%CAV2001.
%Trimmed version for conference

\documentclass{cav2001}
\usepackage{epsfig}

\begin{document}
\papernum{A5.006}
\title{Fission of collapsing\\ cavitation bubbles}
\author{Christopher E. Brennen}
\address{California Institute of Technology, Pasadena, CA 91125}

\maketitle

\section*{Abstract}

High-speed observations (for example, Lauterborn and Bolle 1975, 
Tomita and Shima 1990, Frost and Sturtevant 1986) clearly show that 
though a collapsing cavitation bubble
approaches its minimum size as a coherent single volume, it usually 
reappears in the first
rebounding frame as a cloud of much smaller bubbles or as a highly 
distorted single volume (see, for example, figure \ref{F2}).
This paper explores two mechanisms that may be responsible for that 
bubble fission process,
one invoking a Rayleigh-Taylor stability analysis and the other 
utilizing the so-called
microjet mechanism. Both approaches are shown to lead to 
qualitatively similar values for the number of fission fragments and 
the paper
investigates the flow parameters that effect that number. Finally, we 
explore the effective damping of the Rayleigh-Plesset single bubble 
calculation which that fission process implies and show that it is 
consistent with the number of
collapses and rebounds which are observed to occur in experiments.



\section{Introduction}
\label{S0}

Rayleigh-Plesset calculations for cavitation bubbles are now commonly 
embedded in
efforts to computationally simulate cavitating flows. The implicit 
assumption is
that the bubble remains sufficiently spherical for this equation to adequately
represent its dynamic volumetric behavior. When the latter is 
compared with experimental
observations, there are many respects in which this approximation 
proves acceptable,
particularly during the growth of the bubble to its maximum size and 
the initial part
of the collapse phase. However, most high-speed observations of the 
collapse show
that the bubble fissions during passage through its minimum volume 
and, thereafter,
the Rayleigh-Plesset analysis fails to accurately predict the dynamic 
behavior. In part
this is because the Rayleigh-Plesset equation fails to represent the energy
dissipation associated with the fission process. As a consequence, the number
and strength of the rebounds observed in the experiments are much smaller than
predicted by the calculations.

\section{Collapse Relations}
\label{S1}

However, before investigating the fission process, it is necessary to 
establish the essential features of the growth and collapse of a
cavitating bubble as it passes through a low pressure region in the 
flow. For this purpose we use the approximate relations that are 
derived
from the classical Rayleigh-Plesset model and presented in Brennen (2001).
It is assumed that
thermal effects may be neglected, that the mass of non-condensable 
gas in the bubble
remains constant and that the behavior of that gas can be represented 
by a polytropic constant, $k$.
Then the Rayleigh-Plesset equation (see, for example, Brennen 1995) 
connecting the time-dependent bubble radius, $R(t)$, to the pressure 
in the liquid, $p_\infty(t)$, is
\begin{equation}
\frac{p_V  - p_\infty (t)}{\rho} + \frac{p_{Go}}{\rho}
\left\{ \frac{R_o}{R} \right\}^{3k} = \ R \ddot R + \frac{3}{2} (\dot R)^2
+ \frac{4 \mu_e \dot R}{\rho R} + \frac{2S}{\rho R}
\label{BE26} \end{equation}
where $p_V$ is the vapor pressure at the prevailing temperature, 
$p_{Go}$ is the partial
pressure of non-condensable gas in the initial cavitation nucleus,
$\rho$ and $S$ are the liquid density and the surface tension and the 
overdot denotes $d/dt$. For the moment, $\mu_e$, can be considered to 
be  the liquid viscosity.

For the purposes of the present investigation we consider a single 
cavitation nucleus of radius $R_o$ initially at equilibrium at the
initial liquid pressure $p_\infty^i$. It follows that the initial
partial pressure of gas is given by
$p_{Go} = p_\infty^i - p_V  + \frac{2S}{R_o}$.
This nucleus is then subjected to an episode in which the liquid
pressure is decreased below the vapor pressure, $p_V$, causing explosive
cavitation growth of the bubble to a radius much larger than $R_o$.
The ambient pressure eventually increases again, causing the bubble 
to collapse violently.

