ON THRESHOLDS FOR SURFACE WAVES ON RESONANT BUBBLES

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ON THRESHOLDS FOR SURFACE WAVES ON RESONANT BUBBLES

R. Nabergoj, A. Francescutto

To cite this version:

R. Nabergoj, A. Francescutto. ON THRESHOLDS FOR SURFACE WAVES ON RESONANT BUB- BLES. Journal de Physique Colloques, 1979, 40 (C8), pp.C8-306-C8-309. �10.1051/jphyscol:1979854�.

�jpa-00219560�

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ON T H R E S H O L D S FOR S U R F A C E W A V E S ON R E S O N A N T B U B B L E S NABERGOJ R., and FRANCESCUTTO A.

Istituto di Fisica dell'Universita di Trieste, 34 127 - Trieste, Italy

Abstract. - The problem of the stability of gas bubbles in a sound field has been investigated using a model for the motion of a nearly spherical bubble in a slightly viscous liquid. The radial motion amplitude and the pressure amplitude thresholds for the instability of the spherical shape have been obtained by the theory of parametric resonance. The comparison with available experimental data for resonant bubbles appears to be rather satisfactory.

1. Introduction. - The mechanism of acoustic cavi- tation is closely influenced by the behavior of gas bubbles in the liquid. In real situations, a pulsating bubble does not maintain the spherical shape, due to the onset of surface waves at a particular threshold. When the amplitude of the deformations is sufficiently high, microbubbles can be generated at the bubble surface /!/. It appears that certain combinations > of bubble radius and drinving frequency are especially effective in bubble formation. This fact is related to the parametric excitation of sur^

face waves by radial oscillations.

The aim of the present work is to obtain the threshold in terms of the amplitude of the radial motion or in terms of the corresponding pressure

amplitude of the driving fields. The viscous effects are directly introduced in the equations of motion for the non-spherical bubble, and this represents the main difference from the preceding theories of Hsieh-Plesset /2,3/ and Benjamin /4,5/.

The general theory of the parametric excitation of surface waves on pulsating gas bubbles is considered in Part 2. Simplified formulas of the thresholds for resonant bubbles are given in Part 3.

2. Parametric excitation of surface waves. - Let us consider a nearly spherical bubble in an incompres- sible, unbounded liquid of small viscosity. As usual / 6 , 7/, the free surface may be described in terms of spherical harmonics Y , that is

where a (t) are the amplitudes of the surface deformations and R(t) is the mean radius, obtained from the Rayleigh-Plesset equation. The behavior of the surface for |a | « R can be obtained in the limit of small amplitude oscillations, and in the linearized theory, the equations describing the different amplitudes become uncoupled. Insofar as small viscosity liquids are considered, elsewhere / 8 / we obtained, by using the Lagrangian procedure with a dissipation function, the equation for the a 's in the form

where p is the density of the liquid, a the surface tension and u the viscosity. In the exact so- lution the equation of motion is more involved / 9 / , having an integro-differential structure, except in the case of negligible boundary layer effects, where Eq.l is recovered /10/.

By constant radius R(t) = R , that is for surface oscillations about the equilibrium position, Eq.l reduces to the equation for a damped harmonic oscillator of natural frequency

and damping constant ? 3n = (n+2l(2n+l)y/pR^ .

JOURNAL DE PHYSIQUE Colloque C8, supplément au n°ll, Tome 40, novembre 1979, page C8-306

Résumé. - Le problème de la stabilité de bulles gazeuses dans un champ sonore a été étudié à l'aide d'un modèle décrivant le mouvement d'une bulle presque sphérique dans un liquide légèrement visqueux.

L'amplitude du mouvement radial et les seuils d'amplitude de pression pour l'instabilité d'une forme sphérique ont été obtenu au moyen d'une théorie de résonance paramétrique. La comparaison avec les résultats expérimentaux pour les bulles en résonance apparaît comme assez satisfaisante.

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979854

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JOURNAL DE PHYSIQUE C8-307

I n t h e study o f the surface d i s t o r t i o n s i t i s expedient t o s e t

where Ri = R(O), so t h a t Eq.1 takes t h e form

w i t h

For understanding surface phenomena i t i s important t o consider t h e mechanism o f e x c i t a t i o n o f surface deformations. Clearly, t h e r e s u l t s o f t h e l i n e a r i z e d approximation cannot be v a l i d f o r surface o s c i l l a t i o n s o f h i g h ampl i tude when t h e breakup o f t h e bubble takes place. However, f o r o s c i l l a t i o n s o f small ampl i tude, Eq. 1 i s adequate f o r d e s c r i b i n g several f e a t u r e s o f the motion and i n p a r t i c u l a r t h e development o f an i n s t a b i l i t y o f the s p h e r i c a l shape. The surface d i s t o r t i o n s o f a v i b r a t i n g bubble can, i n p a r t i c u l a r , be para- m e t r i c a l l y e x c i t e d by t h e r a d i a l pulsations.

