# Power law singularity for cavity collapse in a compressible Euler fluid with
Tait-Murnaghan equation of state

Daniels Krimans<EMAIL_ADDRESS>Physics and Astronomy
Department, University of California Los Angeles, Los Angeles, California
90095, USA  Seth Putterman  Physics and Astronomy Department, University of
California Los Angeles, Los Angeles, California 90095, USA

###### Abstract

Motivated by the high energy focusing found in rapidly collapsing bubbles that
is relevant to implosion processes that concentrate energy density, such as
sonoluminescence, we consider a calculation of an empty cavity collapse in a
compressible Euler fluid. We review and then use the method based on
similarity theory that was previously used to compute the power law exponent
$n$ for the collapse of an empty cavity in water during the late stage of the
collapse. We extend this calculation by considering different fluids
surrounding the cavity, all of which are parametrized by the Tait-Murnaghan
equation of state through parameter $\gamma$. As a result, we obtain the
dependence of $n$ on $\gamma$ for a wide range of $\gamma$, and indeed see
that the collapse is sensitive to the equation of state of an outside fluid.

This article may be downloaded for personal use only. Any other use requires
prior permission of the author and AIP Publishing. This article appeared in D.
Krimans and S. Putterman, "Power law singularity for cavity collapse in a
compressible Euler fluid with Tait–Murnaghan equation of state", Phys. Fluids
35, 086114 (2023) and may be found at https://doi.org/10.1063/5.0160469.

##  I Introduction

Rayleigh [1] calculated that the radius $R$ of an empty cavity in an
incompressible ideal fluid collapsed to zero at a finite time $t_{0}$ as:

$R(t)=A(t_{0}-t)^{2/5},$ (1)

where $A=(5/2)^{2/5}(E/2\pi\rho)^{1/5}$, $\rho$ is the mass density of the
fluid and $E=(4\pi/3)p_{0}R_{m}^{3}$ is the energy of the fluid, which is the
energy to form the initial cavity of radius $R_{m}$ in an otherwise stationary
fluid, where $p_{0}$ is the ambient externally applied pressure. If there are
$N$ atoms inside the cavity the collapse will arrest at a finite radius
$R_{c}$ and at this moment the energy per particle will be $E/N$. A typical
experiment [2] reaches $R_{m}=50$ $\upmu$m with $N=1.2\times 10^{10}$. Due to
the finite size of the atoms the collapse arrests at $R_{c}\approx 0.5$
$\upmu$m and at this stagnation point the average energy delivered to each
particle is $E/N\approx 25$ eV. As the emission of ultraviolet photons, with
an energy of $6$ eV, can be observed from the interior of the collapsed bubble
[3], the Rayleigh bubble dynamics is widely regarded as providing a zeroth
order picture of sonoluminescence.

As the moment of the Rayleigh collapse is approached, from Eq. (1) the
velocity of the cavity wall approaches infinity as
$\dot{R}(t)=(-2A/5)(t_{0}-t)^{-3/5}$. Consider again the situation where gas
is contained in the cavity. On the one hand it will arrest the collapse prior
to reaching zero radius. On the other hand the speed of the gas at the
boundary of the cavity $r=R(t)$ can become supersonic relative to the medium
in the bubble which, in contrast to the external fluid, is compressible [2].
If the supersonic motion occurs well before arresting of the collapse then an
imploding shock wave can form. The shock can focus to the origin, $r=0$,
independent of the presence of matter. In this case $E/N\approx 1000$ eV has
been theoretically predicted [4]. Realization of this handover in the focusing
of energy density would raise prospects for the use of acoustics to achieve
thermal fusion [5]. Ramsey [6, 7] has emphasized that a small compressibility
of the surrounding fluid might slow down the collapse and affect the
attainment of a next stage in energy focusing. In particular, one notes that
in 1960 Hunter [8] calculated that for water (which is the fluid of choice for
most experiments) near the moment of the collapse:

$R(t)=A_{n}(t_{0}-t)^{n},$ (2)

where $n=0.5552\approx 5/9$. Hunter used the Tait-Murnaghan [9, 10] form of
the equation of state for the fluid’s pressure $p$:

$p(\rho)=B\left(\left(\frac{\rho}{\rho_{0}}\right)^{\gamma}-1\right),$ (3)

where $B=3000$ atm, $\gamma=7$, $\rho_{0}=1000$ kg/$\textrm{m}^{3}$. The
compressibility of water slows down the collapse and preliminary analysis
indicates that the handover to an imploding shock wave can be suppressed [11].

