-- ,.- . - - ""'-~~-----------.

1 0 nr;r 1973Bibl, Nat. lab.

PHILIRESEARCH·REPORTS

\ SUPPLEMENTS

~! I

PHlllPS RESEARCH LABORATORIEShlllps Res. Repts Suppl.led In the Netherlands

1973 No. 3

© N.V. Philips' Gloeilampenfabrieken, Eindhoven, Netherlands, 1973.Articles or illustrations reproduced, in whole or in part, must be

accompanied by full acknowledgement of the source:PHILlPS RESEARCH REPORTS SUPPLEMENTS

ON THE INSTABILITYOF A TRANSLATING GAS BUBBLE

UNDER THE INFLUENCEOF A PRESSURE STEP *)

BY

w. A. H. J. HERMANS

*) Thesis, Technical University, Eindhoven, June 1973.Promotors: Prof. Dr Ir G. Vossers and Prof. Dr Ir L. van Wijngaarden (Techn, Univ,Enschede).

Philips Res. Repts Suppl. 1973, No. 3.

'-,-, _._--._-~~.- '.

AbstractThe dynamics of a translating gas bubble due to a sudden pressure risein a liquid is studied in this paper. We have assumed a bubble in a liquidwhich is infinite, incompressible and non-viscous. The bubble shape isgiven by a series of Legendre polynomials. The velocity potentialofthe liquid is determined as a function of translational velocity and therate of deformation of the bubble. The kinetic and potential energiesof the system are calculated, and using Lagrange's principle a set ofdifferential equations for the motion of the bubble has been obtained.From these equations the equilibrium state of the bubble is determined.If the equilibrium pressure of the liquid is changed, the bubble behav-iour may be stable or unstable. The domains of instability as a functionof the pressure rise and radius are determined by solving equations ofthe Hill type, these being achieved by transformation of the equationsof motion. Including the viscosity of the liquid the instability domainwill change. This domain is approximated by a simple equation.Because a pressure step will be accompanied by a change in velocity ofthe liquid, this results in a change in both the translational velocity andthe bubble surface velocity. These changes, which depend on the pressurepulse and the bubble shape, are given in the paper. The shape of thebubble after it is hit by a pressure pulse is calculated numerically andcompared with photographs taken in an experimental set-up. Bothresults are given and we may conclude that the theory is confirmed bythe experiment. It is shown that an initial translational velocity of thebubble can lead to the formation of a liquid jet and even to bubblesplit-up.

- _.~. --_. --_._----

CONTENTS

1. INTRODUCTION

2. DYNAMICS OF A TRANSLATING CAVITY2.1. Introduetion2.2. The coordinate system . . . . .2.3. The velocity potentialof the liquid

2.3.1. Introduetion .2.3.2. Solution of the Laplace equation

2.4. The kinetic and potential energies2.5. Equations of motionAppendix AAppendix BAppendix C

3. STABILITY CRITERION OF A CAVITY3.1. Introduetion3.2. The equilibrium state of a translating cavity3.3. Stability of the equilibrium state . . . . .3.4. Influence of a pressure step on the stability of a cavity

3.4.1. Introduetion .3.4.2. Solution of the modified Rayleigh equation3.4.3. The range of stability

3.5. Effects of viscosity3.6. ConclusionAppendix D ....

4. DYNAMICS OF A BUBBLE IN A QUIESCENT LIQUID SUB-JECT TO A STEP CHANGE IN LIQUID PRESSURE4.1. Introduetion4.2. Numerical method . . . .4.3. Rayleigh's equation4.4. The unstable bubble shape4.5. Liquid jet of a translating void

5. COMPARISON BETWEEN EXPERIMENTAL AND THEORET-ICAL RESULTS5.1. Introduetion5.2. The initial conditions of a bubble after passage of a pressure pulse5.3. Experimental set-up . . . . . . . .

1

4447791419222328

3333343840404143525658

595959626273

75757579

5.4. Results of the experiment ' .5.5. Numerical results

8385

6. CONCLUSION . . . 97

LIST OF FREQUENTLY USED SYMBOLS 98

REFERENCES 102

-1-

1. INTRODUCTION

The problem of a collapsing bubble in a liquid was first formulated by Besantand it took over fifty years till in 1917 Rayleigh solved the problem of a spheri-cal void collapsing in an incompressible liquid under constant-pressure condi-tions. Thereafter, numerous authors have studied the behaviour of sphericalbubbles, under a wide range of physical conditions. Por instance, allowancefor compressibility of the liquid, as a refinement of the Rayleigh theory, wasaccomplished by Gilmore (1952) and modified by Mellen (1956) and Plynn(1957), (1964).From observations it was already known that an initially spherical bubble

does not always remain spherical, but may become highly distorted duringcollapse or expansion. This unstable behaviour became very interesting dueto a suggestion given by Kornfeld and Suvorov (1944) that cavitation damagemay be caused by the impact of liquid jets formed by involution of collapsingcavities near a solid surface. A perturbation theory by Rattray (1951) suggestedthat the presence of a solid wall during the collapse of an initially sphericalbubble could cause the formation of a liquid jet directed towards the wall.High-speed photographs taken by Benjamin and Ellis (1966) later indeedconfirmed that jets were formed on bubbles collapsing near a solid wall. Onthe other hand, PIesset and Mitchell (1956) considered the instability of astationary bubble with an initially non-spherical form in an infinite liquid.They solved the linearized problem of a collapsing and expanding bubbleunder constant pressure difference between cavity and liquid. A reason for jetformation was not found by that formulation.Although it is true that jets are formed it is curious that from observations

on the rate of pitting in an aluminium test section exposed to a cavitation cloudin a water tunnel, Knapp (1955) estimated that only one in 30000 of thetransient cavities swept into the region of the test section caused a damagingblow.

In this respect it is interesting that in a recent theoretical study by PIesset andChapman (1970) a numerical solution of a collapsing stationary bubble at ashort distance from a solid wall gave an estimated jet velocity of only about100 m/so The formation of a jet is not contingent on the presence of a wall butcan also be effected by the translational motion of the bubble, as was shownby Ivany (1966). He took high-speed photographs of bubbles collapsing in aventuri. The combination of translation and pressure change caused a hollowvortex ring to be formed by involution of the rear of the cavity. It seems thatin the problem of jet formation the main factor will not be the solid wall butthe translatory movement of the cavity.

On the other hand there is a great interest in the chemical process industryin the behaviour of a liquid-bubble mixture. This concerns the change of sound

-2-

speed of the mixture and the influence of pressure changes during steady andunsteady motion through pipes and vessels. An important aspect in this fieldmight be the translatory motion of the bubbles with respect to the liquid andthe instability of collapse. It is necessary, therefore, to know the behaviour ofa single bubble.In this thesis the influence of a pressure pulse on a translating bubble is

investigated. The liquid is taken as an infinite, incompressible, non-viscousmedium. Thus the influence of a boundary wall is not taken into account. Asa result ofthe translating velocity u ofthe cavity, its shape will be non-spherical.The deviation from the spherical form is represented by perturbation coeffi-cients of a series of Legendre polynomials. In chapter 2 the liquid motion isrepresented by a velocity potential as function of the bubble coordinates andvelocities. The kinetic and potential energies of the system are calculated, a setof differential equations determining the motion of the bubble shape is derived,using Lagrange's principle. These equations are non-linear; when linearizedthey are still coupled by the velocity u.At constant velocity u the cavity will have an equilibrium shape depending

on this velocity. A deviation from this shape will result in shape oscillations,which might be either stable or unstable. A characteristic criterion for thisis the Weber number. Oscillations of another type occur when the liquidpressure is changed. This is the main aspect that will be investigated. Inchapter 3 the influence of a sudden pressure step on the stability of a cavitywall is studied analytically. The equations of motion are transformed to aset of second-order differential equations of the Hill type. In this case athreshold pressure step depending on the cavity radius can be calculated toinitiate instability.Ifthe liquid viscosity is included, the system will have a damping factor, too.

Asymptotic stability is possible. The threshold pressure then needed for theonset of shape oscillations is approximated in an elegant and practical equation.Of course, the shape of a bubble after it is hit by a pressure step above thethreshold pressure is very interesting. This shape is calculated numerically.In chapter 4 some examples are given and it is shown that under certain circum-stances a liquid jet is formed. One example is a collapsing void. A liquid jetwith high velocity is formed which arises from a high local pressure spot at therear ofthe bubble. The duration ofthe pressure pulse is found to be ofthe orderof 1 fLS; and fairly appreciable evidence exists that the maximum pressure atthe centre of collapse can reach 104 bar. When such a high pressure arises, thecompressibility of water is bound to become a vital factor in the motion towardsthe end of collapse.At the Technical University in Eindhoven a special test rig has been built to

generate pressure steps in a water-filled pipe. The interaction of the pressurestep and a released air bubble can be observed and filmed. A comparison ofthe

-3-

numerical solution with empirical results is thus possible. The initial conditionsfor the numerical programme are then determined by the experiment. Ananalytical approximation for the initial conditions is given.The validity of the present theory is confirmed by the experiment.

-4-

2. DYNAMICS OF A TRANSLATING CAVITY

2.1. Introduetion

A translating cavity differs from a stationary cavity in two main respects:its shape is not spherical and the liquid-pressure field is spherically asymmetricalaround the bubble centre. The behaviour of a rising bubble has been the subjectof many experiments. A bubble rising due to gravitational forces has the shapeof an oblate spheroid; it may rise vertically or along a zig-zagor spiral path.Most of the work done in this direction has been for the purpose of determiningthe drag coefficient and rise velocity on the one hand and the reasons for thedeflection from rectilinear motion on the other. Experimental and theoreticalwork bySaffman (1956) and by Hartunian and Sears (1957)made them supposethat above a critical Reynolds or Weber number the flow becomes such thatthe rectilinear motion changes into a periodic one.

Although there are still many questions to be answered about this phenom-enon, it may be neglected here. Our investigations are concerned with a muchsmaller time scale (less than one millisecond) than is the casewith a rising bubble,for which a rectilinear motion may be considered in our case.

Theoretical studies of a translating cavity are rather rare, probably becauseof the inherent problems posed by its non-spherical behaviour. In only a fewinitial conditions can its shape be treated as spherical. Using a perturbationtheory, Yeh (1967) studied the dynamics of a translating gas bubble in aninviscid liquid with pressure gradient. The analytical results were comparedwith the photographs of Ivany et al. (1966). The agreement was poor and noexplanation could be given for the behaviour of the bubbles in the photo-graphs.A very interesting study was made by Eller (1966). The eigen-frequencies

of a non-spherical cavity were calculated and the interaction of the cavitywith an acoustic pressure field was given.In this thesis the cavity will be assumed to be in translating motion. The

shape of the non-spherical cavity will be described with the aid of Legendrepolynomials.

We assume that the liquid velocity satisfies a potential which will thereforebe a function of the bubble shape and bubble surface velocities. The kineticand potential energies of the system are calculated.In the system used here it is possible on the basis of Lagrange's principle to

obtain a set of second-order differential equations for the generalized coordi-nates of the bubble.

2.2. The coordinate system

The surface of the cavity is treated as a free surface whose shape and position

-5-

are to be determined as functions of time. Due to the translating velocity it isassumed that the shape of the cavity is symmetrical about the direction oftranslation. The cavity surface is given by the function r = rs (B, t) or by thefunction F (r, B, t) = r - "s= 0, where r and e are spherical coordinates ina coordinate system with origin o. This centre, which will be defined later,moves with respect to the rest frame and origin 0' in the x-direction at avelocity x = u(t). See fig. 2.1.

y

Iy

r

~o~,--_'~----------~~----~~_J--~--~~~------Xlx

x

Fig. 2.1. The coordinate system of a moving cavity.

The rest frame has the same velocity as the liquid at infinity, which is zero.The cavity shape is described with respect to the moving coordinate system.The function "s may be expanded in a series of Legendre polynomials as

co

rsCB, t) = R(t) + ~ kn(t) Picos B).n=l

(2.1)

In this expression R(t) is the coefficient of the zero-order Legendre polynomialPo = 1. The coefficients kn(t) are the perturbation terms, so that

eo

~ kn(t) Pn(cos B)n=l

expresses the amount by which the bubble deviates from the spherical shapewith radius R.Although it seems rather obvious to use the expression of eq. (2.1) to describe

-6-

the bubble surface, difficulties will arise in adding the coefficient k1 of the first-order Legendre polynomial. These are connected with the definition and choiceofthe centre O. Many authors who have used the Legendre expansion are oftenin doubt about the addition of k1 in bubble dynamics with and without atranslating motion. This is why we shall first examine the suitability of theexpansion given in eq. (2.1).

Fig. 2.2. A sphere with centre 0 moved over a distance zlx, The sphere with centre A' willbe described with regard to the centre 0 by rs(8) = R + LlR(8).

In fig. 2.2 a spherical cavity with radius R and origin 0 is moved over adistance L1x< R to the new origin a'. The circle with origin 0' may be de-scribed with respect to the centre 0 by the expression rs(8) = R + L1R(8),where

L1R(8) = L1x cos 8 + (R2 - L12x cos" 8)1/2 - R. (2.2)

Expanding the second term on the right-hand side of eq. (2.2) and introducingthe Legendre polynomials Pn(cos 8) (see Lense (1954)) we get the expression

L1R(8) = L1x PI + (L1X)2 (~_ ~Pl) +R R R 3 3

(L1X )4 ( 1 2 1) ((L1X )6)+]i 15- 21 P 2 + 35P4 + 0 ]i . (2.3)

To the coordinates of the moved circle or sphere with centre 0 so derived, wemay add a perturbation

eo

~ k; Pn(cos 8),n=1

describing the non-spherical deviation of the bubble surface. If the coeffi-cient k1 of the first-order polynomial equals

-7-

(2.4)

the surface of a moved perturbated sphere is given by

rs(e) = R [1+~(;Y+ 115(;YJ++P2 {k2- R GC:Y + :1 (;YJ} + k3 P3 + (2.5)

+P4[k4+ :5R(~rJ+n~5knPn+ o((L1;Y)R,which may be written as

rs(e) = R + ~ an Pn(cos e).n=2

(2.6)

This shows that a bubble surface given by (2.1) may be described with (2.6),in which an are new coefficients with al equal to zero. The new origin hasbeen moved with respect to the first one over a distance -kl, Ikll < R. Inother words, it is always possible to eliminate the coefficient al by the choiceof centre. The coordinate x appears as a new, independent variable definingthe centre of a translating bubble. At this moment there is no necessity forother definitions of this centre. At the end of this chapter we shall, however,return to this subject.In the following the bubble shape will be given by

rs(e, t) = R(t) + ~ an(t) Pn(cos e)n=2

(2.7)

and the coordinates to determine are x, R, a2, a3' a4' ....

2.3. The velocity potentialof the liquid

2.3.1. Introduetion

In sec. 2.2 we have chosen a coordinate system do define the shape of acavity. Our purpose is to determine the shape and motion of the bubble wall.The problem is one of the type known as free-boundary-value problems, be-cause the position and shape ofthe bubble, that is the boundary, are not knownuntil the complete solution is obtained.In this study we shall assume that the fluid outside the bubble is a liquid,

while that inside it is a gas. The motion of the wall is mainly controlled by theinertial, the thermal and the diffusive effects. If all these aspects were takeninto account, the problem would be too complicated, so we have to simplify

-:-8-

it in order to find a model to describe the dynamics of a translating bubble.In the following i~will be assumed that(a) the liquid is incompressible,(b) the flow is non-viscous,(c) the bubble contains an ideal gas,(d) there is no mass di~usion across the bubble wall.The compressibility of the liquid may become important if the bubble-wall

velocityapproaches the sonic speed of the liquid. This will be the case in bubbleimplosions, for instance, where the radius becomes very small. But even thenthe liquid may be treated as incompressible down to a radius of about 10%of its original radius, as was shown in the studies by Gilmore (1952), Mellen(1956) and Flynn (1957). In this thesis it is not expected that the liquid velocitywill be so high that compressibility effects will be important. A check on thiswill, however, always be necessary.Concerning points (b) and (d), it may be supposed that viscosity and diffusion

effects will take place on a time scale of greater order than that appropriateto the problem we shall describe. The time necessary to build up the viscousstresses will be of the order of magnitude of (R2/V) s, where v is the kinematicviscosity of the liquid. Parkin et al. (1961) showed that the time necessary foran air bubble to dissolve in water is of the order of magnitude 109 R2 s. Sincethere will be no mass diffusion we assume that the bubble gas will satisfy anunequivocal relation between pressure and volume. The motion of the gas inthe bubble is not of interest because of the low density in comparison withthe liquid density.The present problem is then reduced to solving the Laplace equation for the

velocity potential in a liquid

(2.8)

It is convenient to define the potential in a frame in which the fluid at infinityis at rest, see for instance Lamb, section 40. The same will apply in our system,in which the potential will be determined in the fixed frame, the liquid velocitybeing defined by

v = -\lep. (2.9)

We can write the solution of the velocity potential as a series of Legendrepolynomials. The potential will have the form

co

r ;;,::r; (2.10)

n=OThe problem now is to evaluate the coefficients cn(t) as functions of X, R, R.,an, á; (the dot denotes the derivative to time); this means that the liquid

-9-

velocity at every moment is a function of the instantaneous bubble shape andits surface velocity.

2.3.2. Solution of the Lap/ace equation

This goal may be accomplished once the normal velocity at the cavity bound-ary has been specified. The whole motion of the liquid is determined uniquelyat any particular instant, say t = to, by the normal velocity of the internalboundary. The normal velocity may be taken to comprise two parts, the firstcorresponding to a translation at axial velocity x = u(t) of the instantaneousform at t = to and the second to the rate of deformation.

The cavity surface is given by the time function F = 0, in which

F(r, 0, t) = r- R(t)- 1: an(t)Pn(cos 0).n=2

(2.11)

Since r is a spherical coordinate from the centre of the moving cavity, thesurface condition

dF-=0dt

(2.12)

is taken in the moving system, too.This results in

bF- + Vd' \!F = 0,bi

(2.13)

where Vd is the surface velocity due to the rate of deformation. The surfacevelocity corresponding to the translating motion is, according to fig. 2.3, givenby

u (cos 0) I, - u (sin 0) io. (2.14)

Fig. 2.3. The velocities at the cavity surface.

