Collapse and rebound of a gas bubble

Guillermo H. Goldsztein

School of Mathematics

Georgia Institute of TechnologyAtlanta, GA 30332-0160

Abstract

In this paper we study the collapse and rebound of a gas bubble. Ourgoals are twofold: 1) we want to stress that different mathematical modelsmay lead to extremely different results and 2) we introduce a new class ofsimplified reliable models. We accomplish our first goal by showing thatthe results obtained from two of the simplest and most widely used models(the isothermal and adiabatic approximations) are very different while thebubble is highly compressed. This period of time is short but it is of cru-cial importance in most phenomena where bubble collapses are relevant.To accomplish our second goal, we identify a non-dimensional parameterthat is a quantification of the strength of the bubble collapse and we showhow to use this large parameter to obtain new simplified models throughthe use of standard asymptotic techniques. Illustrative examples and dis-cussions on the wide range of applicability of the approach introduced inthis work are given.

Keywords: Fluid Mechanics, Bubble dynamics, Cavitation, Asymptotic Ex-pansions, Boundary Layers.

1 Introduction

In this paper we study the collapse and rebound of a gas bubble. To describe thephenomenon to which this expression makes reference to, consider the followingsituation. A gas bubble is immersed in a liquid. The pressure in the liquidbecomes and remains negative for enough time for the bubble to grow manytimes its equilibrium size. Once the bubble has expanded, the pressure in theliquid increases to a positive value. As a consequence, the bubble (if it stillexists) decreases in size violently, overshooting its equilibrium size and bouncingback (whenever it does not break). This process is usually referred to as thecollapse and rebound of the bubble.

Bubble collapses are frequently found in practice and easy to produce. Cav-itation damage, sonochemistry, shock wave lithotripsy, ultrasonic imaging andsonoluminescence are examples of technological and scientific importance wherebubble collapses play a central role.

1

Cavitation damage makes reference to the phenomenon that can be describedas follows. Strong pressure variations in the liquid are produced by the jet andthe shock emitted after the bubble collapses. The combined effect of severalbubbles undergoing this process is believed to cause erosion in nearby solids(see [4, 23, 53]). This is an undesirable effect.

During a strong collapse, bubbles emit light. This phenomenon is knowas sonoluminescence. This light emission is believed to be due to chemicalreactions induced by the extremely high temperatures attained inside the bubbleduring its compression. Multi-bubble sonoluminescence was observed over halfa century ago (see [11, 32, 4, 52, 53]), but it was the most recent discovery ofsingle bubble sonoluminescence that sparked a lot of interest within the scientificcommunity (see [2, 12, 16, 1, 3, 13, 5, 46]).

Chemical reactions result from the high temperatures and pressures in thebubble during the collapse. This phenomenon is known as sonochemistry andhas a wide variety of applications (see [27, 28, 49])

It is clear that any successful quantitative theory that describes any of thephenomena mentioned above, should be able predict the dynamics of the collapseand rebound of a bubble. A large number of different mathematical models tostudy the dynamics of single spherical bubbles can be found in the literature.These models range from simple ODEs (RayleighPlesset equations) to complexsystems of PDEs (NavierStokes equations). Some of the main contributions tothe field of bubble dynamics include [47, 33, 10, 34, 35, 36, 7, 8, 9, 38, 39, 40,41, 37, 45, 43].

Whenever we consider phenomena in which bubble collapses are relevant,we are faced with the dilemma of choosing an appropriate model. On one hand,we would like to use a model that can be easily implemented and producesresults easy to interpret. On the other hand, we would like to make sure thatno important physical effects is neglected by our model so that the results weobtain are reliable. In this paper we address some of these issues.

The goals of the present work are twofold. Our first goal is to stress thatdifferent models can lead to extremely different results. On the other hand,we note that the bubble collapse is a violent and fast motion. This violence ofthe collapse can be quantify with a large non-dimensional parameter that weidentify. Our second goal is to show that asymptotics on this parameter is areliable source of new simplified models.

To keep our analysis as simple as possible, we will assume that the liquidunder consideration is incompressible and inviscid, the gas inside the bubble isinviscid, the mach number inside the bubble and the effect of surface tensionare negligible and the system posses spherical symmetry.

Some of the physical effects neglected in this paper become important inthe short period of time while the bubble is compressed. For example thecompressibility of the liquid slows down the speed at which the bubble collapses.Thus, if our goal is to accurately predict the value of different variables suchas temperature or pressure during the collapse, the analysis to be presentedhere has to be extended to include some of the physical effects mentioned andpossibly others.

2

However, the exact values of the variables in the short period of time whilethe bubble is compressed is not always the goal pursued. An example is givenin 5, where the goal is to find the applied sound field that, while satisfyingcertain restrictions, maximizes the strength of the collapse. The model presentedhere leads to the correct result when applied to this example.

The objective of this work is to show how to obtained a dimensionless largeparameter that quantifies the strength of the collapse and how to use it toobtained reduced models. Thus, to keep the analysis as clear as possible wehave chosen to make the physical assumptions mentioned above. We remarkthat part of the value of this analysis is that it can be extended to includephysical effects neglected here (such as liquid compressibility). These extensionsare currently being developed.

We will study a system consisting of a single gas bubble immerse in a liquidthat extends to infinity. To model the bubble collapse and rebound, we willsimply study the evolution of this system when the initial radius R(0) is muchlarger than the radius at equilibrium conditions Re (R(0) Re), the initialvelocity of the bubble wall is zero (Rt(0) = 0) and the pressure in the liquidfar away from the bubble remains at all times equal to the equilibrium pressurepe. Under these conditions, the bubble behaves as follows. Since R(0) Re,the pressure inside the bubble at time t = 0 is smaller than the pressure in theliquid far away from the bubble pe. As a consequence, the bubble size starts todecrease. Since much inertia is gained, the bubble overshoots its equilibrium sizebecoming very small. Once enough pressure inside the bubble builds up due tocompression, its size stops decreasing and starts to increase. This motion occursin two different time scales. Most of the time the bubble is expanded and fora much shorter time the bubble is compressed. Figure 1 of 2 shows a typicalplot of R vs. t under the present conditions.

To describe the evolution of this system, it is necessary to determine thepressure inside the bubble. In this regard, the polytropic approximation hasbeen widely used by models in the literature. More precisely, the pressure insidethe bubble is assumed by these models to be proportional to R3, where isa constant. In particular, = 1 corresponds to the isothermal approximation(i.e. the temperature in the bubble is constant in space and time) and = ( being the ratio of the specific heats of the gas) corresponds to the adiabaticapproximation (i.e. the entropy in the bubble is constant along particle paths).

