4.1 Cavity shapes

Figure 4 shows images of the shapes of the cavity, starting from its maximum depth and diameter (figure 4 e), to the singular shape at $t_{0}$ just before jet emergence. The forcing amplitude here is $A/R=0.0223$ and $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ , $2R=15~\text{cm}$ .

Figure 3. (a) Coexistence of two different modes in the breaking regime. Modes (2,1) and (0,1) coexist in the breaking regime of axisymmetric (0,1) mode ( $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.954$ , $A/R=0.0281$ ). The dominant mode is (0,1). The three images show the last wave amplitude (stable wave before breaking), cavity formation and jetting ( $T$ is the period). The (2,1) mode interference results in loss of axisymmetry. (b) Coexistence of (0,1) and (3,1) modes at $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=1.04$ , $A/R=0.0208$ . The (3,1) mode is more dominant and if the forcing amplitude $A/R$ is increased, breaking occurs in (3,1) mode.

Composite images of the wave-depression cavity and of jet formation (upper part of each image) are shown in figure 5, obtained in container $2R=15~\text{cm}$ . When $A/R=0.0223$ and $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ (figure 5 a), the cavity collapses without pinch-off at the bottom, whereas when $A/R=0.0224$ pinch-off occurs (figure 5 b), resulting in a lower jet velocity because of dissipation of some of the energy in the form of a bubble and the resulting cavity depth is less.

Figure 4. Sequence of images showing the shape of the cavity during collapse ( $A/R=0.0223$ and $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ ) in a near singular event in the $2R=15~\text{cm}$ container. The shape at $t=t_{0}$ here is the singular state. Images (a–e) are respectively at times 1, 1.5, 2, 3 and 6 ms before the collapse time $t_{0}$ . Time at $Z_{c}$ (maximum depth of cavity after last stable wave amplitude) here is 20 ms before $t_{0}$ .

The bubble pinch-off in the collapsing cavity, when the wave depression is created by a large-amplitude wave, forms an interface similar to vapour bubble collapse/drop impact. Based on potential flow theory of collapsing cavities by Longuet-Higgins (Reference Longuet-Higgins1983), the critical value of vertex angle is $109.5^{\circ }$ . For cavity collapse with bubble pinch-off, the vertex angle ( $\unicode[STIX]{x1D6FE}$ ) is obtained close to the theoretical value (see figure 6 b–d) with an exception when the pinched-off bubble is small (figure 6 a) or very close to the singular event. For near-singular collapse the vertex angle is found to be less than $90^{\circ }$ .

The decay rate $\unicode[STIX]{x1D705}$ of capillary waves (internal damping) is in the linear theory limit and for a clean surface, $\unicode[STIX]{x1D705}=2\unicode[STIX]{x1D708}k^{2}$ (Lamb Reference Lamb1932) and is valid when viscosity effects are weak, which is the case when the wavenumber $k\ll \sqrt{\unicode[STIX]{x1D714}/\unicode[STIX]{x1D708}}$ (Denner Reference Denner2016). Krishnan, Hopfinger & Puthenveettil (Reference Krishnan, Hopfinger and Puthenveettil2017) showed that this damping rate is a good approximation for damping of small-amplitude capillary waves on the cavity boundary in bubble collapse in water and even in glycerine–water. Adding detergent may, besides lowering the surface tension, give rise to surface tension gradients (hence induced fluid motion) due to straining by waves, in a way similar to that shown by Hürhnerfuss, Lange & Walter (Reference Hürhnerfuss, Lange and Walter1985) for other additives. In the present situation, the amplitude of capillary waves on the cavity boundary is small so that wave straining is most likely also small. Decay experiments (figure 7) are conducted with capillary waves in water and with 1 % detergent water solution. As seen in figure 7, the wave amplitude decay is exponential with constant $\unicode[STIX]{x1D705}$ , but in both water and 1 % detergent solution the decay rate is slightly larger than given by the classical expression. Writing $\unicode[STIX]{x1D705}=C\unicode[STIX]{x1D708}k^{2}$ , experiments give $C=2.1$ for water and $C=2.4$ for water with 1 % detergent. The wavenumbers $k$ are determined by measuring the wavelengths in the experiments and are respectively $k=1100$ and $1220~\text{m}^{-1}$ . A prefactor larger than 2 is plausible for real fluids as is shown by Rajan & Henderson (Reference Rajan and Henderson2018) (their equation (75c)). To study capillary wave damping in more detail goes beyond the scope of the present work. The interest here is to show the scaling for wave damping and get an estimation for complete damping of capillary waves.

