FLUIDS Revista Mexicana de Fısica S59 (1) 77–83 FEBRUARY 2013

Effect of liquid density on the Rayleigh-Taylor instabilityof sonoluminescing bubbles

F. A. Godınez and M. NavarreteInstituto de Ingenierıa, Universidad Nacional Autonoma de Mexico,

Ciudad Universitaria, Coyoacan 04510, Mexico D. F., Mexico,e-mail- fgodinezr@gmail.com

Received 30 de junio de 2011; accepted 30 de noviembre de 2011

Rayleigh-Taylor instability of a radially oscillating gas bubble immersed in water is numerically studied. The model includes density changesin both gas within the bubble and liquid around its wall. The radial dynamics is described by Gilmore equation, which is coupled with thevan der Waals (gas) and Tait (liquid) equations of state for simulating density variations. Shape instability is analyzed by the classical Hill’sequation, which describes the time evolution of the distortion amplitude. During the last moments of collapse, a sudden rise in gas and liquiddensities is predicted; this exerts a stabilizing effect on the Rayleigh-Taylor instability. An instability borderline set by a new overwhelmingcriterion is in good agreement with an experimental boundary.

Keywords: Rayleigh-Taylor instability; single bubble sonoluminescence; surface distortion.

Se estudia numericamente la inestabilidad de Rayleigh-Taylor de una burbuja de gas que oscila radialmente y esta inmersa en agua. El modeloincluye cambios de densidad tanto del gas contenido en la burbuja como del lıquido en su superficie. La dinamica radial se describe con laecuacion de Gilmore; la cual, esta acoplada a las ecuaciones de estado de van der Waals (gas) y Tait (lıquido) para simular las variaciones dedensidad. La inestabilidad superficial se estudia mediante la evolucion de la amplitud de una distorsion descrita con la ecuacion de Hill dela teorıa clasica. Durante losultimos instantes del colapso, se predice un aumento subito en las densidades del gas y del lıquido ejerciendoun efecto estabilizador sobre la inestabilidad de Rayleigh-Taylor. Una lınea fronteriza de inestabilidad calculada con un nuevo criterio deaplastamiento se aproxima bien a una frontera experimental.

Descriptores: Inestabilidad de Rayleigh-Taylor; sonoluminiscencia de una burbuja; distorsion superficial.

PACS: 78.60.Mq; 43.25.+y; 47.20.Ma

1. Introduction

In typical experiments of single-bubble sonoluminescence(SBSL), a gas bubble in a liquid is acoustically levitated andmade to oscillate so violently that pulses of light are emit-ted at the time of collapse [1]. Stable SBSL is characterizedby light emission in each period of the acoustic driving atprecisely the same bubble oscillation phase and precisely thesame brightness for millions of cycles [2]. However, stableSBSL occurs in a narrow region of the parameter space (Pa,R0); wherePa is the driving pressure amplitude andR0 isthe equilibrium bubble radius [3]. This region is delimited byprocesses as: dissolved gas diffusion, chemical reactions, andspherical shape instabilities. This paper is concerned with thestudy of the later. In particular, the Rayleigh-Taylor instabil-ity (RTI) is considered. The RTI occurs near the minimumbubble radius when gas is accelerated towards the fluid andacts on the timescale of nanoseconds [3]. In Ref. 4, it is sug-gested that the RTI of the bubble surface is responsible forthe extinction of the sonoluminescence whenPa exceeds acertain threshold. In fact, the onset of the RTI is connectedwith the destruction of the bubble into fragments.

Some numerical studies on shape stability of oscillatinggas bubbles have shown that the shape and location of theRTI threshold lines in the (Pa, R0) space are strongly influ-

enced by diverse assumptions as: the value of the bound-ary layer thickness (δ) [5] and the type of gas compressionthat depends on the polytropic index (γ) [6]. Also, the trendand/or location of the RTI boundaries are very sensitive to pe-culiarities of the radial dynamics model; simplified versionsof Rayleigh-Plesset equation (RPE) might result in bad pre-dictions of the stability zone [7]. Thus, it is desirable to im-prove the models by introducing some corrections. In Ref. 6and 7, this was done by taking into account the variation ofgas density inside the bubble; and it was found a consider-able suppression of the RTI and the threshold lines with thisimprovement approach better to some experimental data. InRef. 5, the RTI lines were computed by assuming a smallnon-spherical perturbation in the driving sound field; the nu-merical results are in good agreement with many experimen-tal data.

