2.1 The case of $m=N$ ; the isothermal case

In the case where $m=N$ (the isothermal case), and assuming a non-constant solution, equation (2.3) has the first integral

where $\unicode[STIX]{x1D702}_{1}$ is an integration constant. From the initial conditions (2.5), we find $\unicode[STIX]{x1D702}_{1}=2p/N$ . In order for this solution to exist, we assume that the right-hand side is always non-negative. Clearly, if $p=0$ , this need not be true. So, we need $p>0$ large enough to ensure this quantity is always positive. Under the convention that a bubble expands on $t<0$ and collapses on $t>0$ , we take the positive root for $t<0$ near the origin, while the negative root is taken for $t>0$ near the origin. Separating variables, we have that, for $t>0$ ,

This equation implicitly relates the isothermal bubble radius to time, and hence captures the temporal dynamics of the bubble. Conversely, one may view this relation as the time it takes for a bubble of non-dimensional radius 1 to shrink to some value $0<r<1$ . By the form of (2.6), the process is time reversible or symmetric from the origin.

Note that the logarithmic term means that the radius of the bubble will never be zero, hence the isothermal bubble will not collapse. Indeed, one may calculate the minimal possible radius from (2.7). To do this, one notes that the integrand is singular at some positive value, say $r(t)=r_{min}$ , where $0<r_{min}<1$ . Then, the value of the integrand will become complex for $r<r_{min}$ . As such, $r_{min}$ denotes the minimal value of the bubble radius. At this point, we then take the branch positive of the square root from (2.6), until the scaled maximal value of 1 is attained, after which point we switch again to the negative branch. This process continues, and we see that the bubble radius must oscillate between $r_{min}$ and 1.

From (2.7), we see that $r_{min}$ satisfies

Note that 1 is always a root. For $0<r_{min}<1$ there is at most one root, given by

where $W$ denotes the Lambert $W$ function. Note that for $0\leqslant p\leqslant 1$ , the value of $r_{min}(p)=1$ , while for $p>1$ , $0<r_{min}(p)<1$ . So, there is a solution with radial oscillation for $p>1$ . If, instead, $0\leqslant p\leqslant 1$ , there is no root $0<r_{min}<1$ and the solution never takes values below $r=1$ . Either the solution is constant, or a solution does not exist. In the case of a constant solution, see § 2.4. Therefore, if $p>1$ , the bubble oscillates between the value given by (2.9) and 1, otherwise we do not have a time-variable solution.

We remark that we could have solved (2.2) numerically, and obtained periodic solutions for both $p>1$ and $0<p<1$ . However, in the case of $0<p<1$ , the solutions would have a minimum at $t=0$ rather than a maximum. This simply means that the bubbles would not be appropriately scaled in radius. If we were to re-scale such a bubble so that the maximal value would be one, we would have to re-scale $a$ , which in turn would re-scale $p$ . Therefore, any periodic solution can be described under a scaling which gives the bubble maximal radius at $t=0$ and requires $p>1$ . Similar comments hold for the cases considered subsequently throughout this paper.

2.2 The case of $m<N$

We next consider the $m<N$ case. While this is physically not a common scenario, the results are included for completeness. We should note, however, that polytropic indices less than one have been considered for gas or vapour bubbles undergoing phase change (Fuster & Montel Reference Fuster and Montel2015).

Proceeding with the case of $m<N$ , we assume a non-constant solution and integrate (2.3) subject to the initial conditions (2.5), to obtain

As before, we take the positive root for $t<0$ near the origin, and the negative root for $t>0$ near the origin. For the $t>0$ case, we obtain by separation of variables the implicit formulation

For $t<0$ , we simply have the time reversal of this equation.

In order for the representation (2.11) to make sense, we must have that the square root term is real. However, it is clear that this depends on $p$ . Define the function

We find that there is a critical value of $p$ , $p=p_{crit}$ , such that $F$ will remain positive. In particular,

Since $N>m$ , $F^{\prime }(x)<0$ for $p>p_{crit}$ yet $F(1)=0$ , so $F(x)$ must be positive on $[0,1]$ . Therefore, the integral (2.11) always exists and is finite, for $p>p_{crit}$ . If $p=p_{crit}$ , then the highest-order term is removed, leaving

and this again is positive and decreases on $(0,1)$ until $F(1)=0$ . Therefore, when $p=p_{crit}$ , the integral (2.11) always exists. If $p<p_{crit}$ , then $F\rightarrow -\infty$ as $x\rightarrow 0^{+}$ since $N>m$ . As such, the integrand fails to be real valued as $x\rightarrow 0^{+}$ . There will then exist some value $0<r_{min}<1$ at which $F(r_{min})=0$ . Hence, if $p<p_{crit}$ , then the bubble will not collapse but rather will tend toward some minimal radius, before oscillating back to 1. If $p\geqslant p_{crit}$ , then the bubble will collapse in finite time.