In Brennen (2001) we develop approximate analytical expressions
for some of the main features of the bubble dynamics.
The focus is on that very brief instant at the heart of the collapse
when the bubble radius becomes very much smaller than $R_o$.
We first define a typical duration for the reduced pressure interval, 
$t_R$, and a typical
tension, $p_V-p_\infty^m$, under which growth occurs so that
\begin{equation}
R_{max} \approx \left\{ \frac{2(p_V-p_\infty^m)t_R^2}{3 \rho} 
\right\}^\frac{1}{2}
\label{BE004A}
\end{equation}
During the initial part of the collapse the acceleration,
$\ddot R$ is negative. However later in the collapse this acceleration
changes sign as the noncondensable gas inside the bubble begins to be
compressed. Defining the origin of our time frame, $t=0$, to be the 
moment of minimum
bubble size, we shall refer to the moment at which $\ddot R=0$ as the beginning
of the rebound and denote it by $t=-t_*$. It is readily demonstrated 
that if the
ambient pressure at collapse is $p_\infty^c$ then the minimum bubble 
radius, $R_{min}$,
at $t=0$ is given by
\begin{equation}
\frac{R_{min}}{R_o} =  \left\{ \frac{2 R_o^3 p_{Go}}{3 \rho (k-1) K} 
\right\}^\frac{1}{3(k-1)}
\quad \mbox{where} \quad
K \approx 2 (p_\infty^c - p_V) R_{max}^3 / 3 \rho
\label{BE34} \end{equation}
Moreover, the radius, $R_*$, and radial velocity, $\dot{R_*}$, at the
beginning of the rebound are given by
\begin{equation}
R_*= (k)^\frac{1}{3(k-1)} R_{min}
\quad \mbox{and} \quad
\dot{R_*}= \left\{ \frac{(k-1)}{k} \frac{K}{R_*^3} \right\}^\frac{1}{2}
\label{BE35A}
\end{equation}
To parameterize these results, we define
a Thoma cavitation number, $\sigma$, which describes
how close the inlet pressure is to the vapor pressure and a
non-dimensional pressure distribution characteristic, $\alpha$, as
follows:
\begin{equation}
\sigma= (p_\infty^i-p_V)/(p_\infty^i-p_\infty^m) \quad; \quad
\alpha= (p_\infty^c - p_\infty^m)/(p_\infty^i - p_\infty^m)
\label{BE001}
\end{equation}
The parameter $\alpha$ will be a characteristic of the geometry of the
flow regardless of the vapor pressure and will often taken a value
of the order of unity. Then the surface tension, $S$, and the residence
time, $t_R$, are conveniently represented by the parameters:
\begin{equation}
R_o^*=\frac{R_o(p_\infty^i - p_\infty^m)}{S}
\quad; \quad
C_P = \frac{t_R}{R_o} \left\{ \frac{2 (p_\infty^i-p_\infty^m)}{3 \rho 
} \right\}^\frac{1}{2}
\label{BE002}
\end{equation}
Note that in the expressions \ref{BE001} and \ref{BE002} the pressure
difference $(p_\infty^i - p_\infty^m)$ has been uniformly used as one
of the non-dimensionalizing factors.
In addition, we introduce a combination
parameter which will appear in several places in the results ahead,
namely
\begin{equation}
C_Q =  C_P (k-1)^\frac{1}{3} (\alpha-1+\sigma)^\frac{1}{3}
(1-\sigma)^\frac{1}{2}
\label{BE003}
\end{equation}
Then it follows that
\begin{equation}
\frac{R_{max}}{R_o} = \frac{C_P}{(1-\sigma)^\frac{1}{2}} \quad ; \quad
\frac{R_{min}}{R_o} = (\sigma + 2/R_o^*)^\frac{1}{3(k-1)}
C_Q^{-\frac{1}{(k-1)}} \quad ; \quad
K= \frac{2 R_o^3 C_Q^3 (p_\infty^i - p_\infty^m)}{3 \rho (k-1)}
\label{BE004}
\end{equation}