Figure 5. Composite images of cavity just before collapse (bottom of each image) and the jetting (upper part of each image) in container $2R=15~\text{cm}$ : (a) $A/R=0.0223$ and $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ , near singular case with jet velocity as high as $44~\text{m}~\text{s}^{-1}$ ; (b) $A/R=0.0224$ , $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ ; the velocity is $18~\text{m}~\text{s}^{-1}$ .

The characteristic time can be taken as half the gravity wave (mode 0,1) period $t=\unicode[STIX]{x03C0}/\sqrt{3.832g/R}$ . Disturbances of wavelength $\unicode[STIX]{x1D706}$ can thus be considered fully damped when $\unicode[STIX]{x1D705}t\approx 4$ . Thus, waves of wavelength

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D706}/R\lesssim 3.98\sqrt{C/Re_{I}}\end{eqnarray}$$

will be damped, where $Re_{I}=(R^{3}g)^{1/2}/\unicode[STIX]{x1D708}$ is a Reynolds number defined with the gravitational velocity. For water, in a container of 10 cm in diameter, $Re_{I}=3.5\times 10^{4}$ ; hence $\unicode[STIX]{x1D706}/R<0.073$ . In glycerine–water of $10^{2}$ times the viscosity of water, $\unicode[STIX]{x1D706}/R\approx 0.3$ (assuming $C=2$ ), which is sufficient for obtaining smooth wave-depression contours. However, it is likely that also in glycerine–water $C>2$ , so that in glycerine–water waves $\unicode[STIX]{x1D706}/R>0.3$ are already damped. Composite images of last stable wave amplitude and cavity shape 2.5 ms before implosion are shown in figure 7(b). Water shows the presence of capillary waves on the cavity interface just before cavity implosion. As the decay rate in 1 % detergent solution is larger, the capillary waves are damped resulting in a smoother cavity.

Figure 6. The vertex angles of the interface in four different pinch-off collapses for $2R=15~\text{cm}$ are shown. (a) Just pinch-off for near singular case $A/R=0.02796$ with vertex angle $\unicode[STIX]{x1D6FE}=90^{\circ }$ ; (b) $A/R=0.0281$ , $\unicode[STIX]{x1D6FE}\approx 108.5^{\circ }$ ; (c) $A/R=0.02987$ , $\unicode[STIX]{x1D6FE}\approx 110^{\circ }$ ; (d) $A/R=0.030$ , $\unicode[STIX]{x1D6FE}\approx 105.6^{\circ }$ .

Figure 7. (a) Log–linear plot of capillary wave amplitude verses time after forcing is stopped: $+$ , water, $\unicode[STIX]{x1D705}=2.53~\text{s}^{-1}$ ; $\times$ , water with 1 % detergent, $\unicode[STIX]{x1D705}=3.58~\text{s}^{-1}$ ; $\unicode[STIX]{x1D714}=110~\text{rad}~\text{s}^{-1}$ . (b) Composite images of last wave amplitude (in the upper half) and cavity shape $2.5~\text{ms}$ before cavity implosion (in the lower half) for water with $b/R=0.944$ in the left-hand image and 1 % detergent solution with $b/R=0.956$ in the right-hand image.

4.2 Cavity dynamics

The exponents of the power law in time of cavity radius change are directly related to the relevant forces that act on the cavity (Longuet-Higgins & Oguz Reference Longuet-Higgins and Oguz1995, Reference Longuet-Higgins and Oguz1997). Figure 8 indicates how $(Z(t)-Z_{0}(t_{0}))$ (the axial change of cavity) and the radial changes are measured when jetting conditions are approached from the full-size cavity $Z_{c}$ (after last stable wave amplitude). As indicated in the inset, $Z_{0}(t_{0})$ represents the cavity depth at time $t_{0}$ ( $t_{0}$ corresponds to jet initiation) measured from the free surface and $Z(t)$ refers to the depth at any time instant before the collapse.

Figure 8. Cavity shapes with definitions of $Z_{c}$ , $r_{1}$ , $r_{0}$ and $Z_{0}$ . $Z_{c}$ is the maximum depth following the last stable wave amplitude $b$ and $r_{1}$ is the initial minimum cavity radius. $Z_{0}$ is the depth of the cavity when jetting starts and $r_{0}$ is the radius. Parameters $r_{m}$ and $Z$ represent the minimum radius and depth of the cavity at any instant of time $\unicode[STIX]{x1D70F}$ . The dashed lines correspond to $z=0$ for pinch-off and non-pinching cases separately.