First, for $p=p_{crit}$ , we have from (2.11) that the dynamics of $r(t)$ is given by

Now, the time $t=t^{\ast }$ at which the bubble completely collapses, $r(t^{\ast })=0$ , is given by setting $r(t)=0$ in this equation. We obtain

Similarly, for $p>p_{crit}$ , the time of complete collapse, $t^{\ast }$ , at which $r(t^{\ast })=0$ can be given by the integral

While a closed form for this integral is not forthcoming in general, it can always be calculated for specific $N$ , $m$ and $p$ .

For $p<p_{crit}$ , we see that the dynamics for the bubble is given by (2.11) when the bubble decreases in radius, and the same equation with the negative root taken when the bubble increases in radius. The bubble will therefore oscillate in radius, with $r_{min}\leqslant r(t)\leqslant 1$ . This minimal value corresponds to a zero of $F(x)$ . Note that $\lim _{x\rightarrow 0^{+}}F(x)=-\infty$ , $F(1)=0$ , and $F^{\prime }(x)=0$ at one unique point in the interval $(0,1)$ , say at $x=x_{0}$ , where

Then, $F^{\prime }(x)>0$ on $x\in (0,x_{0})$ and $F^{\prime }(x)<0$ on $x\in (x_{0},1)$ . Therefore, there exists a unique root $r_{min}$ such that $F(r_{min})=0$ and $0<r_{min}<x_{0}$ . Finding this root in closed form involves solving a polynomial equation of arbitrary order. However, $r_{min}$ can be calculated for any fixed $N$ , $m$ and $p$ . Note that $r_{min}$ satisfies the implicit relation

for which there is a unique solution $0<r_{min}<1$ whenever $0<p<p_{crit}$ .

What we have shown here is that, when $N>m$ , the bubble will collapse in a finite time whenever $p\geqslant p_{crit}$ , and the time of collapse is given by (2.17) in general. On the other hand, the bubble persists and oscillates between a scaled radius $r_{min}$ and 1, provided that $p<p_{crit}$ . Therefore, we have classified all qualitative behaviours of higher-dimensional bubbles in the case $N>m$ .

2.3 The case of $m>N$

In the case where $m>N$ , assuming a non-constant solution, and integrating (2.3) subject to the initial conditions (2.5), we obtain

While this implicit solution will appear superficially like that given in (2.11) for $m<N$ , note that the behaviour of the solutions in this case will be qualitatively different. Therefore, we give the $m>N$ derivation for both completeness and to demonstrate some of the differences.

Since $m>N$ , the $r^{-m}$ term is dominant as $r\rightarrow 0^{+}$ . Therefore, the right-hand side of (2.20) tends to $-\infty$ as $r\rightarrow 0^{+}$ . Yet, for real valued dynamics, ${\dot{r}}^{2}\geqslant 0$ , hence the limit $r\rightarrow 0^{+}$ cannot be possible. Therefore, there must exist a minimal possible value of $r(t)$ , $r_{min}$ , such that the dynamics corresponding to (2.20) remains real valued. Assuming the existence of such a minimal value for the radius, the dynamics for $r_{min}<r(t)<1$ corresponds to the positive root of (2.20) for $t<0$ close to the origin, and the negative root for $t>0$ close to the origin. By separation of variables, we obtain the implicit relation

for $t>0$ , with the relevant time-reversed solution valid for $t<0$ .

Since the behaviour of the solution $r(t)$ depends on the lower bound $r_{min}$ , we need to obtain conditions under which such a lower bounded exists. Let us define the term under the square root as the function

Then, $G(1)=0$ while $\lim _{x\rightarrow 0^{+}}G(x)=-\infty$ . Then, there are two possibilities. First, the function $G$ could increase monotonically, and with $G^{\prime }>0$ for all $x\in (0,1)$ , we would have $x=1$ as the only root. We would have $G^{\prime }(1)\geqslant 0$ . In such a case, no $r_{min}$ satisfying $G(r_{min})=0$ and $0<r_{min}<1$ exists. The other possibility is that $G$ increases, becomes positive, and then decreases so that $G(1)=0$ . Therefore, one would have $G^{\prime }(1)<0$ . In this latter case, there would exist $r_{min}$ satisfying $G(r_{min})=0$ and $0<r_{min}<1$ . Therefore, the condition $G^{\prime }(1)>0$ is sufficient for the existence of the desired constant $r_{min}$ . From (2.22), we have

and hence

This implies that $G^{\prime }(1)<0$ only if $p>1$ . Therefore, when $p>1$ , there exists a solution $r(t)$ to the implicit (2.21) which tends to a root $r_{min}$ of $G(x)$ . To find this root, one must solve the polynomial equation $G(x)=0$ , obtaining roots on $x\in (0,1)$ . In the case of multiple roots, one would need the root nearest to, but less than, $x=1$ (which itself is always a root). Since solving $G(x)=0$ involves solving a polynomial equation of degree $m$ with variable coefficients, a closed-form solution for $r_{min}$ in terms of $N$ , $m$ and $p$ is not forthcoming. However, for fixed values of $N$ , $m$ and $p$ , we can always find such a solution, provided that $p>1$ .