V = Vd + (U COS B, -u sin B) (2.15)

-10 -

And since

the bubble surface condition (2.13) becomes

dF ?JF ( 1 ?JF) ( ?JF)- = - + (vo + u sin B)- - + (o, _ U cos B)-dt ?Jt rs ?JB r=rs ör r=r.

co.I IandPn= -R- dnPn- (Vo + usin B) -- + Vr- ucos B= 0, (2.16)

rs dBn=2 n=2

whereco

(2.17)

n=O

co

(2.18)

n=O

After some algebra eq. (2.16) can be written as<Xl cc

n=O n= 1

eo.I u ?JZ dP1= R + d P + u Pt _ - ---n n Z ?JO dB '

(2.19)

n=2

use being made of the following definitions:co

(I an(t) )r.(B, t) = R(t) 1+ -Pn(cos B) = R(t) Z(B, t),

R(t)(2.20)

n=2

b (t) __ C_n(_t)_n - R(t)n+2

At this stage it is very expedient to use a method suggested by Bller (1966).Following his example, eq. (2.19) is multiplied by Z2 Pie) de, in whiche = cos B, and integrated between -1 and 1. The second summation on theleft-hand side of eq. (2.19) is rearranged as

(2.21)

-11-

00 001 n

I I1 dPnez I s, I . dPn ê) ( 1 )b ------PjdC=- - Pj(smO)-- - dO

n zn+1 dO ()8 n dO ()8 Z"-1 0n=1 n=1

I(bn dPn l)n=- -Pj(sinO)-- +

n de zn 0n=1

00

I i, In 1 ê) ( dPn)+ - - - Pj (sin 0) - dO. (2.22)n znê)O dO

on=1

And using the identity

1 ê)( dPn)- - (sin 0) - = n (n + 1) Pmsin 0 ê)O dO

eq. (2.22) results in00 00

1 1

I s, I 1 dPndPj I I 1- ---dC- (n + l)bn -PnPjdC.n z- dO dO z»-1 -1n=1 n=1

(2.23)

With the result of (2.23), the left-hand side of eq. (2.19) changes after it hasbeen integrated, into

001 1

I I b; I 1 dPj dPnho PoPjdC+ - ---dC.

n Z" dO dO-1 -1

n=1

(2.24)

In the right-hand side of eq. (2.19) a similar integration is possible, whichresults in an infinite set of simultaneous equations for the coefficient b.:For j = 0,

1 00

s, = t R J Z2 dC + tI: án J Z2 r, dCn=2

-] -1

(2.25)

and for i > 1

(2.26)

-12 -

From these equations it is seen that bn is a linear combination of the velocitiesu, ft.., dn> written in the general form

b = u b, + ft.. bR + ä B,

which is a vector notation for

(2.27)

(bO) (bOU) . (bOR) (~2)(b02 b03 ••• b~m)~1 = U b.l" + R b~R + ~3 b~2 :.. . . .. .. . . .. .bn bnu bnR dm bn2· . . . . .. bnm

(2.29)

The coefficients bi can be determined from (2.25) and (2.26) as power seriesin all of tlie an by using an iteration procedure, assuming that the an are smallwith respect to R, i.e. anlR = O(e), where 8 is a small number. Products ofthe form an am will then be of order 82, and so on. If the radius R and thedisturbance terms an have a period time t of oscillation, then Rand dn, willbe of the order of magnitude Rit and anlt respectively, and we may say thatdnlR is also a quantity of the order of magnitude e. Later on we shall moreor less drop this last assumption. Now it is possible to develop the vectorsand matrix of eq. (2.27) in series of 8 terms which can be written as

bu = bu(O) + 8 bu(1) + 82 bu(2) + ,bR = bR(O) + e bR(1) + 82 bR(2) + ,B = B(O) + s B(1) + 82 B(2) + ,

(2.28)

in which e.g. (!J bU) is the contribution of b of the order of magnitude e'.The calculation and the resulting expressions for b, are recorded in appen-

dix B. With eqs (2.10), (2.25) and (2.26) and the definitionsco

n=21

J elPn dPJYn/(t) = yk - - dC,

d8 d8-1

(2.30)

the velocity potentialof the liquid will beeo

{I R3 3L an (n + 1 Rn+3 n- 1 Rn+1 )rP=u --cos8+- - ------Pn+1-------Pn-1 +2 r2 2 R 2n+ 1 rn+2 2n + 1 rn

n=2co eo

3L2n + 1 Rn+2 I 2j + 1+ - --- --- Pn --- Y1}1 Yn/ +8 n+l rn+l j+l

n=1 J=1

n=1 j=l k=1

-13-

1 I.2n+ 1 Rn+2 (I3 2j+ 1 I2k+ 1+- ----Pn --- --YlklY·klYj1+2 n + 1 rn+1 8 j + 1 k + 1 J n

3 a 9 a ) 1I 2n+ 1 Rn+22 2 3 2+--Y1 +--Y2 +- ----P x10 R n 14 R n 4 n + 1 rn+1 n

n=1

[ I3 ( 1 aj-l 1 aJ+l)] }X Y1n3-. -j(j+l)Yjn2 --------- +O(e4) +2 2j- 1 R 2j+ 3 R

j=3

n=2 n=2

. {R2( I 1 a/) I 1 anRn+2+R - 1+ --- +2 -----P +

r 2n+ 1 R2 n+ 1R r+ 1 n

00 00

1I 2n+ 1 Rn+2 [1 I(2 a a.)]+- ----Pn jY2PndC+ --_!_Ynl-_!_Yj/ +2 n + 1 rn+ 1 j + 1 R R

-1n=1 J=z00 00

1I 2n+ 1 Rn+2 I 2j+ 1 ( 1+- ----Pn --yjn1 jY2PjdC+4 n + 1 rn+1 j + 1

-1n=1 j=1

k=200

I (2 a R2 1 R2 1 1 Rn+2 )+ à; --~-+--jY2PndC+----Pn +

2n+ 1R r 2 r n+ 1 rn+1-1n=2

n=1 j=2

n=1 j=2 k=1

-14 -

, As is seen the potential cp satisfies the condition cp = 0 (1/r) for r ---* 00. Thesame result up to O(e) is found for instance by Yeh (1967). The great advantageof the method used here is that it permits the potential to be found up to ahigher order of e.

2.4. The kinetic and potential energies

Having obtained the velocity potentialof the liquid it is possible to deter-mine the kinetic energy of the system. Because the density of the bubble maybe neglected with respect to the density of the liquid, the kinetic energy ofthe system may be written as

T =! e J J J 1 '\7cp12 d..,v

where e is the liquid density and d.. the differential volume element of theliquid. With the aid of Green's theorem this equation can be rearranged intothe Kelvin's equation, see Lamb (1932), sec. 44:

T =-!e f f cp :: ds,A

(2.32)

where ds is an element of area A on the cavity surface. An integration over asurface at infinity makes no contribution, because cp has been chosen such thatcp = O(l/r) for r ---* 00, so that that integration contribution vanishes. Themotion of the liquid is expressed in the coordinates x, R, a2, a3' a4 ... or inthe general coordinates qo, ql' q2' q3 .... Hence the fluid kinetic energy isrepresented by

co 00

(2.33)

The symmetric matrix MI} (q) is called the "added-mass" tensor.For the system we are looking for, it is now easier to split up the kinetic

energy into three parts as has been suggested by Benjamin and Ellis (1966), toobtain insight into the influence of the translational velocity u. The velocitypotential may be split up into two parts: the first corresponding to the velocity uand the second to the rate of deformation, written as

(2.34)

According to eqs (2.32) and (2.33) the kinetic energy of the whole system maybe written as

T= !Mu2 + Ju + T', (2.35)

-15 -

where

(2.36)

(using Green's second identity), (2.37)

1 II 'öep2T' = - - e ep2 - ds.2 'önA

(2.38)

As we see, M is the instantaneous induced (added) mass associated with therectilinear motion of a rigid body, T' the energy associated with a deformablebody and J the couple term of the linear motion and deformation. To calculatethese integrals it is necessary to determine the boundary condition 'ö4>l'ön=\lep • D where D is the normal vector on the cavity surface pointed towardsthe liquid (see fig. 2.3). Using eq. (2.11) this vector is

\lFD=--.

I \lFI(2.39)

The surface velocity Vd according to the rate of deformation is now given by

(2.40)

so that using eqs (2.13), (2.39) and (2.40) one gets

'öep2 'öF/'öt-=--'ön I \l FI

(2.41)

The normal velocity associated to the translating motion is according to eq.(2.34) now given by

-u \lep1 • D = U. D, (2.42)

which results in'öep1-=-i".D.'ön

(2.43)

In appendix C the integrals (2.36) to (2.38) have been calculated. The resultsare given here up to O( S2) :

-16 - ,

co

(n+l)2(n+2) anan+2 ]-27 --+ 0 83

(2n+ 1) (2n+ 3) (2n+ 5) R2 ( ) ,(2.44)

n=2

+ n(17n2-22n+9)) (an)2 +

(2n+ 1)2 (2n- 1) Rco

n=2

co

2 [. L ( 18 n an an+1 ) 18 a3J =-n(!R3 -R --+ 0(83) --à2-+3 (2n+ 1) (2n+ 3) R2 35 Rn=2co

L n ( 1 an-1 1 an+1 )]+9 àn-- -------+0(82) ,2n+ 1 2n- 1 R 2n+ 3 R

n=3(2.45)

co

2 {. [ L n - 3 (an )2 ]T' = - n (! R3 R2 3 - 3 - + 0(83) +3 (2n+ l)(n + 1) R

n=2co.L 6 (n + 3) (an )+R à -+082 +

IJ (2n+ l)(n + 1) R ()n=2

co

(2.46)

n=2

To balance the kinetic energy of a moving fluid we should recall the generallaw of conservation of energy which in its application to the system consideredhere can be formulated: the rate of change of the kinetic energy equals thework done by the pressure forces on the volume, written as

(2.47)

The surface integral is taken over the inner and outer surfaces of the liquid;

- J P eo va;,. ds,A <XI (2.48)

-17-

P being the pressure at these surfaces while the normal vector n is directedtowards the liquid, the contribution of this integral at infinity equals

where the index 00 means the value taken at infinity. Since the liquid velocityv = - \lcp is of the order of magnitude l/r2 we have up to O( e2)

r -+- 00. (2.49)

n=2 n=2

With the results of appendix C, where the cavity volume and area are cal-culated, eq. (2.48) results in

dV- JPa;,v<XI.ds=-P<XI-'

A <XI dt

The contribution of the surface integral over the inner boundary, i.e. the cavitysurface, now equals

(2.50)

dV- J Pc vc· ds = Pc-

Ac dt(2.51)

(c indicates that the values are taken over the cavity surface). Since, owing tothe incompressibility we have

J Vc • ds + J Va;,. ds = O.Ac Aa;,

With the results of (2.50) and (2.51) eq. (2.47) can be written now as

(2.52)

which means that the rate of change of the kinetic energy equals the rate ofchange of the potential energy.

The first integral on the right-hand side of eq. (2.52) results in

t dV t t dPa;,- f Pa;,-dt=-Pa;, vi + J V-dt,dt t=O dt

t=O t=o

(2.53)

if the pressure P <XI is explicitly a function of time.With the second integral on the right-hand side of eq. (2.52) we have to be

(2.56)

-18 -

careful because there is a pressure difference due to surface tension betweenthe surface pressure Pc in the liquid and the bubble inner pressure PI. Becausethe pressure PI has an unequivocal relationship with the cavity volume and ais the surface tension between gas and liquid, the second integral equals

(2.54)

where Vo and Ao are reference volume and area at t = O. So far, however,we have concerned ourselves with a system with a liquid pressure P co at infinitywhich will be constant. The potential energy will not then be explicitly a functionof time and will be given by

vU = a (A - Ao) + P co (V - Vo) - J PI dV.

Vo(2.55)

According to the results of appendix C the volume is

n=2

and the cavity surface area

[ I 2 + n (n + 1) (a )2 ]A = 41£ R2 1 + ~ + O( e4) •2 (2n + 1) R

(2.57)

n=2

Comment

The coordinates to describe the place and shape of the bubble arex, R, O2, 03' ...• Because of the x coordinate it is impossible to add theal term since it is always possible by the choice of the centre 0 to eliminateal. In other words, al is not an independent variable. With the results ofappendix C the centre of mass can be determined, resulting in

xz- x 1 f r/-- = - - sin 8 cos 8 sin (oc- 8) ds = 0(e2).R V 2R

A

(2.58)

Only in the case of linearization will the centre of the bubble be the centreof mass. If the bubble mass could not be neglected it would be necessaryto reckon with the true centre of mass in calculating the kinetic energy.

-19 -

2.5. Equations of motion

Having calculated the potential and kinetic energies it is now possible on thebasis of these data to use the powerful equations of Lagrange. Following thismethod, see Pars (1965), we obtain a set of differential equations of the gener-alized coordinates, say qo, ql' q2' ... , qn. In our system the available coor-dinates to describe the place of a liquid particle are x, R, a2, a3, a4' ....The only equation we have at the moment is the velocity potential with thevelocities x, R, d2, d3, d4, ••• , which means that this dynamic system is non-holonomic. Besides it has an infinite number of degrees of freedom, but thenwe can take a finite number of coordinates. It is possible, however, to regardit as an ordinary Lagrangian system, because the liquid is taken as incompres-sible. A proof of this is given in Lamb (1932), sec. 135, and in Birkhof (1960),sec. 109.

The Lagrangian of the system is given by

L=T-U (2.59)

and the equations of Lagrange by

a ez. ia.----=0dt"öx "ÖX '

(2.60)

a ez si.----=0dt "öR "öR '

(2.61)

a ez, "öL----=0,dt ()dn "öan

n = 2, 3, 4, ... , N. (2.62)

Comment

With the present knowledge it is useful to apply the method to a sphere,with an = dn = 0, n ~ 2, with the result that the kinetic-energy terms are

M= tneR3,J =0,TI = 2 «» R3 R2•

With this the Lagrangian is written as

R

L = in e R3 u2 + 2n e R3 R.2_ 4nR2 (J- J (Poo- PI) d(~nR3).Ro

-~(2R2 + ~ 2 + n(n + 1) a/).(! L,; 2n + 1

n=2

(2.63)

- 20-

Using eqs (2.52) and (2.53) we get the following set of equations:

du ft..-+ 3-u=0,dt R

Assuming u = 0 we have the well-known equation of Rayleigh, but ifu =1= 0 a direct coupling between the translational velocity u and the radius Ris given by u = Uo (Rol R)3. In other words, if R< 0 the velocity u willincrease and if R > 0 the velocity u will decrease.

The Lagrangian of the system of a moving cavity in an incompressible liquidis formed up to 0(e2) by using eqs (2.35) and (2.55), giving

[1 3 1 I((n + 6) (n + 1)2L = u2 - R3 - - R2 a + - R +6 10 2 4 (2n + 1)2 (2n + 3)

n=2

n(17n2-22n+9») 9 I (n+l)2(n+2) ]+ an2- -R an an+2 +

(2n + 1)2 (2n- 1) 2 (2n + 1)(2n + 3) (2n + 5)n=2

[.I 6n 6+ u - R R a a - - à R2 a +(2n + 1) (2n + 3) n n+1 35 2 3

n=2

3n (an-1 an+1)]+R2 à -- ----- +n2n+ 1 2n-l 2n+ 3

n=3

. .I n- 3 . I 2 (n+ 3)+ R2 R3 - R2 R an2 + R R2 ànan+

(n + 1) (2n + 1) (n + 1) (2n + 1)n=2 n=2

00 1 JRp -POl (2 IOO 2 )+ R3 à 2 + I d _ R3 + R __ a 2 +(n + 1)(2n + 1) n (! 3 2n + 1 n

Ron=2 n=2

(2.65)

- 21-

And applying eqs (2.60) to (2.62) a set of coupled differential equations up toO(e) is found. These equations for x, R, a2, a3, a4' ... are respectively:

~(5R2- 9Ra2)+ u (15RR- 18Ra2- 9Rà2)= 0, (2.64)

. ( 3 n 3 n ) 2 [ 9 n (n- 1)2u --Ra ---Ra +u a +2n- 1 n-l 2n+ 3 n+l 2 (2n- 3)(2n-l) n-2

and for n ~ 3, with al = 0 we have

9(n+l)2(n+2) ((n+6)(n+l)2 n(17n2-22n+9)) ]+ an+2- + an +

2(2n+ 3)(2n+ 5) 2(2n+ I) (2n+ 3) 2 (2n- 1)(2n+ 1)

(. . 3 (2n+ 1) .) 2 2 ••

+ u 6Ran_1 + 3Ran_l- Ran+! + R an+2n+3 n+l

6. [ n + 3.. 3. P1- Pro+-- R R àn+ 4 an R R + - R2- +n + 1 2 (n + 1) 2 Q

(n (n + 1)) a ]+ 1+ - =0.

2 eR(2.67)

These equations were also derived by Yeh (1967).He used the velocity potentialto calculate the pressure distribution along the cavity. Applying the Bernoulliequation he obtained a set of differential equations by equating terms of thesame power of Pn. But an error in calculating the form of the cavity madehis results erroneous.

- 22-

Appendix A

Recurrence formula for Legendre polynomials; for literature, see e.g. Lense(1954):

(cos B) P;n+l n---Pn+l + --Pn-l'2n+l 2n+l

(AJ)

. dPn n (n + 1)(sin B) - = (Pn+! - Pn-1),

dB 2n + 1(A.2)

1 dPn----sin B dB

[(n-l)/2]I (2n- 4k- I)PII-2k-1,

k=O

(A.3)

1 2J r, e,dC = s.;-1 2n + 1

(~nj: Kronecker symbol). (AA)

Using the above equations the following are obtained:

(A.5)

00 00 001

J '\' a dPn ~ a dPj d = ~ 2n (n + I) a 2

~ n dB ~ j dB C ~ 2n + 1 n,-1

n=2 j=2 n=2

(A.6)

001

J dPl dPnI 2n (n + 1) ( an-1 all+1 )-- ajPjdC= ----- ,dB dB 2n + 1 2n - 1 2n + 3

-1j=2

n ;;::::3, (A.7)

, n=2.

-23 -

Appendix BIn this appendix the b; coefficients which appear in the solution of Laplace's

equation will be calculated. Because the velocity potential is a linear combina-tion of the velocities it is possible with the exception of one, to put all the veloc-ities equal to zero. In this way the contribution to b. with respect to that veloc-ity can be found. Let us consider first the vector b, due to the translationalvelocity u. For j = 0 we find the solution of eq. (2.25), giving

bs;> 0 (B.I)

and for eq. (2.26) for j ;;::::1 we findOCJ

1 1

LIJ dPn dP) 1J dP 1ar,b - (1 + y)-n_-dC = - (1 + y)2 __ drnu n de de 2 de de '0,

-1 -1.=1

(B.2)

where

(B.3)

.=2Because of Y< 1, one can use the expansion

k=O

With this expansion and the expression of b." in eq. (2.28), (B.2) changes into

n=l k=O

= t (Y1Jo + 2 Y1J1 + Y1/). (BA)

Equation (BA) is an infinite set of equations, one for each value of j to besolved simultaneously for all bm, (n ;;::::I). This is possible by equating termsof equal order of magnitude. The zero-order equation is

L1 (0) 0 _ 0~ b.u Yn) - t Y1J .

n=l

According to eq. (A.6)2n (n + 1)

Yn/ = on)'2n+ 1

2 b (2) _ 3 (2n+ 1) "'" 2j + 1 Y 1 y: 1

13 nu - S (n + 1) L j + 1 1) nj'

j=1

(B.S)

- 24-

Therefore the zero-order contribution to bnu gives

b (0) - 1. .<:nu -"2" Un1'

or (B.S)bu(O) = (O,!, 0, 0, ... ).