In 2 we will use both the isothermal and the adiabatic approximation inour modeling of the bubble collapse and rebound. We will show that both ap-proximations give very similar results while the bubble is expanded (which is formost of the time). In fact, the curves R vs. t obtained by both approximationsare indistinguishable to the naked eye when plotted in the time scale that thebubble takes to go from its maximum size to its minimum size (see Figure 1).This plot can mislead to wrong conclusions since, by looking at it, one could betempted to believe that both approximations give the same results and they areboth correct. However, sometimes we are particularly interested in the behaviorof the system during the short period of time that the bubble is compressed.For example, if we are modeling the phenomenon of sonoluminescence, we might

3

want to predict the temperatures attained inside the bubble. Thus, a look atFigure 1 is not enough to draw any conclusions. In fact, we will show in 2 thatthe results obtained from these two approximations are extremely different fora short period of time while the bubble is compressed (see Figure 2).

This discrepancy rises the following question: which one of these two approx-imations (if anyone) is giving us the correct results?. To answer this question,we note that the gas is isothermal while the bubble is expanded, but as thebubble collapses, this approximation eventually breaks down. In fact, the gasis adiabatic while the bubble is compressed. Thus, each approximation is validduring different periods of time but neither is valid at all times. This behaviorsuggests the use of the isothermal approximation while the bubble is expandedand the adiabatic approximation while the bubble is compressed. In 3, weprescribe an informal approach following these ideas. We say that this proce-dure of 3 is informal because we use an ad-hoc rule to decide when the motionceases to be isothermal and becomes adiabatic. However, the presentation ofthis analysis is justified since it is simple and it introduces some of the mainideas behind the more formal procedure that is presented later in this work.

In 4 we undertake a more formal analysis. Our starting point is the completeset of PDEs (conservation of mass, momentum and energy both in the liquidand the gas). As it has been recognized independently by two different groups(see [30, 31] and [43, 42]), under the conditions we consider in this work, thiscomplete set of equations can be reduced to a nonlinear PDE coupled with theRayleigh-Plesset equation. We non-dimensionalize these equations and observethat two parameters characterize our problem. One of them can be considered ameasure of the strength of the collapse and the other one is the non-dimensionalcoefficient of thermal diffusion inside the bubble. We profit from the fact thatthe first of these two parameters is large to simplify the system with the useof asymptotic techniques (boundary layers in time). As a result we split thetime domain in three. While the bubble is expanded, it behaves isothermallyand the dynamics of the bubble radius is not affected by the pressure in thegas. As the bubble becomes small, we need to introduce a boundary layer intime where the isothermal approximation is no longer valid. Eventually, oncethe bubble becomes small enough, the evolution of the bubble radius is affectedby the high pressures inside the bubble. Thus, a new boundary layer (inside theone we already had) develops. There, the gas is adiabatic.

The value of our analysis, becomes clear when we compare our approachwith computing the evolution of the system numerically without the use ofasymptotics. A first advantage of our approach is that its results are easier tointerpret. More precisely, the dependence of the solution as a function of thenon-dimensional parameters of the system (the strength of the collapse and thecoefficient of thermal diffusion in the bubble) is explicitly shown. This is notsurprising since the result of the use of asymptotic techniques is to reduce theoriginal system to simpler systems that do not contain the expansion parame-ters. An other advantage of our method is that it is computationally cheaper.Our simplified model requires the solution of two ODEs and the computation ofa constant. The computation of this constant requires the numerical solution of

4

a simple PDE. These equations are very inexpensive to solve and they do not de-pend on the parameters of our original system. On the other hand, performingnumerical computations on the original system becomes extremely expensive asthe collapse becomes stronger. (Two are the reasons for this increase in com-putational times. The first one is the presence of different time scales in theproblem. The difference in these time scales increases with the strength of thecollapse. The second reason is that the thickness of the boundary layer in spacedeveloped inside the bubble while it is compressed decreases as the strength ofthe collapse increases.) As an example supporting our discussion, we mentionthat the authors of [19] were faced with these difficulties to resolve the bubblecollapse numerically in their study of single bubble sonoluminescence. As a con-sequence, they had a limit on the size of the forcing sound amplitude that theycould used in their numerical computations. Our approach does not have theselimitations. In fact, the problem studied in [19] is a typical example where ourmethod can be used.

In 5 we show that our approach can be used in any situation involvingbubble collapses (not just in the examples presented in this paper). In thisregard, we mention that our asymptotic analysis could be a component of anumerical code that simulates the dynamics of a system composed of one orseveral bubbles in a liquid. This code would detect when a bubble is collapsing,use our asymptotic results and then continue the numerical calculation after therebound, avoiding the computationally expensive task of resolving the bubblecollapse numerically. In 5 we also present two illustrative examples. In thefirst example the system is initially in equilibrium. Suddenly, at time equal zerot = 0, the pressure in the liquid away from the bubble p becomes negativeand remains constant for a period of time [0, t0]. This period of time is longenough so that the bubble size at t = t0 is many times its equilibrium value.At this time t = t0, the pressure p becomes positive again. In our secondexample, the pressure away from the bubble p is not constant but is of theform p = pe(1+a sin(t)+b sin(2t+)), where pe is the equilibrium pressure.We use our method to find the constants a, b and that maximize the strength ofthe bubble collapse under the condition that the acoustic intensity and frequencyare constant (i.e. a2 + b2 and are constants). These two examples illustratethe general applicability and some of the advantages of our method.

Bubble collapses are sometimes desirable. In sonochemistry for example,chemical reactions are produced by the extremely high temperatures generatedin the interior and in the vicinity of bubbles when they collapse. In thesesituations, bubble collapses are induced by ultrasound. Clearly, it is desirableto be able to induce bubble collapses with low acoustic intensities. Efforts inthis direction have been undertaken (see [17, 20, 50] for example). In particular,the objective of the work in [20, 50] is to develop a sonodynamic approach tocancer therapy. These authors found experimentally that the acoustic intensityrequired to produce some chemical reactions can be much lower if the ultrasoundhas two frequency components instead of one. This work is the motivation ofour second example of 5.

To conclude the introduction we mention that while a large number of studies

5

on the dynamics of collapsing bubbles exists (including the works previouslymentioned and [24, 14, 21, 26, 43, 42, 48, 51, 25, 22, 29, 44, 18]), the analysisintroduced here is new and will lead to new simplified and reliable models.

2 The polytropic approximations

In this section we will use both the adiabatic and the isothermal approximationsand we will compare the results obtained.

Here and in the rest of this paper, we assume that the liquid is incompressibleand inviscid, the gas inside the bubble is ideal and inviscid, the effect of surfacetension is negligible, the mach number inside the bubble is negligible (i.e. thebubble wall speed is much smaller that the sound speed inside the bubble) andthe system posses spherical symmetry. Under these conditions, the evolutionof a gas bubble immerse in a liquid that extends to infinity is described by theRayleighPlesset equation

`

(R Rtt +

3

2R2t

)+ p(t) = p(t) (1)

where ` is the density of the liquid, R is the radius of the bubble, t is the time,p(t) is the pressure in the liquid at infinity and p(t) is the pressure at thebubble wall on the side of the gas. This equation was first obtained by Rayleigh(see [47]) and its derivation can be found in many books on bubble dynamics(see [4, 53]).