We have shown that in the case of $N<m$ , there exists a solution with oscillating radius satisfying $r_{min}\leqslant r(t)\leqslant 1$ provided that $p>1$ . Here, $r_{min}$ is the largest root of $G(x)=0$ satisfying $0<r_{min}<1$ . On the other hand, when $p\leqslant 1$ , a solution does not exist, since the integrand in (2.21) will become complex valued for such a case.

2.4 Time-constant solutions

Let us finally consider the distinct case in which a solution is constant in time. For such a case, any of the three first integrals given above would not be useful. Clearly, a constant solution $r(t)\equiv p^{-1/m}$ exists for the scaled (2.2), provided $p>0$ . From the dynamics discussed above, we saw that the bubble radius would oscillate in time, or the bubble would collapse at some finite value of time. Therefore, it makes sense to inquire about the stability of any constant solution $r(t)\equiv p^{-1/m}$ to (2.2). Let us convert (2.2) into the form of the two-dimensional first-order system,

At the equilibrium corresponding to $r(t)\equiv p^{-1/m}$ , we have $y_{1}\equiv p^{-1/m}$ and $y_{2}\equiv 0$ . Then, the Jacobian for the system (2.25) evaluated at this equilibrium is

and $J$ has eigenvalues

As $n>2$ and $p>0$ , the sign of the term $1-(m+1)p$ will completely determine the dynamics. When $p<1/m+1$ , we have $\unicode[STIX]{x1D706}_{-}<0<\unicode[STIX]{x1D706}_{+}$ , hence the equilibrium is unstable, with deviation from the equilibrium locally like $O(\text{e}^{\unicode[STIX]{x1D706}_{+}t})$ . If $p=1/m+1$ , then we have $\unicode[STIX]{x1D706}_{-}=0=\unicode[STIX]{x1D706}_{+}$ , so the equilibrium is weakly unstable (the instability is order $O(t)$ ). Finally, if $p>1/m+1$ , $\unicode[STIX]{x1D706}_{-}$ and $\unicode[STIX]{x1D706}_{+}$ are purely imaginary, so the equilibrium is a centre. A small perturbation of the equilibrium in this case will result in a solution which oscillates with amplitude of the order of the small perturbation. This is completely consistent with the result we have derived above, with either oscillations or collapse possible, depending on the value of the pressure term.

2.5 Summary

When $N\leqslant m$ , we have that an oscillating solution exists for $p>1$ , which in terms of the original physical parameters means that an oscillating solution is found when

If this condition is not satisfied, an oscillating solution does not exist for $N\leqslant m$ . Meanwhile, when $N>m$ , an oscillating solution exists for $p<p_{crit}=N/N-m$ , which in terms of physical parameters means that

In each case, the oscillating solution is periodic, and both the period of oscillations, $\unicode[STIX]{x1D70F}$ , and the amplitude of oscillations, $A$ , of such a solution can be calculated in terms of the physical parameters. In the isothermal ( $N=m$ ) case we have

and

where $W$ denotes the Lambert $W$ function. When $N\neq m$ , and one of the relevant conditions for the existence of oscillating solutions holds, there always exists a solution $0<\unicode[STIX]{x1D708}^{\ast }<1$ of the polynomial equation $(R(0)/R_{0})^{m}P_{ext}/P_{0}(\unicode[STIX]{x1D708}^{m}-\unicode[STIX]{x1D708}^{N+m})-N/N-m(\unicode[STIX]{x1D708}^{m}-\unicode[STIX]{x1D708}^{N})=0$ . The period of the oscillations is then

while the amplitude is $A=1-\unicode[STIX]{x1D708}^{\ast }/2$ .

A less interesting constant radius solution exists when the pressures exactly balance one another, and this constant solution is given by $R(T)=R_{0}(P_{0}/P_{ext})^{1/m}$ .

Solutions which collapse in finite time $t=t^{\ast }$ were shown to exist for $N>m$ , provided $p\geqslant p_{crit}$ (in contrast to the case when $N\leqslant m$ , for which no solution exists when the inequality involving the initial strain is reversed). While given in terms of scaled parameters in (2.17), we can write the time of collapse as a function of physical parameters, obtaining

Figure 1.