\section{Stability to Spherical Harmonic Distortion}
\label{S2}

The stability of cavitating bubbles to nonspherical disturbances has 
been investigated
analytically by Birkhoff (1954), Plesset and Mitchell (1956), Brennen 
(1995) among others.
These analyses examined the spherical equivalent
of the Rayleigh-Taylor instability. If the inertia of the gas in the 
bubble is assumed to
be negligible, then the amplitude, $a(t)$, of a spherical harmonic 
distortion of order $n$
$(n > 1)$ is governed by the equation:
\begin{equation}
\frac{d^2 a}{dt^2} + \frac{3}{R} \frac{dR}{dt} \frac{da}{dt}
- \left\{ \frac{(n-1)}{R} \frac{d^2 R}{dt^2} -
(n-1) (n+1) (n+2) \frac{S}{\rho R^3} \right\} a = 0
\label{BE77} \end{equation}
Note that the coefficients require knowledge of the global dynamic 
behavior, $R(t)$.
The fact that they are not constant in time causes departure from the 
equivalent
Rayleigh-Taylor instability for a plane boundary. However, the 
coefficient of $a$
within the parentheses, $\{ \}$, of equation \ref{BE77}
is not greatly dissimilar from the case of
the plane boundary in the sense that instability is promoted when
$\ddot R > 0$ and surface tension has a stabilizing effect.
Plesset and Mitchell (1956) examined the particular case of
a vapor/gas bubble initially in equilibrium that is subjected to a
step function change in the pressure at infinity. Later calculations 
by Brennen (1995),
incorporated the effect of the noncondensable gas in the bubble. The effect
of the gas is essential for present purposes since its compression
causes the rebound and the instability that is addressed here.

It is clear from equation \ref{BE77} that the most unstable
circumstances occur when $\dot R <  0$ and $\ddot R \geq 0$.
These conditions are met following the beginning of the rebound (as 
defined earlier) and result in very rapid growth in non-spherical 
distortion.
This leads to the very rapid disintegration of the bubble
and its metamorphosis into the cloud of smaller bubbles which is
seen in experiments to emerge from the collapse of the bubble.
The growth rate of the distortions is controlled by the magnitude of 
the term in the parentheses, $\{ \}$,
in equation \ref{BE77}. The larger the value of this term the greater
the growth rate. Note that $n$ occurs only in this term and that the 
functional dependence on $n$ has the form
$(n-1) \left\{ \Gamma - (n+1)(n+2) \right\}$ where
$\Gamma=\rho R^2 \ddot{R}/S$.
It follows that as long as $\Gamma$ is positive, there will be a 
particular value of $n$ (denoted by $n_m$) for which the term has a 
positive maximum.
We would expect this mode of distortion to
dominate and therefore to play a role in determining the number of 
fission bubbles.
Parenthetically we note
that Shepherd (1980) follows a qualitatively similar argument in an effort
to predict the wavelengths of surface distortion seen on the bubbles 
in the same experiments
from which figure \ref{F2} was taken.

One complicating factor is that $\Gamma$ will vary with time within 
the unstable interval
increasing from zero at $t=-t_*$ to a maximum $\Gamma_m$ when 
$R=R_{min}$.  Clearly, however, $\Gamma_m$ is a representative value 
and can be written in terms of the parameters defined in section 
\ref{S1}:
\begin{equation}
\Gamma_m=
\frac{R_o^* C_Q^\frac{3k-1}{(k-1)}}{(\sigma + 2/R_o^*)^\frac{2}{3(k-1)}}
\label{BE010} \end{equation}
Then, taking $\Gamma_m$ as the characteristic
value of $\Gamma$, the most unstable mode is
\begin{equation}
n_m= \left\{ (7+3\Gamma_m)^\frac{1}{2}-2 \right\} /3
\quad \mbox{and} \quad
n_m \approx \left\{ \frac{\Gamma_m}{3} \right\}^\frac{1}{2} =
\frac{(R_o^*)^\frac{1}{2} C_Q^\frac{3k-1}{2(k-1)}}
{3^\frac{1}{2} (\sigma + 2/R_o^*)^\frac{1}{3(k-1)}}
\label{BE012} \end{equation}
provided $n_m \gg 1$.
We note that $n_m \approx (\Gamma_m/3)^\frac{1}{2}$ is functionally 
similar to the
most unstable surface distortion wavelength prediction included in 
the analysis of
Shepherd and Sturtevant (1982).