In this section we also assume that the gas is polytropic (i.e. the pressureis proportional to a power of the density), which, under the present conditions,implies

p(t) = cte R3 (2)

for some constant cte. Equation (2) reduces to the isothermal approximationwhen = 1 and to the adiabatic approximation when = ( being the ratioof the specific heats of the gas). If the gas is isothermal and the temperature isthe equilibrium temperature, the constant cte in equation (2) is given by pe R

3e

(where pe and Re are the pressure and bubble radius at equilibrium conditionsrespectively). If the gas is adiabatic, the constant cte depends on the totalamount of entropy inside the bubble. In other words, this constant dependson the initial conditions. Following most models in the literature, we choosecte = pe R

3e if = .

To model the bubble collapse and rebound, we assume the pressure far awayfrom the bubble to be equal to the equilibrium pressure, the initial radius to bemuch larger than the equilibrium radius and the initial bubble wall velocity tobe 0. Thus, we are interested in the solutions of the system

`

(R Rtt +

3

2R2t

)+ pe = pe

(ReR

)3R(0) = R0 and Rt(0) = 0 (3)

where R0 Re, for = 1 and = . In our calculations we have chosenR0 = 10 Re and = 1.4 (these values are commonly found in practice). Figure 1

6

0 6 12 180

5

10

PSfrag replacements

t/te

R/Re

Figure 1: Plots of R/Re vs. t/te obtained with both the isothermal and theadiabatic approximations. The two curves are indistinguishable.

16x10 36016x10360

0

5x10144

PSfrag replacements

(t ti)/te

R/ReR/Re

(t ta)/teR/Re 10

610 6

0

0.02

PSfrag replacements

(t ti)/teR/Re

(t ta)/te

R/Re

Figure 2: A closer look at the plots R/Re vs. t/te (with the origin of thehorizontal axis shifted) corresponding to the isothermal (left figure) and theadiabatic (right figure) approximations for times where the radius is close to itsminimum value. R attains its minimum at t = ti and t = ta when the isothermaland adiabatic approximations are used respectively.

shows the two curves R/Re vs. t/te obtained with the adiabatic and isothermalapproximations. The constant te is the natural time scale

te = Re

`pe

. (4)

These curves looked like one. They are indistinguishable to the naked eye.However, a closer look at these curves reveals that these approximations giveextremely different results for a short period of time while the radius is closeits minimum (see Figure 2). This disagreement applies not only to the bubbleradius R but also to the other variables in the problem (temperature, pressure,velocity, etc.).

A more precise quantification of the discrepancy between these approxima-tions can be easily obtained. In appendix A we show that the asymptoticbehavior of Rmin (the minimum of R) as , the ratio between the equilibrium

7

and the initial radius

=ReR0

, (5)

becomes small, is given by

Rmin Re

exp

(

1

3 3

)or Rmin

Re

( 1)1

3(1)

1

1 (6)

depending whether the isothermal or the adiabatic approximation is used re-spectively. Thus, evaluating these expressions at = 0.1 and = 1.4, we obtainthat Rmin 1.710

144Re and Rmin 6.8103Re in the isothermal and the

adiabatic case respectively.In the class of phenomena that motivates the present study (cavitation dam-

age, sonoluminescence, sonochemistry, etc.), we are particularly interested in thebehavior of the system while the bubble is compressed. In this regard, the calcu-lations of this section show that different models can lead to extremely differentresults.

3 Failure of the polytropic approximations. An

informal solution

The disagreement found between the isothermal and the adiabatic approxima-tions rises doubts on their validity. In this section we address this issue and weprescribe an informal procedure to describe the evolution of the bubble. Thepurpose of this section is to gain insight on the behavior of our system and tointroduce the main ideas of the more formal asymptotic analysis of 4.

3.1 Thermal diffusion time scale

We start by discussing the criteria of the validity of the isothermal and adiabaticapproximations. To this end, we need to consider the energy equation insidethe bubble, which, under our present conditions, takes the form

cv (Tt + v Tr) +

(r2 v

)r

r2p =

( r2 Tr

)r

r2, (7)

where is the density of the gas, T its temperature, v its velocity, its thermalconductivity, cv its specific heat at constant volume and r is the radial spatialvariable (i.e. the distance to the center of the bubble). We assume that bothcv and are constants. Making use of the estimates

r= O

(R1

)and = O

(g R

3e R

3), (8)

where g is the density of the gas at equilibrium conditions, we obtain (com-paring the first term of (7) with its right hand side) the thermal diffusion timescale

td =cv g R

3e

R. (9)

8

Thus, the gas is isothermal while any other time scale in the problem is muchlarger than td, and it behaves adiabatically for any period of time much shorterthat td. Note that the thermal diffusion time scale depends on the bubble sizetd = td(R).

3.2 An informal model for the bubble evolution

We now go back to our description of the bubble collapse and rebound (i.e. thesolution of (1) with p(t) = pe and initial conditions R(0) = R0 Re andRt(0) = 0). While the bubble is expanded, the pressure in the gas is smalland as a consequence, it does not affect the evolution of the bubble radius.Accordingly, p(t) can be initially neglected from equation (1). Thus, we replaceour system (1) by

`

(R Rtt +

3

2R2t

)+ pe = 0 R(0) = R0 and Rt(0) = 0. (10)

The solution R(t) of this equation decreases monotonically and it becomes 0 ata finite time tc. In fact, the asymptotic behavior of R as t approaches tc is givenby

R A Re t 25e (tc t)

25 with A =

(25

6

) 15

35 , (11)

where te and were introduced in (4) and (5) respectively. This asymptoticformula is due to Rayleigh (see [47]) and its derivation is given in the appendix B.

As the bubble decreases in size, the pressure in the gas increases. Thus,once the bubble is small enough, this built up in pressure affects the evolutionof our system preventing the bubble radius to become 0. As a consequence,the assumption under which equation (10) was obtained (p(t) being negligible)is no longer valid. This implies that, for a period of time while the bubble iscompressed, we can not neglect the pressure inside the bubble. To evaluatep(t), we should, in principle, integrate the energy equation (7) coupled with theRayleigh-Plesset equation (1). To avoid this and motivated by our knowledge ofthe qualitative behavior of our system, i.e. the motion is isothermal while thebubble is expanded and adiabatic while the bubble is compressed, we prescribethe following rule.

Rule: We use the isothermal approximation in the time interval [0, t] andthe adiabatic approximation in [t, tmin], where tmin denotes the time at whichR attains its minimum and t is determined next.