Plot of the normalized bubble radius, $r(t)$ , over time for either (a) $m=N$ (so that the polytropic index is $\unicode[STIX]{x1D6FE}=1$ ; recall that $\unicode[STIX]{x1D6FE}=1$ corresponds to an isothermal bubble) or (b) $m=N+2$ (corresponding to a monatomic ideal gas with $N$ translational degrees of freedom). We also fix the scaled pressure at $p=1.5$ , while varying the dimension of the bubble, $N$ , starting with the most physically relevant case, $N=3$ . As the dimension increases, both the amplitude and period of the oscillation decrease. Therefore, the bubbles contract and oscillate more rapidly in higher dimensions. Identical qualitative trends are found in (b) for the monatomic ideal gas, with the amplitude and period of oscillations both decreased relative to the isothermal case shown in (a).

Figure 2.

Plot of the normalized bubble radius, $r(t)$ , over time for fixed dimension $N=3$ and either (a) $m=N$ (so that the polytropic index is $\unicode[STIX]{x1D6FE}=1$ ; recall that $\unicode[STIX]{x1D6FE}=1$ corresponds to an isothermal bubble) or (b) $m=N+2$ (corresponding to a monatomic ideal gas with $N$ translational degrees of freedom). We vary the scaled pressure $p$ , finding that an increase in $p$ results in an increase in the amplitude of the oscillations. On the other hand, an increase in the scaled pressure, $p$ , results in a decrease in the period of these oscillations. Therefore, as the scaled pressure increases, the bubble will oscillate more rapidly and with greater amplitude.

Figure 3.

Plot of the normalized bubble radius, $r(t)$ , over time for fixed dimension $N=3$ and scaled pressure $p=1.5$ . We vary the parameter $m=N\unicode[STIX]{x1D6FE}$ , where $\unicode[STIX]{x1D6FE}$ is the polytropic index. For small enough $m$ , the bubble will collapse in finite time. For larger $m$ we recover bounded positive solutions which oscillate for all time. As $m$ is increased, the period of the oscillations decreases, so the bubbles oscillate more rapidly. The amplitude of the oscillations is also decreased as the polytropic index increases.

In figure 1 we plot the normalized bubble radius $r(t)$ over time for various choices of the dimension parameter, $N$ . Specific solutions are given for both the isothermal bubble and the bubble containing a monatomic ideal gas with $N$ translational degrees of freedom, although qualitative results will be similar for other choices. We find that the extent of extreme oscillations is restricted as dimension increases, with the dimension parameter $N$ acting to dampen the oscillations, resulting in less fluctuation in the normalized radius of the bubble over time. On the other hand, the frequency of oscillations increases (or, the period between oscillation decreases) as we increase the dimension parameter $N$ . Therefore, the bubbles will oscillate more quickly yet less extremely as the dimension increases. Since the normalization chosen did not depend on dimension, we have that the same qualitative results hold for the non-normalized radius $R(T)$ .

In figure 2 we plot the normalized bubble radius $r(t)$ over time for various choices of the composite scaled pressure parameter, $p$ . Specific solutions are again given for both the isothermal bubble and the bubble containing a monatomic ideal gas with $N$ translational degrees of freedom. As the parameter $p$ increases, the amplitude of the oscillations increases while the period of the oscillations decreases (the frequency of oscillations increases). Therefore, an increase in $p$ destabilizes the bubble, resulting in wilder and more rapid oscillations. Recall that $p\sim P_{ext}$ and $p^{-1}\sim P_{0}$ . Therefore, the influence of an increase in the external pressure constant $P_{ext}$ on the bubble radius will be qualitatively similar to an increase in the composite parameter $p$ , while the reverse behaviour will occur given an increase in the internal pressure constant $P_{0}$ . Hence, an increase in the external pressure will result in more rapid and frequent oscillations, tending to destabilize the bubble from an equilibrium state (which occurs at $p=1$ ). On the other hand, an increase in the internal pressure coefficient has a stabilizing effect on the oscillating bubble, reducing both the amplitude and the frequency of oscillations, and can gradually push the bubble toward the equilibrium at $p=1$ .

We consider the influence of the scaled polytropic parameter, $m$ , in figure 3. Recall that $m=N\unicode[STIX]{x1D6FE}$ , where $N$ is the dimension and $\unicode[STIX]{x1D6FE}$ is the standard polytropic parameter. We see that (holding dimension $N$ fixed) an increase in $m$ results in a decrease in the amplitude of oscillation of the bubbles, yet an increase in the frequency at which such bubbles oscillation (i.e. the period of oscillation decreases with an increase in $m$ ). For very small $m$ , the internal pressure is too weak, and so the external pressure crushes the bubble until collapse. Such small $m$ have been considered for gas or vapour bubbles undergoing phase change (Fuster & Montel Reference Fuster and Montel2015).

Again, figure 3 shows that when internal pressures dominate, the oscillation amplitude will be small, while when external pressures dominate, the bubble can exhibit more extreme oscillations. In the graphic provided, we take $N=3$ corresponding to the most physically interesting case, while we note that similar qualitative behaviours are seen at higher dimensions. The case of $m=3$ in figure 3 corresponds with the isothermal case, while the $m=5$ case corresponds to a monatomic ideal gas with $N$ translational degrees of freedom.