Even for ideal hydrodynamics the compressibility of the fluid has a strong
influence on the extent to which energy density is concentrated. Motivated by
this perspective we present a calculation for $n(\gamma)$ for a wide range of
materials described by different values of $\gamma$ that extends the results
of Hunter for water and of Rayleigh for incompressible fluid. Key results are
displayed in Fig. 1. These calculations might motivate a search for candidate
liquids to achieve greater levels of energy focusing. Our calculations also
provide an asymptotic limit that can be used to evaluate accuracy of more
general numerical solutions, for example, simulations of all-Mach number
bubble dynamics [12].

Figure 1: Predicted values of $n$ as a function of $\gamma$ (black dots), the
proposed fit for the range of $\gamma$ close to the value of water that has
$\gamma=7$ of the form $n=0.4+a\gamma^{-b}$, where $a=0.657424$, $b=0.735226$
(solid red line), and the value for the incompressible fluid $n=0.4$ (dashed
magenta line). In (a), values of $\gamma$ are those close to the water that
has $\gamma=7$. In (b), a larger range of $\gamma$ is considered.

##  II Theory

In this section, we give a summary of the methods used to perform the
computations [8].

We assume a spherically symmetric scenario in which a spherical empty cavity
has its center placed in the origin of the coordinate system and whose radius
is described by a time dependent function $R(t)$. Outside of this radius, we
assume an infinite ideal fluid which is described by mass conservation law and
Euler’s equation, where due to spherical symmetry fluid’s only nonzero
component of velocity is radial component $u$, and both radial velocity
component and mass density $\rho$ are only functions of radial coordinate $r$
and time $t$. We do not consider an equation for entropy as we assume that the
flow is homentropic. The following equations are considered for $r>R(t)$.

$\begin{gathered}\rho\left(\frac{\partial u}{\partial t}+u\frac{\partial
u}{\partial r}\right)=-\frac{\partial p}{\partial r},\end{gathered}$ (4)
$\begin{gathered}\frac{\partial\rho}{\partial t}+u\frac{\partial\rho}{\partial
r}+\rho\left(\frac{2u}{r}+\frac{\partial u}{\partial
r}\right)=0.\end{gathered}$ (5)

To describe pressure $p$, we assume the Tait-Murnaghan equation of state,
where $s$ is the specific entropy.

$p(\rho,s)=B(s)\left(\left(\frac{\rho}{\rho_{0}(s)}\right)^{\gamma}-1\right)$
(6)

For simplicity, as the flow is homentropic, from now on we will not explicitly
write dependence of $B,\rho_{0}$, or other entropy dependent variables on the
specific entropy $s$.

We would like to enforce two boundary conditions for both radial velocity $u$
and pressure $p$, where one is at the interface between the empty cavity and
fluid at $r=R(t)$, and the other one is far from the cavity as $r\to\infty$.
For radial velocity, we assume that the empty cavity is a free surface and
that far away from the cavity the fluid is at rest. For pressure, we assume
that at the surface of the cavity, pressure is zero as the cavity is empty,
and far away it approaches some finite value $p_{\infty}$. Boundary conditions
are summarized next, where the dot represents the derivative with respect to
the time.

$\begin{gathered}u(R(t),t)=\dot{R}(t),\qquad\lim\limits_{r\to\infty}u(r,t)=0.\end{gathered}$
(7)
$\begin{gathered}p(R(t),t)=0,\qquad\lim\limits_{r\to\infty}p(r,t)=p_{\infty}.\end{gathered}$
(8)

Using the assumed equation of state Eq. (6), it is possible to compute the
speed of sound squared $c^{2}$ as a function of $\rho$, and change variables
describing fluid from $u,\rho$ to $u,c^{2}$. This is convenient in order to
apply similarity theory for the later parts of the collapse when $R(t)\to 0$,
as both variables $u$ and $c^{2}$ can be directly compared to the velocity of
cavity’s wall $\dot{R}(t)$.