For considering t h i s

w

assume t h a t , due t o a superimposed acoustic f i e l d , t h e e x t e r n a l pressure o s c i l l a t e s about i t s average s t a t i c value pa w i t h angular frequency

,

t h a t i s

When t h e acoustic f i e l d i s s u f f i c i e n t l y weak, t h a t i s << 1, t h e r a d i a l o s c i l l a t i o n w i l l be appro- x i m a t e l y simple harmonic. A l i n e a r i z e d computation on Rayleigh-Plesset equation then gives

where t h e re1 a t i v e p u l s a t i o n ampl i tude C=AR/Ro i s o f the same o r d e r o f magnitude o f q

,

^and

4

^{i s a}

constant phase s h i f t which f o r convenience may be p u t equal t o zero. By s u b s t i t u t i n g Eq.3 and r e t a i - n i n g o n l y l i n e a r terms i n C, Eq.1 takes t h e form o f a Mathieu equation,

where

and xn = 3(n+2)dpRo. I t i s w e l l known from t h e 2 theory o f t h e Mathieu equation t h a t , f o r p a r t i c u - l a r combinations of the constantsan and

4,

^the

s o l u t i o n s may have t h e form o f modulated o s c i l l a t i o n s w i t h amplitude e x p o n e n t i a l l y growing w i t h time.

This behavior i n d i c a t e s t h a t parametric e x c i t a - t i o n o f surface waves i s possible, g i v i n g r i s e t o t h e i n s t a b i l i t y o f t h e s p h e r i c a l shape.

L i m i t i n g our a n a l y s i s t o t h e f i r s t i n s t a b i l i t y zone, a particulh,. s o l u t i o n n f Eq.4 i s given /11/

by

w i t h

and $,t),( a p e r i o d i c f u n c t i o n subresonant 1/2 t o t h e freauency o f t h e f o r c i n g f i e l d . We have

t h a t i s t h e a m p l i f i c a t i o n o f t h e surface o s c i l l a - t i o n s due t o parametric resonance f o l l o w s t h e law exp (Xn-6,)t

,

t h e boundary o f t h e i n s t a b i l i t y zone being given by Xn-~, = 0

.

A f t e r s t r a i g h t , forward c a l c u l a t i o n s , the p u l s a t i o n ampl i tude t h r e s h o l d f o r the onset o f surface waves i s then obtained i n t h e form

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NABERGOJ R., and a1

I n t h e small v i s c o s i t y approximation 8, <<won and xn<< w

,

^{S O}t h a t a s i m p l i f i e d expression f o r

on

Ct can be obtained. For the subresonance region, where w

-

2won, by r e t a i n i n g t h e l e a d i n g terms, i t r e s u l t s

F o r t h e 1 in e a r bubble o s c i l l a t i o n s , t h e pressure amplitude can be e a s i l y expressed i n terms o f t h e p u l s a t i o n amplitude. The r e l a t i v e pressure amp1 i t u d e t h r e s h o l d i s then given by

where

]'I2

i s t h e + 2 0 ( 3 k - l ) / 3 k p ~ R ~

bubble resonance frequency f o r 1 in e a r o s c i l l a t i o n s , k t h e p o l y t r o p i c exponent o f t h e gas transformation, and 8 = 2peq/pRo 2

,

^{w i t h} ^p t h e e f f e c t i v e v i s -

eq

e

^q

c o s i t y o f t,he l i q u i d t a k i n g i n t o account t h e v i s - cous, thermal and r a d i a t i o n e f f e c t s .

3. Discussion and conclusions.

-

^Fig. ^1.^shows

the curves Cto = Ct(wo) versus t h e r e s t r a d i u s Ro

1 an 1.5~10“

Fig. 1.

-

P r e d i c t e d r e l a t i v e p u l s a t i o n amplitude t h r e s h o l d Cto =4R/R versus r e s t r a d i u s R f o r bubbles i n water d r i g e n a t Lesonance, togeeher w i t h the " e f f e c t i v e " t h r e s h o l d

, . , . C

The t h i n l i n e curves

i n t h e range 5 x 10'~cm sR06 1.5 x 1 0 - ~ c m fo r bubbles i n water d r i v e n a t resonance. For compa- r i s o n we show a l s o t h e experimental data o f H u l l i n /12/. The agreement between t h e o r y and experiment i s q u i t e s a t i s f a c t o r y .

The dashed l i n e j o i n i n g t h e minima c o u l d be assumed as the " e f f e c t i v e " threshold, t h e i n s t a b i - l i t y r e g i o n covers t h e upper p a r t o f t h e plane.