The general dynamics we see is consistent with the specific solutions considered in the papers of Klotz (Reference Klotz2013) and Kudryashov & Sinelshchikov (Reference Kudryashov and Sinelshchikov2014, Reference Kudryashov and Sinelshchikov2015).

Before concluding this section, we shall consider directly the period and oscillation of the oscillating bubble. Making use of the scaled pressure $p=(R(0)/R_{0})^{m}P_{ext}/P_{0}$ , we have the non-dimensional period of oscillation $\unicode[STIX]{x1D70F}_{\ast }$ given by the relation

and explicitly given by

Here, the amplitude $A$ of the bubble oscillation is given by the maximal root $0<A<1/2$ of the equation

for $N\neq m$ , and is given exactly by

for $N=m$ (where again $W$ denotes the Lambert $W$ function). In figure 4 we plot the amplitude of the oscillations as a function of both the non-dimensional pressure $p$ and the parameter $m$ , for fixed $N=3$ . In figure 5 we similarly plot the period of the oscillations as a function of both the non-dimensional pressure $p$ and the parameter $m$ , for of both the non-dimensional pressure $p$ and the parameter $m$ , for fixed $N=3$ . For $p<1$ there are no oscillating solutions and hence $A=0$ . For $p>1$ , the amplitude of the oscillations gradually increases. For larger $m$ the rate of increase in the amplitude of the oscillations with $p$ is slower, and hence for fixed ${>}1$ we see a decreases in the amplitude of the oscillations as we increase $m$ . Note that for small $m$ the amplitude of the oscillations tends toward $A=0.5$ as $p$ increases. When $A=0.5$ , the bubble has reached the maximal (scaled) deflection from the initial radius, corresponding to $r(t)=0$ . Therefore, the bubble will collapse. Hence, for small $m$ and large enough $p$ , we observe bubble collapse, as expected. This is seen in figure 4 when we take $m=2$ and consider $p>3$ . For larger $m$ , the increase in oscillation amplitude increases more slowly, and finite time collapse of a bubble is prevented. From figure 5 we see that the non-dimensional period of oscillation, $\unicode[STIX]{x1D70F}_{\ast }$ , for the bubbles does not strongly depend on $m$ , with only a slight decrease with increasing $m$ for small pressures. Meanwhile, pressure has a strong influence over $\unicode[STIX]{x1D70F}_{\ast }$ , with a sharp decrease in $\unicode[STIX]{x1D70F}_{\ast }$ with pressure for small $p$ , and a more gradual decrease in $\unicode[STIX]{x1D70F}_{\ast }$ with large $p$ . As we take $p\rightarrow 1^{+}$ the period becomes arbitrarily large, as oscillating solutions do not exist for $p<1$ (hence any solutions which do exist would have infinite period).

Figure 4.

Plot of the amplitude $A$ for oscillations of a bubble with normalized bubble radius, $r(t)$ , over time for fixed dimension $N=3$ . The solutions are plotted for various fixed $m=2,3,4,5$ over the non-dimensional pressure $p=(R(0)/R_{0})^{m}P_{ext}/P_{0}$ . For $0<p<1$ oscillating solutions do not exist. We see a decrease in the amplitude of oscillations as the polytropic parameter $m=N\unicode[STIX]{x1D6FE}$ increases.

Similarly, for the case where there can be collapse of the bubble, the non-dimensional collapse time $T_{\ast }$ can be defined by the relation

and is explicitly given by

This is valid only when $N>m$ and $p>p_{crit}=N/N-m$ . In figure 6 we plot this non-dimensional collapse time as a function of the non-dimensional pressure $p$ and the parameter $m$ , for fixed $N=3$ . We see that the influence of $m$ is minimal (there is a slight increase in the collapse time as $m$ increases), while the more pronounced influence is due to the non-dimensional pressure $p$ . In particular, $T_{\ast }\sim 1/\sqrt{p}$ for large enough $p$ . This makes intuitive sense: as the ratio of external to internal pressure is increased, the bubble will collapse more rapidly.

Figure 5.

Plot of the non-dimensional oscillation period $\unicode[STIX]{x1D70F}_{\ast }$ for oscillations of a bubble with normalized bubble radius, $r(t)$ , over time for fixed dimension $N=3$ . The solutions are plotted for various fixed $m=2,3,4,5$ over the non-dimensional pressure $p=(R(0)/R_{0})^{m}P_{ext}/P_{0}$ . For $0<p<1$ oscillating solutions do not exist, hence there is blow-up as $p\rightarrow 1^{+}$ . We see a decrease in the period of oscillations as the polytropic parameter $m=N\unicode[STIX]{x1D6FE}$ increases.

Figure 6.