The first-order equation iseo

I 13 b (1) ~ y: 0 - I b (0) Y. 1 - y: 1nu nj nu nl - njn

n=1 n=1

and using the expression for bnu(O), one obtains for n ~ 1:

. 3 (2n + 1)ebnu(1) = Y1/·

4 (n + 1)

According to eq. (A.7) the first-order contribution to bnu gives

3n ( 1 an-1 1 an+1)ebnu(l)=- -----------, n~l,2 2n-l R 2n+3 R

(B.6)

(B.7)

where an = ° for n ~ 1.The second-order equation is

I 1 I I n+l132 b (2) _ Y. 0 _ 13 b (1) y: 1 + b (0) __ Y. 2 = 1. Y 2nu n nj nil nj nu 2 nj "2 1)

n=l n=l n=1

and using the expressions (B.S) and (B.6), one obtains for n ~ 1:eo

Finally the third-order expression is given for n ~ 1:

2n + 1 ex> ex>

133 bnu(3) = 2 ( 1) Cl y1n3 + ~ 132 bju(2) Yj/- ~ el bju(l) Yjn2). (B.9)n + j=l j=l

With this iteration procedure it is possible to obtain the higher-order contri-butions.

The vector bR

To calculate the vector bR we use the same procedure, but now we put uand dn equal to zero. The equation for j = ° turns out to be

-25 -

bOR = t J (1 + Y)2 dC-1

eo

(B.IO)

,"=2

and for j ~ 1

co eo

n=l k=O

(B.l1)

The zero-order contribution will be

n ~ 1, (B.I2)

giving the first-order equation of eq. (B.l1):co

n=l eo

(B.I3)

,"=2or

2 aneb (1) _

nR -n+IR'

The second-order equation will be

n~2.

eo <Xl

S2 b (2) ~ Y. 0 - Leb (1) Y. 1 - J1y2 P d)"nR n) nR n) - l ..

n -1

n=l n=l

and using the results (B.I2) and (B.l3) the second-order contribution can bewritten as

co

e2 bnR(2) = 2n + 1 ('" _2_ a) Yn/ + j y2 r, dC),2(n+I) ~j+IR -1

)=2

n ~ 1. (B.I4)

In the same way the third-order contribution for n ~ 1 is calculated to be

-26 -

co

2n + 1 L [aj 2j + 183 b (3) = __ Y 2 + XnR 2 (n + 1) R jn 2 U + 1)

j=1co

X Yjn1 ( ; y2 r, de+~ _2_ ak Yjk1)J.-1 ~k+lR

k=2

(B.lS)

The matrix B

Here of course the velocities u and R are taken as zero. For j = 0 eq. (2.25)will then give

1

bom= i J (1+ 2Y + y2) Pm de-1

2 am 1---+i Jy2Pmde + 0(83),2m + 1 R -1 .

and eq. (2.26)for j ~ 1:

(B.16)

co co

L L (n+k-l)'(B(O) + 8 B(1) + 82 B(2) + ...) (_l)k . Yn/ =

n! k!n=1 k=O

1

= J(l +2Y+ y2)PmPjde. (B.17)-1

The results of the zero-order contribution of eq. (B.17) will then beco

n=1

m ~ 2, n ~ 1. (B.lS)

The first-order equation isco co

n=1 n=1

and using the result (B.lS) the first-order contribution for n ~ 1 is

(B.19)

X Yin1(_1_Yim1 +2;YPiPmdC)J.m + 1 -1

With the results calculated up till now it is possible to give the velocitypotential up to an accuracy of third order of magnitude in s,

(B.20)

-27 -

The result of the second-order contribution for n ~ 1 is

2n+ 1 [ L 2j+ 1132 b (2) = - Y. 2 + X

nm 4 (n + 1) nm 2 (j+ 1)i=1

-28 -

Appendix·C

For the kinetic and potential energies the cavity volume and surface haveto be derived. The normal vector n on the surface can be given in the r- andB-directions, as is shown in fig. C.l, by

n = cos 0(, -sin 0( (C.l)

{1

Fig. C.l. Diagram used to calculate the cavity volume and area.

or, using n = VFjl VFI and eq. (2.20), by

_ [ (1 ()Z)2]-1/2 ( 1 ?lZ)n- 1+-- 1,---Z()(} Z()(}

[1(?lY)2 (()Y)2 ]= 1-:2 bi + Y ()8 + O(e4) ,

[ ?lY 1 (?lY)3 ]- - (1- Y + y2) +- - + O(e4) .()e 2 ?lB (C.2)

According to fig. C.l we find with

ds = 21£r, (sin B) [ers dB)2+ (drsYll/2

= 21£rs2(SinB)[1 +(~():eYT/2

dB

= -21£ R2 [1 + 2Y + y2 +~(:;f+ O(é)]dC (C.3)

-29 -

the cavity surfaceco

(I2 + n (n + 1) a 2 )A = 4:TtR2 1+ _n_ + 0(84) .

2 (2n + 1) R2

n=2

The cavity volume is

" [ (1 ez )2J1/2V = f -:Tt 's3 (sin" f}) [sin (f)_ a)] 1+ Z ~ df}o

co

4 ( I 3 a2

)= _ :TtR3 1+ -- _n_ + 0(83) ,3 2n + 1 R2

n=2

where the first third-order term equals

The kinetic energy as given by eq. (2.35) contains the integral form

taken over the cavity boundary.From fig. C.1 and eqs (2.41) and (2.43) it is seen that

(ê)CP1) =-ix• n =-cos (f)- a) =-cos f}cos et- sin f}sin etê)n r=rs

ê)Y [ 1 (ê)Y)2 J-(sin 8)- 1- Y+ y2_ - - + 0(83) ,ê)f} 2 ê)f}

co

-R- ~anPn

C~2 )r=r. =~~;I= {1+ [(1j;)=:Zjê)f}]2}1/2-

(C.4)

(C.5)

(C.6)

(C.?)

- 30-

(C.8)

n=2

giving the following intègrals:

bY )+ (sin (})- [1+ Y + O(e4)] dCb{}

(C.9)

and

+ ~ dnPn [1+ 2Y + y2 + O(S3)]} dC, (C.IO)n=2

where

[1 1I an (5n - 3 n + 1 )(c?1)r=rs= R -cos {}- - - --Pn-1- --Pn+1 +2 2 R 2n+l 2n+I

n=2

3 L an ( n2 (n+ 1)2 ) 3I 2n+ 1+-Y - --Pn-1- Pn+1 +- --Pn X2 R 2n+l 2n+l 8 n+l

n=2 n=1

I 2j + 1 I 3 (n + 1) anX -- Y 1 Y. 1+ y2 X

j + 1 1) n) 4 (2n+ 1) R)=1 n=2

3 L L2j+I I2n+l- - Y (2n+ 1)P; Yd Yn/ + --Pn X8 i+I n+l

n=1 )=1 n=1

- 31-

00 00

(3 a2 9 a3 ) 3I 2n + 1 I 2j + 1x --Yl/+--Y2/ +- --P --x20 R 28 R 16 n + 1 n j + 1

n=1 1=100 00 00

k=l 1=3 r,.n=100

(1 a1-1 1 a}+1) 1L2n+l . ]x -------- +- --Pny1n3+O(e4)

2j- 1 R 2j+ 3 R 4 n + 1

and

(C.Il)

n=l

00 00.{I 1 an2 In - 1 an(..1.2

) = =RR 1+ _ __ P _ y2+~ r t's 2n + 1 R2 n + 1R n

n=2 n=200 00

n=1 1=2

00 0000

I 1 a/ I an II-Y ---+ (n+l)-Pny2-- (2n+l)P Yx2n+ 1 R2 R 2 n

n=2 n=2 n=100 00

1=2 n=100

I[ a 2j+ 1 (1X - _!_ Y 2+ Y 1 J y2 P dC +R t» 2 (j+ l) In _ 1 . 1

1=1

00 00

+~_2 ak Y1k1)J+O(e4)}+R{~ án(_2_· an+_1Pn)+~k+1R ~ ~+IR n+lk=2 n=2

00 00 00

n=2 n=1 1=2

- 32-

I (n+ 2 2 a" I 1 )+ dn --Y2Pn----Y+- fy2PndC +

2 2n+ I R 2 -1

n=2

n=1 j=2

II2n+ 1 I [ I2k+1- - --Pn dj Yn/- _-ykn1 X4 n+1 k+1

11=1 J=2 k=1

(C.12)

- 33-

3. STABILITY CRITERION OF A CAVITY

3.1. Introduction

In chapter 2 we described the shape of a cavity, using a set of n degrees offreedom u, R, a2, a3' ... , say q. By using the method of Lagrange, a set ofdifferential equations is derived, with the time t as the independent variableand the dependent variables u, R, R., am à., n ~ 2. On the other hand theequations contain a set of parameters which determine the physical conditionsof the system we are examining. These are P 'Xl> PI(O), u(O), R(O), y, e, G, say p.If a situation with constant coordinates q and parameters P exists, we call itthe equilibrium state of the system, say the cavity. This equilibrium state atthe time t = 0 is qo + Po and might be given by a point 0 in the space Eq,pn.

The question to be answered now is whether the points starting in the neigh-bourhood of 0 remain near 0 with increasing time or not. In other words, thesystem will be called stable or unstable. For this wemay change the equilibriumvalues of the coordinates or the parameters. In both cases, however, we canuse the same definition to define what is stable or what is not. In general theequilibrium state might be stable, asymptotically stable or unstable. If we havean equilibrium state we take for convenience the origin of the p + q systemin the equilibrium position Po + qo and define the spherical domainlp + q] < R as SeR) and the sphere lp + ql = R with H(R). According tothe definitions given by La Salle and Lefschetz (1967) we say the origin is(a) stable, if there is an r ~ R < A, so that a point p + q starting in the

spherical domain Ser) always remains in the domain SeR) (fig. 3.1);(b) asymptotically stable, when it is stable and when with increasing time it

approaches the origin;(c) unstable, when there is no sphere Ser) small enough to keep the move-

ment of the point inside the sphere H(R).

Unstable

Fig. 3.1. Stability domain in the coordinate system.

- 34-

All these stability problems concern the coordinates and parameters if oneor more of them is changed from its equilibrium condition. To solve thestability problem we make use of the differential equations. In most casesthese equations are linearized and uncoupled. The question may arise if it ispossible to find the right solution in that case, because a negligibly small valuemight become significant just in an unstable situation. This is why in generalit may be stated that if stability is assured under linearized conditions, itremains an open question whether it exists with the exact non-linear equations.In the case which we shall consider, it will be possible to find the influence ofthe higher-order terms although we use linearized equations with respect tothe disturbance terms an'

In this chapter the equilibrium state will be defined. The stability of thisstate will be discussed, making use of the studies which have been done in thisdirection, as e.g. by Eller (1966). The instability caused by changing the param-eters will form the main part of this chapter. It is shown that there is somethreshold pressure P 00 for the onset of shape oscillations. Because we havechosen a dissipationless system stability will never be asymptotic. It will beinteresting to investigate one of the dissipative causes for its damping proper-ties in order to find out when asymptotic stability will exist. This is why theinfluence of viscosity is incorporated, with the result that some remarkableconclusions can be drawn with respect to stability in practice.

3.2. The equilibrium state of a translating cavity

The first case to be determined is the equilibrium shape of a cavity translatingat a constant velocity. This is a relatively simple case to practise the equationsof motion. The constant values of the coordinates and parameters will bewritten as Ro, ano, uo, and the cavity pressure as Po, the liquid pressure as P 000'

This equilibrium situation is found by putting R = án = 0 in the Lagrangianequation (2.63), giving

(3.1)

The equations of motion will then be specified by

( oL ) = 0 and ( 'OL ) = 0,< oR R=Ro 'Oan an=ano

These equations can be found directly from eqs (2.64) to (2.67), putting ü, R,R, ä., án equal to zero and replacing the variables by Ro, ano and ua. Equa-tion (2.65) then changes into

n= 2, 3, .... (3.2)

(3.3)

- 35-

This first equation is used to eliminate the quantity (Po - P 000)/(2 from theequations (bL/ban)an=ano= O.We define the dimensionless Weber number by

(2 U02 s;

We=---- (3.4)a

Substituting eqs (3.3) and (3.4) into eq. (3.2) we achieve a set of equationsdetermining the equilibrium shape:

(3 39 a20 9 a40 ) a20We + __ +0(82) +2-+0(82)=08 70 s; 14 s, s;

(3.5)

and for n ~ 3 (al = 0):

[9 n (n- 1)2 a(n-2)0

We --+(2n- 3) (2n- 1) Ro

((n+6)(n+1)2 n(17n2-22n+9) )ano- + -2 -+(2n + 1) (2n + 3) (2n + 1) (2n- 1) Ro

+ 9 (n + 1)2 (n + 2) a(n+2)0+ 0(82)J + 4 (n- 1) (n + 2) anO+ 0(82) = o.(2n + 3) (2n + 5) Ro . Ro

(3.6)

The equilibrium radius Ro is, for a givenWe number, related to the translatingvelocity by eq. (3.4). This radius Ro is the coefficient of the zero Legendrepolynomial in the expansion of the cavity surface. This is why this radius isan average radius of the equilibrium cavity. Using a simple method it waspossible to determine from the equations of motion a set of equations (3.4)to (3.6) for the equilibrium shape.Various workers have already found solutions for the equilibrium shape:

the work of Eller (1966) merits special mention here. His method ran parallelto the one used here, hence we shall refer to it. Starting from the assumptionthat Weber is in the order of magnitude of a., it is possible to express anOina series of ascending powers of We. From eq. (3.5) it immediately follows thatthe first-order approximation of a20 is equal to

a20 3- = - - We + O(We2). (3.7)Ro 16

From eqs (3.5) and (3.6) we further see that all anOwith odd n are equal tozero. The equilibrium shape is thus an even function with respect to a lineperpendicular to the x-axis (translation direction). This symmetry is causedby D'Alembert's paradox and because the pressure difference between liquidand bubble is compensated only by the surface tension. We know accordingto D'Alembert's paradox: in an irrotational flow of an ideal incompressible

- 36-

fluid passing a body of finite size and in the absence of sources and sinks, theresultant vector of the pressure forces on the body is equal to zero. In the casewe are considering here the absence of sources or sinks is assured because ofthe constant shape in time of the equilibrium cavity. Starting with a spherewe know, the pressure distribution on the surface follows the expression

or written as(3.8)

The angle-dependent part of the pressure Pc on the surface, !e U02 P2, is an

even function. This pressure can only be compensated by the surface tensionby changing the radius of the bubble, because the gas pressure in the bubbleis constant. Since the pressure Pc is a function of P2 the bubble radius has tobe a function of P 2 too, which results in the addition of the a2 perturbationterm. By this the bubble has a shape, which is even about the axis perpen-dicular to the axis of motion, say an even shape. This in its turn will resultin another pressure distribution on the surface but it will be even again. Theshape must therefore be changed again, but in an even way. By this it will beobvious that a cavity moving in an incompressible liquid will have an evenequilibrium shape, also independent of the direction of motion. It will, more-over, be obvious that an equilibrium shape only exists if the surface tensionis included.From the equations given we can determine the equilibrium shape only up

to the order of We2• In chapter 4 we employ equations of motion which arenot fully linearized with respect to the disturbance coordinates an. Those resultswill be used here to determine the equilibrium shape and pressure.The equations of the equilibrium shape then become

Po-Proo 2(a2o)2 [1 3 a20 39 (a2o)2]---Ro-2-- - -We ----+- - +O(We4)=0,a 5 n; 4 10 s; 350 s;

(3.9)

[3 39 a20 9 a40 249 (a2o)2] a20 6 (a2o)2We ----+---- - +2--- - +O(We4)=0,8 70 Ro 14 Ro 392 s, s, 735 s;

(3.10)

(324 a20 2015 a4o) a40We ----- +72-+0(We4)=0,

35 Ro 231 s; s; (3.11)

75 a40 a60--We + - + O(We4) = 0.880 s; s, (3.12)

.---------------------- - --- - --

- 37-

The equilibrium values of an obtained are

a20- =-0·18750We- 0·05080We2- 0·00996We3 + ... ,Ro

a40- = 0·02411We2 + 0·00653We3 + ...,Ro

(3.13)

a60- = -0·00206 We3 + ....Ro

The equilibrium pressure of a translating bubble has the following proportionwith regard to the pressure of a transient bubble:

Po 1-----=I---xP<x>o+ 2a/Ro 1+ IX

X [0·1250We + 0·03515We2 + 0·01339We3 + O(We4)], (3.14)

where use is made of the definition

(3.15)2a

The equilibrium values of ano given by E1ler (1966) do not fully agree withthe results given here concerning the second- and third-order terms. The sameis the case with the third-order term of the bubble pressure.

0·7I~alt

(}6

0·5

D·'

1I!r3rd order)

~order)

0·4 7!:(1st order)

0·3

0-2 '?J9.rJrdorder)Ra

--"5

pu~RaWe=-u-

20·5

Fig. 3.2. Equilibrium values of a20 and a40'

- 38-

In fig. 3.2 the successive approximations for a20 and a40 are plotted. It isseen that up to We = 1 the linear approximation of the equilibrium shapegiven by eq. (3.7) does not introduce a significant mistake.

3.3. Stability of the equilibrium state

Once the equilibrium state has been obtained, the next logical step is todetermine whether it remains stable if one or more of the coordinates orparameters are changed. In this section we shall be concerned with the problemof changing the initial conditions, say the coordinates. In conformity with thedefinitions given in sec. 3.1 the equilibrium state will be said to be unstable ifan infinitesimal deviation of the equilibrium shape gives an unlimited growthof one or more of the coordinates a2, a3, a4' .... Considerable work has, infact, been done by others in this direction, so that our only reason for repeatingit here is to put it in the proper perspective vis-a-vis the unstable behaviourwhich will be examined in this thesis.

Hartunian and Sears (1957) have treated this subject experimentally as wellas theoretically. They calculated the velocity potentialof a liquid containinga translating bubble up to the order e. Using the equation of Bernoulli it ispossible to obtain an expression of the pressure distribution about the bubblesurface. Applying Newton's law, a set of differential equations linear in an wasobtained, but coupled by the velocity u. This set of equations can be solvedfor small oscillations, by taking a general solution of anCl) = exp (À t), whereÀ is a complex number to be determined. If the real part of À is negative, theensuing motion is stable, whereas positive values of À will indicate a divergenceof the mode in question. The procedure to be followed in solving a set ofequations is to solve a determinant. The accuracy depends on the number ofequations and the influence of the terms with a higher number. As the equi-librium shape is a function of the Weber number, the criterion of instabilitywill be a function of the Weber number too.This stability problem has now been solved numerically by Hartunian and

Sears. For a 2X 2 determinant they found instability for We = 4·00 and fora 4x4 determinant We = 2·72. Going to higher-order determinants gaveresults only negligibly different from those obtained from the 4x 4 deter-minant. This method which can be used for linear equations is rather simple,but also unrealistic, because the higher-order terms are neglected. There is,however, another method which can be applied to this system, which is in factconservative. It is stated in the proposition of Lagrange or Ljapunov that anundamped system with one isolated equilibrium point will be stable or un-stable. To find these regions one needs a Ljapunov function, which is usuallyhard to find, but in this case it is simpler to use the Lagrangian instead. How-ever, care is then required as the translating motion plays an important rolein this case. An effective potential energy can be determined with the help of

- 39-

Routh's procedure, as given e.g. in Goldstein (1950). This potential energy isa function of the added mass M, the volume V and surface area A, which intheir turn are a function of an- Depending on whether the second derivativeof this effective potential energy is positive or negative, the :system is stableor unstable. The more accurately the potential energy (i.e. higher order in an)is determined, the better the transition stable-unstable can be calculated. Eller. (1966) solved this problem, which is in fact an eigenvalue problem. He cal-culated the energy up to 0(e2) and found that the third eigenvalue, concerningthe coordinate a3' which has the value À3 = 4 Ro2 Cf (1·429 - 0·2755 We +- 0·1406 We2) gave the criterion We = 2·34.

