Dynamics of a cavitation bubble between oblique plates

Research on cavitation and cavitation-induced erosion has been conducted over the past century. Investigating the causes of erosion damage of screw propellers of ships initiated research of this phenomenon.^1,2 This was followed by mathematically formulating the collapse and rebound of a single bubble and the stability of the associated fluid flow,^3,4 by experimentally studying the collapse of a single spark-induced cavitation bubble and its impact pressure generation near a wall,^5,6 by examining a laser-induced single bubble near a solid boundary,^7,8 and by analyzing cavitation damage by a single bubble.^9–11 In their analytical investigation, Kornfeld and Suvorov¹² concluded that the damage caused by a collapsing bubble is due to the bubble's jet impact, and the experiments of Naudé and Ellis¹³ and Benjamin and Ellis¹⁴ confirmed this finding. Plesset and Chapman¹⁵ were the first to numerically demonstrate the development of a jet during the collapse of the bubble, as confirmed by the experiments of Lauterborn and Bolle.¹⁶ Philipp and Lauterborn⁹ and recently, Sagar and El Moctar¹⁷ conducted a detailed experimental investigation of a single bubble's collapse and its damaging effect on a metal surface. Philipp and Lauterborn⁹ investigated the damage on a metal surface caused by the collapse of a single cavitation bubble and of several hundred cavitation bubbles. They also visually resolved the shock waves during the toroidal collapse. Lindau and Lauterborn¹⁸ experimentally observed the bubble's collapse employing cinematography and long-distance microscopy. They extensively studied shock waves emitted during the bubble's first collapse and the collapse of the toroidal bubble. Li et al.¹⁹ presented experimental and numerical investigations for nonlinear interaction, coalescence, collapse, and rebound of two pulsating bubbles. Their investigations provided further insight into the high-speed liquid jet formation, jet impact, splash, fragmentation, and associated flow field characteristics. Vogel and Venugopalan²⁰ briefly reviewed the process of pulsating laser tissue ablation.

In practice, cavitation in pumps, turbines, bearings, nozzles, etc. occurs not only between parallel plane surfaces but also between oblique plane surfaces. One example is cavitation occurring in ultrasound cleaning devices, where the substrate may have a non-parallel inclination relative to the cleaning horn. Another example is cavitation occurring inside pump impellers or water turbines,²¹ where severe erosion occurs at the blade's suction side and downstream in the blade to blade non-parallel channel. These examples demonstrated that optimized hydrodynamic structural components often consist of complex non-parallel shapes to obtain optimum thrust and mass flow. This motivated us to reveal the dynamics of bubbles or clouds of bubbles collapsing between non-parallel surfaces.

Regarding the application of bubble cavitation, cavitation laser surgery is used for myocardial laser revascularization and laser angioplasty.^22,23 Failure of this method increases risks associated with vessel dissection,²⁴ coronary artery perforation,²⁵ and inadequate tissue removal²⁶ caused by large-amplitude shock waves induced by cavitating bubbles.²⁷ In addition, Zotz et al.²⁸ observed more pronounced cavitation in the high-speed rotational angioplasty, where a diamond-studded burr spins at 50 000–200 000 rpm to excavate calcified or fibrotic plaque, allowing microscopic debris to embolize on the coronary capillary bed. Graf et al.²⁹ observed the mechanical heart valve damage (prosthesis) due to cavitation effects during valve closure, resulting in water hammer effects. Kafesjian et al.³⁰ predicted the locations of erosion damage on the mechanical valve and correlated this with the cavitation implosions that occurred.