In most literature including the above mentioned works,the liquid density at the bubble wall is considered a constantat room conditions. This approximation is well justified inmost of the acoustic cycle, but when the bubble is close to itsminimum radius, it is expected that liquid compressibility ef-fects become important. The aim of this paper is to improvethe shape stability analysis of a radially oscillating gas bubbleby taking into account liquid density variations on its surfaceand gas density changes within it.

78 F. A. GODINEZ AND M. NAVARRETE

2. Model and methodology

2.1. Radial dynamics

The Gilmore model [8] describing the radial motion of a bub-ble is used.

(1− R

C

)RR +

32

(1− R

3C

)R2

=

(1 +

R

C

)H +

(1− R

C

)R

C

dH

dt, (1)

H =

P∫

P∞

ρ−1l dP, (2)

C = [c20 + (m− 1)H]1/2, (3)

P = Pg − 2σ

R− 4µ

RR. (4)

HereR is the bubble radius;C, ρl, andP are the speedof sound in the liquid, its density, and the pressure at the bub-ble wall, respectively.H is the difference between the liquidenthalpy at the bubble wall and at infinity, andc0 is the speedof sound in the liquid at room conditions.

The pressure at infinity isP∞ = P0+Pa sin(2πft), P0 isthe ambient liquid pressure;Pa is the amplitude of the drivingsound field.

It is assumed that gas pressure inside the bubble obeys avan der Waals type equation of state [6]:

Pg =(

P0 +2σ

R0

) (R3

0 − h3

R3 − h3

)γ

, (5)

hereR0 is the ambient bubble radius,h is the hard core vander Waals radius, andγ is the effective polytropic exponent.

Gas densityρg inside the bubble varies according to

ρg =PgMg

<Tg + Pgb, (6)

where< is the universal gas constant,Mg is the mass permole of gas, andb is the excluded van der Waals hard corevolume.

The following version of Tait’s equation [9] is used formodeling the liquid densityρl:

ρl = ρ0

(P + B

P0 + B

)1/m

, (7)

whereB andm are constants depending upon the liquid, andρ0 is the ambient liquid density.

In order to solve (1), it is necessary to knowHas func-tion of P , which at the same time is function ofR. From (2)and (7) it is readily found that

H=m(P0 + B)

1m

(m− 1)ρ0

[(B+P )

m−1m − (B + P∞)

m−1m

]. (8)

2.2. Dynamics of the distortion amplitude

The growth or decay of a small initial shape distortion fromthe spherical bubble interface is expressed in a form of spher-ical harmonics expansion:

r = R(t) +∞∑

n=2

n∑q=−n

aqn(t)Y q

n (θ, φ), (9)

whereR(t) is the undistorted bubble radius,Y qn is a spher-

ical harmonic withq and n denoting its degree and moderespectively, andaq

n is the corresponding amplitude. In thelinearized case|aq

n/R| ¿ 1, and the superscriptq is dropped.Thus, the differential equation that describes the time evolu-tion of the distortion amplitude is [6]:

an(t) + Bn(t)an(t) + An(t)an(t) = 0, (10)

Bn(t) = 3R

R+

[n(n+2)2

1+2δ/R − βn

]2µ

ρlR2(1 + n+1

nρg

ρl

) , (11)

An(t) =

{[(n + 1)(n + 2)

n

ρg

ρl− (n− 1)

]R

R+

βnσ

ρlR3

+[βn − n(n− 1)(n + 2)

1 + 2δ/R

]2µR

ρlR3

}

×{

1/

(1 +

n + 1n

ρg

ρl

)}, (12)

whereβn=(n-1)(n+1)(n+2), dots denote time derivatives,µis the dynamic viscosity andσ the surface tension of the sur-rounding liquid. Equations (11) and (12) employ a bubbleboundary layer type approximation (BLA) [3] with a thick-nessδ defined by:

δ = min(√

µ

ρlω,R(t)2n

), (13)

wheref = ω/2π is the frequency of the driving sound field.