Given the most unstable Rayleigh-Taylor mode, the next question is to
estimate the number of fission fragments which that mode might lead 
to. Assuming that
the fission fragment size is directly related to the wavelength of 
the distortion on
the surface of the whole bubble, a crude estimate would be that the 
fragment radius, $R_F$,
would be given roughly by $R_F=R/n_m$. Then if the original volume is 
equally divided amongst
these fragments it follows that the number of fission fragments is $n_m^3$.
We delay discussion of this result until an alternative approach is
examined.


\section{Jet Breakup}
\label{S3}

The results at the end of the last section assumed a particular model
of bubble fission. In another set of circumstances, it has been
observed that a bubble collapsing close to a wall or free-surface
forms a re-entrant jet which shatters the bubble into many fragments 
when the jet
impacts the other side of the bubble surface. In this section we seek
to estimate the number of fission fragments that would result from 
this mode of bubble disintegration. To do so crudely we estimate that 
the size of the bubbles that survive such a violent process are those 
for which the surface tension forces holding the fission fragment 
together are roughly equal to the shear forces tearing it apart. The 
shear rates involved could be estimated as $\gamma=\dot{R}/R$ at the 
beginning of the rebound and, from
equations  \ref{BE35A}, $\dot{R}$ can be estimated as $\dot{R_*}$ where
\begin{equation}
\dot{R}_*^2 = \frac{\left\{ R_o C_P / t_R \right\}^2
\left\{ C_Q/k^\frac{1}{3} \right\}^\frac{3k}{(k-1)}}{
\left\{ \sigma + 2/R_o^* \right\}^\frac{1}{(k-1)}}
\label{BE012A} \end{equation}
To estimate the fission fragment size, $R_F$, we could then equate the typical
surface tension force, $2 \pi R_F S$, to the typical shearing force,
$6 \pi \mu \gamma R_F^2$ so that $R_F=S/ 3 \mu \gamma$. It would then follow
that the number of fission fragments would be $n_j^3$ where
\begin{equation}
n_j = \frac{R_*}{R_F} = \frac{3 \mu \dot{R}_*}{S}
= \frac{6^\frac{1}{2} R_o^*
\left\{C_Q /k^\frac{1}{3} \right\}^\frac{3k}{2(k-1)}}{
C_\mu (\sigma +2 / R_o^*)^\frac{1}{2(k-1)} }
\quad \mbox{where} \quad
C_\mu = \frac{R_o \left\{ \rho (p_\infty^i-p_\infty^m) 
\right\}^\frac{1}{2}}{\mu}
\label{BE017} \end{equation}
The shearing force, $6 \pi \mu \gamma R_F^2$, used in deriving this result
is a low Reynolds number formulation and requires that
$\rho \dot{R}_* R_*/\mu \ll 1$. If, on the other hand,
$\rho \dot{R}_* R_*/\mu \gg 1$, an appropriate estimate of the shearing force
would be $\pi \rho \gamma^2 R_F^4$. Then it would follow that
\begin{equation}
n_j =\frac{R_*}{R_F} = \left\{ \frac{\rho \dot{R}_*^2 R_*}{2S}
\right\}^\frac{1}{3}
= \frac{(R_o^*)^\frac{1}{3} \left\{ C_Q/k^\frac{1}{3}
\right\}^\frac{3k-1}{3(k-1)}}{
3^\frac{1}{3} (\sigma +2 / R_o^*)^\frac{2}{9(k-1)}}
\label{BE018} \end{equation}
It is fairly easy to demonstrate that
if the viscous terms dominate the inertial terms and lead to
expression \ref{BE017} rather than expression \ref{BE018} then the
viscous terms in the Rayleigh-Plesset equation itself should have been
included in the analysis of section \ref{S1}. Since they were not
so included, it follows that expression \ref{BE017} is of dubious
validity. Consequently, in the interest of brevity, we will not 
pursue the low bubble Reynolds number result any further.