Comparing the results obtained by Hartunian and Sears with those of Ellerwe see that the linear approximation results in We = 2·72, while the second-order approximation of Eller gives We = 2·34.Experiments to determine this instability have been carried out by a number

of workers, e.g. Hartunian and Sears (1957) and Saffman (1956). The relativevelocity of a bubble with respect to the liquid is determined from the buoyancy.The instabilities observed in a rising bubble, are oscillations in the shape anddeviation from the rectilinear path. Hartunian and Sears were the first to finda reasonable criterion. They found that in a great variety of liquids this instabil-ity occurs at a Weber number above 1·59.

Correlation between the theoretical and experimental results seems to berather poor. In the first place, the theoretical results are still unreliable as theapproximation is poor for such high Weber numbers. In fact, even a second-order approximation of the equilibrium state remains poor for We > 1·5.Hartunian already realized this from his first-order approximation. To obtainan estimate of higher effect in the deformation of a bubble he took the pres-sure distribution about an oblate spheroid, knowing the principal radii ofcurvature at the stagnation point and equator to satisfy the equilibrium con-dition of the pressure. In this way he found instability at a Weber numberof 1·51. In the second place the experimental results are not completely com-parable with the theoretical model, due to the gravitational and viscous forces.The experimental equilibrium shape will include odd perturbation terms.It may be concluded that the equilibrium shape is a function of the Weber

number. However, in a real situation, viscous and other forces will playa part.From experimental and theoretical results we may draw the conclusion that

a moving bubble in a liquid will continue to exhibit a stable behaviour if theinitial conditions are changed, provided that the Weber number is less then 1·5.This result is very important because we may now use linear equations, say

for We < 1, with any initial condition, say R(O), R(O), aiO) àn(O), being surethe cavity surface will remain stable according to the definition given insec. 3.1.

- 40-

3.4. Influence of a pressure step on the stability of a cavity

3.4.1. Introduetion

In the preceding section we discussed the stability of the equilibrium shapeat an infinitesimally small distortion. Instability of the bubble surface mayalsooccur if one of the parameters is changed, particularly the liquid pressure P co'

The instability may then lead to such complex phenomena as jet formation andeven to bubble split-up.It has been shown by Taylor (1950) that a plane interface between two fluids

of different densities in accelerated motion is stable or unstable according towhether the acceleration is directed from the heavier to the lighter, or con-versely.The corresponding problem for a spherical interface has been discussedby Binnie (1953), although his analysis appears to be in error because he omittedsome terms of the same order as the one which was used in the equation.PIesset (1954) rectified this and gave the first exact formulation of the insta-bility of a perturbated stationary spheric. PIesset and Mitchell (1956) investi-gated the stability of the perturbation terms of expanding and collapsingbubbles, assuming the pressure inside the bubble to be constant. The con-clusions they drew are that an expanding bubble is stable, while a collapsingone is not. In the latter case the distortion amplitude grows as R-1/4 whenthe bubble radius R tends to zero. This increase in distortion is found withand without surface tension. In the case of an expanding bubble the effect ofsurface tension seems to be unimportant, but small irregularities in the sphericalinterface may grow to significant amplitudes if the surface tension is neglected.In experiments, it was discovered that a gas bubble trapped by an acoustic

standing wave began to dart about in erratic fashion when the sound-pressureamplitude was raised above a certain threshold value. This behaviour may beaccounted for by the instability ofthe spherical surface when it undergoes radialpulsations of sufficient amplitude. Benjamin (1964) gave a theory based on thedifferential equations given by Plesset and Mitchell (1956), but his theory wastoo crude to provide adequate agreement between theoretical and experimentalfindings. Later work done by Crum and Eller (1969) gave a good explanationof this instability.To date, all the theoretical studies in literature concerning bubble instabilities

concern stationary bubbles, while the pressure of the liquid is periodic, or witha constant pressure difference between bubble and liquid.In this section we shall analyse the stability of a moving bubble in the case

of a step change in the ambient pressure. Although the non-linearized differen-tial equations are not exact, a fair degree of accuracy may be assumed for therange of stability with We < 1. The perturbation terms are linearized, butthe bubble-radius variation is described by a series of harmonic oscillations.A set of equations of the Hill type can then be derived, enabling a threshold

-41-

pressure to be determined for the liquid above the equilibrium pressure whereinstability of the perturbation terms occurs. The influence of surface tension,adiabatic or isothermic bubble pressure change, and translational velocity uois studied.

The domain of stability of an undamped system is rather unrealistic. For thisreason the influence of the liquid viscosity is added. In this case it is possibleto approximate the range of stability by a simple equation.

3.4.2. Solution of the modified Rayleigh equation

Equation (2.64) is the linearized differential equation for the translationalvelocity u. A solution of this equation can be found by integration, giving

t • t' •

JUf 15 R R2 - 18 R R a2 - 9 R2 a2-dt =_ dt,u 5 R3_ 9 R2 a° 0 2 (3.16)

This solution is used to eliminate the velocity u in the modified Rayleigh equa-tion (2.66), which is linearized with respect to the perturbation term. Neglectingterms of higher order of magnitude, as O(e) eq. (2.66) changes into

.. 3. PI-Pro 2a 1 (Ro)6( 18 a20 12a2)RR+-R2_ +---U02 - 1---+-- =0.

2 eeR 4 R 5 Ro 5 R(3.17)

Using the linear approximation of a20 for We < 1 an approximation to firstorder of We of eq. (3.17) is given by

.. 3. PI- Pro 2 a [ 1 (Ro )S ]R R +- R2 _ + - 1- - We - + 0(We2) = O. (3.18)2 e eR 8 R

The bubble pressure PI can be given for an ideal gas by the gas law

(Vo )3Y

PI=PO V .The exponent y is taken to be unity if the process is isothermic, and y = cp/cv,the ratio between the specific heats of constant pressure and volume, if theprocess is adiabatic.

When the second- and higher-order terms of a.[R (say We) are neglectedeg. (3.19) is changed to

(3.19)

(3.20)

-42 -

For small oscillations of the bubble radius, eq. (3.18) may be linearized bysubstitution of R(t) = Ra [1 + r(t)], resulting in

.. (3 r Po 2 a 3 ua2) Po - P 00 2 a ua2

r + r e Ra 2 - (! Ra 3 + 2: Ra 2 = e Ra 2 - (! Ra 3 + 4 Ra 2 '(3.21)

where r(t) has a general solution with an exponential term exp (i Wo t). Start-ing with the initial conditions of the equilibrium shape with a pressure-stepchange from P coo -:+ P co, where P 000 is the equilibrium pressure found by put-ting the left-hand side of eq. (3.21) equal to zero, the solution is

r(t) = Aa (-1 + cos wat). (3.22)

The amplitude Aa and the angular resonance frequency Wo are given by

e-lAa = -- (1 - <5), (3.23)

3y3yP

W02 = 000 (1 + <5),

(! R02(3.24)

with

1(3 Y - 1 2- Y)<5=- +We--IX 3y 8y

(3.25)

and

Pooe=--.r.;

IX is given by definition (3.15). For IX» 1 the correction factor <5 may beneglected, and eq. (3.24) is the so-called Minnaert frequency, see Minnaert(1933). The correction factor <5 is directly related to the surface tension a andthe translational velocity ua.For a greater pressure step we may not linearize with respect to the radius R,

so the non-linear eq. (3.18) has to be solved, although second- and higher-orderterms in Weber may be neglected. Starting with the same initial conditions,the solution of R(t) will have an even behaviour. The solution can be givenby the following Fourier series:

R 00

- = 1- ~ An bn (1 - cos nwt), (3.27)Ra n=l

(3.26)

where

(3.28)

(3.29)

-43 -

andbi = 1,

y+2b ----z - 4' (3.30)

6 y2 + 21 Y + 25b -------3 - 64 '

Cl =-2y,Cz = (y - 1) bz - b3 + 2/ y2 - Y - i, (3.31)

dl = 3y + 2,

18 + 27 y- 6 y2dz=------

8

This approximation is only valid if Ao < 1, and because 1 ~ y < ~thesolution is sufficiently accurate for e< 2.

(3.32)

3.4.3. The range of stability

The set of equations determining the bubble shape are (2.64) to (2.67). Fora pressure step the radius R and velocity u are given as a function of time byeqs (3.27) and (3.16). The stability of the bubble shape due to a pressure stepwill, however, depend on the behaviour of the perturbation terms an given bythe linearized differential equations (2.66) and (2.67). The pressure term(Pt- Poo)/e can be eliminated by eq. (2.65) and the translational velocity uby eq. (3.16). In this way we obtain differential equations of the coordinates anwith the known function R and the parameters 11 and uo, where the an are theunknown variables yet to be determined. The known function R is in its turna function of the parameters P 000. e, 11, Uo, y, Ro. If we define a new variable,as has been done by PIesset (1954), by

(3.33)

and

the equation for an can be written as

jin + Yn [11/ (~ y- ~(~y- 2n; 1 ~ - u/ (~ YJ = Fm (3.34)

where

(n + 2) (n2 - 1) 11I1n

z = ,e R03(3.35)

U 2 _ (n + 1) (uo)2 X

n - 4 (2n + 1) Ro

(n+6)(n+l)2 n(17n2-22n+9)

X +-------(2n + 3) (2n- 1)

2 (2n+ 1)), (3.36)

-44 -

_ 3 (n + 1) (4n+ 1) Uo ( RO)5 R. ( Yn-1 Yn+l )Fn- - - - ---3-- +

4 Ro R Ro 2n - 1 2n+ 3

_3(n+l)Uo(RO)4(. _2n+l.)+2 s, R Yn-1 2n + 3Yn+1

_9 (n + 1) (~)2 (RO)8( n (n- 1)2 Yn-2 +4 Ro R (2n- 3) (2n- 1)

(n + 1)2 (n + 2) )+ (2n + 3) (2n + 5) Yn+2 , n ~ 3

and for n = 2

(3.37)

(3.38)

If there is no translational motion, say Uo = 0, F; will be equal to zero. Equa-tion (3.34) is then the same as was found by Hsieh and PIesset (1961). an thenequals the angular resonance frequency of the n-shape oscillation. Let us nowconsider eq. (3.34). Equation (3.34) is a linear, second-order differential equa-tion with periodic coefficients and, thus, is an example of Hill's equation.Substituting the solution of R(t) given by eq. (3.27) and using the transfor-mation

Z = trot, (3.39)

one finds that eq. (3.34) can be written in the standard form for the inhomo-geneous Hill equation:

d2Yn eo • 4- + Yn [Bo + 2 2: B21cos (21Z)]= - Fm (3.40)dz2 1=1 ro2

where the coefficients B2I are given through terms of third order in Ao:

Bo = r-II + (3r- 8lI)A + (9 + 3b2) r- (54- 8b2)II- 4n; 5) A2+

+ (25 + 12b2 + 3 b3) p.: (300 + 72b2 +8b3)II- 4n- 5) A3+0(A4),

(3.41)

-45 -

82= (-:-~r+ 4IJ + 2n + 1) A + (-6r+ 36IJ + 2n + 1) A2 +

(7n+5)+ -(7;é+3b2)r+(225+18b2)IJ+-2--H2n+3)b2 A3 +

+ O(A4), (3.42)

84= (~(1-b2)T- (9- 4b2)IJ + (8n + 4)b2- 4n4

1)A2 +

+ ((11- 6 b2)T- (90- 36 b2)IJ + (8 n+ 4) b2 _ 4 n21)A3+ O(A4

),

(3.43)

(

n-l86 = - Ct - 3b2 + ~b3) r + (15- 8b2+ 4 b3)IJ + -2- + (18n + 9)b3 +

10n+ 1 ). - 2 b2 A3 + O(A4

), (3.44)

where

(20'n)2T= -w

and

(2UII)2IJ= - .

w(3.45)

By this the coefficients 821 are a function of the parameters uo, Ro, Pooo, 8,

0', (! and y.The homogeneous equation CFn = 0) has a solution of the following form:

00 00

Yn=A1exp(,uz) ~ c2rexp(2rzi)+A2exp(-,uz) ~ c2rexp(-2rzi),r=-CO r=-CO

(3.46)

and can be written as

(3.47)

The present notation agrees with that used by Maclachlan (1964). The coef-ficients C2r= C-2r depend on the initial conditions of the coordinates andvelocities. The exponent ,u depends on the parameters and might be eitherrealor imaginary. If ,u is real, negative or positive then exp (-,u z) -)0 corespectively exp (,uz) -)0 co, when z tends to infinity. In the case of homo-geneous equations we can define a solution as being unstable, if for z -)0 co,say t -)0 co, Yn -)0 co, say an -)0 co. The criterion for instability is then givenby ,u is real.To solve eq. (3.40) we have to look likewise for a particular solution. This

(3.48)

- 46-

can be found by using the method of variation of parameters, as is given byInce (1956), and is

z z

o oTherefore the complete solution of the inhomogeneous Hill equation (3.40) is

z z

Yn= Al Ynl + A2 Yn2+ A3 [Ynl(Z) JYn2(U)FnCu)du- Ynz(Z) J YnI (U)Fn(u) du].o 0 (3.49)

Because we are interested in a stability criterion and not in the exact solutionwe are not concerned with determining the coefficients. Our only interest willbe the exponent Il, since a real value of it indicates instability. An importantrole in solving this problem is played by the function E;This function Fm dependent on n, contains the following variables:

F2 = F( Y2' R Y3' Y3' R, Y4),F3 = F(R Y2' R Y4' Y2' Y4' Ys),F4 = F(R Y3' R Ys, Y3' Ys, Y2' Y6),r,= F(R Y4' R Y6' Y4' Y6' 13, Y7),etc.

(3.50)

This shows clearly how the differential equations are coupled by the function Fn.It is still possible to find an analytical solution by using an iteration procedure.First of all the known functions of R(t) and R(t) of F; can always be writtenin the following way:

00

(3.51)

n=-oo

and00

( :0) = t QJ exp (t n i)I:exp (2 n i z), n =F O. (3.52)

n=-oo

Because the function F; is linear in Yn the iteration procedure is then as follows.The homogeneous solution for every Yn can always be found. To obtain a firstcontribution of the particular solution of Y2 we take into account the functionF2(R), since we know only the function R. The first contribution of the par-ticular solution of Y3 is then found with F3(R Y2' Y2), although Y2 is notexactly completed. In its turn the second contribution of Y2 is now calculatedwith Fz(R 13, Y3) and the first contribution for Y4 with F4(R 13, Y3' Y2). It is

,,--------------- -~._.~- .. -

- 47-

obvious that this method is rather tedious, but since we are not interested inthe exact solution it is relatively easy to find the stability criterion.

Starting the procedure we first define the equilibrium values of Yn by

YnO= Ro 3/2 anO' (3.53)

which results in the first contribution of F2 being equal to

9 (U )2 (R )3/2 81 (U )2 (R )7F2=- - _0 _0 Ro5/2+_ _0 _0 120 +

4 s; R 10 s, R

_ 81(~)2(Ro)8 .10 Ro R 12

The last term has to be included in the left-hand side of eq. (3.40) for n = 2.Using eq. (3.51) we obtain the following general expression:

(3.54)

F2 = 2: C2n exp (2 n i z).n=-oo

(3.55)

The particular solution can now be found by integrating eq. (3.48) by substi-tuting eq. (3.55). The first integral of (3.48) then equals

z

exp(ltz) ~ c2rexp(2riz)fexp(-ltu) ~ c2rexp(-2riu) Xr=-(() r=-OO

o

X 2: C2n exp (2n iu) du = exp (,uz) ~ c2rexp (2 r iz) Xn=-oo r=-oo

I a2nX [exp(-ltz+2niz)-1].

-It + 2 ni11=-00

A similar integration of the second integral equals

I I a2n-exp(-Itz) c2rexp(2riz) .[exp(ltz+2niz)-1].

1t+2nlr=-oo n=-OQ

One part of the particular solution of 12 can be included in the homogeneoussolution. By this the first contribution of the solution of 12 is

Y2 =A1exp(1t2Z) 2: c2nexp(2niz)+A2exp(-,u2Z) Xn=-oo

X ~ c2nexp(-2niz)+ 2: cnexp(2niz).n=-OO n=-C()

(3.56)

(3.57)

- 48-

The index 2 of fl indicates that fl2 is a result of the homogeneous solution ofh.The particular part of the solution is a function of R(l) and of the initialvalue Y20. This particular solution has no influence on the stability. The result(3.56) is now used to determine the first contribution of F3' giving

eo

F3 = exp (!:n i) [BI exp (u2 z) ~ n a2n exp (2n i z) + B2 exp (-fl2 z) X-<Xl

co cc

X ~ na2n exp(-2 n iz) + Cl exp (fl2 z) ~ b2nexp(2 n i z)+ C2 exp (-fl2 z) X-00 -00

eo co co

X ~b2nexp(2niz)+ ~nd2nexp(2niz)+ ~f2nexp(2niz)]+-00 -00 -00

eo co

+Dl exp (fl2 z) ~ h2nexp (2n i z) + D2 exp (-fl2 z) ~ h2nexp (2 n i z).-00 -00

Using eq. (3.49) the first approximation of Y3becomesco co

Y3= Al exp (fl3 z) ~ c2nexp (2 n i z) + A2 exp (-fl3 z) ~ c2nexp( -2 n i z) +-00 -00

co eo

+exp(!:ni) ~ a2nexp(2niz)+Blexp(fl2z) ~ b2nexp(2niz)+-00 -00

eo

+ B2 exp (-fl2 z) ~ b2nexp (-2 n i z) + exp (t:n i) X-<Xl

co co

X [Cl exp (fl2 z) ~ d2nexp(2 n i z) + Cz exp( -fl2 z) ~ d2nexp( -2n iz)].-00 -00

(3.58)

The third term is a result of R(l) and Y20 but it now has a phase angle of:n12 rad. The next terms are from the homogeneous part of Y2. For the sta-bility criterion we have to reckon with fl2 and fl3. From the foregoing we seethe linear character of the solution. This will be obvious if we again investigatethe partienlar solution (3.48). The integrals will always be of the followingform:

co z eo

exp(unz) ~ c~rexp(2riz)Jexp(-flnU) ~ c2rexp(-2riu)exp(flmu) Xr=-oo r=-ooo

co

X ~ a2rexp(2riu)du=r=-oo

eo eo

= exp (fllll) ~ a2nexp (2 n i z) - exp (fln) ~ bzn exp (2 n i z).n=-CX) n=-C()

et)

-49-

There will never be an exponential coefficient consisting of more than one !till.Continuing the iteration procedure it may be obvious from the foregoing thatthe total solution of Yn is given by

et) et)

Yn = L [Alii exp (J-tm z) L a2r exp (2 r i z) + Bm exp (-J-tm z) Xm=2 r=-CO

X L b2r exp (-2 n i z)]. (3.59)r=-CO

Although we have linearized equations for Ym the complete solution of Yn

includes terms with the exponential factor /hili' m ~ 2, because the trans-lational velocity Uo gives a coupling of the differential equations. AssumingUo = 0, the linearized equations are uncoupled and then the solution of Yncontains only the exponent /hn. As a result only those perturbation termshaving an initial condition differing from the equilibrium terms can give riseto instability. In other words, without translational velocity the linearizedequations give a stability criterion depending upon the initial conditions. Todetermine the practical stability of this system it would be necessary to involvethe second- and higher-order terms in the equations of motion. In that casethe second- and higher-order terms will give a coupling of the equations, aswould be the case with the velocity uo. Then, a very small disturbance whichwas neglected, may grow exponentially and may dominate the linearized solu-tion.

