Toroidal cavitation by a snapping popper

Cavitation is a phenomenon in which bubbles form and collapse in liquids due to pressure or temperature changes. Even common tools like a rubber popper, which is a children's toy that rapidly changes its shape when snapped, can be used to create cavitation at home. As a rubber popper toy slams a solid wall underwater, toroidal cavitation forms. As part of this project, we aim to explain how an elastic shell causes cavitation and to describe the bubble morphology. High-speed imaging reveals that a fast fluid flow between a snapping popper and a solid glass reduces the fluid pressure to cavitate. Cavitation occurs on the popper surface in the form of sheet cavitation. Our study uses the axisymmetric Rayleigh-Plesset equation and the energy balance to capture the relationship between the bubble lifetime and the popper deformability. The initial distance between the popper and the wall is an important parameter for determining the cavitation dynamics. Presented results provide a deeper understanding of cavitation mechanics, which involves the interaction between fluid and elastic structure.

Figure 1

(a) Side-view images of the underwater popper ($R_p\approx 16$ mm) approaching the glass substrate. The platform height was set at $H\approx 9\mathrm{mm}$ (i.e., $H/R_p\approx 0.56$). (b) Bottom-view images of the same popper and cavitation dynamics. We used a 50% glycerol-water mixture ($\approx 5$ cSt) for visualization purposes. The platform height was set at $H\approx 12\mathrm{mm}$ (i.e., $H/R_p\approx 0.75$). The scale bars represent 10 mm. Both images were edited to enhance the brightness/clarity. The images were originally presented at the Gallery of Fluid Motion competition (November 2022, Indianapolis, IN) and are reproduced from it [30]. We note that the black circle that appears on the center of the popper is a hole on the popper's top surface.

Figure 2

(a) Radius of the cavitation $R_{\mathrm{in}}$ (open markers) and $R_{\mathrm{out}}$ (filled markers) as a function of the platform position $H$. Colors distinguish the popper sizes (blue for $R_p=16\mathrm{mm}$ in the preliminary experiment, black for $R_p=16\mathrm{mm}$ in the main experiment, and red for $R_p=22\mathrm{mm}$ in the preliminary experiment). We note that we have five trials at each condition (marked by dots) to obtain the mean value (square) and standard deviation (error bar). (b) Lifetime of the cavitation $t_{\mathrm{life}}$ as a function of the platform position $H$. Trend lines are $t_{\mathrm{life}}=-0.089H+2.118$ ($R$-squared value: 0.7306) for a smaller popper (a black dashed line, preliminary and main experiments combined) and $t_{\mathrm{life}}=-0.094H+2.687$ ($R$-squared value: 0.5891) for a larger popper (a red dashed line).

Figure 3

(a) Schematic of the bottom-view measurement by employing two high-speed cameras (not to scale). (b) Schematic of the side-view measurements with and without tracer particles [see also the left- and right-hand side portions of (c), not to scale]. Some of the measured quantities ($H,h,U_{\mathrm{popper}},$ and $V_{\mathrm{particle}}$) are presented in (c), while they were calculated in separate measurements.

Figure 4

High-speed images of cavitation phenomena upon popper slamming ($R_p\sim 16$ mm). (a1) Angled bottom-view of the toroidal bubble at $H/R_p\sim 0.69$. The dark dots and tilt are used to measure 3D shapes. The frame rate was 6000 frames per second. (a2) Side-view images of the toroidal bubble separately filmed at the same condition with a frame rate of 100 000 frames per second. (b1),(b2) High-speed images of the bubble in the transition regime ($H/R_p\sim 1$). (c1),(b2) High-speed images of the vortex-ring type bubble formed at the tip of the popper ($H/R_p\sim 1.13$).

Figure 5

The bottom shape of the popper ($R_p\sim 16\mathrm{mm}$) is estimated by the three-dimensional imaging at a frame rate of 6000 frames per second. $r=0$ represents the location of the popper center. Colors distinguish the times from $t=-3$ to 1 ms. The reference time $t=0$ ms represents the one frame earlier than the cavitation onset. Solid lines are computed based on the popper height at each distance $r$ (marked by dots) from the center by adopting the fourth-order polynomials while assuming that the popper shape is axis-symmetric. Stars represent the popper center and $R_{\mathrm{rim}}$ at $t=0$ ms when available. The height of the substrate $z=0$ is set arbitrarily but remains the same for all three panels. All three panels share the legend shown in a box underneath. The experimental conditions were $H/R_p\sim 0.69$ for (a), $H/R_p\sim 1.0$ for (b), and $H/R_p\sim 1.13$ for (c). We note that the arrows in (a) indicate the direction of the speeds of popper center $U_{\mathrm{popper}}$ and seeded particles $V_{\mathrm{particle}}$ that we present in Fig. 6 [see also Fig. 3].