In the next section we consider the consequences of the result \ref{BE018}.

\section{Fission Fragments}
\label{S5}


In assessing the results of the last two sections, namely
expressions \ref{BE012}, \ref{BE017} and
\ref{BE018} for the number of fission fragments, we note that all three results
have quite similar forms.  This is because they all involve fission 
forces which
are inertial in origin and a resistance to fission governed by surface tension.
Moreover, since $\sigma$ is often of
order unity, the magnitude of $n_m$ or $n_j$ is primarily determined
by the numerator and is therefore a function of $R_o^*$ and $C_Q$
(or, effectively, $C_P$), though $C_\mu$ also appears in the
expression \ref{BE017}. Concentrating on those numerators
involving $R_o^*$ and $C_Q$ we see that
$n_m$ and $n_j$ sensibly decrease with increasing $S$. The variation 
with nuclei size is more complex and requires consideration of the
factor $(\sigma+2/R_o^*)$ in the denominators. For very small nuclei
sizes such that $R_o^* \ll 2/\sigma$ the number of fission fragments 
increases with the nuclei size (provided $k$ takes some reasonable 
number). However, for larger nuclei such that $R_o^* \gg 2/\sigma$ 
the number of fission fragments {\it decreases} as the nuclei size 
increases. This
slightly non-intuitive trend occurs because the maximum bubble size
becomes essentially independent of the nuclei size; however, the
larger nuclei contribute more non-condensable gas to the collapse and
the collapse is therefore {\it less} violent leading to fewer
fission fragments.
\begin{figure}
\vspace{.75in}
\centerline{\Large\bf (Figure goes here)}
\vspace{.75in}
%\centerline{
%\epsfig{file=figsplit1.eps,width=2.5in}
%\hspace{0.1in}
%\epsfig{file=figsplit2.eps,width=2.5in}}
\caption{Left: values of
$n_m$ from the Rayleigh-Taylor instability
analysis. Right:
values of $n_j$ from the re-entrant jet breakup analysis. Both 
plotted against $R_o^*$ for  $k=1.4$ and various values of the 
parameter, $C_Q$, and the cavitation number, $\sigma$.}
\label{F20} \end{figure}

We will now illustrate the results for $n_m$ and $n_j$ with some
numerical examples. In figure \ref{F20}
values for $n_m$ and $n_j$  from the expressions \ref{BE012} and
\ref{BE018} are plotted against the dimensionless nuclei size, $R_o^*$,
for various values of the parameter, $C_Q$, and the cavitation index,
$\sigma$. Note that for a typical surface tension, $S$, of $0.07 
kg/s^2$ and a typical pressure
difference, $(p_\infty^i - p_\infty^m)$, of $10^5 kg/m\ s^2$, nuclei 
of radii, $R_o$, ranging from $1 \mu m$ to $100 \mu m$ would yield
$R_o^*$ values ranging from $1.4$ to $140$, roughly in the middle of the
horizontal scale. Moreover, cavitation indices in the range used in
the figures are commonly experienced. Perhaps the greatest uncertainity
lies in estimating typical values of $C_Q$ occuring in practice.
The simplest way to estimate this is to use the first of equations
\ref{BE004}. Common values of $R_{max}/R_o$ range from 10 up to 100
and higher and this provides an estimate of $C_P$. The definition
\ref{BE003} suggests that $C_Q$ will typically be about an order
of magnitude smaller than $C_P$ and this leads to an estimate of
$C_Q$ of order unity or greater. Figure \ref{F20} indicates
that for $C_Q=1$ the smaller nuclei may lead to collapses without
fission if the cavitation number is large enough. However, large
nuclei at lower cavitation numbers will lead to breakup into large
numbers of fission fragments.


\section{Some Comparison with Observations}
\label{S10}

While many of the photographs of cavitation bubbles before and
after the first collapse (for example,
Lauterborn and Bolle 1975, Tomita and Shima 1990) show that the
bubble has fissioned into many fragments, the photographs rarely have
the kind of resolution that would allow a count of the number of
those fragments.  On the other hand, though they are not
normal cavitation bubbles, the beautiful photographs of Frost and 
Sturtevant (1986) showing the breakup of ether vapor bubbles in 
glycerol are of sufficient resolution to allow comparison with the 
present
analysis.