Plot of the non-dimensional collapse time, $T_{\ast }$ , for bubbles under the conditions $N>m$ and $p>p_{crit}=N/N-m$ . In these plots, we take the dimension to be $N=3$ . Similar plots can be obtained in higher dimensions. We plot the collapse times over $p$ , choosing values $m=1.5,2,2.5$ for the polytropic parameter. We find that the dominating parameter is $p$ , while collapse times vary little with $m$ . For this reason, we can take the limit $m\rightarrow 0$ and use the resulting exact solution for $T_{\ast }$ in order to obtain a qualitatively accurate approximation to the collapse time for all $p>p_{crit}=N/N-m$ . This approximation slightly underestimates the collapse time $T_{\ast }$ , and hence we conclude that the smaller the polytropic parameter, $m$ , the more rapid the collapse of the bubble (although this influence is relatively small). As the non-dimensional pressure $p$ is increased, we see a more pronounced decrease in the collapse time, finding that the collapse time decreases as $T_{\ast }\sim 1/\sqrt{p}$ .

Since the collapse time of the bubble appears to vary little with $m$ , let us take the limit $m\rightarrow 0$ . In this limit, the non-dimensional collapse time $T_{\ast }$ can be calculated exactly:

We take the $N=3$ case and plot it in figure 6. Note that this exact limiting solution is in good agreement with the results obtained for positive values of $m$ . Therefore, from (2.41), the collapse time of a bubble scales like $T_{\ast }\sim 1/\sqrt{p-1}$ when the conditions $N>m$ and $p>p_{crit}=N/N-m$ are satisfied. While such collapse solutions are only possible for $m<N$ (i.e. $\unicode[STIX]{x1D6FE}<1$ ), recall that polytropic indices less than one have been considered for gas or vapour bubbles undergoing phase change (Fuster & Montel Reference Fuster and Montel2015) and therefore such collapse solutions may be physically relevant for such scenarios.

3.1 The case of $P_{0}=0=P_{ext}$

When $P_{0}=0=P_{ext}$ , (1.1) reduces to

This equation has an exact solution for any dimension $N$ and any initial conditions $R(0)$ and $R^{\prime }(0)$ , which is

for $R(0)\neq 0$ . If this solution were to be re-scaled so that $R(0)=0$ , we would recover equation (29) of Klotz (Reference Klotz2013). Without any balancing pressures, this bubble gradually expands or contracts in time for $R^{\prime }(0)\neq 0$ , with the rate of expansion when $R^{\prime }(0)>0$ moderated by the dimension. Meanwhile, if $R^{\prime }(0)<0$ , the bubble will contract until it collapses at time

For a solution with $R^{\prime }(0)=0$ , we obtain a bubble of constant radius over time.

3.2 The case of $P_{0}=0$

If $P_{0}=0$ yet $P_{ext}\neq 0$ (corresponding to an empty bubble), then (1.1) reduces to

Note from the form of (3.4) that there can be no constant solutions. If we assume a solution has a local maximal or minimal value at $t=0$ so that $R^{\prime }(0)=0$ (with this choice of $R^{\prime }(0)$ motivated by the fact that we will be interested in the time from maximal radius to collapse), we obtain a first integral for (3.4) as has been done for similar cases above, and this takes the form

This equation holds for positive $R(T)$ only if $R(0)\geqslant R(T)$ , and one may show that for $P_{ext}>0$ the solution must decrease in radius from $t=0$ , so manipulating this first integral gives the implicit relation between $R(T)$ and $T$ , viz.,

The solution will monotonically decrease in time, with eventual collapse at finite time $T=T^{\ast }$ given by

One may verify that $T^{\ast }$ decreases as the dimension $N$ increases. Actually, noting that

we obtain the interesting series relation

Let us also note that we may express the integral in (3.7) using gamma functions, viz.,

and this gives

For large $N$ , we have the asymptotics

As the dimension $N$ increases, we see a decrease in the collapse time of the bubble. Note that this formula is exactly that obtained for the Rayleigh collapse in equation (14) of Klotz (Reference Klotz2013).

Figure 7.

Plot of the bubble radius, $R(t)$ , over time for fixed dimension $N=3$ in the case where $P_{0}=0$ so that only the external pressure is acting on the bubble. This corresponds to a hollow bubble under the traditional Rayleigh dynamics. We take $P_{ext}/\unicode[STIX]{x1D70C}=1$ and $R(0)=1$ while varying the value of the initial rate of change of the radius, $R^{\prime }(0)$ . Naturally, the time to collapse is decreased when the bubble initially is decreasing in radius, and the time to collapse is increased if the radius is initially increasing. The plot corresponding to $R^{\prime }(0)=0$ corresponds to the Rayleigh collapse in equation (14) of Klotz (Reference Klotz2013).