To select t, we argue as follows. We expect that p(t) affects the evolutionof our system for a period of time so small that the following assumption holds:

Assumption 1: The effect of p(t) on R(t) is negligible in the time interval[0, t].

Accordingly, we take R to be the solution of (10) in [0, t]. On the otherhand, we expect that the bubble decreases in size substantially before the

9

isothermal approximation fails and thus, we expect that the following assump-tion is also valid:

Assumption 2: Equation (11) is valid at t = t.Now the choice of t is natural, namely, since the characteristic time scale

of (10) is tc t, the isothermal approximation fails when tc t is not muchlarger than the thermal diffusion time scale td (see (9)), thus, we take t to bethe solution of

tc t = td(R(t)) (12)

where td is given by (9) and R is given by (11). Once we solve (12), we have

t = tc

(6

25

) 17

te 57

37 , (13)

where denotes the non-dimensional coefficient of thermal diffusion inside thebubble

=

cvgRe

`pe

. (14)

Having chosen t, it only remains to prescribe p(t) in the time interval[t, tmin]. According to our rule, the gas behaves adiabatically during this periodof time and thus, the pressure inside the bubble is given by

p(t) = C R(t)3 if t [t, tmin] (15)

for some constant C. To compute this constant C we simply require the pressurep(t) to be continuous at t = t. Thus, since our rule assumes the isothermalapproximation (p(t) = pe(Re/R)

3) for t < t and equation (15) for t > t, wehave

p = C R3 = pe

(ReR

)3at t = t, (16)

from where C can be easily computed once R is replaced by its asymptoticapproximation (11)

C =

(25

6

) 37 (1)

97 (1)

67 (1)pe R

3e . (17)

To conclude our informal description of the bubble collapse and rebound, weshould verify the assumptions 1 and 2. In this regard, we note that a typicalvalue of (corresponding to the liquid being water, the gas being air, theequilibrium radius, temperature and pressure being 10m, 300K and 1 atmrespectively) is 0.5. Thus, we assume that is a parameter of order 1, andas a consequence, since is small, we have that equation (13) implies thattc t tc (since tc = O(

1te)), which confirms the validity of assumptions 2.On the other hand, to verify assumption 1, we note that it is only necessary toconfirm this assumption at t = t which can be easily done using the asymptoticformula for R (equation (11)).

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3.3 Summary and characteristics of the informal model

In summary, according to the method presented in this section, the bubbleradius R is given by

R(t) =

{Ri(t) if 0 t tRa(t) if t t tmin

(18)

where t was defined in (13), tmin is the time at which Ra(t) attains its minimum,Ri(t) is the solution of (10) and Ra(t) is the solution of

`

(Ra Ratt +

3

2R2at

)+ pe = C R

3a , (19)

with C given by (17) and (since (11) is valid at t = t) the initial conditions tointegrate (19) are

Ra(t) = A Re t 25e (tc t)

25 and Rat(t) =

2

5A Re t

25e (tc t)

35 (20)

(where A is given in (11)). Note that given our conditions, we have R(2 tmin t) = R(t) for any 0 t tmin.

The non-dimensional parameters of our system are defined in (5), and thenon-dimensional thermal diffusion coefficient inside the bubble , which wasintroduced in (14). Figure 3 shows a plot of R/Re vs. t/te (R given by (18))with = 0.1, = 0.5 and = 1.4. As expected, the plot in Figure 3 (a) looksexactly like the one in Figure 1. The time range of the plot in Figure 3 (b) is[t, 2 tmin t]. During this period our present model assumes the adiabaticapproximation. The plot in Figure 3 (c) shows a closer look of the curve R/Revs. t/te around the time where the radius attains its minimum. These plotsclearly indicate the presence of three different time scales: 1) most of the timethe gas inside the bubble is isothermal and its pressure does not affect the radiusevolution (Figure 3 (a)); 2) the time interval [t, 2 tmint], where the isothermalapproximation fails (Figure 3 (b)); and 3) a shorter period of time where thepressure inside the bubble affects the dynamics of the radius (Figure 3 (c)).

To compare the results of this approach with the ones obtained in the pre-vious section, we remind that the values of the minimum radius are Rmin 1.7 10144Re and Rmin 6.8 10

3Re when the isothermal and adiabaticapproximations are used respectively (with = 0.1 and = 1.4). On the otherhand, we show in the appendix C that the asymptotic behavior of Rmin as 0according to the present model is

Rmin Re

(25

62

) 17 1

( 1)1

3(1)

1

137 . (21)

Thus, for the example under consideration ( = 0.1, = 1.4 and = 0.5), wehave Rmin 0.027Re.

We end this section with some comments. Although our rule to compute thepressure inside the bubble is not a formally correct approximation, our results

11

0 6 12 180

5

10

PSfrag replacements

t/te

R/Re

(a)

(t tmin)/te(b)

(t tmin)/te(c)

0.5 0.50

2

4

PSfrag replacements

t/te

R/Re

(a)

(t tmin)/te

(b)

(t tmin)/te(c)

0

0.06

14x10614x106

PSfrag replacements

t/teR/Re

(a)(t tmin)/te

(b)

(t tmin)/te

(c)

Figure 3: Plots R/Re vs. t/te corresponding to = 0.1, = 1.4 and = 0.5obtained by equation (18).

12

have the right order of magnitude (this will be confirmed later in this paper).Finally, we mention that two key characteristics of our system make possiblethe present analysis and its more formal version that will follow, namely, is asmall parameter and the transition from isothermal to adiabatic motion occurswhile the pressure inside the bubble p(t) does not affect the dynamics of theradius R(t).

4 Formal asymptotic analysis

In this section we undertake a more formal analysis of the bubble collapse andrebound. We will first present the system of equations that govern the evolu-tion of our system, then we will solve this system with the use of asymptotictechniques and finally we will discuss some of the characteristics of the solutionobtained.

4.1 Governing equations

We assume that the liquid under consideration is incompressible and inviscid,the gas inside the bubble is ideal and inviscid, the mach number inside thebubble is negligible (i.e. the bubble wall speed is much smaller than the soundspeed inside the bubble), the system posses spherical symmetry, effects dueto vaporization, condensation and surface tension, as well as variations in thetemperature of the liquid are negligible and the thermal diffusion coefficient andthe specific heats of the gas are constants.

We remind the reader that to model the bubble collapse and rebound, wesimply study the evolution of our system when the initial radius R(0) is muchlarger than the radius at equilibrium conditions Re (R(0) Re), the initialvelocity of the bubble wall is 0 (Rt(0) = 0) and the pressure in the liquid faraway from the bubble remains at all times equal to the equilibrium pressure pe.