Frost and Sturtevant (1986) (see also, Shepherd and Sturtevant (1982),
Shepherd (1980), Frost (1985)) allowed drops of ether in gycerol to
explosively evaporate and then examined the surface appearance as the
bubble oscillated. In addition they measured the pressures radiated as
a result of these oscillations. Sample photographs just after either
the first or second collapses are shown in figure \ref{F2}. We counted
the individual bubbles observable on the half-surface facing the 
camera. Doubling that number to account for the back side yielded 
values of
320, 400 and 410 respectively for the three cases. In the framework
of section \ref {S2} this number should correspond to $n_m^2$ and hence the
photographs indicate $n_m \approx 20$.

To compare the theory of section \ref{S2}, we note that the pressure
radiated from an oscillating bubble has an amplitude, $\tilde{p}$,
given roughly by
\begin{equation}
\tilde{p}= \frac{\rho}{4\pi {\cal R}} \frac{d^2V}{dt^2}
\approx \frac{\rho R^2}{{\cal R}} \frac{d^2R}{dt^2}
\label{E101} \end{equation}
where $V$ and $R$ are the volume and radius of the bubble and ${\cal 
R}$  is the distance from the bubble center to the point of pressure
measurement (see Brennen 1995). Substituting for $R^2 d^2R/dt^2$ in
the definition \ref{BE008} and using the approximation in equation \ref{BE012}
yields
\begin{equation}
n_m= \left\{ \Gamma_m/3 \right\}^\frac{1}{2}
\approx  \left\{ {\cal R} \vert \tilde{p} \vert /3S \right\}^\frac{1}{2}
\label{E102} \end{equation}
where $\vert \tilde{p} \vert$ is the amplitude of the radiated pressure.
>From the pressure traces given by Frost(1985), we estimate the values
of $\vert \tilde{p} \vert$ for the three photographs in figure \ref{F2}
to be $6 \times 10^4 kg/m s^2$. With this, ${\cal R}=6mm$ and $S=0.07 N/m$,
equation \ref{E102} yields $n_m=41$.
\begin{figure}
\vspace{.75in}
\centerline{\Large\bf (Figure goes here)}
\vspace{.75in}
%\centerline{
%\epsfig{file=photoD.eps,width=1.4in}
%\hspace{0.1in}
%\epsfig{file=photoA.eps,width=1.4in}
%\hspace{0.1in}
%\epsfig{file=photoB.eps,width=1.4in}
%\hspace{0.1in}
%\epsfig{file=photoC.eps,width=1.4in}}
\caption{Photographs of ether vapor bubbles in glycerol. From left to right:
a bubble before the first collapse and three examples of bubbles
after the first collapse (reproduced with permission from Frost (1985)).}
\label{F2} \end{figure}

Though the evidence is limited, the
qualitative agreement between the theoretical value of $n_m=41$ and 
the experimental values around $n_m=20$ is encouraging.
Clearly, however,
further comparisons are necessary to validate the theory.