This is why the stability criterion of a gas bubble with We < 1, subjectedto a sudden pressure rise of the liquid, is given by the following definition:

the set of differential equations

d2Yn et) •-- + Yn L 2 (jn21 cos (2 1 z) = Fm n ~ 2dz2 1=0

has homogeneous solutions of the formet)

Yn = L Yn) exp (P-n) z) cos (j Z + cp),)=1

n~2,

where/hn) = /hn) (uo, e, P et)0, Ro, tb y, a, n).

The bubble shape is called unstable, if for a given combination of theparameters Uo, e, P et)0, Ro, (!, a, y there exists at least one /hnJ real.The exponent /hn) is related to the coefficients (jn21. The index n is related to

the nth coordinate and the index jto the jth frequency of the solution. Accordingto the theory of Hill's equations the solution can be represented in stable andunstable regions, where each value of j corresponds to a separate region. Theseregions can be given in an (e- I)-Ro plane, where the dependence ofthe other

- 50 _.

parameters e, uo, (J, P 000, Y can be indicated in this plane. Thus we have asequence of instability regions for every Yn coordinate. At the moment only thefirst three regions (j = 1, 2, 3) are considered. On the boundary of theseregions the value of ft] is equal to zero. Approximate equations of ft] on theseboundaries are obtained from Hayashi (1964), and are given in appendix D.The instability criterion of an undamped system, which we are considering atthe moment, is then given by f-tnl > O.The calculations for this criterion were performed by a computer, according

to the following procedure. First, the parameters e, Uo, (J, y, P coo were chosen,and a value for n was selected. Next, a sequence of values of Ro and the pres-sure step (e- 1) were chosen and tested to see which pair of values satisfieseq. (D.14) for j = 1, 2, 3. This procedure was then repeated for a new valueof n.A sample ofthe results of these computations is given in fig. 3.3,which shows

the location of the first, second and third unstable regions of shape oscillationswith n = 4. The values used for hydrostatic pressure, density, surface tension

2 5 2 5 2 5

la'

2

~'3

~ ~ 1-/ .r'l=/~·~.'/. '/.

"7"7771"//// '/. '/1'l"///..-0 v/-~v

\ ~,~,IV

II';

2- -

5

E-l

5

2

--- Ra (m)

Fig. 3.3. The first three regions of shape instability with 11 = 4 for an air bubble in water;Proo = 5.104 N/m2, Uo = 0 mis, y = 1'41.

.---~----~-----'

- 51-

and translational velocity are Pa;,o = 0·5 bar, f2 = 103 kg/m", (J = 0·072 Nimand Uo = 0 mis. The curves are for the adiabatic case with y = 1·41. Thesevalues are for an air bubble in water. In the shaded regions the shape oscilla-tions will be unstable with angular frequencies of tw, w and ~w.

Of course, it would be possible to calculate and draw the regions of the fourthand higher ones, but as is seen, these will become more and more sharp andsmall. In the next section it will be demonstrated that they have no real mean-ing. The accuracy of the regions has an upper limit at e = 2 and all over theboundaries 821 (i ;;:::1) has to be small. As we see from fig. 3.3, the configurationof the regions corresponds closely to the stability chart of Mathieu, althoughwe have a Hill equation and other plane coordinates.We can also investigate the influence of the translational velocity uo. Because

we have linear equations in Yn the admissible velocity Uo cannot be greater, asit would give a Weber number of approximately unity. The results of thesecalculations for n = 4 are given in fig. 3.4 with the same data as in fig. 3.3except for the velocity uo. For convenience, only the first unstable region isgiven because it is the most important. The influence of this velocity is rather

10'

5

nj=4·1

/'u =1·5 /' ,...

\ \ 1·0 / /' :;..-1{}5 '/

\ /110

\1\ 1 'fI

I

- -5

5

€-T

5

2

2

5 104 2 5

--- Ra {m}Fig. 3.4. The first region of shape instability with 11 = 4 for an air bubble in water at severaltranslational velocities; Pooo= 5.104 N/m2, l' = 1,41, IlO= 0, 0'5, 1'0, 1·5 m/so

- 52-

TO'

2

'////. r/h /,

'l/.'4 V.'/. '//,(2., 1 'l::'l::1?1 V///..-0~ ~ :%:1% ~ ~~ ~ 3·l ~ ~~ /,

,I~ ~%~

~~~~~ -t ~~~ :/'/,

6·1'//. r/,

r/. rr: '//,

8·' 0!{/)<. -Vh~9" ,fi V.a,

~~/

~ ~?t ~~~I/i

~~~~J~~

15·

2~ V- ~

5

E-T

5

2

TfT

5

TU 10-5 2 5 5 TO-3 2 5

- Ro{m)

Fig. 3.5. Threshold pressure step for the onset of non-spherical oscillations of an undampedair bubble in water; Pooo = 5.104 N/m2, IlO = 0 mIs, y = 1·41.

small and if any at all, it seems to have a stabilizing effect because the area issmaller.A composite region of instability is obtained by combining the unstable

regions for all values of j and n. Any point within this composite region repre-sents a bubble whose spherical shape is not stable, according to the definitiongiven. In general, the bubble shape is stable when the pressure step is sufficientlysmall and becomes unstable when the pressure step is increased above a certainthreshold.An example of this threshold for the onset of shape oscillations is given in

fig. 3.5 as a function of the bubble radius Ro. Again the same data have beentaken as in fig. 3.3. To keep the figure within convenient limits only the firstregions have been taken into account. Unstable regions with different n-valuescan overlap, in which case more than one unstable oscillation could be excited.

3.5. Effects of viscosity

In a real liquid, the presence of viscosity tends to damp out the surfaceoscillations of a bubble. So it is reasonable to assume that stable oscillations

2 (2 (n + 2) (2n + 1) ')1)2ftn) > R 2 •

o w(3.62)

- 53-

for small pressure steps might be asymptotically stable. To determine the do-main of asymptotic stability it is necessary to take into account the non-lineari-ties of the equations of motion. Although we have used linear equations wehave seen that it is possible to give a stability criterion for small Weber numbers.It was possible to do this because the translational velocity plays the same roleas the higher-order terms. This method will be continued in the case whereviscosity is included.

According to Lamb (1932) a surface oscillation on the interface of a sphericalbubble of radius Ro and a viscous liquid with kinematic viscosity ')I decaysexponentially as exp (-()( t) where

(n + 2) (2n + 1) ')I()(=------------- (3.60)

Using this damping factor, it is assumed that instability can occur wheneverthis decay rate is less than the growth rate t ft co t of unstable solutions inthe absence of viscosity. This assumption has been investigated by Benjamin(1964), supposing that the free oscillations have the frequency co and dampingby viscous dissipation will change co only slightly. This means that a may beonly a small fraction of os, or that

(n + 2) (2n + 1) ')I< _1_ (3YPooO)1/2Ro2 s; e

Provided n is not too great this will mostly be the case. The criterion forinstability is now

(3.61)

If ftn/ is less than zero we have asymptotic stability.An example of the threshold for the onset of shape oscillations is shown in

fig. 3.6 with n = 4 and for different viscosities. We see that the higher theviscosity the less sharp the regions will be, while even for ')I = 10-6 the regionstwo and higher completely disappear for 8, < 2.

The composite region of instability for a damped system with ')I = 1·6. 10-6m2/s, the viscosity of water, is given in fig. 3.7. Calculations including the veloc-ity uo for We < 1 shows only a very small deviation of that region withoutvelocity.

From fig. 3.6 it follows that the lowest point of region j = 1, is approxi-mately at that radius Ro where 8, - 1 = 0 of the undamped system. Thetangent of the composite region will thus contact the separate regions at thesepoints. This will be a help in finding a simple criterion for instability as func-tion of the parameters.

- 54-

la'

5

~·3

~~M

V.v.v. //, ~~ 1//. '/.

I'//)Ç<'=104,f'l"//~I7/? ~

=0 V=O I' ~ W '" 3.10-410-5

~ ~1O-6

, WIlO-

O

2

5

lOO

5

2

2

5 5 10-4 5

- Ro(m)

Fig. 3.6. The regions ofshape instability with II = 4 at several viscosities; Pwo = 5.104 N/m2,

Uo = 0 mis, y = 1·41.

It follows from eqs (D.l) to (D.5) that the first-order approximation of #in region 1 is

(3.63)or

#2 = t eo e2,where eo is approximately unity and e2 « eo if A «1 and n »2.

Using eq. (3.62) we find instability if

(3.64)

4 (n + 2)(3 Y e 'jJ2)1/213-1> ,s; Pwo

(3.65)

where the value of Ro is taken at the eigen-frequency of the coordinate Yn'This radius is

(J

Ro = 4 (n + 2) (n2 - 1) .3 yPwo

(3.66)

- 55-

E-T

t 2 I--t-+-\--H-H-H-I---+-H-+H+l-l--H-+++++I

Too~I~~'l~~~~~~~tu~n~st~O~bk~~~t!~~~~~~~~~~2·1

5 'i:~-':.' 1%

- RO (f!lJ

Fig.3.7. Stable and unstable regions of an air bubble in water when viscosity is included;Pooo = 5.104 N/m2, IlO = 0 mIs, y = 1·41, v = 1.6.10-6 m2/s.

Because of its small contribution the velocity u is not incorporated in thisapproximation. From eqs (3.65) and (3.66) it is possible to find an equationat which pressure step the bubble will be unstable. This equation is

16 v3 ((3y)5 e3)1/2(8-1)3 >--

Ro2 a r.; (3.67)

This equation only holds with sufficient accuracy if

v3 «( R2 )1/2.a2 3ye3Pct:Jo

If we introduce a velocity v in eq. (3.67) and define the Euler and Reynoldsnumber respectively as e v2/Pct:Jo and v Ro/v we see that

(We EUl/2)

(8- 1)3>0 ,Re3

(3.68)

which is a relation between forces of momentum, viscous stresses, surface ten-sion and pressure forces.

- 56-

In fig. 3.7, eq. (3.67) is drawn. It is an elegant approximation to obtain anidea when bubble instability occurs after a sudden pressure step.

3.6. Conclusion

The shape of a gas bubble with translational motion is given by the coordi-nates R, an (n ~ 2). We have determined the equations of motion for an in-compressible non-viscous liquid and a gas bubble with an ideal gas. Althoughlinearized in an these equations are strongly coupled. The equilibrium state ofthis system has a bubble shape which deviates from the sphere. This shape isstable or unstable depending on the Weber number. This stability domain hasbeen calculated analytically by other authors for We < 2'34; it has beenfound experimentally for We< 1·59. Taking this into account it is obviousthat instability will not occur, as a result of the initial conditions, if the equa-tions of motion are used for We < 1. The error which will then be made isrelatively small.

Using the equations under this condition we have investigated the stabilityof the bubble shape if one of the parameters Ro, P 00o, e, Uo, (J, a, y is changed,particularly for the case of a sudden pressure rise of the liquid.

The velocity u could be expressed in Rand a2, the radius R in a Fourierseries with coefficients which are a function of the parameters. In this way itwas possible to give the equations of an the form of a Hill type. With thenormal procedures it was possible to calculate a stability domain and to repre-sent it in an (s=- 1)-Ro plane.

Including the liquid viscosity as a damping factor we could calculate a do-main of asymptotic stability and of instability. Under some circumstances thisboundary can be given by a simple equation.

From the foregoing we can draw several conclusions. First of all we cannever use linearized equations if We > 1 and even the second-order termswill not give any more information.

If we use the equations for We < 1 in the case of a pressure step, we canagain have a stable or unstable behaviour. Under practical circumstances thisstability will be asymptotic and a part of the unstable domain will be asymp-totically stable too, as a result of the liquid viscosity. This is why the oscillationof the perturbation terms will be damped out and tends to their equilibriumvalue. These perturbation terms are, however, rather small, because We < 1.We may therefore draw the conclusion that in a situation where asymptoticstability is expected, given by eq. (3.67), we can ignore the perturbation terms anand reckon with a spherical bubble. The equations to use in that case are (3.16)with a2 = a20 = 0 for the velocity u and (3.27) for the radius R. Because thevelocity does not play an important role in the behaviour of the radius R wecan ignore it and treat the bubble as a stationary one. All the other aspects

- 57-

such as viscous and thermal damping, diffusion and so on can then be incor-porated.

For the case that the pressure step is so high that instability will occur, theperturbation terms may not, of course, be neglected. These terms will acquirea significant value and may lead to such complex phenomena as bubble split-up. But the second- and higher-order terms have to be included to correct theresult of the calculation. Because of the short duration of this behaviour wemay neglect the viscous, thermal and diffusion effects.

From fig. 3.7 it appears that in the case of bubbles smaller than 10 (J.minwater high pressure steps are necessary to cause instability. To cause cavitationdamage the bubbles would have to be larger. In the next chapters we shallinvestigate this unstable behaviour in larger bubbles.

- 58-

Appendix D

In this appendix we shall be concerned mainly with the characteristic expo-nents ft1associated with solutions in the first, second and third unstable regionsof the Hill equation.For each region Hayashi (1964) gives expansions for ft and 80 as a function

of 821• Since we are concerned with the domain in the neighbourhood of theboundary !1-1 = 0 a closer evaluation of the exponent !1-1 will be necessary,when the stability condition of a higher-order approximation is desired. Weassume that the parameters 82, 84, (}6 ••• are sufficiently small. This meansthat (}2 < 2 and (}21 < 1 for i ~2, since these parameters are a function ofthe amplitude A, which is accurate enough up to e = 2.With these simplifications the value of the exponent !1-1 can be given by the

following equation:

(2 x 11.2)22 2 2 4 t-s 2 2

X2 !1-1 + Xl X3 - X4 + X/ - Xl X2 = 0, (D.l)

where Xl ••• X4 are functions of (}21. In the first unstable region thesefunctions are

Xl = t 82+ à (}2 (}4+ -ds82 842 + lïh 8/86- &s (}23 + iï 8486, (D.2)

x2 = (}2+ i8284 - 1!4 82842 + ï82 (}22 (}6- iï 823 + fz 84 86, (D.3)x3 = (}o- 1+ i(}22 + Nz 8/ 84 +~(}42 + to 862 + lri\ 82 84 86, (D.4)x4 = i8/ + Ii(}22 84• (D.5)

In the second one they are

Xl = i84 - to 822 + lh 82 86-.Jz (}22(}4' (D.6)

x2 = (}4- i822 + ~8286- à 822 (}4 - ïfu (}2 84 86, (D.7)X3 = 80- 4- ~822 + to 8/ + iö (}6

2- fz 8/ 84 - iö 828486, (D.8)

X4 = i2 842 + iï (}22 84 + 2~88284 (}6. (D.9)

In the third region they are approximated by

Xl = ~(}6- iï 82 (}4+ 3~4 823,X2 = (}6- i82 (}4+ i4 823,x3 = (}o- 9- to (}Z2- iö8/ + lii (}62 + räo 8/84,

X4 = .Jz8/.

(D.IO)

(D.ll)(D.12)(D.13)

For an undamped system the condition !1-/ > 0 holds for the unstableregions. !1-1 vanishes at the boundary of this region. Consequently it followsfrom (D.l) that the boundary is given by the condition

(D.14)

.--~----------,.----~~-- - -~- -. ~

- 59-

4. DYNAMICS OF A BUBBLE IN A QUIESCENT LIQUID SUBJECTTO A STEP CHANGE IN LIQUID PRESSURE

4.1. Introduction

The equations of motion given by eqs (2.64) to (2.67) are linearized withrespect to the perturbation terms an'As we have seen in chapter 3, a thresholdpressure step exists above which the bubble becomes unstable. In other words,the perturbation terms will grow to a rather large value, in which case thesecond- and higher-order terms become important. As a result of these higher-order terms the equations will be extremely coupled. Under these circumstancesit will be impossible to obtain an analytical solution. The only alternative is totry to arrive at the approximate solution by means of a numerical method.

In this chapter we shall investigate the influence of the translational velocityon the shape of the bubble when it becomes unstable. We shall see that theformation of a liquid jet will be the result of this instability which in turn givesrise to rather high liquid velocities. For convenience all the results quoted wereobtained for an air bubble with a radius Ro = 10-3 m in water with pressurePooo = 0·5 bar.

4.2. Numerical method

The equations (2.64)to (2.67) werenumerically integrated by the Runge-Kuttamethod using an EL-X-8 Philips digital computer. Since it is impossible to solvean infinite number of equations simultaneously, only the first eleven equationsfor the an were integrated. All other equations for n > 12 were regarded asnot playing an important role in the early history of the bubble. Therefore,together with eqs (2.64) and (2.65) a total of thirteen equations were pro-grammed for u, Rand a2 to a12•All terms of the equations may be non-dimensionalized through the fol-

lowing definitions:

dimensionless velocity (u*, R*, dn*)

length

Ro

velocity V .!,Po

dimensionless length (R*, an*)

dimensionless acceleration (u*, R*, an*)Ro

acceleration f2 - ,Po

(4.1)

dimensionless pressure (P!*, P*)pressure

Po

dimensionless time (t*),/ Po

= time V f2 Ro 2 '

- 60-

where Ro is the initial radius and Po the initial bubble gas pressure. The intro-duetion of these equations does not cause a change in the form of the equa-tions except the terms (P1- Poo)/e and 2a/e which change respectively into(R*3 - e*) and l/a*, where

Poo PoRoe* = - and a* = -- (4.2)

Po 2a! ,The initial conditions of the problem are the equilibrium state of the bubbleand a sudden pressure rise of the liquid, which can be illustrated in the fol-lowing way:

translational velocityradiusperturbation,

U* = uo*,R* = Ro* = 1,a2* =- i\We,an* = 0, n >2,_R* = dn* = 0,e*.

velocitiespressure step

For a pressure step in the unstable region the computations showed a limitationto the use of the linearized equations with respect to the accuracy of the per-turbation terms, although the Weber number in the beginning was smaller thanunity. Because the process is unstable the perturbation terms rapidly acquire arelatively high value with respect to the radius R. The order of magnitude ofthe terms was soon as follows:

ü* = 0(10),_R* = 0(10),ä2* = 0(10),än* = 0(1),

u* = 0(1),R* = 0(1),d2* = 0(1),dn* = 0(0'1),

R* = 0(1),a2* = 0(0'1),an* = 0(0'01).