Figure 9. (a) Log–log plot of radial change of cavity, without pinch-off, as a function of time $\unicode[STIX]{x1D70F}^{\ast }=\unicode[STIX]{x1D70F}/\sqrt{(R/g)}$ for three different forcing amplitudes $A/R$ in container $2R=15~\text{cm}$ . At the beginning, radial cavity shrinkage follows $r_{m}/R\propto \unicode[STIX]{x1D70F}^{\ast \unicode[STIX]{x1D6FC}}$ with $\unicode[STIX]{x1D6FC}=0.5$ over a short time; then, $\unicode[STIX]{x1D6FC}$ is slightly larger, $\unicode[STIX]{x1D6FC}=0.53$ , with a viscous cut-off of $\unicode[STIX]{x1D6FC}\approx 1$ at small $\unicode[STIX]{x1D70F}^{\ast }$ . *, $A/R=0.0223$ , $U_{j}=45~\text{m}~\text{s}^{-1}$ ; ○, $A/R=0.0273$ , $U_{j}=36~\text{m}~\text{s}^{-1}$ ; $\times$ , $A/R=0.0368$ , $U_{j}=28~\text{m}~\text{s}^{-1}$ ; ▫, axial change of cavity, $(Z(t)-Z_{0}(t_{0}))$ , corresponding to $A/R=0.0223,U_{j}=45~\text{m}~\text{s}^{-1}$ with ‘a’–‘e’ corresponding to the cavity images in figure 4. (b) Radial change of cavity with pinch-off as a function of $\unicode[STIX]{x1D70F}^{\ast }$ for $A/R=0.0224$ (see figure 5 b). As in (a), $\unicode[STIX]{x1D6FC}$ changes from 0.5 to $\unicode[STIX]{x1D6FC}\approx 1$ .

Figure 9(a) shows the radial and axial shrinkage of the cavity as a function of dimensionless time $\unicode[STIX]{x1D70F}^{\ast }=\unicode[STIX]{x1D70F}/\sqrt{(R/g)}$ , where $\unicode[STIX]{x1D70F}=(t_{0}-t)$ , for three singular ( $b$ close to singular wave amplitude $b_{s}$ ) non-pinch-off regimes, respectively for $A/R=0.0223$ , $A/R=0.0273$ and $A/R=0.0368$ . Clearly, two exponents $\unicode[STIX]{x1D6FC}$ in $r_{m}/R\propto \unicode[STIX]{x1D70F}^{\ast \unicode[STIX]{x1D6FC}}$ exist. At larger times, $\unicode[STIX]{x1D70F}>1~\text{ms}$ , the exponent is 0.5, for both radial and axial shrinkage. However, close to the singularity, when $\unicode[STIX]{x1D70F}<1~\text{ms}$ , the time exponent of radial shrinkage is $\unicode[STIX]{x1D6FC}\approx 1$ which is characteristic of viscous effects. The radial shrinkage of the cavity with pinch-off is shown in figure 9(b), where the change in exponent from $\unicode[STIX]{x1D6FC}\approx 0.5$ to 1 also occurs. The small deviation of $\unicode[STIX]{x1D6FC}$ from $0.5$ is highlighted when the radial and axial changes are compensated by $\unicode[STIX]{x1D70F}^{-1/2}$ as shown in figure 10(a). Furthermore, with logarithmic correction $r_{m}/R(-\log (r_{m}/R))^{1/4}$ as proposed in bubble pinch-off (Gordillo et al. Reference Gordillo, Sevilla, Rodríguez-Rodríguez and Martinez-Bazan2005; Bergmann et al. Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006) and axisymmetric bubble collapse (Eggers et al. Reference Eggers, Fontelos, Leppinen and Snoeijer2007), the exponent is $\unicode[STIX]{x1D6FC}=0.5$ (figure 10 b). Since both radial and axial length scale follow the same power-law exponent ( $\unicode[STIX]{x1D6FC}\approx 1/2$ ), the cavity collapse is self-similar. Figure 11(a) shows the surface profiles of the collapsing cavity at different instants of time. When the radial and axial changes are normalized by $\unicode[STIX]{x1D70F}^{-1/2}$ (figure 11 b) all the profiles collapse to one curve confirming the self-similar cavity collapse. The self-similarity is lost when crossover from $\unicode[STIX]{x1D6FC}=0.5$ to 1 is approached. This crossover is similar to the one observed by Burton & Taborek (Reference Burton and Taborek2007) for coalescence of liquid lenses where they observed viscous scaling of neck radius ( $\unicode[STIX]{x1D6FC}\approx 1$ ) until 1 ms and thereafter inertial scaling ( $\unicode[STIX]{x1D6FC}\approx 0.5$ ). In capillary-driven flow, onset of viscous effects is related to a critical value of Ohnesorge number $Oh_{c}=\unicode[STIX]{x1D707}/\sqrt{\unicode[STIX]{x1D70C}R\unicode[STIX]{x1D70E}}\approx 0.03$ which is the inverse of a Reynolds number defined with the capillary velocity. In the present case, the transition is inertial–viscous, hence determined by a critical value of the inertial Reynolds number $Re_{I}=(R^{3}g)^{1/2}/\unicode[STIX]{x1D708}$ . A time of 1 ms corresponds to $Re_{I}\approx 500$ .