Figure 6

(a) Circles show the vertical velocity of the popper, $U_{\mathrm{popper}}$, measured through the three-dimensional imaging tracking method as shown in Fig. 5. Open circles represent the mean value of five trials under the same condition, while small dots denote individual trials. Triangles show the speed of particles in the radial direction, $V_{\mathrm{particle}}$, measured through particle tracking below the popper's bottom surface. The fastest particles for larger $H/R_p$ are found outside the popper's bottom surface ($r>R_{\mathrm{rim}}$), marked by crosses. (b) The ratio of speeds $V^{*}=V_{\mathrm{particle}}/U_{\mathrm{popper}}$ calculated from the values in Fig. 6 is plotted as a function of $H/R_p$. (c) Circles show the location of the popper's surface closed to the substrate (e.g., Fig. 5), $R_{\mathrm{rim}}$. Markers and error bars are the mean and standard deviation calculated from five trials, respectively. A gray shaded region outlines the maximum and minimum values for each condition. A solid line shows the first-order approximation $R_{\mathrm{rim}}\sim R_p\sqrt{1-{(H/R_p)}^2}$. The discrepancy from the data may suggest that the popper stretches beyond its original length due to elasticity. (d) The thickness of the gap $h$ was measured through side-view imaging (marked by the diamonds). Markers and error bars are the mean and standard deviation calculated from five trials, respectively. A gray shaded region outlines the maximum and minimum values for each condition. A solid line denotes $h\sim 0.4\mathrm{mm}$, which was calculated from the data for $0.56\le H/R_p\le 0.81$. It is noted that we did not measure $h$ when $H/R_p>1$, as the vortex-ring-type bubbles formed regardless of the distance from the substrate.

Figure 7

Comparison between the normalized bubble lifetime $t_{\mathrm{life}}/\left[R_p\sqrt{\rho /(p_0-p_v)}\right]$ vs the bubble size $(R_{\mathrm{out}}^2-R_{\mathrm{in}}^2)/R_p^2$. The same data set was used as that in Fig. 2. Squares represent the mean value over five trials that are marked by dots. The error bars are showing the standard deviation. Colors distinguish the typical popper size at rest. The fitting line shows a trend, where the equation was estimated to be $t_{\mathrm{life}}/\left[R_p\sqrt{\rho /(p_0-p_v)}\right]=1.075{[(R_{\mathrm{out}}^2-R_{\mathrm{in}}^2)/R_p^2]}^{\frac{1}{2}}$ ($R$-squared value: 0.702). (b) Comparison between the normalized bubble lifetime and the timescale based on the elastic potential energy, which was found in Eq. (10). The same representation of the data as that in (a) holds. The fitting line is found to be $t_{\mathrm{life}}/[R_p\sqrt{\rho /(p_0-p_v)]}=0.179{[E/(p_0-p_v)]}^{\frac{1}{2}}{(h_p/h)}^{\frac{1}{2}}{(h_p/R_p)}^{\frac{3}{4}}{(1-H/R_p)}^{\frac{3}{4}}+0.335$ ($R$-squared value: 0.686).

Figure 8

(a) The experimental setup involves elevating a glass tank and filling it with approximately 10 cm of deionized water. A cantilever structure, including a 3D-printed platform, is submerged to precisely position the rubber popper within the water. Three synchronized high-speed cameras are placed to record both side-view and bottom-view videos of the experiment. All cameras are set to a frame rate of 5000 frames per second. (b) This part of the setup shows a 3D-printed platform, designed to secure the popper at a specific height. The diagram shows a cross-section of the inverted popper. The initial height of the platform sets the parameter $H$ for the experiment. (c),(d) Actual images of the cut popper with (c) and without (d) a hole at the apex. A hole in (c) penetrates the popper top surface, while a dent in (d) is filled at the inner surface of the popper.