Under these conditions, and introducing the change of variables

u =

g

(R

Re

)3and x =

r

R, (22)

the complete the complete set of equations governing the dynamics of our systemreduces to the Rayleigh-Plesset equation

`

(RRtt +

3

2R2t

)+ pe = pe

(ReR

)3u(x = 1) R(0) = R0, Rt(0) = 0 (23)

coupled with the nonlinear partial differential equation

ut =(x2f)x

x2f(x = 0) = f(x = 1) = 0, (24)

where f is given by

f =

[1

3

ut(x = 1)

u(x = 1)+

( 1)

RtR

]xu +

cv g R3eR

uxu

. (25)

13

The system (22-25) was obtained independently by two different groups(see [30, 31] and [43, 42]). For completeness, we include its derivation in theappendix D.

4.2 The asymptotic model

The fact that the initial bubble radius is much larger than the radius at equi-librium conditions (R0 Re), makes it possible to solve the system (23-25) bymeans of asymptotic techniques. While this analysis is one of the main resultsof this paper, it can be somewhat technical for the reader not familiar withasymptotic methods. Thus, we have chosen to display and discuss the resultshere, and describe their derivation in the appendix E.

The asymptotic behavior of the bubble radius as 0 is given by

R(t)

Re

{1Ri() if c and

107 +

52(1) c

K 1

(1) 37 Ra() if 0 and

107 5

2(1)(26)

where Ri satisfies

Ri Ri +3

2R2i + 1 = 0 Ri(0) = 1, Ri (0) = 0; (27)

Ra is the solution of

Ra Ra +3

2R2a = R

3a Ra(0) =

(25

6( 1)

) 13(1)

, Ra(0) = 0; (28)

the parameter was defined in (5); is related to t through

t = R0

`pe

; (29)

c is the time at which Ri becomes 0 (i.e. Ri(c) = 0); is given by

= a57

256(1) b

56(1)

57

107 +

52(1) + c; (30)

the constant b is defined in appendix E; is given by (14); we have defined

a =

(25

6

) 15

(31)

and the constant K is given by

K = a57

53(1) b

13(1)

27 . (32)

(We mention here that in the appendix E we have used a different notation.)Formula (26) describes R for t [0, tmin], where tmin is the time at which Rattains its minimum. Given our conditions, we have R(t) = R(2 tmin t) iftmin t 2 tmin.

14

0 6 12 180

5

10

PSfrag replacements

t/te

R/Re

t/teR/Re 9.146787 9.14679

0

0.015

0.03

Outer Solution

Inner SolutionPSfrag replacements

t/teR/Re

t/te

R/Re

Figure 4: Plots R/Re vs. t/te corresponding to = 0.1, = 1.4 and = 0.5.

Our analysis provides us not only with the asymptotic form of the bubbleradius R, but also with the asymptotic behavior of the other variables in theproblem (such as density, pressure and temperature of the gas, etc..). Theseexpressions can be obtained following our analysis in the appendix E. The con-stant b that appears in the above formulae, does not depend on . This constanthas to be computed numerically and, when the thermal diffusion coefficient in-side the bubble is constant, we obtained b = 0.30. If the thermal diffusioncoefficient of the gas depends on the temperature, the value of b changes, but itis always an order 1 constant independent of .

Figure 4 shows plots of R/Re vs. t/te (te was defined in (4) and R is givenby (26)) corresponding to the parameter values = 0.1, = 0.5 and = 1.4.We have plotted the inner and outer solutions. As expected the left plot looksexactly equal as the ones obtained in 2 with the polytropic approximations andin 3 with our informal model.

In agreement with our observations of 3, the analysis of appendix E showsthe presence of three different time scales in our problem. While the outersolution is valid (most of the time) the gas is isothermal and the bubble radiusevolution is not affected by the gas pressure. For a shorter period of time,once the bubble has decreased in size substantially, the gas inside the bubbleis neither isothermal nor adiabatic (the gas pressure does not affect the radiusdynamics during this period). Finally, for a much shorter period, while thebubble is highly compressed, the gas is adiabatic and the pressure inside thebubble does affect the evolution of the radius, preventing it to become 0.

To compare the results of our asymptotics with the ones obtained in theprevious sections, we remind that values of the minimum radius are Rmin 1.7 10144Re and Rmin 6.8 10

3Re when the isothermal and adiabaticapproximations are used respectively and Rmin 0.027Re when the informalapproach of 3 is used (these values are obtained when = 0.1, = 1.4 and = 0.5). On the other hand, we show in the appendix E that the asymptotic

15

behavior of Rmin as 0 according to the present model (equation (26)) is

Rmin Re

(25

62

) 17

(b

1

) 13(1)

1

137 . (33)

Thus, our asymptotic analysis gives Rmin 0.01Re.Note that our informal approach gave the right order of magnitude of Rmin

(compare equation (21) and (33)). Also note the explicit dependence of ourresults on the parameters of the system (i.e. and ). If we change our initialconditions or the fluids, we do not have to solve the equations (27) and (28)again (unless changes). Another convenient feature of our approach is that theresults obtained are easy to interpret, for example, formula (33) shows explicitlythe dependence of Rmin on and .

5 Applications, examples and extensions

So far we have only considered a simple example in which we have a bubblecollapse. In this section we will show that the analysis of 4 can be appliedto most situations where bubble collapses are relevant, and we will present twoillustrative examples.

5.1 General applicability of the asymptotic expansions

In any situation (not only in the example of 4), while the bubble size decreasesfast, the dynamics of the bubble radius is governed only by inertia (i.e. the firsttow terms of the RayleighPlesset equation (1)). Thus, during this period oftime, R satisfies

R A Re t 25e (tc t)

25 (34)

for some constant A. The dimensionless constant A is large and depends on thehistory of the bubble evolution. This parameter A is a measure of the strengthof the bubble collapse. In our example of 4, we have A = (25/6)1/53/5.

For the short period of time while the bubble is highly compressed, its dy-namics is independent of the pressure in the liquid at infinity. Thus, the asymp-totic behavior of R during this short period of time depends only on A and itcan be easily obtained from our analysis of 4 and appendix E. We just have toreplace (25/6)1/53/5 by A. As a result, we obtain

R(t) Re A57

53(1) b

13(1)

27 Ra() for || A

57+

256(1) (35)

where Ra is the solution of (28) and is now given by

t = A57

256(1) b

56(1)

57 te + tc. (36)

In this sense our analysis is very general. The constant A of equation (34)depends on the particular problem under consideration, but once A is obtained,the behavior of R while the bubble is highly compressed is given by equa-tions (35-36).

16

5.2 Slowly varying external pressure

We now illustrate our discussion with an example. More precisely, we willassume that the pressure in the liquid at infinity is not constant, but it is givenby

p(t) =

{pe p if 0 < t < t0pe otherwise,

(37)

where p > pe, t0 te, p/pe1 = O(1) and the system is at equilibrium initially(i.e. R(0) = Re and Rt(0) = 0).