Zeff et al. (Reference Zeff, Kleber, Fineberg and Lathrop2000) in their experiments on wave-depression cavity collapse in a container of size similar (12.5 cm in diameter) to that used in the present experiments, assumed that the collapse is driven by surface tension which is related to a power law $(Z(t)-Z_{0}(t_{0}))\propto \unicode[STIX]{x1D70F}^{2/3}$ . The choice of surface tension and inertia as the competing forces, based on the analysis of Keller & Miksis (Reference Keller and Miksis1983) in breaking of liquid sheets, limits their analyses to an exponent $2/3$ . Clearly, as shown in figure 9(a), in the present experiments, cavity shrinkage $(Z(t)-Z_{0}(t_{0}))$ scales as $(Z(t)-Z_{0}(t_{0}))\propto (t_{0}-t)^{\unicode[STIX]{x1D6FC}}$ , with $\unicode[STIX]{x1D6FC}=0.527$ for container size $2R=15~\text{cm}$ with a similar value in the case of container size $2R=10~\text{cm}$ . The considerably more viscous liquid used by Zeff et al. (Reference Zeff, Kleber, Fineberg and Lathrop2000) limits the time range of existence of exponent 0.5 and hence might lead to a larger apparent value of the exponent.

In the case of a large cavity generated by a moving circular disc (Bergmann et al. Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006), collapse always results in bubble pinch-off. In these experiments the time evolution of radial length scale follows a time exponent $\unicode[STIX]{x1D6FC}\approx 0.6$ . The cavity collapse with bubble pinch-off, as shown in figure 5(b), is similar. It occurs in the present experiments when the last stable wave amplitude $b$ exceeds the singular wave amplitude $b_{s}$ . As in the case of no pinch-off, the radial change of the collapsing cavity follows a power law $r_{m}/R\propto \unicode[STIX]{x1D70F}^{\ast \unicode[STIX]{x1D6FC}}$ up to certain time with a crossover from 0.5 to nearly $\unicode[STIX]{x1D6FC}\approx 1$ at $\unicode[STIX]{x1D70F}\approx 1~\text{ms}$ as is seen in figure 9(b). This differs from the results of Bergmann et al. (Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006) where the exponent is closer to 0.6, tending towards 0.5 only at large Froude number, defined with the disc radius and speed. In the present experiments, a similar Froude number $Fr_{c}$ can be defined with cavity radius $r_{1}$ and wave velocity $V_{w}=b\unicode[STIX]{x1D714}$ in the form $Fr_{c}=(b\unicode[STIX]{x1D714})^{2}/gr_{1}$ which has a value $Fr_{c}\approx 10$ . The difference in exponents for similar Froude numbers in the Bergmann et al. (Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006) experiments is attributed to the initial shapes of the cavities. In the present experiments, the initial shape is already close to self-similar shape which is not the case in with the cavity created by a moving disc.

Figure 10. (a) Log–log plot of radial change of cavity compensated with $\unicode[STIX]{x1D70F}^{\ast -1/2}$ as a function of $\unicode[STIX]{x1D70F}^{\ast }$ . The left-hand axis shows the radial evolution of cavity and the right-hand axis (of different scale) shows the axial change of collapsing cavity. The symbols $\ast$ , $A/R=0.0223$ , $U_{j}=45~\text{m}~\text{s}^{-1}$ ; $\times$ , $A/R=0.0273$ , $U_{j}=36~\text{m}~\text{s}^{-1}$ ; and $\circ$ , $A/R=0.0368$ , $U_{j}=28~\text{m}~\text{s}^{-1}$ represent the radial changes similar to figure 9(a), and the symbols ▫ and ♢ show the axial change of cavity for $2R=15~\text{cm}$ and $2R=10~\text{cm}$ , respectively. (b) Log–log plot of radial change of cavity with time with logarithmic correction $(r_{m}/R)(-\text{log}(r_{m}/R))^{1/4}$ for the same conditions as in (a). The dashed line shows the 0.5 slope.

Figure 11. (a) Cavity shapes of the near singular collapse for $A/R=0.0223$ , at times 20, 18, 14, 10, 5, 2 and 1 ms before $t_{0}$ for container of $2R=15~\text{cm}$ . The radial and axial changes follow the power law $\unicode[STIX]{x1D70F}^{1/2}$ . (b) Cavity shapes scaled with the power law $\unicode[STIX]{x1D70F}^{1/2}$ . The axes are non-dimensionalized with container radius and compensated with $\unicode[STIX]{x1D70F}^{-1/2}$ . The surface profiles at different times collapse well into one self-similar form.