These results show that when $P_{0}=0$ , the constant external forcing acts destructively on the bubble, resulting in finite time collapse. While we have taken $R^{\prime }(0)=0$ above, note that for $R^{\prime }(0)>0$ the radius will initially increase before decreasing to zero. If $R^{\prime }(0)<0$ , then the radius will decrease more quickly to zero. We provide numerical simulations for these scenarios in figure 7. The qualitative behaviours observed above will remain similar to those for the $R^{\prime }(0)=0$ case, with the exception that the quantitative time to collapse will be modified as follows. By the same approach we have used above to obtain (3.6), when $R^{\prime }(0)\neq 0$ we find

Note that we must be careful with how we choose the $\pm$ root. For $R^{\prime }(0)<0$ , we have a solution with monotone decrease in the radius, so we shall pick the ‘ $+$ ’ root. The time to collapse for such a bubble is then

Due to the appearance of the positive constant $R(0)^{N}(R^{\prime }(0))^{2}$ , this collapse time is less than was found for the $R^{\prime }(0)=0$ case in (3.7). Meanwhile, for $R^{\prime }(0)>0$ , the bubble first grows until it attains a maximal radius, $R=R_{max}$ , after which it collapses to zero radius in finite time. On the initial temporal region until the maximal radius is obtained, we take the ‘ $-$ ’ root, while on the temporal region corresponding to collapse we take the ‘ $+$ ’ root. Therefore, when $R^{\prime }(0)>0$ , we have the time to collapse as

Here, the maximal possible radius of the bubble, $R_{max}$ , can be found analytically by setting $R^{\prime }(T)=0$ . Doing so, the exact closed-form value of $R_{max}$ is found to be

The collapse times (3.14) and (3.15) constitute generalizations of the Rayleigh collapse given in equation (14) of Klotz (Reference Klotz2013) which account for various initial configurations of the bubble.

3.3 The case of $P_{ext}=0$

In the case where the external pressure satisfies $P_{ext}=0$ , while $P_{0}\neq 0$ , one can look at the scaling of § 2, and take $p=0$ . If we consider $p=0$ in the definitions of $F(r)$ and $G(r)$ in §§2.2 and 2.3, respectively, we see that one has $F(r)<0$ and $G(r)<0$ for every $0<r<1$ . Since this $r$ represents a scaling of $R(T)$ such that $\max r(t)=1$ , there can be no oscillating solution in such cases. Therefore, when $P_{ext}=0$ , the solutions $R(T)$ do not oscillate. Furthermore, if we assume a constant solution when $P_{ext}=0$ , we find that (1.1) implies $1/R(T)=0$ , which is not possible. Hence, no constant solution can exist when $P_{ext}=0$ . As such, when $P_{ext}=0$ , the solutions can only collapse or grow in radius asymptotically.

If $N>m$ , we can obtain a particular solution in closed form. When $P_{ext}=0$ , (1.1) reduces to

Assuming a solution of the general form (3.2), we obtain

When the ‘ $+$ ’ root is taken, the solution grows without bound, while when the ‘ $-$ ’ root is taken, the solution monotonically decreases in radius until collapse at a finite time $T=T^{\ast }$ given by

Note that this solution exists only for very specific $R^{\prime }(0)$ , unlike the solution (3.2) in which both $R(0)$ and $R^{\prime }(0)$ are free parameters. Therefore, the bubble collapse is a rather particular case, and we must be more careful with how we investigate this case.

In contrast to the case where $P_{0}=0$ yet $P_{ext}\neq 0$ in which the negative external forcing resulted in bubble collapse, here we have that the positive pressure will most often result in asymptotic growth of the bubble in the absence of a balancing external pressure ( $P_{ext}=0$ ). The exception is when the initial change in bubble radius is a large enough negative value, so that the small time dynamics results in collapse before the slower dynamics can kick in and increase the bubble radius.

From (3.17), we obtain

We may again separate variables and obtain an implicit representation for $R$ in terms of $T$ . For $m=N$ we get

while for $m\neq N$ , we have

The choice of sign again will depend on whether a solution is increasing or decreasing over some interval of time. Again, if $R^{\prime }(0)>0$ , then the solutions will monotonically increase without bound in all cases for $T\geqslant 0$ . If $R^{\prime }(0)=0$ , the solutions will still monotonically increase without bound in all cases for $T>0$ . However, if $R^{\prime }(0)<0$ , then the solutions must initially decrease in radius, before increasing without bound. If the rate of initial decrease is strong enough in magnitude, then collapse is possible. Otherwise, the bubble will attain some minimal radius, before unbounded expansion occurs. As we shall prove below, collapse will be possible only for $m<N$ . In figure 8 we plot these behaviours in the sample cases of an isothermal gas ( $m=N$ ) and a monatomic ideal gas with $N$ translational degrees of freedom ( $m=N+2$ ).

Figure 8.