Figure 9

A schematic illustrating the popper snapping process. For simplicity, the cross-section of the popper along its centerline is depicted. Step $①$ shows an initially hemispherical popper being inverted and positioned on a platform. In step $②$, the popper is allowed to remain until it spontaneously snaps, illustrated by the transition from the orange to the red state. Please note, the time taken for the popper to snap was not controlled. The platform facilitates triggering a toroidal cavitation when it comes sufficiently close to the bottom surface of the glass tank. The approximate state of the popper at $t=0$ is represented by the brown curve.

Figure 10

High-speed images (a)–(c) capture a small popper snapping from a height of approximately $H\sim 18\mathrm{mm}$ ($H/R_p\sim 1.13$), highlighting a rapid snap, particularly in the final stage. Please note that $t=0$ ms is set as one frame before the bubble onset. As $H/R_p\sim 1.13$, the bubble is classified as the vortex-ring-type bubble [refer to Fig. 4 for more details]. (d) The tracked location of the popper center in the $z$-direction. The green line indicates the start of the video, at a point where the snapping has already begun [corresponding to image (a)]. The blue line denotes the timing of the bubble onset [corresponding to image (c)]. The timescale for the popper snapping is estimated as the interval between these two points, roughly around $t^{*}\ge 10.8$ ms. A triangle illustrates a swift snap, especially in the last snapping stage [$\sim O\left(10\right)$ ms]. Despite the formation of the vortex-ring-type bubble, the popper continues to oscillate. The magnitude of the $z$-direction value remains fairly consistent, but the popper may form a secondary toroidal bubble as it approaches the bottom of the glass container again.

Figure 11

(a) Shape reconstruction was performed along four different angles; each line is separated by approximately ${90}^{\circ }$ for $H=12\mathrm{mm}$. The image taken at $t=-15.5$ ms is shown as an example. Data at $\theta ={\theta }_1$ (equivalent to the condition in Fig. 5) are denoted by circles. The other three lines are marked by triangles ($\theta \approx {\theta }_1+{90}^{\circ }$), squares ($\theta \approx {\theta }_1+{180}^{\circ }$), and diamonds ($\theta \approx {\theta }_1+{270}^{\circ }$). (b) The reconstructed shape is based on 30 data points combining all four lines shown in (a). Different marker shapes indicate four different angles as indicated in (a). The colors indicate different times relative to the onset of cavitation. Solid and dashed lines show the fourth-order polynomial fits for the combined data and the data for $\theta ={\theta }_1$, respectively.

Figure 12

(a) A comparison between the inner radius $R_{\mathrm{in}}$ [squares, see Fig. 2] and the location of the extreme of the polynomials for the popper bottom shape $R_{\mathrm{rim}}$ [circles, see Fig. 5] as a function of the platform height $H/R_p$. Both radii match well as long as cavitation maintains the toroidal shape ($H/R_p\le 0.81$). (b) The lifetime of the bubble as a function of $H/R_p$ with a trend line of $t_{\mathrm{life}}=-1.502(H/R_p)+2.186$ ($R$-squared value: 0.683).

Figure 13

A comparison between the bubble's inner radius $R_{\mathrm{in}}$ and the inverted popper rim's radius $R_{\mathrm{rim}}$ immediately prior to the onset of cavitation. Circles represent the data taken from the three-dimensional imaging performed at 6000 frames per second. A blue line shows $R_{\mathrm{in}}=R_{\mathrm{out}}$ for comparison. The dotted line and arrowheads help delineate the threshold between fully toroidal cavitation ($H/R_p\le 0.81$) and the transition to vortex ring-type cavitation ($H/R_p\ge 0.88$), as discussed in the main text. The inset shows a comparison of $R_{\mathrm{in}}$ and the outer radius of the bubble $R_{\mathrm{out}}$.