5.1 Forcing amplitude dependency

When the wave-depression cavity collapses, a jet is formed, a phenomenon observed by Longuet-Higgins (Reference Longuet-Higgins1983) and analysed in terms of finite-time singularity by Zeff et al. (Reference Zeff, Kleber, Fineberg and Lathrop2000). As discussed in § 4, in the present experiments the collapse is driven by gravity as is indicated by the time dependency of the cavity shrinkage. The cavity aspect ratio, $Z_{c}/r_{1}$ (see figure 8 for definition), plays an important role in generating high-velocity jets. A small diameter and deep cavity give rise to high jet velocity.

Figure 12. (a) Jet velocities plotted against $A/R$ as indicated in figure 1, at $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ for the container $2R=15~\text{cm}$ . The highest values correspond to near singular events. The non-pinching (*) is followed by pinching (○). For higher bands, pinching is followed by splash or incomplete cavity collapse ( $0.031<A/R<0.033$ ) which is again followed by a non-pinch-off zone. The highest velocity obtained is $45~\text{m}~\text{s}^{-1}$ . The dotted lines show the transition from non-pinch-off to pinch-off cavity collapse. (b) The dimensionless last stable wave amplitude $b/R$ as a function of forcing amplitude $A/R$ .

At a given frequency, $b/R$ depends (nonlinearly) on the forcing amplitude $A/R$ (figure 12 b) which suggests representing the jet velocity as a function of $A/R$ . This is shown in figure 12(a) where the non-dimensional jet velocity is plotted as a function of $A/R$ for a range of forcing amplitudes $0.0198\leqslant A/R\leqslant 0.038$ and at constant frequency close to the natural frequency ( $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ ) to obtain a clear mode (0,1) wave. The results would be the same if the frequency were different as long as there is no contamination by other modes of the type indicated in figure 3. Jetting starts above a certain forcing amplitude and when it is increased further the jet velocity increases rapidly. The aspect ratio $(Z_{c}/r_{1})$ of the fully grown cavity increases with forcing amplitude until pinch-off occurs when $b>b_{s}$ ( $b_{s}$ is referred to as singular wave amplitude). Then, there is a range of $A/R$ over which pinch-off persists and the wave amplitude decreases, until, at larger forcing amplitudes, the last stable wave amplitude starts to increase again. This pattern is repeated with the bands having varying width (figure 12 a). At higher $A/R$ , splashing with incomplete cavity collapse can occur. This incomplete cavity collapse is again followed by proper cavity collapse with no bubble pinch-off. As it is very difficult to continually operate the shaker with high precision, the bands cannot be determined with greater accuracy. Nevertheless, their existence is well established. From figure 12 it is clear that the transition from non-pinch-off to pinch-off is very sensitive to $A/R$ . Nearly perfect singularity (perfect collapse of cavity) needs a very fine adjustment of $A/R$ . At large values of $A/R$ , the axisymmetric mode starts to coexist with the asymmetric mode (2,1) and the wave motion tends to become chaotic.

In figure 13 wave-amplitude growth rates are shown for different forcing amplitudes. It is seen that the last stable wave amplitude is reached in less than 10 wave periods and the growth rate increases with $A/R$ . Symbols $\times$ and $\circ$ represent non-pinch-off cases at $A/R=0.0216$ and $A/R=0.027$ (see figure 12 a) with the same value of $b/R$ , and hence result in a similar jet velocity. This also holds for the pinch-off cases. Since the growth rate increases with $A/R$ , the bands of non-pinching to pinching get narrower with forcing amplitude (figure 12 a).

Figure 13. Wave-amplitude growth as a function of $t/T$ : ▫, $A/R=0.0198$ (no pinch-off), with very low jet velocity; $\times$ , $A/R=0.0216$ and ○, $A/R=0.027$ with same jet velocity, $U_{j}\approx 13~\text{m}~\text{s}^{-1}$ ; $+$ , $A/R=0.0224$ (pinch-off) and $\ast$ , $A/R=0.0281$ with similar jet velocity. In the two no-pinch-off and pinch-off cases the last stable wave amplitudes ( $b/R$ ) are the same, but $t/T$ is less when $A/R$ is larger. Dotted horizontal line indicates the same last stable wave amplitude for different bands.

5.2 Power-law scaling

The jet velocity has been measured just above the free surface and each experimental run has been started with the fluid interface at rest. As demonstrated here, the jet velocity is also given by the final, near singular cavity collapse rate. The cavity starts to shrink rapidly in the radial and to a lesser extent in the vertical direction and after the cavity radius has decreased to $r_{0}$ and $Z_{0}$ there is vertical retraction at the jet speed.