$\begin{gathered}c^{2}=\frac{\partial
p}{\partial\rho}=\frac{B\gamma\rho^{\gamma-1}}{\rho_{0}^{\gamma}},\qquad\rho=\left(\frac{\rho_{0}^{\gamma}c^{2}}{B\gamma}\right)^{1/(\gamma-1)}.\end{gathered}$
(9)

Using the expression for $c^{2}$ in terms of density $\rho$ as in Eq. (9),
spherical Euler’s equation Eq. (4) and mass conservation law Eq. (5) are
rewritten in terms of variables $u,c^{2}$ as follows.

$\begin{gathered}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial
r}+\frac{1}{(\gamma-1)}\frac{\partial c^{2}}{\partial r}=0,\end{gathered}$
(10) $\begin{gathered}\frac{\partial c^{2}}{\partial t}+u\frac{\partial
c^{2}}{\partial r}+c^{2}(\gamma-1)\left(\frac{2u}{r}+\frac{\partial
u}{\partial r}\right)=0.\end{gathered}$ (11)

Using Eqs. (6), (9), it is possible to compute boundary conditions for $c^{2}$
given by constants $c_{0}^{2},c_{\infty}^{2}$, from the corresponding boundary
conditions for pressure as in Eq. (8).

$\begin{gathered}c^{2}(R(t),t)=\frac{B\gamma}{\rho_{0}}=c_{0}^{2},\qquad\lim\limits_{r\to\infty}c^{2}(r,t)=c^{2}_{\infty}.\end{gathered}$
(12)

Instead of solving the system of partial differential equations given by Eqs.
(10), (11) which would mean that we have to supply initial conditions, we
consider similarity theory that is motivated by numerical results [8]. As we
approach the last phase of the collapse where $R(t)\to 0$, we assume that the
length scale of the problem is given by $R(t)$ and the scale for velocities is
given by $\dot{R}(t)$. So, we seek solutions of the following form, where we
are interested in finding functions $f$ and $g$. The goal is to reduce the
problem to a system of ordinary differential equations for $f$ and $g$.

$\begin{gathered}\frac{u(r,t)}{\dot{R}(t)}=f\left(\frac{r}{R(t)}\right),\qquad\frac{c^{2}(r,t)}{\dot{R}^{2}(t)}=g\left(\frac{r}{R(t)}\right).\end{gathered}$
(13)

Motivated by the numerical results of the full hydrodynamic equations given in
Eqs. (10), (11) in the case of water[8], we additionally assume the power law
form $R(t)=A_{n}(t_{0}-t)^{n}$, where $t_{0}$ is a time at which collapse
happens and $n$ is a power law exponent that we would like to compute. Using
such power law assumption and similarity approach for $u,c^{2}$ as in Eq.
(13), we can rewrite Eqs. (10), (11) as two coupled ordinary differential
equations for functions $f$ and $g$, where we introduce variable $x=r/R(t)$.
Then, differential equations have to be solved in the range $x>1$.

$\begin{gathered}f^{\prime}(x)\left(f(x)-x\right)+\left(1-\frac{1}{n}\right)f(x)+\frac{g^{\prime}(x)}{(\gamma-1)}=0,\end{gathered}$
(14)
$\begin{gathered}g^{\prime}(x)\left(f(x)-x\right)+2\left(1-\frac{1}{n}\right)g(x)+g(x)(\gamma-1)\left(\frac{2f(x)}{x}+f^{\prime}(x)\right)=0.\end{gathered}$
(15)

As these equations are ordinary differential equations, we do not have to
worry about what kind of initial conditions to choose for $u,c^{2}$, as
functions $f$ and $g$ can be solved only by the boundary conditions. To
compute boundary conditions from those of $u,c^{2}$ given in Eqs. (7), (12),
we use definitions of $f,g$ in terms of $u,c^{2}$ as in Eq. (13). However, for
the assumed form of $c^{2}$ in Eq. (13), boundary conditions cannot be
satisfied. Instead, because $\dot{R}(t)\to-\infty$ as $R(t)\to 0$, to get
finite speeds of sounds at boundaries, we assume that $g$ is zero at the
boundaries if all we are interested in is the late stage of the collapse.

