Time-dependent cavitation in a viscous fluid

Kinetics of nucleation and growth of empty bubbles in a nonvolatile incompressible fluid under negative pressure is considered within the generalized Zeldovich framework. The transient matched asymptotic solution obtained earlier for predominantly viscous nucleation is used to evaluate the distribution of growing cavities over sizes. Inertial effects described by the Rayleigh-Plesset equation are further included. The distributions are used to estimate the volume occupied by cavities, which leads to increase of pressure and eventual self-quenching of nucleation. Numerical solutions are obtained and compared with analytics. Due to rapid expansion of cavities the conventional separation of the nucleation and the growth time scales can be less distinct, which increases the role of transient effects. In particular, in the case of dominant viscosity a typical power-law tail of the quasistationary distribution is replaced by a time-dependent exponential tail. For fluids of the glycerin type such distributions can extend into the micrometer region, while in low-viscosity liquids (water, mercury) exponential distributions are short lived and are restricted to nanometer scales due to inertial effects.

Figure 1

Dimensionless representation of the main background functions. $w\left(r\right)=3r^2-2r^3$—the scaled work $W/W_{*}$ to form a cavity, and $\dot{r}\left(r\right)=r-1$—the growthor decay rate, time scaled by $\tau$ in Eq. (3), with no inertial effects (solid lines). Inertial corrections, which can be included iteratively at $r\sim 1$, are practically negligible for water and are visible but still small for mercury, as indicated by dashed lines for $B\equiv W_{*}/T=50$. The vertical dashed lines locate the “critical region” where the change of the work $W$ is comparable to thermal energy $T$. The stationary distribution of cavities over sizes $f^{\mathrm{st}}\left(r\right)$ drops from near-equilibrium $f^{\mathrm{eq}}\left(r\right)\sim exp\left[-Bw\left(r\right)\right]$ to the left of the critical region to the growth asymptote $f^{\mathrm{st}}\left(r\right)\simeq I/\dot{r}\left(r\right)$ to the right of that region, $I$ being the nucleation rate. The corresponding solid line is Eq. (7).

Figure 2

Reduced flux $j(r,t)/I\left(0\right)$ at $r=2$ as a function of reduced time $t-t_i\left(r\right)$ for two different initial barriers $B$. Solid line: Eqs. (13) and (15). Symbols: Numerics. The vertical dashed lines indicate quenching of nucleation by growing cavities, Eq. (25) for $B=50$ (right) and $B=20$ (left).

Figure 3

Reduced adiabatic nucleation rate from Eqs. (10) and (25) (solid line) and from numerics (symbols). Note the large-time scaling once time $t$ is reduced by the nucleation time $t_n$.

Figure 4

Exponential tail of the scaled distribution in Eq. (28) at $t\approx t_n$ for the initial barrier $B=20$ (solid line). Symbols: Numerics. he arrow indicates sizes which give maximum contribution to the nucleated volume. Dashed line: Tail of the quasi-steady-state distribution, a power law.

Figure 5

Same as in Fig. 4 for initial barrier $B=50$.

Figure 6

Incubation time including inertial effects in growth for $B=50$. All times are scaled with respective $\tau$ and sizes with respective $R_{*}$ (see Table 1). Solid line: From integration of Rayleigh-Plesset equation. Dashed line: From interpolating Eq. (34). For the sizes considered, inertial effects are invisible for glycerin at room temperature, where the $r$ dependence remains logarithmic.

Figure 7

Transition to quasistationary flux in the case of viscous growth. The curves are labeled by dimensionless time $t$ (in units of $\tau \approx 34$ ns). The size is scaled by the critical size $R_{*}\approx 8.4\AA$ for $B=50$.

Figure 8

Same as in Fig. 7 but for inertial growth. Time is scaled by $\tau \approx 0.024$ ns; size is scaled by $8.4\AA$.

Figure 9

Estimations of the dimensionless nucleation time (in units of $\tau \approx 4.8\sqrt{B}$, ns) as a function of the initial barrier $B$. Symbols: From Eq. (35) with numerical integration of the distribution of nuclei over sizes. Dashed lines: $B/3$ and Eq. (37) for viscous and inertial growth, respectively.

Figure 10

Same as in Fig. 9 for water with $\tau \approx 3.4\sqrt{B}$, ps, and with predominantly inertial growth.