\section{Effective Damping}
\label{S4}


Calculations of the global dynamics, $R(t)$, which use the 
Rayleigh-Plesset equation
and therefore assume spherical symmetry require an estimate of the 
energy dissipation
in order to yield realistic results. The dissipation can have a 
number of physical
origins including viscous dissipation in the liquid, thermal effects 
at the bubble
surface and acoustic radiation (Chapman and Plesset 1971,
Nigmatulin {\it et al.} 1981, Prosperetti 1991, Matsumoto and Takemura 1994).
However, when any of these techniques are applied to cavitating 
bubble dynamics such as are the
subject of this paper, estimates of the damping or effective 
viscosity always produce
far larger and more numerous rebounds than are observed 
experimentally. It is often
remarked that this is because the bubble almost never remains 
spherical through the
first collapse. Experimental observations such as those of Lauterborn 
and Bolle (1975)
and Tomita and Shima (1990)
show that though the bubble may remain relatively intact prior to the 
first collapse.
But what emerges from that collapse is either a highly distorted 
bubble mass or a cloud
of bubble fragments. It seems likely that this fragmentation process 
dissipates substantial
energy and therefore may contribute in a major way to the effective 
damping of the
collapse and rebound cycle. Perhaps this is why the observed number 
of these cycles rarely
exceeds three or four. Consequently,
the first step is to recall
that the viscous term being used in the usual Rayleigh-Plesset 
equation \ref{BE26}
(where $\mu_e$ is now an effective viscosity that incorporates 
several mechanisms of dissipation) implies a rate of dissipation of 
energy given by
$16 \pi \mu_e R (\dot{R})^2$.
If, using the results of section \ref{S1}, this is integrated
over the collapse interval, $-t^*<t<t^*$, an expression for
the total energy dissipated during that interval emerges:
\begin{equation}
\frac{64 \pi \mu_e}{9} \left\{ \frac{R_{min} K (k-1)}{(k+1)^3}
  \right\}^\frac{1}{2}
\label{BE021}
\end{equation}
Turning this around, if we estimated the energy, $E_D$, dissipated in 
a collapse as a result of fission then the appropriate value of 
$\mu_e$
which should be used to represent the fission dissipation is
\begin{equation}
\mu_e= \frac{9 E_D}{64 \pi} \left\{ \frac{(k+1)^3}{R_{min} K (k-1)}
\right\}^\frac{1}{2}
\label{BE022}
\end{equation}


The next step is to consider the fission process and to estimate the
absorption of energy that occurs during that process. As a part of that process
in the next section $E_D$ will be connected
to the number of fission fragments, $n_m^3$ or $n_j^3$.


\section{Dissipation mechanisms during fission}
\label{S11}


There are several mechanisms by which the energy associated with the radial
motion may be deflected or dissipated. First, the formation of jets or
surface waves will channel kinetic energy in directions in which the energy
must end up being dissipated by viscosity as a result of the fluid mixing
and turbulence. Second, the bubble fragments that emerge from the fission
process must necessarily contain more surface free energy than the single
bubble prior to collapse. This deflection of energy will
decrease the kinetic energy associated with the volumetric or radial motion
and therefore also contribute to the damping. We shall attempt 
estimates of both
of these contributions to the fission damping.

First we frame the discussion by observing that the kinetic energy, 
$E$, of a single
spherical bubble in an infinite fluid is
$2 \pi \rho \dot{R}^2 R^3$. Thus, the kinetic energy, $E^*$, at the 
start of the rebound
is
\begin{equation}
E^* = 2 \pi \rho R_*^3 \dot{R}_*^2 = 2 \pi \rho \frac{(k-1)}{k} K
\label{BE031}
\end{equation}
To estimate the kinetic energy which might be dissipated due to mixing,
we suppose that a single microjet penetrates the single
bubble just after the beginning of the rebound when $R=R^*$ and 
achieves a velocity
that is some multiple, $C_J$, of $\dot{R}_*$ (the work of Blake and 
Gibson 1987 and others suggests that $C_J$ may be as high as $10$ - 
see Brennen 1995). Then, if we  assume
that all the kinetic
energy, $E_J$, in the jet is dissipated during collapse and that the 
jet radius is
$1/C_J$ of the bubble radius at the beginning of rebound, it would follow that
\begin{equation}
E_J \propto \pi \rho R_*^3 \dot{R}_*^2
\label{BE032}
\end{equation}
This simply states that the loss is fraction of the total kinetic energy at the
start of the rebound. Consequently the radial kinetic energy that remains is
also a fraction, say $\nu$, of that total kinetic energy. This, in 
turn, implies
that the maximum volumetric radius achieved after the rebound will be
$\nu^\frac{1}{3}$ of that achieved before the collapse.
If $\nu$ is some moderate fraction, this implies that only a very small number
of collapse and rebound cycles will be observed.
Experimental observations such as those of Ellis (1952), Lauterborn 
and Bolle (1975),
Vogel, Lauterborn and Timm (1989) and Tomita and Shima (1990) clearly show that
the number of rebounds usually lies between one and three. The plots of Shima
and Tomita (1981), for example, suggest that $\nu^\frac{1}{3}$ lies 
between $0.5$
and $0.3$ which yields values of $\nu$ between $0.03$ and $0.1$.
Clearly, in order to quantify this process more accurately requires 
considerably
more information than can be developed here. However, it is 
encouraging to recognize
that the analysis is qualitatively consistent with the experimental 
observations.