Experiments show that $Z_{c}(t)$ is proportional to the last stable wave amplitude and when there is no pinch-off $Z_{0}(t_{0})$ is proportional to $Z_{c}(t)$ , thus

(5.1) $$\begin{eqnarray}Z_{0}(t_{0})/b=B_{1},\end{eqnarray}$$

where $B_{1}\approx 0.62$ (no pinch-off) is a constant.

After the singular time $t_{0}$ , the cavity retracts vertically at a constant velocity so that the jet velocity is given by

(5.2) $$\begin{eqnarray}U_{j}=Z_{0}(t_{0})/t_{j}=B_{1}b/t_{j},\end{eqnarray}$$

where $t_{j}$ is the time from the start of vertical retraction (from $Z_{0}(t_{0})$ ) to jet emergence at the free surface. A constant jet velocity inside the cavity implies that the impulse that initiates the jet happens over a distance $\unicode[STIX]{x0394}z\ll Z_{0}$ . The dimensionless jet velocity is then given by

(5.3) $$\begin{eqnarray}\frac{U_{j}}{\sqrt{Rg}}=\frac{B_{1}}{t_{j}\sqrt{g/R}}\frac{b}{R}.\end{eqnarray}$$

Experiments show that the dimensionless time $t_{j}\sqrt{g/R}$ decreases rapidly with $b/R$ , i.e.  $t_{j}/(\sqrt{R/g}\sim (b/R)^{-n})$ , and figure 11 a, in which is plotted $t_{j}^{\ast }=t_{j}/(\sqrt{R/g}(b/R)^{-9})$ as a function of $b/R$ , indicates that $n\approx 9$ and $t_{j}^{\ast }\approx 0.0104$ . Substitution of these values in (5.2) gives

(5.4) $$\begin{eqnarray}\frac{U_{j}}{\sqrt{Rg}}=\frac{0.62}{0.0104}\left(\frac{b}{R}\right)^{n+1}\approx 60\left(\frac{b}{R}\right)^{10}.\end{eqnarray}$$

In figure 14(b) the dimensionless jet velocity $U_{j}/\sqrt{Rg}$ is plotted as a function of $b/R$ in a doubly logarithmic plot. It is seen that the measured jet velocities are in good agreement with the power-law model. The square symbols in figure 14(b) represent jetting with bubble pinch-off.

Figure 14. (a), Dimensionless time $t_{j}^{\ast }=t_{j}/(\sqrt{R/g}(b/R)^{-9})$ as a function of $b/R$ . The solid line shows the best least-squares fit of the data. (b) Log–log plot of dependency of jet velocity ( $U_{j}/(gR)^{1/2}$ ) on last stable wave amplitude ( $b/R$ ). The dashed line shows the power-law fit with an exponent 10 and prefactor 60 in (5.4). Symbol ▫ shows the pinch-off cavity collapse, where the jet velocities are reduced due to the loss of energy by the downward jet.

5.3 Finite-time singularity scaling

The behaviour shown in figure 12(a) suggests a finite-time singularity scaling. Zeff et al. (Reference Zeff, Kleber, Fineberg and Lathrop2000) proposed a constant Weber number behaviour, $We=\unicode[STIX]{x1D70C}U_{j}^{2}(b-b_{s})/\unicode[STIX]{x1D70E}$ , where $b_{s}$ is the singular wave amplitude beyond which pinch-off occurs and the jet velocity decreases rapidly as is seen in figure 12(a). This capillary scaling is of dimensionless form:

(5.5) $$\begin{eqnarray}\frac{U_{j}}{\sqrt{\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}R}}=\left[We_{c}\frac{R}{|b-b_{s}|}\right]^{1/2},\end{eqnarray}$$

where $We_{c}$ is determined from the best fit of the data. As mentioned above, the diameter in their experiments is in the range of those of the present cylindrical containers used. The collapse should, therefore, be driven by inertia as indicated by the value of the exponent $\unicode[STIX]{x1D6FC}$ (see § 4.2). Furthermore, the ratio of static pressure $\unicode[STIX]{x1D70C}gZ_{c}$ to the surface tension force $\unicode[STIX]{x1D70E}/r_{1}$ gives a modified Bond number $(\unicode[STIX]{x1D70C}gR^{2}/\unicode[STIX]{x1D70E})(r_{1}Z_{c})/R^{2}\approx 40$ indicating that inertial forces are dominant and suggesting a Froude number scaling.