In view of these results it seems necessary to take at least the second-orderterms of a2 into account. The method we follow is that the non-linear termsof a2 with the same order of magnitude as the linear terms are not neglected.As a result the next terms are taken into account in the equations of motion:

Ü (R3 + R2 an + R a22),R (R3 + R2 an+ R a22),än R3 + ä2 R2 a2,

U2 (R2 + Ran + a22), R2 (R2 + Ran + a22),UR (R2 + R an + a22), R (dn R2 + d2 R a2),U (dn R2 + d2 R a2), d22 R2.

The second-order terms of a2 are calculated with the method given in chapter 2.It is possible" of course, to do the same for a3 but then also the combinationof a2 and a3 has to be taken into account. The large amount of work necessaryto do this, does not give commensurate information, so we have limited our-selves to a2•The equations of motion we shall use in future are thus

(4.4)

- 61-

it (5 R2 - 9 R a2 + W a22)+ u R (15 R - 18 a2) + u d2(-9 R + ~\fa2)= 0,(4.3)

R (50R3 + 2R2 a2)+ 10ä2 R2 a2- u2Of R2_ 9 Ra2 +Wa22) +

+ R2 (45 R2 + a/) + 4 R d2R a2+ 7 d/ R2 +

+ U2(Ell! R2 - 3j!9 R a + 54 R a - fjJ a 2) - 45 Ud R2 +2 62 422 3

(4.5)

it (9R2 a2- 5R2 a4- 20Ra22) + 15RR2 a3+ ~ä'3R3 +

+ u2 (2ff R as - 1~4R a3)+ u R (30 R a2- 4 a22)+ Ud2 (15 R2 - 24 R a2) +

(4.6)

and for n;;;:: 4

. ( 3n 3n ) [9 n (n - 1)2u --Ran-1---Ran+1 +u2 an-2+

2n- 1 2n + 3 2 (2n- 3)(2n- 1)

9(n+ 1)2(n+2) ((n+6)(n+ 1)2 n(17n2-22h+9)) ]+ an - + a +

2(2n + 3)(2n + 5) +2 2(2n + 1)(2n + 3) 2 (2n + 1)(2n _ 1) n

(. 3(2n+l)) 2 .. 6.

+ u 6Ran_l +3Rdn_1- Rdn+1 +--R2an+--RRdn +2n+3 n+l n+l

[n + 3.. . r.-:»; ( n (n+ 1)) a ] '+ 4 an R R + !R2 - + 1+ - = 0; (4.7)

2 (n + 1) e 2 eR

- 62-

except for n = 5, eq. (4.7) will make an extra contribution of

4.3. Rayleigh's equation

Before we solve the complete set of equations, we shall first study eq. (4.4)determining the radius R, because this equation is part of every equation. Inthe case of a stable behaviour of the bubble shape we may neglect the per-turbation terms. Even in the unstable situation these terms will be small in thebeginning. If we assume u = 0 and no deformation eq. (4.4) turns out to beRayleigh's equation. For a stable bubble this equation will mainly determinethe bubble behaviour and for an unstable bubble it will determine the frequencyof the perturbation terms and the growth rate of these terms.

Many authors have studied Rayleigh's equation. Mention should be madehere of the work done by Lauterborn (1968) who compared the validity of aquasi-linear approximation with respect to the non-linear solution, which holdsfor small pressure steps. In chapter 3 we have given an approximate solutionfor pressure steps with e < 2. For higher pressure steps we have to follow anumerical method. We investigated Rayleigh's equation

.. . Pi- Pao 2 aRR + iR2- +- = 0, (4.8)e eR

for an air bubble in water with the initial conditions Po = 0·5 bar, Ro =10-3 m, a = 0·073 Njm, y = 1·41.In fig. 4.1 the behaviour of R*, R*, _R* is given for e* = 6. Because of the

non-linear character even the numerical method will give problems for highpressure steps (e* > 10), so this will be another limitation to the use of theequations of the bubble motion. The computations show some interestingresults concerning the frequency of the bubble radius. They are given in fig. 4.2.For e* = 1 the frequency is of course the same as given by Minnaert.

4.4. The unstable bubble shape

In this section we shall give some examples of the bubble shape after it hasbeen hit by a sudden pressure step. We can vary the initial conditions withrespect to the radius, pressure, pressure step, translational velocity and fluidconditions. Because our interest is the unstable behaviour of the bubble wehave to take a pressure step above the threshold pressure in the unstable region.For practical reasons we shalllimit ourselves to an air bubble of 1 mm radiusand pressure of 0·5 bar in water. The process will be taken adiabatically. Thevariables left are pressure step and velocity uo, while the initial bubble shapemight be the equilibrium shape according to Uo or another shape at t= O.

-----------------~-'-

- 63-

R*

t 0·5

00 t(f.Ls}50 100 150

4

R*

t2

0 t (ps)150

-2

-4~--------L---------~------~

o~50

/-00__ 150

I---' '-""

t(f.Ls}

150

R*

t 100

50

'0;:

-50

Fig. 4.1. History of R*, R* and ii* for an air bubble in water with pressure step e* = 6;Ro = 1 mm, y = 1,41, a = 0'073 NIm, Po = 5.104 N/m2•

4 6__ o-*_Pro

... _ Po

Fig. 4.2. Period time of oscillation at different pressure steps for an air bubble of 1 mm radiusin water with initial bubble pressure Po; Y = 1'41, (J = 0·073 Nim.

2 8 10

- 64-

1000

80

'" <,<,<,<,

<, Po=O·l bar

........ <,<. <,

I'-.

<:<, I'--- <,Po=0'5bar

<,I"--

<,Po~I'-........<, I'--- <,<, <,<, r-,<,

<,

i'-

800

200

100

.60

To obtain a better idea of the influence of the velocity uo, we shall take theequilibrium shape for a bubble with Uo = 0·20 mis. This shape is calculatedwith eq. (3.13), a third-order approximation for a2, a4 and a6' The processeswhich will be investigated are 8* = 2·0 with Uo = 0·20 mis, 8* = 2·0 withUo = 0·40 mis, 8* = 4·0 with Uo = 0·20 mis.With a numerical programme eqs (4.3) to (4.7) are integrated giving the

values of R and am n ,:;;;12, for t ~ O.These values are illustrated in at-an *plane. The bubble shape is calculated by

12rsCt)= R(t) + ~ an(t)Pn(Cos 0).

n=2

This shape is represented at various times by a plotter.The first example is given for 8* = 2·0 with Uo = 0·20mis. The history of

an* is drawn in fig. 4.3a. Starting with a2 *, a4* and a6* we see, because the

- 65-

equations are coupled by the velocity u, how the higher terms in an* are gener-ated by the lower ones.The unstable character of the process is clear. The perturbation terms show

an exponential growth. The higher terms in an have even a faster growth ifgenerated. This agrees with the analytical results, because the starting pointe - 1 = 1 results in a higher positive value of f-tm with increasing m. After alonger time (t > 600 (Ls) even the an terms for n > 12 will attain a significantvalue. The amplitude of the perturbation terms are getting so high that thereliability of the solution becomes dubious. The condition an < R does notsatisfy anymore.The history of the bubble shape is plotted in fig. 4.3b. As we see, for this

pressure step and velocity Uo the bubble remains approximately spherical uptill t = 600 (Ls. The unstable character of the bubble surface is then shown bya sharp involution at the rear. A liquid jet has started.In fig. 4.4a the history of an* is given for the same pressure step, but at a

higher velocity Uo. Since the coupling of the equations is much stronger now,the generation of the higher terms is also faster. Although the shape is nearlythe same, the liquid jet occurs much earlier. Having initial conditions causingbubble instability, the translational velocity Uo will hasten this process.The next example has a higher pressure step, e*= 4·0. Because this process

starts much higher in the unstable region than in the foregoing examples, it isclear that the unstable behaviour may be expected much earlier. We see fromfig. 4.5 that the bubble shape differs from that at the lower pressure step.

Comparing these instabilities with the instability mentioned by Taylor (1950),it seems that instability occurs for an expanding bubble. However, this doesnot agree with the theory of Piesset and Mitchell (1956). One needs to be care-ful in interpreting results, because the initial conditions and the process con-ditions can be so different that no uniform conclusion can be given. The onlyconclusions we will give are that if a translating bubble is subjected to a sud-den liquid-pressure rise which gives rise to instability, then(a) the character of this instability can result in an involution at the rear of

the bubble; a liquid jet is started;(b) this instability occurs earlier and more rapidly with increasing pressure

steps and velocity Uo;(c) in spite of the enormous change in shape the bubble radius R still corre-

sponds, within a few per cent, to the solution of the Rayleigh equation.

The history ofthe translational velocity u is given by the solution of eq. (4.3).For the analytical approximation of the instability problem we took the solu-tion given by eq. (3.16), and the solution of the modified Rayleigh equationfor R. Since we know that the radius R agrees with the original Rayleigh equa-tion to within a few per cent, the translational equation can be given to zero

Fig. 4.3a. History of R* and an* at 8* = 2·0 and IlO = 0·20 mIs for a translating air bubble inwater; y = 1'41, a = 0·073 NIm.

- 66-

a;at a*/ ~

a71

I If

/~ 7.: IN /a;

V \/ rA}; 1/I'-.....

f v: "'--v f17/ a1t---- -100"", 200~CX)/7 t\: .:~ N,\700 800

)<, V ~ ~\ ~ 1\

1/..__.. I~\

/ \\ I\aiil/ 1\

: \1\1\ l\/6aiO

at as

0-20

0·15a*nt 0·10

005

o

-005

-0,10

-0,15

-0-20

t(flS)

-----------~--~-~---- - -

t :U:

'OOps'40mls

t : 166'02psU: 'BBmIs

t : 332'03psU: '64mls

67 -

t: 55'34psU:. '5lm/s

t : 221'36ps,U: '49m/s

t : 387'37psU: 1'28m/s

t: IIO'6BpsU: t-Olrn/s

t : 276'69psU: '44m/s

t: 442'71psU: 'B2mls

Fig. 4.3b. Thetransientshapeofanairbubblemovingin water; Po = 5.104N/mz, Ro = 1 mm,e* = 2'0, az* = -0'118, a4* = 0'007.

o

5*2II/I

// /a7rI /

-: <, / / *_aIO-: "X- ~'/

/00"", '---200300 ~-

""'- _ 500"",,~a9_-, /A ~a8

3

~f...,_ _ _/ Va~ ~f-._ _.

/ '"i.> \~\\\

a*6

t(flS)

- 68-

0-2

0-20

a;

t 0-/5

0-/0

005

-005

-0-10

-0-15

-0-20

Fig. 4.4a. History of R* and an* at e* = 2·0 and IlO = 0-40 mIs for a translating air bubble inwater; y = 1'41, a = 0-073 NIm.

t:U:

·OOJlS·20m/s

t : 247·49JlsU: ·24m/s

t : 494·97JlsU: ·31m/s

-69 -

t: 82·50psU: ·34m/s

t :329·98psU: ·35m/s

t : 577·47psU: ·36m/s

t: 164·99JlsU: ·47m/s

t : 412·48JlsU: ·69mls

t: 659·97JlsU: ·77m/s

Fig. 4.4b. Thetransientshapeofanair bubblemovingin water; Po = 5.104 NJm2, Ro = 1mm,e* = 2·0, U2* = -0·118, u4* = 0·007.

-70-

T-OO

~ / <, ,

\ / \1\ / \\ / \\ / \R*

'-"

R*

t OSO

0.60

0-40

0-40

o

*,03

I/I

V~ II

/ <,"'// l\ O~

~/<, * l/~r-, 06

'50 A TOO~T50 ~jtar 250 3;10

/ \ ,1\~ \ / \

\ / \\ / \0;

/ \1"-./

t(ps)

0-30

o~

t 020

O-TO

-O-TO

-020

-0-30

Fig. 4.5a. History of R* and an* at e* = 4'0 and Uo = 0-20 mIs for a translating air bubble inwater; y = 1,41, a = 0·073 NIm.

t :U:

'OOps'20mls

t: 89'32psU: 2'48mls

t: 178'64psU: '37mls

-71-

t: 29'77psU: '25mls

t : 119'09psU: '65mls

t :208'41 psU: '55mls

t: 59'55psU: '60mls

t: 148'86psU: '39mls

t : 238'18 usU: l'67mls

Fig.4.5b. The transientshapeofan air bubble moving in water; Po = 5.104 N/m2, Ro = 1mm,

8* = 4'0, a2* = - 0·118, a4* = 0·007.

order in 8 by

-72-

u(t) = Uo ( Ro )3,R(t)

(4.9)

where R(t) is the solution of the Rayleigh equation. Itmight be interesting tocompare this approximation with the actual velocity in the case of a highpressure step. In fig. 4.6 this comparison is given in a uJUo-tjTR plane, whereTR is the period time of oscillation according to the solution of Rayleigh'sequation (see fig. 4.2). The first conclusion is that the velocity reaches its max-imum if R is minimal. For 8* = 2, eq. (4.9) is accurate within a few per cent,but for the higher pressure step the actual maximum velocity is much higher.Since during the first collapse the bubble remains spherical, it might be sur-prising to get a higher velocity u, as the sum of kinetic and potential energiesmust be constant. At this higher velocity the kinetic energy (2.35) will have agreater contribution to the iMu2 term, so must be compensated for by areduction of the Ju term. This illustrates again the importance of the trans-lational velocity with respect to the deformation of the bubble, as the defor-mation would otherwise intensify the maximum velocity u.

15

uua

tla

II,1, 1

1

I-----,k---- --- according to eqs (4·3) to (4·7)----- according to eq. (4·9)

I: \ \ / :51-----H-~+_---~~---+.~-------_,I.: \\ / !

I g*=4·0f-----H-I+1----+-----tuo=0·20m/s-

TR=165ps'

: \ I I

/1 ,.\ / I E!"=2·0l! \ I /" uo=O'40m/s

j/ / "\\ -, / ~/ '~=265JlS

°O~----~O.~5--------~1.0--------~I~.5~------~èO- t/TRayleigh

Fig. 4.6. History of u(/)/uo for an air bubble in water at different pressure steps and velocitiesuo; Ro = 1 mm, y = 1'41, a = 0·073 N/m, Po = 5.104N/m2•

-73 -

For small pressure steps within the stable region the perturbation terms anmay be neglected; a very good approximation for the translational velocity uis then given by eq. (4.9).

4.5. Liquid jet of a translating void

In the case of a cavitation bubble it is possible that with condensation or apressure rise in the liquid a constant pressure difference LIP exists betweenbubble and liquid. In this way, we have a different approach to the real situa-tion from that given in the preceding section, where condensation and diffusionwere neglected.The collapse of a single spherical cavity without translation, the problem

studied by Rayleigh, has an asymptotic solution valid near the end of the col-lapse. It is known, however, that a highly distorted shape may occur in a col-lapsing bubble at the slightest deviation from spherical symmetry. PIesset andMitchell (1956) gave a solution of a linear formulation for a collapsing cavitywith an initial distorted shape with a constant pressure difference. They foundthat the distortion amplitude grows as R-I/4 when the bubble radius R tendsto zero.A translational motion ofthe cavity, however, introduces a newphenomenon

to the collapsing cavity problem: the liquid jet. An elegant manner of approxi-mating this phenomenon is given in an unpublished paper by Eller (1967). Heassumed that, if a cavity begins the collapse as a sphere, no significant depar-tures from the spherical shape occur until after the collapsing cavity has enteredthe asymptotic solution of Rayleigh. In this manner he found the shape of thecollapsing bubble which shows the formation of a liquid jet at the rear of thecavity; this jet will strike the front of the cavity. Eller defined RI as an estimateof the radius at the time of jet impact and Uj as the velocity of the jet at thattime. In the liquid a very high pressure P rnax will occur at impact.

We define the cavitation number as

LIPC=---,

te ua2(4.10)

where LIP is the pressure difference between the liquid at infinity and the cavity.The results found by Eller at jet impact are then

r.;-= 1·18 C,LIP

UJ- = 2·16 C,Ua

R_a = 0.88 CI/3.

Rl(4.11)

For example if LIP = 105 N/m2, e = 103 kg/m" and Ua = 0·20mis the resultsare Pmax = 5·9. lOs N/m2, uJ= 2160 mis and Ra/Rl = 15.

-74-

The same problem can, of course, be solved numerically with the equationsof motion given by eqs (4.3) to (4.7). We calculated the same example for avoid with a radius Ro of I mm. The numerical results are: Uj = 2570 mis andRo/RI = 10·64.Comparing this with the results of Eller wemay conclude thateq. (4.11) is a close and elegant approximation. The numerical calculationsshowed indeed that the bubble remains spherical for a long time. In fig. 4.7

RFt =- 1·94 1-54 1·37 1·27 1·00

Fig. 4.7. The final collapse of a void with translational velocity IlO = 0·20m/so The radius RIat jet impact equals Ro/10'64; LIP = 105 N/m2, e = 103 kg/m",

we show the cavity during the final part of the collapse. The life time of thebubble is at that moment about 90 !Ls. The entire development of the jet afterthis is quite rapid. The time span is less than 1·5 (.Ls.In combination with thehigh pressure, the pressure pulse will take the form of a shock wave.

-75 -

5. COMPARISON BETWEEN EXPERIMENTAL ANDTHEORETICAL RESULTS

5.1. IDtroductionFor some years experiments have been performed in the hydrodynamics

laboratory at the Technical University in Eindhoven, on the response of abubble to a pressure pulse. The experimental set-up is in principle a verticallymounted air shock tube, the lower part of which is filled with water. Pressuresteps of approximately 4 ms duration can be generated in the water section.The interaction of the pressure pulse with an air bubble released earlier can beobserved and filmed through a special observation section. The first results ofthese experiments were reported by Smulders and Vossers (1968). The experi-mental results of a renewed set-up are now available and can be comparedwith those of the theory developed earlier.

In the theory, however, we assumed the pressure rise takes place in the wholeliquid at once. In the experiment this occurs by a pressure pulse travelling at thevelocity of sound through the liquid. Although this velocity is rather high it stilltakes a few microseconds to raise the pressure in the environment of the bubble.This effect will not lead to many complications, but as the pressure pulse passesthe bubble, the bubble meets with a pressure gradient. This results in a changeof the impulse. Therefore the influence of a pressure pulse on the velocity andshape of the bubble is studied. The different assumptions making it possibleto determine these changes after passage of a pressure pulse can be calculatedaccording to a solid body or a cavity in a compressible or incompressible liquid.Itappears that a good approximation can be found by assuming a non-distortingcavity in an incompressible liquid. This results in initial translational and sur-face velocities, different from those before the bubble is passed by the pressurepulse, and in a pressure rise in the whole liquid.

Prior to comparing the results of the experiments with the theory extendedfor a pressure pulse, we give a short description of the experiment to provethat the comparison is valid.