2.3. Rayleigh-Taylor instability

The RTI is studied by following the Bogoyavlenskiy ap-proach [10]; which consists of calculating the amplificationfactor an(t)/an(0) of an initial distortion of nano-metricsizes only during the primary collapse, which is defined as theperiod from the moment of maximum bubble radiustR max,when an initial perturbation (an(0), an(0)) starts to grow, tothe second bounce moment (as indicated by our previous nu-merical experiments) when the distortionan(t) stops grow-ing.

The following criterion to determine the RT borderlinesis adopted [3]:

max{tR max < t < tR min + tRT }

( |an(t)|R(t)

)≥ 1, (14)

Rev. Mex. Fis. S59 (1) (2013) 77–83

EFFECT OF LIQUID DENSITY ON THE RAYLEIGH-TAYLOR INSTABILITY OF SONOLUMINESCING BUBBLES 79

FIGURE 1. Time dependence of the bubble radius in a typical sin-gle bubble sonoluminescence cycle, calculated by (1). The bubblehas an equilibrium radius ofR0=4 µm, pressure and frequency ofthe driving are:Pa=0.142 MP andf=32.8 kHz respectively.

which states that the distortion can overwhelm the bubble ra-diusR(t) within the collapse stage.

HeretR minis the moment of minimum bubble radius, andtRT is set to 2µs to cover the moment when the distortionstops growing. In Fig. (1) are indicatedtR max, tR min andtR min with the corresponding radial positions. It is worthnoticing that both the displacement and velocity of an initialperturbation will affect the location of the RT borderline de-termined by (14).

As in Ref. 6, an initial perturbation of (an(0)= 10 nm,an(0)= 0) represents well the actual imperfections that mayoccur,e.g., thermal fluctuations of molecules at the liquid-gasbubble wall [11].

Before calculating the RT borderlines, Eq. (1) was solvedfor the first 10 cycles to make sure that the bubble oscilla-tions reached a steady state [12]. Then the RT boundaries arethe result of an incremental search on the (Pa, R0) param-eter space covering the ranges:0.05 < Pa < 0.253 MPa,0.6 < R0 < 7 µm. The computational resolution to calculatethe (Pa, R0) space was:∆Pa=0.001 MPa,∆R0= 0.05µm.

The numerical results are compared to experimental dataof Ketterling and Apfel [13], for argon bubbles in water. Theliquid temperature and ambient pressure are at room condi-tions (P0=101.325 kPa,T0=293.16 K) and the frequency ofthe acoustic forcing is 32.8 kHz. Heat exchange between thebubble and the surrounding water is not taken into accountand for simplicity the gas compression during whole bubbleoscillation is assumed isothermalγ =1 [3]; i.e., the heat trans-fer is faster than the time scale of the bubble motion, thus thetemperature of the gas in the bubble is the same as that of theliquid. It is worthy of note that the isothermal approximationholds in most of the oscillation cycle without affecting theglobal dynamics of the bubble. But, when the bubble reachesthe final stages of its first and subsequent collapses-rebounds,the assumption of an adiabatic gas compression (γ ≈5/3) ismore suitable since the bubble wall moves much faster thanthe heat transfer rate. Nevertheless, the implementation of a

numerical scheme by switchingγ would imply more difficul-ties in programming and longer computing time.

For modeling the distortion amplitude, the quadrupletmode (n=2) is considered. The remaining parameters for nu-merical simulations are set to:c0=1483 m/s,σ =0.073 N/m,µ=0.001 Pa s,B=304.7 MPa,m=7, b= 0.03219 l/mole,h = R0/8.86 m,Mg=0.0399 kg/mole,<=8.3145 J/mole K.

3. Results and discussion

Three study cases are considered:

Case I:The density ratioDR=ρg /ρlin (11) and (12)is equal to zero, employing the same formulas used inRef. 3.

Case II: Gas density varies as a function of time andis modeled by (6), liquid density is taken as a con-stantρl=1000 kg/m3; of courseDR is also a functionof time.

Case III: The gas and liquid densities are varying asfunctions of time, so they are modeled by (6) and (7)respectively; in this caseDR is also a function of time.