This external pressure p becomes negative at t = 0 and remains negativefor a period of time t0 much longer than te. Thus, the bubble size will be manytimes its equilibrium value by the time p becomes positive again (t = t0) andas a consequence, the bubble will experience a collapse and rebound. In theappendix F we derive the asymptotic behavior of R for large values of t0/te,obtaining

R(t)

Re

1

2 (ppe)3 pe

if 1 and

1Ri() if 1 c and 107 +

52(1) c

K 1

(1) 37 Ra() if 0 and

107 5

2(1)

(38)

where Ri satisfies

Ri Ri +3

2R2i + 1 = 0 Ri(1) = Ri (1) =

2 (p pe)

3 pe; (39)

Ra is the solution of (28); the parameter is defined as

=tet0

; (40)

is related to t through

t = t0 ; (41)

c is the time at which Ri becomes 0 (i.e. Ri(c) = 0); is now given by

= k57

256(1) b

56(1)

57

107 +

52(1) + c (42)

(where b = 0.30 was defined in appendix E and in (14)); k denotes the constant

k =9 p 5 pe

8 pe

3

2

(p pe

pe

) 32

; (43)

and the constant K is now given by

K = k57

53(1) b

13(1)

27 . (44)

17

0 15 300

5

10

15

PSfrag replacements

t/te

R/Re

Figure 5: Plot R/Re vs. t/te where R is given by (38), = 0.5, = 0.05 and = 1.4.

(We mention here that in the appendix F we have used a different notation.)Figure 5 shows a plot of R/Re vs. t/te, where R is given by (38), = 0.05, = 1.4 and = 0.5.

In this example, the strength of the bubble collapse is A = k 3/5. In fact, itis easy to check that equations (35) and (38) give the same results. This agreeswith our previous discussion, namely, the calculation of the outer solution (andthus the constant A) depends on the particular problem under consideration,but the behavior of R in the boundary layer around its minimum is always givenby (34-36). We remark that this example also illustrates another feature of ouranalysis that applies in general, namely, our asymptotics gives us the explicitdependence of the solution on the properties of the fluids and on the externalpressure.

5.3 Two frequency forcing pressure

Bubble collapses can be easily produced by applying an acoustic field. Theexperiments in [20, 50] suggest that cavitation can be more efficiently inducedif the acoustic field has two frequencies instead of one. Motivated by theseexperiments, we will consider the following example. We now assume that thepressure in the liquid at infinity is given by

p(t) = pe (1 + a sin(t) + b sin(2t + )) . (45)

Our goal is to find a, b and that maximize the violence of the bubble collapses(and thus they also maximize the rate of chemical reactions that we want thecavitating bubbles to produce), given that the acoustic intensity and frequencyare constant (i.e. a2 + b2 and are constants).

The first step towards the solution of this problem is to state it mathemat-ically. This is naturally done thanks to our asymptotic analysis. Namely, thestrength of the bubble collapse (the constant A of equation (34)) is now a func-tion of a, b and . Thus, our problem is to maximize A = A(a, b, ) under

18

the restriction a2 + b2 = r2, with r and constant. For convenience, we haveparameterized a and b as a = r cos() and b = r sin(). Thus, our problem inhand reduces to find and that maximize A, i.e. solve the problem

Amax = max0pi2

A() where A() = max02pi

A(, ). (46)

For definiteness, we have used the values r = 1.5 pe and w = 0.05/te. Thecalculation of A() and Amax is done in appendix G. We found that Amax = 10.8and A is maximized at = 0.19 pi and = 0.4 pi. Figure 6 (a) shows a plot of Avs. . This plot clearly shows that the use of single frequency acoustic pressuresis not optimal. Figure 6 (b) shows a plot of R/Re vs. t/te when the externalpressure is the one that maximizes A. Figure 6 (c) shows a plot of this optimalpressure p/pe vs. t/te. This plot shows that the valleys of p are deep. Thisfeature of p is expected, since the bubble growths while p is negative andthus, the longer p remains negative and the more negative it is, the larger thebubble is before it collapses and consequently the stronger it will collapse.

6 Discussion

We have shown that different models lead to extremely different results duringa short but important period of time while the bubble is compressed.

We have quantified the strength of the bubble collapse by detecting a largenon-dimensional parameter (i.e. the constant A of equation (34)) that is presentwhenever we are in the presence of a bubble collapse. We have presented twoexamples (one in 4 and the other one in 5.2) where we were able to find Aanalytically, but in general, the evaluation of A may require numerical compu-tations. This was the case in our example of 5.3.

We have shown how to use this large parameter to simplify our systemof equations with the use of asymptotic techniques. This asymptotic analysisdivides the time domain in 3 regions. The outer solution (while the bubble isexpanded), depends on the particular problem under consideration. The othertwo regions in time corresponds to two nested boundary layers around the timewhere the bubble radius attains its minimum. There, the evolution of the bubbleradius is given by equations (34-36) and the specific conditions of the particularproblem under consideration enter only through the constant A.

We have discussed the general applicability and advantages of our methodand we have also presented some illustrative examples. Our analysis can beextended to include effects neglected here (such as liquid compressibility). Theseextensions and applications of our method to the study of different problemswhere bubble collapses are relevant will be pursued in the future.

19

0 15

11PSfrag replacements

A

2/pi

(a)

R/Ret/te(b)

p/pet/te(c)

60 1100

15

30PSfrag replacements

A2/pi

(a)

R/Re

t/te

(b)

p/pet/te(c)

0 3502

1

PSfrag replacements

A2/pi

(a)R/Re

t/te(b)

p/pe

t/te

(c)

Figure 6: Figure (a) shows a plot of A vs. (see equation (46)). Figure (b)shows a plot of R/Re vs. t/te when p is the one that maximizes A. Figure (b)shows a plot of this p/pe vs. t/te

20

A The isothermal and adiabatic approximations

In this appendix we derive the formulae in (6). To this end, we multiply equa-tion (3) by R2Rt and integrate once to obtain

`R3 R2t

2+ pe

R3

3= pe

R303

+ pe R3e log

(R

R0

)(47)

and

`R3R2t

2+pe

R3

3+

peR3e

3( 1)

(ReR

)3(1)= pe

R303

+peR

3e

3( 1)

(ReR0

)3(1)(48)

in the isothermal and adiabatic case respectively.At the time where R attains its minimum, we have R = Rmin and Rt =

0. Evaluating equation (47) at this time, neglecting the left hand side sinceRmin R0, and using the identity R0 = Re/, we immediately obtain the firstformula in (6).

Similarly, replacing R by Rmin and Rt by 0 in equation (48) and, keepingonly the third term of the left hand side and the first term of the right hand side(since the rest of the terms are negligible in comparison), we obtain the secondapproximation in (6).

B Collapse of an empty cavity

In this appendix we derived the formula (11). We first multiply equation (10)by R2Rt and integrate the result to obtain

`R3 R2t

2+ pe

R3

3= pe

R303

. (49)

Then we plug the anzats R = A Re t 25e (tc t)

into equation (49) and take thelimit t tc to conclude that = 2/5 and that A is given by (11).