$\begin{gathered}f(1)=1,\qquad\lim\limits_{x\to\infty}f(x)=0.\end{gathered}$
(16)
$\begin{gathered}g(1)=0,\qquad\lim\limits_{x\to\infty}g(x)=0.\end{gathered}$
(17)

The goal now is for each value of $\gamma$ describing the equation of state of
the fluid to find the value of $n$ so that differential equations given by
Eqs. (14), (15) are satisfied with the appropriate boundary conditions given
by Eqs. (16), (17).

##  III Results

It is convenient to rewrite differential equations given by Eqs. (14), (15) in
the following way so that each equation contains a derivative of only one of
the functions. To do this, insert the expression for $g^{\prime}(x)$ from Eq.
(15) to Eq. (14), or insert the expression for $f^{\prime}(x)$ from Eq. (14)
to Eq. (15).

$\begin{gathered}f^{\prime}(x)\left((f(x)-x)^{2}-g(x)\right)=-\left(1-\frac{1}{n}\right)f(x)(f(x)-x)\\\
+2\left(1-\frac{1}{n}\right)\frac{g(x)}{(\gamma-1)}+\frac{2f(x)g(x)}{x},\end{gathered}$
(18)
$\begin{gathered}g^{\prime}(x)\left((f(x)-x)^{2}-g(x)\right)=\left(1-\frac{1}{n}\right)(\gamma-1)f(x)g(x)\\\
-2\left(1-\frac{1}{n}\right)g(x)(f(x)-x)+2(\gamma-1)f(x)g(x)\frac{(x-f(x))}{x}.\end{gathered}$
(19)

Notice that the Eqs. (18), (19) are singular at $x=1$ with the boundary
conditions chosen as Eqs. (16), (17) as at $x=1$ it is not possible to solve
for $f^{\prime}(1),g^{\prime}(1)$ which then would be used to numerically
approximate values of $f,g$ for some $x>1$. So, instead of solving equations
numerically from $x=1$, we start from $x=1+\varepsilon$, where
$0<\varepsilon\ll 1$. To do that, we have to understand what are the new
boundary conditions at such a point. To compute them, we expand both functions
in $\varepsilon$ around $x=1$ as given next, where we use boundary conditions
at $x=1$ as in Eqs. (16), (17).

$\begin{gathered}f(1+\varepsilon)=f(1)+f^{\prime}(1)\varepsilon+O(\varepsilon^{2})=1+f^{\prime}(1)\varepsilon+O(\varepsilon^{2}),\\\
f^{\prime}(1+\varepsilon)=f^{\prime}(1)+f^{\prime\prime}(1)\varepsilon+O(\varepsilon^{2}).\end{gathered}$
(20)
$\begin{gathered}g(1+\varepsilon)=g(1)+g^{\prime}(1)\varepsilon+O(\varepsilon^{2})=g^{\prime}(1)\varepsilon+O(\varepsilon^{2}),\\\
g^{\prime}(1+\varepsilon)=g^{\prime}(1)+g^{\prime\prime}(1)\varepsilon+O(\varepsilon^{2}).\end{gathered}$
(21)

Consider differential equations given by Eqs. (18), (19) up to first order in
$\varepsilon$. As equations are singular at $x=1$, no information is obtained
from the zeroth order in $\varepsilon$. However, from the first order in
$\varepsilon$, it is possible to compute values of
$f^{\prime}(1),g^{\prime}(1)$ and to approximate boundary conditions as
follows.

$\begin{gathered}f(1+\varepsilon)\approx
1+\frac{1}{\gamma}\left(3-2\left(1-\frac{1}{n}\right)-2\gamma\right)\varepsilon,\end{gathered}$
(22)
$\begin{gathered}g(1+\varepsilon)\approx\left(1-\frac{1}{n}\right)(1-\gamma)\varepsilon.\end{gathered}$
(23)