Figure 14

(a)–(d) Zoom-in images of a popper ($R_p\sim 16\mathrm{mm}$) released from a height of $H\sim 13\mathrm{mm}$. Relative time with respect to the cavitation onset was $-0.3$ ms (a), $-0.2$ ms (b), $-0.1$ ms (c), and 0 ms (d). The silver-coated ceramic particles ($d\sim 85$ $\mu \mathrm{m}$) mixed in the deionized water enabled particle tracking. The analyzed particles are marked by dots, where the colors are consistent throughout the images. We note that both axes are in pixels to imply the resolution of the image ($\sim 32$ pixels/mm). (e) The displacement of the particles from their original position at $t=-0.3$ ms. Numbers in the legend represent the characteristic particle position $r_{\mathrm{rep}}$ over the entire course of motions, which is measured at $t=-0.05$ ms. Colors convey the same meanings as those in (a)–(d). (f) The averaged particle velocities over 0.1 ms before the cavitation onset ($V_{\mathrm{particle}}$) as a function of the particle position $r_{\mathrm{rep}}$ over the entire course of motions.

Figure 15

Here we provide a different visualization of the particle tracking data, as previously shown in Fig. 14. The paths and trajectories for selected particles [specifically the 3rd, 6th, 9th, and 11th from the left as seen in Figs. 14] are shown.

Figure 16

Particle speeds $V_{\mathrm{particle}}$ for various cases vs the characteristic particle location $r_{\mathrm{rep}}$. Each panel (a)–(e) contains the data from five different runs for each release height $H$. Each marker represents an individually tracked particle, with the shape of the markers differentiating each experimental run. It is important to note that the number of tracked particles varied with each attempt. Solid and gashed gray lines (when available) represent the mean and standard deviation of $R_{\mathrm{rim}}$ data. We observed a formation of either toroidal cavitation or the transition cavitation upon the first stretch of the popper for (a)–(d) ($0.56\le H/R_p\le 0.94$), while a ring bubble was formed when $H/R_p\sim 1.13$.

Figure 17

(a) A calibration curve between the signal output from the force sensor (S.O.) and the force estimated by the weight ($F=mg$). The fitting line is estimated to be $\text{S.O.}=0.1292F-2.1757$. (b) Measured forces as a function of the platform position $H$, where the colors distinguish the popper sizes. The trend lines are estimated to be $F=-1.531H+26.02$ ($R^2=0.9188$) for $R_p=16\mathrm{mm}$ (blue) and $F=-1.427H+32.4$ ($R^2=0.8869$) for $R_p=22\mathrm{mm}$ (red). We note that we excluded the data for $H=21\mathrm{mm}$ when obtaining the fitting curve for a smaller popper ($R_p=16\mathrm{mm}$). We note that we have five trials at each condition (marked by dots) to obtain the average value (square) and standard deviation (error bar).

Figure 18

A comparison between the normalized force $F^{\prime }=\frac{1}{T}\int Fdt{R}_p^{\frac{1}{2}}/\left(E{h}_p^{\frac{5}{2}}\right)$ (measured in air) vs the stand-off parameter ${(1-H/R_p)}^{1/2}$. The squares are the mean value derived from five trials, indicated by the dots. The error bars represent the standard deviation. Different colors differentiate between the typical sizes of the popper at rest, consistent with those in Fig. 17. The dashed line shows a fitting $F^{\prime }\sim 0.637{(1-H/R_p)}^{\frac{1}{2}}-0.031$ ($R$-squared value: 0.716).

Figure 19

(a) A schematic of the curbed beam system with the popper radius $R_p$, local width $b_{\mathrm{beam}}$, and $h_{\mathrm{beam}}$ of the cut popper near the tip, where the force $F$ is being applied continuously to record deflection ${\delta }_{\mathrm{beam}}$. (b) A typical result with a cut popper, where Young's modulus might be scaled by taking the slope of the $F-{\delta }_{\mathrm{beam}}$ curve.

Figure 20

(a) Comparison of bubble lifetime $T_{\mathrm{life}}$ as a function of the initial platform height $H$ for the poppers with and without a hole at the apex. A dashed line represents the empirical fit for the popper with a hole, as witnessed in Fig. 2. (b) Comparison of the inner bubble radius $R_{\mathrm{in}}$ as a function of $H$ for two different poppers. (c) Typical images of a popper without a hole activated at $H=18\mathrm{mm}$ ($t=t_a$), where only the secondary cavitation ($t=t_a+11.3$ ms) was observed. Note that a dent at the popper center is sealed on the back side by design, meaning that the leakage of the fluid through the popper center is assumed to be negligible.