High-speed photographs, taken at different pressure steps will be presentedand compared with numerical results which have the same initial conditionsas used in the experiment.A more complete account of the observations and of the experiment will be

reported in the near future by the Technical University in Eindhoven.

5.2. The initial conditions of a bubble after passage of a pressure pulse

Up till now we were concerned with the equations of motion of a bubblefor a sudden pressure rise in the liquid. In practical circumstances there willmostly be a pressure gradient or a pressure pulse accompanied by a change in

-76-

velocity of the liquid. Because our interest is the relative motion of the bubbleto the liquid, it is easier to use a coordinate system moving at the same velocityas the liquid. The equation of motion in a moving coordinate system for anincompressible non-viscous liquid without external forces equals (seeKotschin (1954))

dvo 1-=-- \lP,dt e (5.1)

where Vo is the velocity of the centre 0' of the coordinate system of fig. 2.1.If the motion is uniform in the x-direction with the x component of Vo equal

to Vo we have

( Ovo)\l P + x e ---;;t = 0, (5.2)

with the solution throughout the fluid

P(x, t) = Pooo(x= 0) - e x voet). (5.3)

The effect of this fluid acceleration extending over the environment of a bubblewhose instantaneous volume is V may, of course, be interpreted as a force=e voet) V in the x-direction. Because this force is independent of x and x it issimilar to the buoyancy force e g V due to a hydrostatic pressure gradient e gin the x-direction. The potential energy of the bubble we have calculated ineq. (2.55) in a system with a fluid velocity equal to zero at infinity may nowbe extended in this moving coordinate system with this extra force in the x-direction, which results in

v xU=a(A-Ao)+Pooo(V- Vo)-fPtdV- f e~o Vdx. (5.4)

Vo x=oWith this potential energy in the Lagrangian (2.59), the equations of Lagrangeare now determinable. One of the Lagrangian equations (2.60) shows now that

d .- (M u + J) = e Vo V = F.dt (5.5)

Because the force F will cause a change of impulse of the system, the termM u + J is identified with the Kelvin impulse.

The set of equations of motion for a bubble in a pressure gradient is now

(5.6)

-77-

and for n ;;::::3, with al = 0 we have

. ( 3n 3n ) [9 n (n- 1)2u ---Ran-l- --Ran+l + u2 an-2+

2n- 1 2n+ 3 2 (2n- 3)(2n- 1)

9 (n + 1)2 (n+ 2) ( (n+ 6) (n+ 1)2+ 2 (2n+ 3) (2n+ 5) an+2- 2 (2n+ 1) (2n+ 3) +

n(17n2-22n+9)) ] (. 3(2n+l))+ an + U 6Ran_1 +3Ran_l- Ran+l +2 (2n- 1) (2n+ 1) 2n + 3

The history of U, R and anare found by integrating these set of equations.If the pressure gradient, i.e. the acceleration VA' is very great it will take

the form of a shock wave. The impulse of the system is then suddenly changed.Taylor (1942) solved the problem of motion of a spherical body in a compres-sible fluid when subjected to a sudden impulse I. This impulse makes the bodystart moving with a velocity equal to I/Me, M; being the mass of the sphere.This motion is immediately resisted by a pressure in the compressible fluid whichin its turn gives rise to a sound wave or pulse in the surrounding fluid. If weuse his results for an air bubblein water with g, ~ e we find the velocity of thesphere to be

with

I (1+ {Jexp( -2 (J c t/R))u(t) =- ,

M; 1+ {J

M{J=- u,

and c the sound velocity of the liquid. For t -- 00 the velocity is I/M if M;is neglected. For a bubble with a radius of 1 mm, the velocity change deter-mined by (5.10) takes place in less than 10-8 s.

A pressure pulse passing a bubble changes the impulse in a time dependingon the dimension of the bubble and of the sound velocity of the liquid. Thistime of passing is

(5.10)

(5.11)

-78 -

(5.12)

On the other hand the time necessary to change the shape ofthe cavity dependson the period time of oscillation and is found to be

Thus

(5.13)

_!_ = 0 (~V~).T e (}

For an air bubble in water at a pressure of 1 bar eq. (5.14) equals 0(10-2).These facts will allow for the construction of a composite approximation forthe initial values of a bubble hit by a pressure pulse. Because the velocitychange occurs on a time scale of much smaller order than the time scaleappropriate to distortion effects, we may suggest that during the time of pas-sing ofthe pressure pulse over the bubble, the bubble shape will not be changed.This statement translated to our problem for an incompressible fluid is

(a) the pressure pulse will raise the pressure of the whole liquid in the timebetween - e and +e;

(b) the whole liquid will get in the same time scale a velocity change accordingto the acoustic theory;

(c) during this time the shape and position of the bubble is not changed; onlythe surface velocity can alter.

Points (a) and (b) can be written for a pulse with a pressure L1P moving innegative x-direction (as found in an experiment discussed later, the directionis negative) by the following equations:

(5.14)

Pro = Proo + L1P H(t),L1P

L1vo=-,(le

Vo = -L1vo H(t),

where use is made of the Heaviside function H(t), which includes

(5.15)

(5.16)

(5.17)

(5.18)with

c5(t) = { 0,00,

t*Ot=O

and

J c5(t) dt = 1.-.

3n(n+I)LlVo( 1 an-l 1 an+1)1- ~a2/R 2n-I R- 2n+ 3R .

Thus for a sphere it results only in another translational velocity, but for adisturbed bubble also the perturbation terms will get an initial velocity.

(5.23)

-79 -

Substituting eq. (5.18) in the equations of motion (5.6) to (5.9) and integrationbetween - s and e results in the following:

2 Llvou(e)- u(-e) =- ,

I-~a2/R

R(e)- R(-e) = 0,

(5.19)

(5.20)

(3n 3n ) 2[u(e)-u(-e)] --Ran_1---Ran+1 +__ R2 [àn(e)-ànC-e)]

~-I ~+3 n+I= O. (5.21)

For the time t > e the terms Vo R x and Vo an x change into the pressureterms -(LIP/e) R and -(LIP/e) an and can be included in the left-hand sideof the equations (5.7) to (5.9), thus leading again to eqs (2.64) to (2.67); butduring passage of the pressure pulse the surface velocities of the bubble willchange relative to the liquid in the following way:

2 LlvoLluo =- 1 9 /R' (5.22)

- i)a2

5.3. Experimental set-up

The shock tube comprises a thick-walled steel tube (length: 11 m; innerdiameter: 77 mm; outer diameter: 125 mm). The high-pressure end is separatedfrom the low-pressure end by a Mylar diaphragm. The lower part of the low-pressure end is filled with water. See fig. 5.1. When the diaphragm is ruptureda shock wave travels down the tube and on reflection at the water surfacefinally transmits a pressure pulse into the water section. The length of the high-pressure end, the low-pressure air end and the length of the water-filled partare determined in such a way that it is possible to achieve a pressure step ofapproximately 4 ms duration in the water section just below the surface. Thelow pressure P 000 is always taken at 0·5 bar and not lower, to minimize theinfluence of the vapour in the gas section. For convenience, air is used in theexperiment. Therefore, the pulse pressure in the water is limited. The pressurestep e = P oo/P 000 will then be a maximum of 6. This step is measured by apressure pick-up. The rise time of the pressure step just below the water sur-face is less than 4 (Ls.A display from the output of the pick-up for differentpulses is given in figs 5.2 and 5.3. The rise time of the pressure is probably

~ l:~~:t9000

o

- 80-

_-J----'-"High-Rressure air

_-t---,L""o,-"W,--R.ressureair

Camera

Electronicflash unit

Fig. 5.1. Experimental shock tube with bubble generator and camera.

less than the 4 !Lson the display, since the measurement of this time is restrictedby the dimensions of the pick-up and its response time. The pulse itself travelswith a velocity of approximately 1400 m/so Comparing this speed with thevelocity of sound in water of 1450mis, the result is slightly lower due to theinfluence of the elastic wall, dispersion, etc.

- 81-

Fig. 5.2. Response of the pressure pick-up in the shock-tube wall after passage of the shockwave for a pressure step s = 4·0. Scales: lower beam: horizontalSO f.Ls/div, vertical 0·5 bar/div ;upper beam: horizontal 20 f.Ls/div, vertica l O 5 bar/div. The horizontallines start at the timet = 0 and represent the pressure Pooo.

Fig. 5.3. Response of the pressure pick-up in the shock-tube wall after passage of the shockwave for a pressure step e = 5·2. Scales upper beam: horizontal 50 f.Ls/div, vertical 0·5 bar/div.

Although a constant pressure may be expected behind the pressure pulse wenotice an oscillation with a relatively low frequency and a composite of smalleroscillations with much higher frequencies. The cause of the main oscillation isa result of the interaction between the bubble oscillation on one hand and thepipe wall and the free water surface on the other hand. The higher-order fre-quencies are affected by the elastic pipe wall. Reflected waves from obstacles,i.e. bubble generator and the observation-section wall, also cause a pressurefluctuation in the liquid behind the pressure pulse. At a greater depth in theliquid the effect of all these fluctuations is so considerable that it results in a

- 82-

badly conditioned pressure step. This is why the observation section is situatedjust below the free water surface, in spite of the effect of this surface on thepressure during the oscillation of the bubble volume.

The experiments are carried out for an air bubble produced by a bubblegenerator at rates varying between 0·2 and 400 per second, and in a range ofdiameters from 0·5 to 3 mm. For convenience the radius used is always about1 mm (in the experimental results discussed later the radius will be 1·05 mm± 2%). The velocity of rise due to the buoyancy of an air bubble is a functionof the bubble radius and the liquid. Experiments of Haberman and Morton(1953) already showed different rise velocities in pure and impure water, i.e.distilled and tap water. Saffman (1956) gave an analytical approximation ofthe velocity of rise as a function of the bubble radius. However, in the experi-ments the tap water in the shock tube is mixed with NaN02 to prevent cor-rosion of the steel tube; so the velocity of rise in the tube was determinedexperimentally. The results are shown in fig. 5.4. In our case it is 0·16 mis fora bubble of 1·05 mm radius. Calculating the Weber number of these bubbleswith Uo = 0·16 mis, Ro = 1·05 mm, (}= 103 kg/m'' and a surface tension(J = 0·068 Nim we find We = 0·395. According to Saffman (1956) this willnot result in a spiral motion; in the experiment, however, this spiral motion

10

r-...........aI .............. ---I ................... _-_-I -----~--: I----- -_- _-__---

__! ~ ......- --_--- --I ""

~I // ./ / ""/..... A

///'/1//1/j/

/Ijl

1

2 3- Radius Ra (mm)

4

Fig. 5.4. The velocity of rise of spiralling bubbles.A: Experimental results shock tube.Dashed curves: experimental results by Haberman and Morton; a: pure water, b: impurewater.B: theoretical results by Saffman.

- 83-

was observed. This means the motion has an unstable character. Nevertheless,the velocity of rise and the bubble shape were always the same as far as itwas possible to determine. By this, the initial conditions of the air bubblebefore it is hit by the pressure pulse is always the same.The observation section provided in the pipe permits filming of the bubble.

The' following filming equipment was used:Ca) Dynafax camera (max. 26 000 fps), total of 200 frames;(b) Cranz-Chardin camera (max. 106 fps), total of 12 frames.One film from the Dynafax camera gives a good overall idea of the bubblebehaviour over at least 8 ms. With the Cranz-Chardin camera it is possibleto get more-detailed information on certain stages of the bubble response witha minimal attainable interval of 1 !,-S between successive frames.The operation sequence of the experiment is as follows.

(1) An air bubble is released and travels upwards. The time taken to pass twophotocells is measured and used to set various delays.

(2) Another bubble of identical dimensions is again released, and the signalfrom one of the photocells is used to trigger an electrical pulse to a wirethat bursts the diaphragm.

(3) The shock wave travels down the tube and passes a pressure pick-up inthe wall, triggering the flash or flashes for the photographs.

Several experiments are necessary to complete one series of photographswhen the Cranz-Chardin camera is used. This is why great attention is givento the reproducibility of all parameters, a typical parameter being the pressurepick-up, see figs 5.2 and 5.3.

5.4. Results of the experiment

The conditions pertaining to the results here published are the following:(a) the initial bubble radius R = 1·05 mm;(b) the velocity of rise of the bubble before passage of the pressure pulse is

0·16 m/s;(c) the initialliquid pressure is 0·5 bar;(d) the series of photographs to be discussed are taken for a pressure step

e = 4·0 and e = 5·2.The pressure steps registered by the pressure pick-up are shown in figs 5.2

and 5.3. The initial steps are e = 4·0 and e = 5·2. The pressure fluctuates asa result of several reflected pressure waves. The average pressure is somewhatlower. From figs 5.2 and 5.3 we measure e = 3·6 and e = 4·8 respectively.As the pressure pulse passes the bubble, the impulse changes, resulting in

another velocity of the bubble surface relative to the liquid as is showed insec. 5.2. Assuming a spherical bubble with a radius R = 1·05 mm with a risevelocity of 0·16 m/s, the velocity of the bubble relative to the liquid will beaccording to eq. (5.22) after passage of the pressure pulse:

- 84-

Uo = -0·065 mIsUo = -0,14 mIs

for e = 4,0,for e = 5'2.

Although the bubble has an initial movement in an upward direction, afterpassage of the pressure pulse the initial velocity of the bubble in the experimentshown in the photographs, moves in a downwards direction, with a velocitydependent on the pressure pulse. For interpreting the results this is of coursevery important. This velocity is up to now not measured.

Under the above-mentioned circumstances, a series of high-speed photo-graphs has been taken with the Cranz-Chardin camera to achieve a detailedstudy of changes in the bubble shape. In fig. 5.5 they are given for e = 4,0,ë = 3·6 and in fig. 5.6 for e = 5,2, 8 = 4·8. The time elapsed since the pas-sage of the pressure pulse is shown below the photographs, together with theseries number of the experiment and the flash number. Observing the bubbleshape immediately before and after passage of the pulse, we do not noticeany change in shape. The assumption made that the new initial velocities maybe calculated under the assumption that the bubble shape does not change, iscorrect. From these photographs we can measure the coordinates of the equi-librium shape of an air bubble in water with a velocity of rise of 0·16 mIs.Under the assumption that this shape is determined by the coordinates Ro,a20 and a40, we obtain the values a20 * = -0·030 and a40 * = 0·006. However,due to the small dimensions and difficulty in reading photographic details theaccuracy of this measurement is not so high.According to the theory we would expect for a We number of 0·359 as cal-

culated in sec. 5.3 a shape with a20 * = -0·082 and a40 * = 0·0024. This showsa rather bad agreement between theory and experiment. The effect of viscosityand gravity is apparent. Undoubtedly the terms with an odd number, such asa3' will probably also have a value unequal to zero.Studying the transient shape of the bubble in time we see that the bubble

shape is highly distorted after a few oscillations. The unstable character of thebubble is clear at these pressure pulses. During the implosion stage the bubbleshape seems to stabilize to a sphere, but in the expansion stage the distortionincreases. In the series for e = 5·2 for t >250 fLsthis leads even to bubbledivision. The assumption is valid that a jet strikes the front side of the bubble,that is at the left-hand side in the pictures, ejecting air particles along with it,which shows as small deformed bubbles on the front of the bubble.There are still some difficulties in reading the photographs taken with the

Cranz-Chardin camera, e.g. the following.(a) The rising bubble has a spiral movement. The velocity effected by the pas-

sage of the pressure pulse is vertically downwards. If both velocities are ofthe same order of magnitude, the resultant velocity is small and then itsdirection is doubtful (this is the case for e = 4·0).

- 85-

(b) Because it is necessary to make several experiments to complete a series,the velocity mentioned in point (a) has always a different direction. In addi-tion, the individual photographs are taken with different flashes underdifferent angles.

(c) Because they are shadow photographs it is impossible to see jet penetrationor bubble involution.

5.5. Numerical results

To compare the theoretical with the experimental results there is still oneimportant parameter we have not discussed. The parameter y, which deter-mines the thermodynamic behaviour of an oscillating bubble is taken equalto 1 for isothermal and to cp/cv for adiabatic behaviour. These thermal effectshave been studied by PIesset and Hsieh (1960) and PIesset (1964) for smalloscillations of a gas bubble. Their results are given in terms of the characteristiclengths Ro, the acoustic wavelength in the gas phase Ag, and ag/wRo, whereag = kg/egcp, w being the angular frequency of the wave, kg the thermal con-ductivity of the gas and eg Cp the specific heat of the gas. The thermodynamicbehaviour is summarized in table 1.

TABLE I

The thermodynamic behaviour of an oscillating gas bubble in a liquid

frequency range comparison of lengths thermodynamicbehaviour

very high Ag «ag/ wRo «s; isothermal

high ag/wRo < Ag< Ro adiabatic

moderately high ag/wRo < s,< Ag adiabatic

low n; < aofwRo < Ag isothermal

In the case of a bubble hit by a pressure step, the angular frequency is mainlydetermined by the frequency of Minnaert given by eq. (3.24). If we use thedata from table I for air bubbles in water oscillating at Minnaert's frequency,we can construct fig. 5.7 for small oscillations. Then the behaviour will be adia-batic for a bubble with 1 mm radius at a pressure of 0·5 bar. In the actualsituation ofthe experiment however, the oscillations are not small and the bubbleshape is extremely distorted. The thermodynamic behaviour will be a functionof the time and be different in the collapsing and expanding phase of the bubble.This is why several numerical calculations have been carried out for different yto prove which results apply best to the photographs, since y affects not onlythe frequency of oscillation, but also its amplitude. We found that for

t : - 10 f1.S117~- 1

t: 10 f1.S117 - 2

t: 26 f1.s119 - 3

- 86-

t: 59 f1.S118 - 6

t: 70 f1.s118 - 7

t: 93 f1.s118 - 9

t: 110 f1.s118 - 10

t: 134 f1.s120 - 4

&,

t : 154 f1.s120 - 6

Fig. 5.5. Photographs taken of an air bubble in water after it is hit by a pressure step ê = 4·0.The time elapsed since the passage of the shock wave is shown below the photographs, togetherwith the series number ofthe experiment and the flash number. Velocity ofrise 0·16 mis, radiusRo = 1 mm, pressure Pooo = 0·5 bar. After passage of the shock wave the relative velocity ofthe bubble is probably downwards, i.e. in the frames to the left (see also Hermans, Smuldersand Vossers, Rep. Techn. Univ, Eindhoven (1971»).

t: 174 fls120" 8

t: 215 fls125 " 1

t: 250 fls77 " 8

t: 300 fls78 " 8

87 -

t: 194 fls120 - 10

t: 227 fls126 " 3

t: 265flS77 " 11

t: 320 fls78 " 12

Fig. 5.5 (continued).

t: 204 fls120 " 11

t: 240 fls126 " 5

t: 285 fls78 " 3

t: 345 fls79 " 5

t: 5 iJ-s57 - 3

t: 25 iJ-s57 - 4

t: 45 iJ-s57 - 5

- 88-

•t: 60 iJ-s61 - 1

t: 65 iJ-s57 - 6

t : 75 iJ-s61 - 4

t: 85 iJ-s57 - 7

t: 95 iJ-s61 - 8

t: 110 iJ-s61 - 11

Fig. 5.6. Photographs taken of an air bubble in water after it is hit by a pressure step 6 = 5·2.The time elapsed since the passage of the shock wave is shown below the photographs, togetherwith the series number ofthe experiment and the flash number. Velocity ofrise O·J 6 mis, radiusRo = 1 mm, pressure P 000 = 0·5 bar. After passage of the shock wave the relative velocity ofthe bubble is probably downwards, i.e. in the frames to the left.

t: 125 [Ls

SI - 4

t: 165 [Ls

SI - 8

t : 195 [Ls

SI - 11

t: 340 [Ls

60 - 10

- 89-

t: 135 [Ls

51 - 5

t: 175 [Ls

51 - 9

t : 210 [Ls

62 - 5

t: 370 [Ls

60 - 11

Fig. 5.6 (continued).

t: 155 [Ls

51 - 7

t: 185 [Ls

51 - 10

t: 235 [Ls

62 - 10

- 90-

3-0

1·0

\

1\ 1\Adiabatic

\ Transition range \\ \

1\ 1\

~th"ï'\ \

Uo = 0·16mis,Ro = 1·05mm,a2 * = -0·030,a4* = 0·006,p 000 = 0·50 bar,G = 0·068 Njrn,£! = 1000 kg/m".