T h i s t h r e s h o l d

Go

i s suggested by t h e f a c t t h a t i n t h e r e g i o n between t h e minima o f subharmonic resonance t h e r e w i l l presumably appear o t h e r minima, corresponding t o zones o f i n s t a b i l i t y , which can n o t be taken i n t o account i n t h i s f i r s t o r d e r ana-

l y s i s . A simple approximate formula i s r e a d i l y obtained from Eq.5 i n t h e l i m i t o f l a r g e values o f n, namely

w i t h A

z

0.25 c.g.s. u n i t s .

I n F i g . 1 we a l s o g i v e a comparison o f t h e present theory w i t h t h a t g i v e n by Hsieh-Plesset /2,3/. 1t appears t h a t t h e l a t t e r , based on zero damping assumption, does n o t a l l o w an e v a l u a t i o n of an e f f e c t i v e threshold.Moreover, i n t h e considered range o f r a d i i t h e Benjamin's t h r e s h o l d /5/, where the viscous e f f e c t s a r e considered r a t h e r a r t i f i c i a l l y "a p o s t e r i o r i " and n o t d i r e c t l y i n - troduced by t h e equation o f motion, i s p r a c t i - c a l l y c o i n c i d e n t w i t h t h e present one.

F i g . 2 shows t h e pressure amplitude t h r e - shold pto = ntopm versus t h e r e s t r a d i u s Ro r e l a - t i v e t o several s u r f a c e modes, f o r bubbles i n water d r i v e n a t resonance. For t h i s e s t i m a t i o n t h e e f f e c t i v e v i s c o s i t y c o e f f i c i e n t and t h e p o l y t r o p i c index were taken from Ref. 13. I t appears t h a t t h e " e f f e c t i v e " pressure t h r e s h o l d f o r l a r g e r a d i i i s given by t h e approximate law

where K ~ 5 c.g.s. 0 u n i t s .

r e p r e s e n t t h e t h r e s h o l d according t o Hsieh-Plesset A q u a n t i t a t i v e i n d i c a t i o n g i v e n by Storm /14/

f o r zero v i s c o s i t y . The experimental p o i n t s , r e l a -

for a bubble Roe 0.5 cm lies one order of t i v e t o standing t a p water, a r e taken from Fig.6b

o f Ref. 12. magnJ tude above t h e value g i v e n by t h i s formula.

Many p e r t u r b i n g f a c t o r s c o u l d account f o r t h i s d i f f e r e n c e . For example, experiments /12/ i n

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JOURNAL DE PHYSIQUE

F i g

.

^2.

-

P r e d i c t e d pressure amp1 i tude thresh01 d pto versus r e s t r a d i u s Ro f o r bubbles i n water d r i v e n a t resonance.

he-number

o f each curve r e - f e r s t o the p a r t i c u l a r surface mode.

standing water show an increase o f t h e t h r e s h o l d probably due t o t h e i m p u r i t i e s c o l l e c t e d by t h e bubble surface. F u r t h e r i n v e s t i g a t i o n s could shed more l i g h t on t h i s .

REFERENCES

/1/ W i l l a r d , G.W., J. Acoust, Soc. Am.

2

(1954) 933 (A).

121 Hsieh, D.Y., and Plesset, M.S., J. Acoust.

SOC. Am.

-

33 ' (1961) 206.

131 Hsieh, D.Y., J. Acoust. Soc. Am.

56

⁽¹⁹⁷⁴⁾

392.

141 Benjamin, T.B.

,

and Strasberg, M., J . Acoust.

Soc. Am.

-

30 (1958) 697 (A).

/5/ Benjamin, T.B., i n C a v i t a t i o n i n Real L i q u i d s

(R,

Davies ed., E l s e v i e r Pub1 i s h i n g Co., Amsterdam, 1964) pp. 164-180.

/6/ Plesset, M.S., J. Appl. Phys.

-

25 (1954) 96.

/7/ Hsieh, D.Y., J. Basic Eng.

2

(1972) 665.

181 Ceschiz, M. and Nabergoj, R., Phys. F l u i d s 2 1

(1978) 140/

-

/9/ P r o s p e r e t t i

,

A., Quart. Appl

.

^Math.

2

⁽¹⁹⁷⁷⁾

339.

1101 Plesset, M.S., and P r o s p e r e t t i , A., Ann. Rev.

F l u i d Mech.

2

(1977) 145.

/11/ Hayashi, C,, Nonlinear O s c i l l a t i o n s i n Phys5cal Systems (Mc Graw-Hill Co., New-York, 1964)?

1121 H u l l i n , Ch., Acustica

37

(1977) 64.

/13/ Chapman, R.B., and Plesset, M.S., J. Basic Eng. 93 (1971) 373.

-

/14/ Storm, D.L., J. Acoust. Soc. Am.

52

(1972) 152 (A).