5 1869.5

Table A.4Tabulated values of the glycerine surface tension r as a function of the ambienttemperature T1.

T1 ð�CÞ 20 90 150r ðN=mÞ 0.0634 0.0586 0.0519

Table A.1Tabulated values of the glycerine density qL as a function of the ambient temperatureT1 .

T1 ð�CÞ 0 10 15 20 30 40 54qL ðkg=m3Þ 1272.7 1267.0 1264.4 1261.3 1255.1 1249.0 1239.7

T1 ð�CÞ 75.5 99.5 110 120 130 140 160qL ðkg=m3Þ 1225.6 1209.7 1201.8 1194.5 1187.2 1179.5 1164.4

Table A.2Tabulated values of the glycerine viscosity lL as a function of the ambienttemperature T1.

T1 ð�CÞ 0 10 20 30 40 50 60lL ðPa sÞ 12.07 3.9 1.41 0.612 0.284 0.142 0.0813

T1 ð�CÞ 70 80 90 100 110 120 130lL ðPa sÞ 0.0506 0.0319 0.0213 0.0148 0.0105 0.00780 0.00599

F. Heged}us, K. Klapcsik / Ultrasonics Sonochemistry 27 (2015) 153–164 163

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