In § 5.2 it is shown that $t_{j}$ , in (5.2), scales as $\sqrt{R/g}$ , with a prefactor that depends strongly on $b$ . In finite-time singularity scaling, when the singularity is approached, i.e.  $b\rightarrow b_{s}$ , the time $t_{j}\rightarrow 0$ . Thus, the expression of the time $t_{j}$ can be written as

(5.6) $$\begin{eqnarray}t_{j}=B_{2}\left[\frac{|b-b_{s}|}{b}\right]^{1/2}\sqrt{R/g}.\end{eqnarray}$$

In figure 15(a), the dimensionless time $t_{j}/(\sqrt{(R/g)|b-b_{s}|/b})$ corresponding to $B_{2}$ is plotted as a function of $b/R$ . The solid line is the best fit and is $B_{2}=0.252$ . Substituting the expression for $t_{j}$ , equation (5.6), in (5.2) gives for the dimensionless jet velocity

(5.7) $$\begin{eqnarray}\frac{U_{j}}{\sqrt{Rg}}=\frac{B_{1}}{B_{2}}\left[\frac{b}{R}\right]^{3/2}\left[\frac{R}{|b-b_{s}|}\right]^{1/2},\end{eqnarray}$$

where $B_{1}/B_{2}\approx 2.5$ . In figure 15(b) the dimensionless jet velocity $U_{j}/\sqrt{Rg}$ is compared with the singular scaling (5.7). There is good agreement and the singular amplitude $b_{s}$ at which the jet velocity could theoretically be infinite is well identified.

Figure 15. (a) Dimensionless time $t_{j}/(\sqrt{(R/g)|b-b_{s}|/b})$ as a function of $b/R$ . The error bars show that the experimental error gets larger close to the singular cases when $t_{j}$ gets close to the frame resolution. The constant line (——) corresponds to the best least-squares fit. (b) Singular scaling for the first band in $A/R$ ( $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{0}=0.995$ ) for two different container sizes $2R=15~\text{cm}$ ( $0.0198<A/R<0.0257$ , figure 12 a) and $2R=10~\text{cm}$ . Symbols $\times$ , $\ast$ show the non-pinching cavities and ▫, $\circ$ show the pinch-off cavities for $2R=15~\text{cm}$ and $2R=10~\text{cm}$ , respectively. The dashed line shows the theoretical line plotted with $B_{1}/B_{2}=2.5$ .

5.4 Impulse model

It is of interest to develop an impulse model in an attempt to clarify how the jet velocity is related to $U_{r}$ , the radial cavity shrinkage velocity, and to determine the vertical extent of the pressure impulse. As the cavity collapses, fluid particles move towards the centre of the cavity and then accelerate axially, resulting in a thin, high-velocity jet. In other words, strong radial momentum creates a high-pressure peak at the centre whereas pressure at the free surface is constant. This gives rise to strong acceleration of the fluid particle in the form of a jet as observed by Longuet-Higgins (Reference Longuet-Higgins2001). The pressure is localized (flip-through) in a way similar to wave impact in shallow water (Cooker & Peregrine Reference Cooker and Peregrine1995). The pressure builds up in a very short time, so that the momentum equation is dominated by the time-dependent pressure gradient. The viscous and surface tension terms are negligible (large Reynolds and Weber numbers) and the gravity term is of the order of the convective term or less (Froude number of order one or larger). The importance of the convective term with respect to the unsteady terms in the momentum equation is of order $U\unicode[STIX]{x1D6FF}t/L$ and taking for $U$ the jet velocity $U_{j}$ , for $\unicode[STIX]{x1D6FF}t$ the impulse time and for the spatial gradient scale $L=R$ , the ratio of convective term is of order $U_{j}\unicode[STIX]{x1D6FF}t/R$ . It has been shown above that $U_{j}t_{j}\leqslant 0.6R$ and, since the velocity inside the cavity is constant, it is necessary that $\unicode[STIX]{x1D6FF}t\ll t_{j}$ so that $U_{j}\unicode[STIX]{x1D6FF}t/R\ll 1$ . This also applies in the radial direction where $L\sim r_{1}$ , but $U$ is also less than $U_{j}$ . Hence, the governing equation reduces to (Cooker & Peregrine Reference Cooker and Peregrine1991)

(5.8) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{i}}{\unicode[STIX]{x2202}t}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p(r,z,t)}{\unicode[STIX]{x2202}x_{i}},\end{eqnarray}$$

where $i=(r,z)$ and $z=0$ is at the free surface (see figure 8). Integrating (5.8) over the time interval $t_{a}$ to $t_{b}$ , we get

(5.9) $$\begin{eqnarray}u_{ib}-u_{ia}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}P}{\unicode[STIX]{x2202}x_{i}},\end{eqnarray}$$

where $P(x_{i})$ is the pressure impulse:

(5.10) $$\begin{eqnarray}P(r,z)=\int _{t_{a}}^{t_{b}}p(r,z,t)\,\text{d}t,\end{eqnarray}$$

where $(t_{b}-t_{a})$ is the pressure impulse time interval $({\sim}1~\text{ms})$ , and $r$ and $z$ are of range $r\approx r_{0}$ and $z\approx (Z_{0}+\unicode[STIX]{x1D6FF}z)$ . At the free surface, the pressure is constant and we take as reference $P=0$ . In the radial direction, at time $t_{b}$ the radial velocity $u_{rb}=0$ and $\unicode[STIX]{x2202}P/\unicode[STIX]{x2202}r=0$ . At time $t<t_{a}$ $(r>r_{0})$ , the radial velocity $u_{ra}$ scales with the cavity collapse velocity $U_{r}=-C_{r}\sqrt{gR}$ . The coefficient $C_{r}$ can be evaluated from figure 10(a), which shows that $r_{m}/R\approx 0.4\unicode[STIX]{x1D70F}^{\ast 1/2}$ for $\unicode[STIX]{x1D70F}^{\ast }>10^{-2}$ , recalling that $\unicode[STIX]{x1D70F}^{\ast }=\unicode[STIX]{x1D70F}/\sqrt{R/g}$ , $\unicode[STIX]{x1D70F}=(t_{0}-t)$ and $r_{m}$ is the minimum radius of cavity at any instant of time. The radial velocity is then

(5.11) $$\begin{eqnarray}U_{r}=-\frac{\text{d}r_{m}}{\text{d}\unicode[STIX]{x1D70F}}\approx -\frac{0.2}{\unicode[STIX]{x1D70F}^{\ast 1/2}}\sqrt{Rg}\approx -0.08\frac{R}{r_{m}}\sqrt{Rg},\end{eqnarray}$$

giving $C_{r}=0.08(R/r_{m})$ . Equation $(5.11)$ indicates that $U_{r}$ increases with decreasing $r_{m}$ and is constant in the viscous regime when $\unicode[STIX]{x1D70F}^{\ast }<10^{-2}$ (see figure 10 a).

Figure 16. Comparison of experimental (non-pinch-off data in figure 12 a) and theoretical prediction (impulse model) of jet velocities for container $2R=15~\text{cm}$ . The prefactor in (5.14) is 5. The error bars indicate the measurement error of radius $r_{0}$ .

The vertical velocity at $t_{a}$ is $u_{za}=0$ and at time $t_{b}$

(5.12) $$\begin{eqnarray}u_{zb}=U_{j}=-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}P}{\unicode[STIX]{x2202}z}.\end{eqnarray}$$

Equation (5.12) shows that the jet velocity is determined by the vertical gradient of the pressure impulse, which is of order $\unicode[STIX]{x1D6FF}P/\unicode[STIX]{x1D6FF}z$ . From (5.9) we get $\unicode[STIX]{x1D6FF}P\sim -\unicode[STIX]{x1D70C}U_{r}\unicode[STIX]{x1D6FF}r$ , hence

(5.13) $$\begin{eqnarray}U_{j}\sim U_{r}\frac{\unicode[STIX]{x1D6FF}r}{\unicode[STIX]{x1D6FF}z}\sim -0.08\frac{R}{r_{m}}\sqrt{Rg}\frac{\unicode[STIX]{x1D6FF}r}{\unicode[STIX]{x1D6FF}z}.\end{eqnarray}$$

Taking $\unicode[STIX]{x1D6FF}r\approx r_{m}$ and the axial impulse variation order $\unicode[STIX]{x1D6FF}z\sim r_{0}$ , and noting that $z$ is downwards, we get

(5.14) $$\begin{eqnarray}\frac{U_{j}}{\sqrt{Rg}}\sim 0.08\frac{R}{r_{0}}.\end{eqnarray}$$

The experimentally obtained dimensionless jet velocities (non-pinch-off data in figure 12 a) are plotted as a function of $A/R$ and compared with impulse model in figure 16. The theoretical predictions of jet velocity (5.14) with a prefactor of $5$ , i.e.  $U_{j}/\sqrt{Rg}\approx 0.4(R/r_{0})$ , are in good agreement with experiments. At the largest jet velocity, agreement tends to be somewhat less good, which is attributed to measurement errors because, in this case, the singular cavity radius $r_{0}$ is very small. The singular radius is a function of the last stable wave amplitude $b$ , container radius $R$ and viscosity. The dependency on $b$ and $R$ can be obtained from the singular scaling (5.7) which gives

(5.15) $$\begin{eqnarray}\left[\frac{R}{r_{o}}\right]_{s}\approx \frac{b}{R}\left[\frac{b}{b-b_{s}}\right]^{1/2}.\end{eqnarray}$$

The ratio $(r_{0}/R)/(r_{0}/R)_{s}\approx 0.22$ . This gives a prefactor of $0.36$ in (5.14), which is close 0.4 used in figure 16.

The main interest of the impulse model is to demonstrate the parameters that determine the jet velocity. Further theoretical development of the model would be of interest but goes beyond the scope of the present work.