In Brennen (2001), the other drain for kinetic energy,
namely the additional surface free energy, $E_S$, in the fission fragment cloud
is briefly examined and found to be significantly smaller than the 
estimate of the
drain due to fission mixing and turbulence for typical values associated with
cavitation in water. However, both energy sinks are orders of 
magnitude larger than
the dissipation due to the conventional spherical bubble mechanicisms due to
viscous, thermal and acoustic damping. For example, the surface free 
energy estimate
leads to typical effective viscosity values in the range $0.025-0.08 
kg/ms$ compared
with the typical
dynamic viscosity of water of $0.001 kg/ms$.

In summary, we find that both the absorption of energy into surface 
free energy as a result
of fission and the dissipation of energy due to the mixing produced by the
non-spherically symmetric motions are much larger than the 
traditional liquid viscosity contribution to the damping. Moreover, 
for typical applications to cavitation resulting
from micron-sized nuclei in water, the mixing contribution dominates 
the surface free
energy contribution. Only a qualitative examination of this dominant
contribution seems possible and that suggests a specific ratio of the 
maximum bubble
size after collapse to that before. At present, it therefore seems practical to
examine the experimental observations for an appropriate value of this ratio and 
to
use this in the Rayleigh-Plesset module in larger codes.


\section{Conclusions}

Spurred by the need for multiphase flow models of cavitating flows 
which incorporate
accurate Rayleigh-Plesset models for the bubble dynamics, this paper 
has examined
the deficiencies in the Rayleigh-Plesset calculations which occur 
during cavitating
bubble collapse. Numerous experimental observations have revealed 
that though a single
coherent bubble may enter the very rapid collapse phase, almost 
invariably what emerges
is either a highly distorted bubble or a cloud of smaller bubbles. 
This paper seeks to
elucidate this fission process and begins by qualitatively evaluating 
the number of
fission fragments that would result from (a) a fission process resulting from a
Rayleigh-Taylor instability of the interface and (b) a fission 
process resulting from
a re-entrant jet collapse mechanism. Both mechanisms lead to 
qualitatively similar
fission fragment numbers which depend on several global parameters 
which are identified.

In seeking some experimental verification of these fission fragment 
numbers we turn, not
to pictures of cavitation bubbles (which almost never have the 
resolution that would allow
the counting of bubbles), but to the beautiful photographs by 
Shepherd, Frost and Sturtevant of the fission of collapsing ether 
vapor bubbles in glycerol. The theory and
observation of the number of fission fragments are qualitatively 
similar. However, the
observed number of fragments is significantly smaller perhaps because 
viscous effects
on the instability were not incorporated in the theory.

We end the paper by evaluating how the fission process will 
contribute to the damping
of the global volumetric motions that the Rayleigh-Plesset equation 
is often used to
simulate. It is shown that the kinetic energy which is dissipated by the mixing
and turbulence associated with fission is much greater than the energy which is
deflected into the additional free surface energy. But both are much 
greater than
the energy dissipated by the classical viscous, thermal and acoustic mechanisms
normally used to evaluate the bubble damping. Thus the paper concludes that for
a Rayleigh-Plesset method to adequately model the global volumetric motions of
the bubble or bubble cloud, it is neccessary to introduce an energy dissipation
mechanism associated with the fission process. The present analysis 
shows that the best
way to do this is to set the kinetic energy emerging from the 
collapse to some fraction
of that entering collapse. However, comparisons with experimental observations
are neccessary to establish the best value for that fraction.


\section*{Acknowledgements}

This paper is
dedicated to the memory of my colleague Brad Sturtevant whose
enthusiasm and dedication inspired all who knew him. I would also
like to thank Joe Shepherd, Tim Colonius and Steve Hoestler for their
contributions to this paper.


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