C Asymptotics on the informal model

In this appendix we derive equation (21). To do so we multiply equation (19)by R2Rt, integrate the result and use the information from the initial condi-tions (20) to obtain

`R3R2t

2+ pe

R3

3+

C

3( 1)R3(1) `

R3(t)R2t (t)

2=

`3

R5et2e

3. (50)

In the derivation of this equation we have used the fact that both pe R3(t)

and C R3(1)(t) are much smaller than ` R3(t) R

2t (t). Finally, evaluating

this expression at R = Rmin and Rt = 0, and noticing that the first two termsof the left hand side are negligible when R attains its minimum, we obtainequation (21).

21

D Derivation of the system (23-25)

In this appendix we derive the system of equations (22-25). Under our assump-tions (stated in 4), the complete set of NavierStokes equations (both in theliquid and the gas) reduce to the RayleighPlesset equation

`

(R Rtt +

3

2R2t

)+ p(t) = p(t) (51)

(where p is the pressure in the liquid at infinity) coupled with the conservationequations inside the bubble (this coupling is through the pressure in the bubblep(t))

t +(r2 v)r

r2= 0 (52)

cv (Tt + v Tr) +

(r2 v

)r

r2p =

( r2 Tr

)r

r2. (53)

Equation (52) and (53) are the conservation laws of mass and energy respectivelyand we have not displayed the conservation of momentum since it reduces topr = 0 under the zero mach number approximation. Given that the gas is ideal,it satisfies the equation of state p = RT , where R = cp cv = ( 1)cv . Thissystem of equations is completed once we prescribe the boundary conditions

T (r = R) = T` Tr(r = 0) = 0 v(r = R) = Rt, (54)

where T` is the temperature of the liquid.To derive the equations (22-25), we make use of the equations of state and

mass conservation, and the fact p is spatially uniform to rewrite the energyequation as

pt +

(r2 v

)r

r2p = ( 1)

( r2 Tr

)r

r2. (55)

This equation can be integrated once to obtain the velocity as a function of thepressure and the temperature

v =( 1)

Trp

r pt3 p

. (56)

Using the equation of state and the boundary condition for the temperatureT (r = R) = T` we can express the temperature and pressure inside the bubbleas functions of the gas density

p = ( 1) cv T` (r = R) and T =(r = R)

T`. (57)

Thus, we can now write the velocity as a function of the density

v =

cv 2

[t(r = R) + Rt r(r = R)]

3 (r = R)r. (58)

Finally, plugging this expression for the velocity into the equation of conserva-tion of mass and making the change of variables (22), we obtain, after somealgebraic manipulations, the system (22-25).

22

E Derivation of the asymptotic model of 4.2

In this appendix we will reduce the set of equations (23-25) to its asymptoticapproximation. We will first introduce non-dimensional variables. As a conse-quence, the parameter will appear in the dimensionless equations. The factthat is small, will naturally lead us to an asymptotic analysis. More precisely,we will introduce two nested boundary layers and as a result we will distinguishthree regions in time. In a first region (where the outer solution is valid) thegas is isothermal and the evolution of the bubble radius is not affected by thepressure inside the bubble. A second region corresponds to a boundary layerin time where the isothermal approximation is no longer valid. And a thirdregion corresponds to an other boundary layer (inside the one where the gas isnot isothermal) where the pressure inside the bubble affects the evolution of thebubble radius. We now proceed with the details of the analysis.

E.1 Outer solution. Isothermal behavior

Given the initial conditions, the choice of the dimensionless variables is natural:

R = R0 R and t = R0

`pe

t. (59)

In these new variables, our system (23-25) becomes

R Rtt +3

2R2t + 1 =

3 R3 u(x = 1) R(0) = 1 and Rt = 0 (60)

and

ut =(x2f)x

x2f(x = 0) = f(x = 1) = 0 (61)

where f is given by

f =

[1

3

ut(x = 1)

u(x = 1)+

( 1)

RtR

]xu +

2R

uxu

. (62)

As previously mentioned, we will assume that is a parameter of order 1 and issmall. Thus, we can simplify our system (60-62) by keeping only the dominantterms. More precisely, we approximate R and u by R0 and u0 respectively,where these new variables satisfy

R0 R0tt +3

2R20t + 1 = 0 R0(0) = 1 and R0t = 0 (63)

and

0 =

(x2

u0xu0

)x

with u0x(x = 0) = u0x(x = 1) = 0. (64)

23

From this last equation we infer that u0 is a constant and, going back to therelation between u and the gas density (equation (22)), we immediately concludethat this constant should be 1

u u0 = 1. (65)

On the other hand, as already discussed in 3, R0 decreases monotonically, itbecomes 0 at a finite time tc and the asymptotic behavior of R0 as t approachestc is given by

R0 a (tc t)25 where a =

(256

) 15 . (66)

E.2 Boundary layer. Failure of the isothermal approxima-tion

As the bubble becomes small, the approximation (62) fails because some ne-glected effects become important. To take these effects into account, we needto introduce a boundary layer in time. Let be the boundary layer thickness( is yet to be determined). Then, given the equation (66), we introduce thechange of variables.

R = a 25 R and t = tc + t. (67)

Thus, plugging these expressions into equations (60-62), our system becomes

R Rtt +3

2R2

t+ a2

65 = a5 3 R3 u(x = 1) (68)

and

ut =(x2f)x

x2f(x = 0) = f(x = 1) = 0 (69)

where

f =

[1

3

ut(x = 1)

u(x = 1)+

( 1)

RtR

]xu +

a

2

75 R

uxu

. (70)

From these equations it is clear that the approximation for u (equation (65))is the first one to fail. This fact leads us to the natural choice of that makesthe coefficient multiplying R ux/u in the formula (70) an order one quantity

=

[2

a

]5/7. (71)

To solve (68), note that a is an order one constant and both and are smallparameters. Then, neglecting the small terms in equation (68) (i.e. we onlykeep the first two terms of the left hand side), solving for R and matching theresult with the outer solution, we obtain the asymptotic behavior of R (for smallvalues of )

R (t)2/5 (for negative t). (72)

24

This approximation fails as t 0. Thus, we will need to include a newboundary layer. To this end, we first need to study the behavior of u as t goesto 0. In this limit, the motion becomes adiabatic. This means that the pressurebecomes proportional to R3 and the entropy becomes constant along particlepaths. These facts and the asymptotic behavior of R (equation (72)) imply thatu satisfies

u(x = 1) b (t)6(1)

5 and u u0(x) (for x 6= 1 ) as t 0 (73)

for some constant b and some function u0(x). To compute the constant b and thefunction u0, we need to replace R by its asymptotic approximation (t)

2/5 inequation (69) and solve that equation with u subject to the asymptotic behavioru 1 as t . We have computed b numerically and we obtained

b = 0.30 (74)

(The details of the numerical code to compute b and u0(x) will not be discussedhere).