Plot of the bubble radius, $R(t)$ , over time for fixed dimension $N=3$ when (a) $m=N=3$ (so that the polytropic index is $\unicode[STIX]{x1D6FE}=1$ ; recall that $\unicode[STIX]{x1D6FE}=1$ corresponds to an isothermal bubble) or (b) $m=N+2=5$ (corresponding to a monatomic ideal gas with $N$ translational degrees of freedom). We scale the physical quantities like $P_{0}/\unicode[STIX]{x1D70C}=1$ , $R(0)=1$ , and $R_{0}=1$ , while varying the initial rate of change of the bubble radius, $R^{\prime }(0)$ , as a control parameter. We see for $R^{\prime }(0)<0$ that the bubble radius initially decreases, before increasing without bound due to the lack of any external pressure to balance the gas pressure inside the bubble. On the other hand, for $R^{\prime }(0)\geqslant 0$ , the growth of the radius is immediate and again continues in an unbounded manner. Similar qualitative results are observed for both the isothermal ( $m=N$ ) and monatomic ideal gas with $N$ translational degrees of freedom ( $m=N+2$ ), with the only difference being that the decay or growth in the latter case is somewhat moderated as the pressure term is smaller in this case.

Let us assume that $R^{\prime }(0)<0$ so that the bubble radius initially decreases. We find that the minimal radius is given by

for $m=N$ , while when $m\neq N$ we must have

for $m<N$ , and

for $m>N$ . Now, in order for a bubble to collapse in finite time, we would need $R_{min}\leqslant 0$ , and this is clearly only possible for $m<N$ . When $m\geqslant N$ , the bubble will decrease in size to the relevant minimal radius value given above, before then increasing in radius without bound.

The minimal value of the bubble radius in the case where $R^{\prime }(0)<0$ clearly depends on the value of $R^{\prime }(0)$ , with the value decreasing with an increase in the magnitude of $R^{\prime }(0)$ . Therefore, we can calculate the critical value of $R^{\prime }(0)$ for which the bubble will collapse rather than eventually increase in size. Since this is only possible when $m<N$ , we pick the corresponding formula above and set $R_{min}=0$ , finding that

Yet, upon taking the derivative of (3.18) and evaluating at $T=0$ , we find the exact same value. Hence, at the critical value $R^{\prime }(0)=R^{\prime }(0)_{crit}$ , the bubble collapses and the governing equation is exactly solvable. The time to collapse is again given by (3.19). When $R^{\prime }(0)<R^{\prime }(0)_{crit}$ , the bubble will still collapse, yet more rapidly. While such a case is not exactly solvable, we can obtain the collapse time through (3.22) by integrating from the initial radius value to a zero radius value, viz., the time of collapse is given by

Again, the collapse time formula (3.27) is valid only for $R^{\prime }(0)<R^{\prime }(0)_{crit}$ and when $m<N$ . For all other parameter regimes, the bubble will undergo eventual expansion.

4.1 Additional extensions

In addition to adding surface tension and viscosity effects, there other ways to extend the results we have presented. We list some of these extensions here.

It is natural to wonder about the utility of obtaining analytical expressions rather than simply numerically integrating the Rayleigh–Plesset equation (1.1) in order to find the bubble radius as a function of time. If one knows exactly what they are looking for, that is concerned with a very specific physical scenario which fixes all model parameters, the numerical solution would be easiest. However, the analytical solutions are useful for better understanding the possible qualitative behaviours of solutions. Such solutions are particularly useful if one is more interested in a range of parameters, rather than some fixed parameter values.

One could consider other time-dependent pressure terms. This would make the most physical sense for the external pressure term, although the internal pressure could be taken as time dependent in order to approximate thermal effects. In such cases, one would not expect the existence of first integrals as discussed in the present paper, so any results would be limited to numerical simulations or asymptotic approximations. Still, the qualitative results outlined in this paper would likely serve as a good prediction of the types of dynamics possible in the time-dependent pressure cases where the variation in time was sufficiently gradual. One could also consider variable internal pressures for the purposes of obtaining new and possibly interesting Navier–Stokes solutions.

While we have considered a bubble immersed in an incompressible fluid, it is possible to model the oscillations of a gas-filled bubble in a compressible fluid. As mentioned in Klotz (Reference Klotz2013), it should be possible to obtain a Herring equation which treats compressibility as a small damping term. See Herring (Reference Herring1941) and also the more accessible references Trilling (Reference Trilling1952) and Vokurka (Reference Vokurka1986). Prosperetti & Lezzi (Reference Prosperetti and Lezzi1986) and Lezzi & Prosperetti (Reference Lezzi and Prosperetti1987) give a systematic treatment of the subject and derive governing equations for such bubble dynamics in a compressible fluid. Prosperetti (Reference Prosperetti1987) obtains a third-order ordinary differential equation governing the bubble radius $R(T)$ as an analogue of (1.1) which accurately accounts for compressibility of the fluid. The analysis of such models, along the lines presented here, will be considered in a future work.