Figure (2) shows the computed RTI borderlines in the(Pa, R0) space for the three study cases. To the left of a RTIline the bubble is stable. The RTI line forcase Ihas a neg-ative slope, as well; it is located on the left and far from theexperimental boundary. Clearly, this line significantly under-predicts the experimental threshold.

When including gas density variations,case II, the pre-diction shows a borderline located to the right of the line for

FIGURE 2. Calculated Rayleigh-Taylor instability borderlines forthe three study cases. Symbols represent stability thresholds fromexperiments by Ketterling and Apfel [13], for argon bubbles peri-odically driven atf=32.8 kHz in liquid water at 293 K and at anambient pressure of 0.101 MPa.

Rev. Mex. Fis. S59 (1) (2013) 77–83

80 F. A. GODINEZ AND M. NAVARRETE

FIGURE 3. (a) and (b) show the time development of the radius, and acceleration of a bubble during the brief stage of collapse and rebounce,for R0=4µm andPa= 0.147 MPa.t= 0 indicates the moment of minimum bubble radius. (c) and (d) show the time evolution of the gas andliquid densities respectively, (e) shows theDR profile, and (f) shows the behavior ofA2(t). Dotted lines correspond tocase I, dash-dottedlines are forcase IIand solid lines are forcase III.

Rev. Mex. Fis. S59 (1) (2013) 77–83

EFFECT OF LIQUID DENSITY ON THE RAYLEIGH-TAYLOR INSTABILITY OF SONOLUMINESCING BUBBLES 81

case I, indicating a bigger stability region in the (Pa, R0)space and better agreement with the experiments, howeverthis result remains conservative. The position change of linefor case II, and the enlargement of the stability region areconsequences of the stabilizing effect of gas density vari-ations on the RTI according to [7]. Additionally, if liquiddensity varies, an extra widening of the stability zone in the(Pa, R0) space is found, see thick solid line in Fig. (2).

Although the line forcase III indicates a stabilizing ef-fect due to changes inρg andρl, it does not fit well with theexperimental boundaries. In fact, it over predicts the stabilityzones. Now, the effect of gas and liquid densities on the RTIfrom theAn(t) coefficient in the third term of (10), is ana-lyzed with major detail. As stated in Ref. 6, the density ratioDR strongly affectsAn(t) changing the contribution ofR, adominant factor in the RTI.

In most time of a cycle or atPa < 0.1 MPa,DR is small.But for largerPa and when the bubble radius is near to itsminimum, it becomes significant. Without considering gasdensity variations,case I, A2(t) resembles a negative pulseimplying instability, see dotted line in Fig. (3f). Figure (3c)shows the gas density profile forcase II, which is identical totheDR curve (but dimensionless); ifR(t) equals theRmin,the gas density reaches a maximum (∼ 1200 kg/m3 limitedby the hard core van der Waals radius for argon bubbles)and A2(t) becomes a positive pulse decreasing the ampli-tude of shape distortions, see dash-dotted line in Fig. (3f).In Fig. (3e) theDR curve shows a peak with a local maxi-mum nearRmin , this behavior is associated with the suddenrise of liquid density predicted by present model. Fig. (3d)shows the liquid density behavior, this resembles a pulse witha maximum∼ 2000 Kg/m3 that even suggests transient so-lidification of water around the bubble wall,i.e., ice particlesformation [14]. A2(t) pulse shape curve forcase III, solidline in Fig. (3f), is similar to that found forcase II, but withless amplitude. As before, this implies suppression of theRTI.

It is important to note that the change in behavior ofA2(t)due to gas and/or liquid density variations are only true at theend of a violent collapse; in fact the RTI will occur during thesubsequent afterbounces,i.e., after the first collapse [11].

This is true for the three study cases, Fig. (4) depicts howthe RTI amplifies an initial distortion over the first bounce un-til reaching a maximum when bubble collapses for a secondtime. But at the same time, Fig. (4) shows that the maximaldeformationMax |a2| is attenuated by the increment in bothgas and liquid densities near the collapse.

Figure (4a) indicates that without taking into account gasdensity changesMax |a2|≈1.12µm, now if gas density vari-ations are consideredMax |a2|≈ 98 nm, this is a reduction≈ 91%. By including liquid density changes the predictedMax |a2| ≈ 40 nm; this is a reduction≈ 96 % with respectto case I.