E.3 Thinner boundary layer. Adiabatic behavior

Equations (72) and (73) imply that the right hand side of (68) becomes ofthe same order of left hand side of that equation when t = O(5/(2(1))) andthus, the approximation used to obtained (72) fails for such small values of t.Consequently, we need to introduce a new boundary layer (inside the one wealready have). To do so we define

=(3 b a5

) 56 (1) (75)

and make the change of variables

R = 2/5 R , t = t (76)

and, for convenience, we introduce the function h(t) defined by

u(x = 1) = b 65 (1) h. (77)

The equations (73-75) now become

R Rtt +3

2R2t + a

2 ( )65 = R3 h (78)

ut =(x2f)x

x2f(x = 0) = f(x = 1) = 0 (79)

f =

[1

3

hth

+( 1)

RtR

]xu +

75

R

uxu

. (80)

25

Thus, matching the solutions in both boundary layers as t goes to andt goes to 0 and neglecting the small terms in (78-80) we obtain the followingapproximations for R and u

R R0 , u u0(x) and h R3(1)0 (81)

where R0 satisfies

R0 R0t t +3

2R20t = R

30 and R0 (t)

2/5 as t . (82)

Note that the asymptotic behavior of R0 does not uniquely determine R0. In or-der to implement our results, it is convenient to have R0 uniquely determine andhave initial conditions instead of its asymptotic behavior. Thus, we arbitrarilychoose R0t(0) = 0 and to determine R0(0), we note that

E(t) =R30 R

20t

2+

R3(1)0

3( 1)(83)

is constant and consequently E(0) = E() from where R0(0) can be obtained.In summary, R0 satisfies

R0 R0t t +3

2R20t = R

30 R0(0) =

(25

6(1)

) 13(1)

and R0t(0) = 0, (84)

which completes our asymptotic analysis.

F Asymptotics for the example of 5.2

In this appendix we show how to extend our asymptotic analysis to the casein which the external pressure is given by (37) and the initial conditions areR(0) = Re and Rt(0) = 0. The equations to solve now are:

`(RRtt +

32R

2t

)+ p(t) = pe

(ReR

)3u(x = 1);

R(0) = Re, Rt(0) = 0(85)

and equations (24-25). Our first step is to non-dimensionalize these equations.Given the external pressure, the natural choice of the dimensionless variables is

t = t0 t =te

t and R = t0

pe`

R =Re

R, (86)

where we have introduced the notation

=tet0

. (87)

Thus our system becomes

R Rtt +3

2R2t +

ppe

= 3u(x = 1)

R3R(0) = and Rt(0) = 0 (88)

26

and

ut =(x2f)x

x2f(x = 0) = f(x = 1) = 0 (89)

where f is given by

f =

[1

3

ut(x = 1)

u(x = 1)+

( 1)

RtR

]xu +

2uxu

. (90)

A simply asymptotic analysis with as the small expansion parameter, showsthat

R

2 (p pe)

3 pet and u 1 for t 1. (91)

To solve these equations for t > 1, we follow similar arguments of appendix E.We approximate R and u by R0 and u0 respectively, where these new variablessatisfy

R0 R0tt +3

2R20t + 1 = 0 R0(1) = R0t(1) =

2 (p pe)

3 pe(92)

and

0 =

(x2

u0xu0

)x

with u0x(x = 0) = u0x(x = 1) = 0. (93)

Equation (93) impliesu u0 = 1. (94)

On the other hand, R0 becomes 0 at a finite time t0 and the asymptotic behaviorof R0 as t approaches t0 is now given by

R0 k (t0 t)25 where k = 9 p5 pe8 pe

32

(ppe

pe

) 32

. (95)

To obtain k, we used the fact that

E(t) =R30 R

20t

2+

R303

(96)

is constant and thus, k can be obtained from the equation E(1) = E(t0). Notethat we are now in exactly the same situation as in the appendix E once wereplace by and the constant a of equation (66) by k (see equation (95)).Thus, the rest of the asymptotic analysis follows from that appendix.

G Asymptotics for the example of 5.3

In this appendix we complete the calculations of 5.3.

27

G.1 Positive slowly varying external pressure

If the external pressure p is positive and varies slowly (i.e. p = p(t) withte 1) and the initial radius R(0) is of the order of the equilibrium radius Re,the bubble will behave quasitalically after a transient time of the order te. Moreprecisely, the gas will be isothermal and the bubble radius will approximatelysatisfy

p(t) = pe

(ReR

)3(97)

G.2 Non-positive slowly varying external pressure

Suppose now that p is of the form p = p(t) with te 1 and thatp is positive for t satisfying 0 < t < t0 or t1 < t but it is negative for t inthe time interval (t0, t1). In this case, the bubble radius will satisfy (97) for0 < t < t0, but for larger values of t the bubble will grow. If we further assumethat (t1 t0) = 0(1) and that the minimum of p is of the order of pe, theevolution of R for t > t0 can be described follows.

We first introduce the dimensionless variables

= t = t

teand Ri =

R

Re. (98)

where = te. The RayleighPlesset equation (see (23)) now becomes

Ri Ri +3

2R2i + P () =

3 u(x = 1)

R3i(99)

where P () = p()/pe. Since 1, we neglect the right hand side of (99).To find the initial conditions of Ri at = 0 = t0, we need to match thebehavior of R for t < t0 (equation (97)) and t0 < t. Clearly, in the limit of smallvalues of , this matching leads to the initial conditions Ri(0) = Ri (0) = 0.In summary, in the limit 0, the asymptotic behavior of the bubble radiusis given by

Ri Ri +3

2R2i + P () = 0 and Ri(0) = Ri (0) = 0. (100)

for > 0.Since P becomes positive, Ri eventually becomes 0 at a finite time = c.

In fact, the behavior of Ri as c is given by

Ri B(c )25 (101)

for some constant B. In other words, if we define tc = c/, the asymptoticbehavior of R as t tc in the limit te 0 is

R A Re t 25e (tc t)

25 (102)

28

where the constants B and A of equations (101) and (102) respectively arerelated by

A = 3/5B. (103)

It is clear that this constant B depends only on P ().

G.3 Slowly varying two frequency external pressure

We are now ready specialize our analysis to the example of 5.3. Namely, weset P () = 1 + a sin() + b cos(2 + ), where a = r cos() and b = r sin(), wechoose r = 1.5 and = 0.05 and we now follow the procedure described aboveto compute A = A(, ) numerically.

References

[1] B. P. BARBER, R. A. HILLER, R. LOFSTEDT, S. J. PUTTERMAN andK. R. Weninger, Defining the unknowns of sonoluminescence, Phys. Rep.,281:66-143 (1997).

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