2·5

1·5

(}5

10-5 10-4 10-3 10-2-

- Radius Ra (m)

Fig. 5.7. Thermodynamic behaviour of an oscillating air bubble in water at an angular fre-quency w = (1IRo) (3Pa>ole)1/2.

1·30< y < 1·41 this parameter did not affect the numerical results noticeablyenough to make comparisons with the photographs. This is why we approxi-mate the thermodynamic behaviour by y = 1·35. Now it is possible to cal-culate the history of the air bubble in the experiment. We will do this for thetwo series from sec. 5.4 for e = 4·0 and 5·2.

Before the bubble is hit by the pressure step the initial conditions are always

- 91-

At t = 0 the bubble is hit by a pressure pulse. This changes the velocities,which may be calculated by eqs (5.22) and (5.23) with the velocity of soundin water taken to be c = 1400 mis. For t > 0 we use the equations of motion(4.3) to (4.7) with the new initial conditions and an average pressure step ofrespectively ë = 3·6 and ë = 4·8. The initial condition and parameters of thetwo examples are compiled in table 11.We neglect the small effects of bubblesurface tension in our calculations of the pressure step and therefore use theaverage pressure step e* as ë.

TABLE 11

Initial conditions and parameters for the numerical programme to comparethe theoretical with the experimental results

fig. 5.8 fig. 5.9

e* = 3·60 e* = 4·80Uo =-0·046 mis Uo =-0·129 misRo = 1·05mm Ro = 1·05mmp 000 = 0·5 bar P 000 = 0·5 bara = 0·068 Nim (J = 0·068Nime = 103 kg/m" e = 103 kg/m"a2* =-0·030 a * =-0·0302

a4* = 0·006 a * = 0·0064

. * =-0·0042 . * =-0·0059a3 a3

. * = 0·0008 . * = 0·0012as asy = 1·35 Y = 1·35

The results of the transient shapes are given in figs 5.8 and 5.9. Comparingthese results with the photographs of figs 5.5 and 5.6, we see very close con-formity to the pressure step e = 5·2. We shall first discuss the result fore = 5·2.The shapes are of the same configuration: a spherical cap at the front (on

the left-hand side in the picture), and a flattening at the rear. Because thisshape clearly indicates direction of travel, we can see that after the pulse thebubble motion in the experiment is indeed in the downward direction. Theduration of the collapse in the experiment is shorter than the theory predicts.This could be corrected by taking e* = 5·2, but then the radius would be-come too small. Varying the exponent y did not affect this non-conformation.Damping could be the main reason although it is a second-order effect.

t : 'OOpsU: '05m/s

t : 49'50psU: '08m/s

t : 98'99psU: '39m/s

-92 -

t: 16'50psU: '05m/s

t : 56'OOpsU: '13m/s

t : 175'49psU: -Iêrn/s

t : 33'OOpsU: '05m/s

't : 82'50psU: '30m/s

t : 131'99psU: '10m/s

Fig. 5.8. The transient shape of an air bubble hit by a pressure step B = 3·6. The initial condi-tions are: Ro = 1·05mm, Q2* = -0,030, Q4* = 0,006, IlO = -0,046 tols, á3* = -0'0042,ás* = 0'0008, 'Y = 1·35. The shapes are comparable with the photographs of fig. 5.5.

t: 148·49psU: ·07m/s

t : 197·99psU: ·06m/s

t : 247·49psU: ·12m/s

-93 -

t : 164·99psU: ·06m/s

t :214·49psU: ·06m/s

t :263·99psU: ·24-m/s

Fig. 5.8 (continued).

t: 181·49psU: ·05m/s

t: 230·99psU: ·OBm/s

t: 2BO·49psU: ·30m/s

t: 'OO/1SU: '13m/s

t : 40'50/1sU: '21m/s

T: 81'00/1sU: 1'91m/s

- 94-

t: 13'50/1sU: '14m/s

t : 54'00/1sU: '36m/s

t: 94'50/1sU: '61m/s

t : 27'00/1sU: '16m/s

t: 67'50/1sU: 1'04m/s

t :107'99/1sU: '31m/s

Fig. 5.9. The transient shape of an air bubble hit by a pressure step B= 4·8. The initial condi-tions are: Ra = 1·05 mm, a2* = -0'030, a4* = 0'006, ua = -0·129 mIs, á3* = -0'0059,ás* = 0'0012, Y = 1·35. The shapes are comparable with the photographs of fig. 5.6.

t: 121'49{Jsu: '22m/s

t: 161'99{Jsu: '16m/s

- 95-

t: 134'99{Jsu: '18m/s

t: 17S'49{Jsu: '18m/s

t : 148'49{Jsu: '16m/s

t: 188'99{Jsu: '22m/s

t: 202'49{Jsu: '33m/s

t: 21S·99J.lsu: '66m/s

Fig. 5.9 (continued).

t: 229·49J.lsu: '98m/s

-96 -

For t >200 !Lswe see a jet starting at the rear of the bubble and a con-striction in the middle, which divides the bubble into two toroidal shapes. Thisagrees with the photographs. We see that the theory is a big help here in inter-preting the experimental results and reverse.The calculated transient shape of the bubble for the pressure step e = 4·0

does not agree completely with the experiment. It is too symmetric. This isprobably due to the low initial velocity of -0·046 mis. The measured risevelocity and the calculated velocity correction due to the pressure pulse arenearly the same, which results in a very low effective velocity. If the velocitywere a few centimetres per second higher, then the shapes would be similarto those in the photographs. Nevertheless we may conclude that the theory isconfirmed by the experiment.The preceding theory can also be used to explain a series of photographs

presented by Benjamin and Ellis (1966). Their experimental set-up was a re-inforced Perspex box partially filled with water at a pressure of about 0·04 atm.A nucleus with a radius of about 0·1 mm could be formed. When this nucleushad reached the centre of the volume of water, the box was struck downwards.As a result of this blow, the sudden downwards acceleration of the box pro-duced a lower pressure in the water causing the nucleus to grow to a cavityof 12 mm radius before it collapsed again. After the collapse in the expandingstage a jet was seen on one ofthe photographs. The vitalfactorforthis behaviourwas explained to be gravity, because in gravity-free conditions, which were alsopossible, the jet was not seen.If we study the various stages in the experiment we conclude that after an

expanding wave has passed the nucleus, this nucleus grows until a compres-sion wave passes. The velocity of rise of the cavity after passage of the pres-sure pulse will be about 0·10 mis because of the gravity and the time spanbetween expansion and the compression wave. This velocity, as a result of thegravity, before the collapse starts is, as explained and shown in chapters 4 and 5,the vital factor causing a liquid jet. When the initial translational velocity wasnot present, as when the experiment was conducted under gravity-free condi-tions, the pressure pulse still caused an unstable bubble. However, this instabil-ity would not cause a liquid jet at such an early stage.

-97 -

6. CONCLUSION

In this thesis we have investigated the behaviour of a bubble with translationalvelocity in a liquid. Special attention was paid to the situation in which a pres-sure pulse in the liquid caused an unstable behaviour of the bubble surface.The changes in bubble shape due to a pressure step, occur on a time scaleconsiderably smaller than the changes of a so-called "unstable movement" ofa translating bubble at equilibrium with a Weber number larger than about1·50. This is why zig-zag or spiral movement is neglected. On the other hand,velocities of a steady translating bubble, resulting from gravity and viscosity,are still important and therefore used.We found that a pressure pulse may cause a stable or an unstable bubble

surface; a threshold pressure as a function of R and the physical propertiesof the liquid is calculated for the onset of shape oscillations and given by eq.(3.67). For pressure steps below this threshold, behaviour is stable, resultingin a variation of bubble diameter but maintaining a spherical shape. Theproblem is reduced to the equation of Rayleigh with a solution given by eq.(3.27). If the bubble has a translational velocity then the change in diameterwill cause a change in velocity and be determined by the simple relation (4.9).Thus, for a pressure step in the liquid within the stability domain, the shape,

size and velocity of a bubble are completely determinable. This might bevery interesting, for instance, for bubbly mixtures in which it is possible tofind a relationship between bubble concentration and number with the helpof rather simple equations of motion for a single bubble.A more complicated situation occurs if the pressure step is so high that the

bubble motion will be unstable. In that case the bubble shape and its velocityare no longer determinable in a simple way, but have to be calculated numeri-cally. Although the bubble shape is unstable, the size (volume) does not deviatemuch from that calculated with the Rayleigh equation, but the maximum trans-lational velocity at minimum size of the bubble will be higher than in the caseof a stable behaviour. A very interesting result is that for a certain combinationof pressure step and initial translational velocity, the instability results in aliquid jet at high velocity. The resulting velocity after passage of a pressurepulse is one of the most important criteria that control whether or not a jet isformed. This criterion is not given, but from experiments reported in this thesisand from experimental results of Benjamin and Ellis (1966)it seems that alreadya velocity of 0·10mis and a pressure step, resulting in an unstable behaviourof the bubble surface, is sufficient to give a liquid jet.

- 98-

LIST OF FREQUENTLY USED SYMBOLS

ag coefficient of thermometrie conductivity kgleo cpan coordinate describing the cavity shape, eq. (2.7)an* dimensionless coordinate ani RoLlàn velocity change of an during passage of the shock wave, eq. (5.13)ano equilibrium value of am eq. (3.13)b vector used in eq. (2.27)b; coefficient in the expansion of the velocity potential

coefficient in the expansion of the bubble radius R, eq. (3.27)bnlll element of the matrix BbnR element of the vector bRbnu element of the vector b,bR vector of bb, vector of bC speed of sound in the liquidCn coefficient in the expansion of the velocity potential

coefficient in the expansion of the amplitude of the radius R, eq.(3.28)constant

cp specific heat of the gas at constant pressureCv specific heat of the gas at constant volumed; coefficient in the expansion of the angular frequency of oscillation,

eq. (3.29)i, unit vector in radial directionix unit vector in x-directioniy unit vector in y-directionio unit vector in tangential directionko heat transfer coefficient of the gask; coordinate describing a cavity shape, eq. (2.1)n unit normal vector at cavity surfacep vector describing the parametersq vector describing the coordinatesq, generalized velocities qo = U, ql = R., qn = ä; (n ~ 2)r spherical coordinate measured from centre of cavity

dimensionless radius of small radial oscillations, eq. (3.21)rs location of cavity surface, eq. (2.7)

timeu translational velocity'u* dimensionless translational velocityUj liquid velocity at jet impact, eq. (4.11)

-99 -

Un correction of the angular resonance frequency of the n-shape oscil-lation by the translational velocity Uo, eq. (3.36)

Uo translational velocity at equilibrium shapeLluo change in translational velocity during passage of the shock wave,

eq. (5.30)V velocity of liquid at a pointVc velocity of the cavity surfaceVd velocity of the cavity surface due to distortionVo velocity of the centre 0' or of the liquid at infinityLlvo velocity change of the liquid during passage of the shock wave, eq.

(5.17)Vr velocity in radial directionVo velocity in tangential directionx coordinate of the system with centre 0'zlx small displacement in x-directionXn coefficient in eq. (DJ)XZ

coordinate of the centre of mass of the cavityY coordinate of the system with centre 0'Yn coordinate describing the cavity shape, eq. (3.33)YnO equilibrium value of YnYnl first solution of Hill's equation, eq. (3.47)Yn2 second solution of Hill's equation, eq. (3.47)z variable twtA surface of cavityAo some reference surface of cavityAc surface of cavity, eq. (2.51)A(() surface of a sphere approaching infinityB general matrix, eq. (2.27)C cavitation number LlP/teuo 2

F function describing surface of cavity F = r- rsforce on cavity in x-direction, eq. (5.3)

I Kelvin impulseJ part of the kinetic energy TL LagrangianM added mass of a solid according to the translational velocityM; mass of the cavityMij generalized masso centre of the cavity0' centre of a coordinate system, fig. 2.1P pressure at a point in the liquidPc pressure in liquid at cavity surfaceP i pressure of gas in cavity

-100 -

P; Legendre polynomialPo equilibrium value of PIP 00 pressure in liquid at infinityP 000 value of P 00 at equilibrium stateR radius of cavityL1R small displacement of RL1R velocity change of radius during passage of the shock wave, eq.

(5.28)R* dimensionless radius R/RoRo equilibrium value of RR, radius at jet impact, eq. (4.11)T kinetic energy of the system, eq. (2.35)

period time of oscillationT' part of the kinetic energy TTR period time of oscillation of Rayleigh's equationU potential energyV volume of cavityVo some reference volume of cavityWe Weber number e U0

2 Ro/aYk a summation, eq. (2.29)YnJ an integral, eq. (2.30)Z function describing the cavity shape, eq. (2.20)a angle, eq. (C.l)

dimensionless quantity P 000 Ro/2adecay exponent, eq. (3.60)

a* dimensionless quantity Po Ro/2a{J dimensionless quantity M/Mcy parameter in the equation of state, eq. (3.19)t5 dimensionless quantity, eq. (3.25)e a small quantity

a pressure step P oo/P000

ë average pressure step in the experimente* a pressure step P oo/P0ç; cos ()() spherical coordinate()21 coefficients of Hill's equation, eq. (3.40)Ag acoustic wavelength in gasf-tnJ exponent of nth coordinate and the jth frequency of the solution

of Hill's equation'JI kinematic viscosity of liquide density of liquideo density of gas

- 101-

a surface tensionan angular resonance frequency of the n-shape oscillation, eq. (3.35)W angular frequency of oscillation, eq. (3.29)Wo angular resonance frequency of radial pulsations, eq. (3.24)4> velocity potential, eq. (2.34)4>1 part of the velocity potential 4>4>2 part of the velocity potential cpr dimensionless quantity, eq. (3.45)IJ dimensionless quantity, eq. (3.45)

-102 -

REFERENCESBenjamin, T. B., in Davies, R. (ed.), Cavitation in real liquids, Elsevier, Amsterdam,New York, 1964.

Benjamin, T. B., and Ellis, A. T., Phil. Trans. Roy. Soc. London A260, 221,1966.Binnie, A. M., Proc. Camb. phil. Soc. 49, 151, 1953.Birkhoff, G., Hydrodynamics, Princeton University Press, New Yersey, 1960.Crum, L. A., and Eller, A. I., Acoust. Res. Lab., Harvard Univ. Cambridge No. 61, 1969.Eller, A. I.Acoust. Phys. Lab., Univ. of Rochester, 1966.Eller, A. I., Acoust, Res. Lab., Harvard Univ., Cruft Lab. Cambridge, Massachusetts, 1967.Flynn, H. G., Acoust, Res. Lab., Harvard Univ., Tech. Mem. 38, 1957.Flynn, H. G., Phys. Acoust., Academic Press, New York, 1964, vol. IB.Gilmore, F. R., Hydrodyn. Lab., Calif. Inst. Tech., Report 26-4, 1952.Goldstein, H., Classical mechanics, Addison-Wesley Pub!. Comp. Inc., Massachusetts,

1950.Haberman, W. L., and Morton, R. K., David Taylor Model Basin, Rep. 802, 1953.Hartunian, R. A., and Sears, W. R., J. Fluid Mech. 3, 27, 1957.H ayashi, C., Nonlinear oscillations in physical systems, McGraw-HiIl, Inc., New York, 1964.Hermans, W. A. H. J., Smulders, P. T., and Vossers, G., Techn. Univ., Eindhoven,Rep. R-131-D, 1971.

Hsieh, D. Y., and PIesset, M. S., JASA 33,206, 1961.Ince, E. L., Ordinary differential equations, Dover Pub!., New York, 1956.Ivany, R. D., Hammitt, F. G., and Mitchell, T. M., Trans. ASME 88, 649, 1966.Knapp, R. T., Trans. ASME 77, 1045, 1955.Kornfeld, M., and Suvorov, L., J. app!. Phys. 15, 459, 1944.Kotschin, N. J., Kibel, I.A., and Rose, N. W., Theoretische Hydromechanik, Akademie-Verlag, Berlin, 1954, Band I.

Lamb, Sir H., Hydrodynamics, Dover Pub!., New York, 1932, 6th edition.Lauterborn, W., Acustica 20, 14, 1968.Lense, J., Kugelfunktionen, Geest und Portig, Leipzig, 1954.MacLachlan, N. W., Theory and application of Mathieu functions, Dover PubI., Inc.,New York, 1964.

Mellen, R. H., JASA 28, 447, 1956.Minnaert, M., Phi!. Mag. (Ser. 7) 16, 235, 1933.Parkin, R., Gilmore, F., and Brode, H. L., Rand. Corp. Mem., RM-2795-PR, 1961.Pars, L. A., A treatise on analytical dynamics, Heinemann, London, 1965.PIesset, M. S., J. app!. Phys, 25, 96, 1954.PIesset, M. S., in Davies, R. (ed.), Cavitation in realliquids, Elsevier, Amsterdam, NewYork, 1964.

PIesset, M. S., and Chapman, R. B., Univ. of Michigan, CoIl. of Eng., Rep. 85-49, 1970.PIesset, M. S., and Hsieh, D. Y., Phys. Fluids 3, 882, 1960.PIesset, M. S., and Mitchell, T. P., Quart. appl. Math. 13, 419, 1956.Ra t tr ay, M., Ph. D. Thesis, Calif. Inst. of Techn., 1951.Rayleigh, Lord, Phi!. Mag. 34, 94,1917.Saffman, P. G., J. Fluid Mech. 1, 249, 1956.Salie, J. Ia, and Lefschetz, S., Die Stabilitätstheorie von Ljapunov, Hochschultaschen-bücher, Mannheim, 1967, no. 194.

Smulders, P. T., Paper at Euromech. Coli. 7, Grenoble, 1968.Tayl o r, G. I., Scientific papers of Sir G. I. Taylor, Camb. Univ. Press., 1942, Vol. Ill,No.33.

Taylor, G. I., Proc. Roy. Soc. London, A201, 192, 1950.Yeh, H. C., Ph. D. Thesis, Univ. of Michigan, 1967.