%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^*