Figure (5) shows a comparison between the RTI bound-aries as computed with two different models of radial oscil-lations. Dash-dotted line, calculated by considering gas den-

FIGURE 4. At the top,R(t) versus time during collapse stage andthe radial bounces following the collapse, for aR0=4 µm bubbledriven atPa=0.132 MPa andf=32.8 kHz. Zero time correspondsto the moment of minimum bubble radius. The corresponding timehistory of the distortion amplitude a2(t), for: (a)case I, (b) case IIand (c)case III.

FIGURE 5. Influence of the modeling of the radial bubble dynamicson the determination of the Rayleigh-Taylor instability borderlines.Lines for cases IIand III were computed from Gilmore model.Curves labeled as: isothermal R-P and reduced R-P were calcu-lated from Rayleigh-Plesset model [7]. Symbols indicate stabilitythresholds from experiments [13] as explained in Fig. (2).

sity changes and using a simplified RPE with an isothermalassumptionγ=1[7], is compared with line forcase II.

Clearly the RTI line calculated via RPE has a verticalslope and is located to the left of line forcase II, implyingstronger collapses and more RT instable bubbles. This resultis explained by Gilmore equation, which contains correctionterms for liquid compressibility (with a damping effect on theradial oscillations) that make it a better description than the

Rev. Mex. Fis. S59 (1) (2013) 77–83

82 F. A. GODINEZ AND M. NAVARRETE

FIGURE 6. Comparison between the Rayleigh-Taylor instabilityborderlines as computed with different values of the new over-whelming criterion defined by (15). Symbols indicate stabilitythresholds from experiments [13] as explained in Fig. (2).

RPE. Dashed borderline was calculated with a RPE but in-cluding radial damping mechanisms as heat transfer. Now,the line is less conservative and shows a positive slope inagreement with the experimental trend at least for bubbleswith R0 ≤ 3.5µm.

According to Fig. (5), the present model forcase III,predicts a RTI line with positive slope for bubbles withR0 ≥ 3 µm. However, the experimental RTI threshold sug-gests the contrary behavior, that is to say a negative slopecurve especially for bubbles in the range 3≤ R0 ≤ 5.5µm.

As stated before, the RTI borderlines determined by crite-rion (14) imply the rupture of the bubbles. Now, we proposean overwhelming criterion defined as

max{tR max < t < tR min + tRT }

( |an(t)|R(t)

)= α

0 < α ≤ 1, (15)

which, in principle forα < 1, it would allow calculation

of RTI threshold lines that imply shape deformed bubblesdisplaying radial oscillations over many acoustic cycles andeven emitting light pulses. In other words, the bubbles wouldremain after collapse with no break-up and dissolution.

Figure (6) illustrates RTI lines calculated from three val-ues of the new overwhelming criterion,i.e., α = {1, 0.5,and 0.3}. When α=1 the RTI line corresponds to that ofcase III in Fig. (2), forα = 0.5 a RTI line slightly differentfrom that of case III is obtained.

Nevertheless, forα = 0.3 the prediction is a RTI linewhose slope has changed from positive to negative matchingwell the experimental data.

4. Conclusions

During the last instants of a micrometric bubble implosion,the presented simulations have revealed a sharp rise in gas(inside) and liquid (around the surface) densities suppressingsignificantly the Rayleigh-Taylor instability.

In recent years researchers have focused their attention onproducing SBSL in acid–water binary liquid mixtures. Themain reason is that bubbles in these liquid media emit lightpulses more intense than those produced in water. However,only few theoretical works on spherical stability of these bub-bles have been published and these are based on simplifiedRPE models [15-17].

It is expected that the stabilizing effect of gas and liq-uid densities on shape distortions of typical sonoluminesc-ing bubbles found here, appear and behave similarly in SBSLacid-water systems.

Acknowledgments

The first author wishes to thank CONACyT-Mexico for thescholarship granted during his doctoral studies. The authorsthank J. A. Ketterling and R. E. Apfel for making their exper-imental data available. The authors also thank Jose FranciscoHernandez Valle for his useful comments.

Work supported by Instituto de Ingenierıa/UNAM undergrant number 9152 and DGAPA/UNAM under grant numberIN107509.

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