For a given choice of $\gamma$, we search through values of $n$ and for each
guess of $n$ we numerically integrate Eqs. (18), (19) starting from
$x=1+\varepsilon$, where boundary conditions given by Eqs. (22), (23) are
used, until $x=5$. The maximum value of $x$ is chosen from practical
considerations as then it is clear whether boundary conditions as $x\to\infty$
given by Eqs. (16), (17) are satisfied or not. If they are, we report this
value of $n$ as the predicted value for the power law exponent of the cavity
wall’s collapse. Value of $\varepsilon$ used in numerical calculations is
$\varepsilon=10^{-3}$. It was checked that if this value is taken to be
smaller then the results do not change significantly. For example, if
$\gamma=8$, then predicted values for $\varepsilon=10^{-3},10^{-4},10^{-5}$
are $n=0.540799,0.540801,0.540801$, respectively. As an example of a search of
$n$ for $\gamma=8$, consider results in Fig. 2 which show how solutions of the
function $g$ look like if $n$ is smaller, equal, or greater than the predicted
value.

Figure 2: Numerically obtained solutions for $g(x)$ shown here from
$x=1+\varepsilon$ until $x=2$, where $\varepsilon=10^{-3}$ (solid black line),
$\varepsilon=10^{-4}$ (dashed blue line), $\varepsilon=10^{-5}$ (dotted red
line). In (a), the guessed value of $n$ is smaller than the predicted one by
$10^{-4}$. In (b), it is equal to the predicted value
$n=0.540799,0.540801,0.540801$ for $\varepsilon=10^{-3},10^{-4},10^{-5}$,
respectively. In (c), the guessed value of $n$ is larger than the predicted
value by $10^{-4}$. The guessed solutions in (a) and (c) are not satisfactory
because they are not smooth at around $x=1.55$ and do not satisfy the boundary
condition given in Eq. (17) as $x\to\infty$.

First, we consider obtained results for $n$ as a function of $\gamma$, where a
range of $\gamma$ is chosen to be close to the value for water, $\gamma=7$. We
predict that the majority of materials have $\gamma$ values close to the one
of water, so for this range, we propose the following fit that might be useful
for practical applications, $n=0.4+a\gamma^{-b}$, where $a=0.657424$,
$b=0.735226$. For this range of $\gamma$, the results are shown in Fig. 1(a).
We have checked that our calculation for the case where $\gamma=7$ that
corresponds to the water agrees with the obtained value of $n$ of Hunter [8].
However, the described method can also be used to compute $n$ for large values
of $\gamma$, results for which are given in Fig. 1(b). We see that as
$\gamma\to\infty$, values indeed approach the incompressible limit $n=0.4$
[1].

##  IV Conclusion

Sonoluminescence is a phenomenon whereby a hydrodynamic singularity causes
energy density to be concentrated by many orders of magnitude [13]. The
irreducible and fundamental case in which this singularity is realized is when
an empty cavity is surrounded by an ideal Euler fluid. Due to the presence of
a singularity many additional physical factors are excited into action. These
include surface tension, viscosity, thermal conduction, vapor pressure and
other gases that may be present in the cavity. Here we have endeavored to
study the general ideal Euler limit, which is the source of the singularity,
by including the compressibility of the fluid. We find a spectrum of dynamic
singularities that are determined by the thermodynamic equation of state. Our
results agree with previous work of Rayleigh for the incompressible fluid [1]
when $\gamma\to\infty$, and with Hunter [8] for the case of water where
$\gamma=7$. From the obtained quantitative results, we see that in the late
stages of collapse, the collapse is faster if $\gamma$ is larger. Therefore,
the selection of materials with high $\gamma$ will facilitate the attainment
of higher levels of focusing of energy density in a bubble. Using the proposed
fit of our numerical data, one can readily use our results for practical
applications. Our numerical results can also be used to check two fluid
numerical solvers that use compressible Euler’s hydrodynamic equations by
comparing the full solutions to the asymptotic empty cavity result provided in
this paper.

Gas or vapor inside the cavity will influence the final stages of collapse.
Furthermore, excitation of the gas by the collapsing cavity provides an
experimental marker for this singular process. Investigation of this
phenomenon is left for future papers.

###### Acknowledgements.

This research has been funded by the DoD via HQ0034-20-1-0034 and the AFOSR
under FA9550-22-1-0425 (20223965). We thank Steven Ruuth and Seth Pree for
many valuable discussions.

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