Jetting and vorticity have their own immediate practical applications, such as in microfluidic pumping,³¹ vigorous mixing,³² and surface cleaning.³³ Reuter and Mettin³⁴ investigated the dynamics of collapsing bubbles near a flat surface and the associated potential application for the removal of surface attached particles. Jetting, intense vortex flow, shear stress, and second collapse-induced flow are the main factors associated with surface cleaning. For this process to be effective, the bubble needs to be close enough to the surface.^34,35 Wu et al.³⁶ concluded that viscosity significantly influences surface cleaning performance, whereby the liquid jet plays a significant role. In addition, Vyas et al.³⁷ investigated biofilm cleaning for dental care applications, postulating that enhancing the cavitation generation process is possible by increasing the CO₂ concentration in the water to more effectively remove the biofilm. The important aspect of these applications is that they can be utilized at microscopic spatial and low temporal resolutions. Lab-on-chip (LOC) devices are the promising applications of microfluidics used for biological and chemical analyses, where microscopic mixing in a laminar flow is highly desirable. The noninvasive generation and flexible positioning of a cavitation bubble is an added benefit. Zwaan et al.³⁸ investigated the behavior of the laser-induced cavitation bubble in a light-absorbing liquid filled microfluidic channel. They reported the generation of multiple jets when a bubble collapses near complex geometric shapes. Zhang et al.³⁹ demonstrated the laser-induced bubble generation technique in microfluidic with its application for a micro-valve to block the flow and micro-pumping to achieve the required flow rate in a specific direction. Wu et al.⁴⁰ demonstrated the application of laser-induced microbubbles in microfluidic chips. The bubble and jet mechanisms actuate the small and controlled volume required for sample sorting. The bubble's collapse time is important to specify the so-called switching or actuation time. By varying the bubble's size, the flow of the control volume can be controlled over time, and multiple sorting is easily possible. Chen et al.⁴¹ reported that a bubble's dynamics controls the accurate timing in the range of milliseconds of the flow passing cells and particles to enable high purity sorting. Shin et al.⁴² presented the on-demand micro-manipulation of micro-objects via the bubble by integrating optothermal and acoustical actuation. Along with its application, adverse effects of the cavitation bubble also exist, and these effects need to be investigated. At a larger scale, Zhong et al.⁴³ proposed a novel method using an ultrasonic-enhanced submerged cavitation jet to clean and suppress ship fouling. The increase in mass loss caused by the submerged jet in combination with an ultrasonic device was of the order of 13%.

Apart from the microscopic application of methods relying on a collapsing bubble, the extracorporeal shock wave lithotripsy (ESWL) is a commonly applied medical treatment procedure for getting relief from kidney stones. High-energy shock waves are used in this therapy to fragment a kidney stone into pieces. The resulting tiny kidney stone residuals can pass through the urinary tract, but this method may cause bleeding, bruising, or pain. The two left images in Fig. 1 show the anatomy of the kidney and the position of a kidney stone obtained from a computerized tomography (CT) scan. In both graphs, a kidney stone is located in the renal pelvis at the mouth of the ureter. The right image in Fig. 1 shows a schematic of a pump impeller with cavitation occurring on and between impeller blades.²¹ 

The calyces, renal pelvis, and ureter are shaped like a nozzle/diffuser, and shock waves generated by the cavitation bubble significantly affect the neighboring healthy tissues, causing pain when applying this “ESWL” procedure. Nevertheless, cavitation-based methods are increasingly being used in many biomedical applications, and such methods generate cavitation bubbles between two surfaces that are not parallel to each other. Thus, the purpose of our investigation was to provide basic insight into bubble dynamics generated between two oblique non-parallel surfaces, thereby possibly contributing to the improvement of biomedical devices employing cavitation-based methods.

Until now, attention has focused on bubble dynamics in the far-field or near a solid surface to reveal the physics behind the bubble's behavior and the reason behind the associated damage. Nevertheless, Shima and Sato⁴⁴ experimentally and theoretically investigated the bubble dynamics also in narrower spaces. They observed prolate-shaped bubbles collapsing between narrow parallel plates, with higher collapse times than for bubbles near a single wall. For a bubble located nearer to one of the parallel walls, its behavior is similar to its collapse near a single solid surface. Ueki et al.⁴⁵ photographed spark-induced cavitation bubbles collapsing between parallel plates and concluded that the gourd shape of a collapsing bubble occurs, if it collapses at a smaller distance from the parallel plates, and they also observed that further splitting of bubbles occurs. Kucherenko and Shamko⁴⁶ investigated the dynamics of a spark-induced bubble between two parallel plates. Indeed, bubble dynamics between two solid walls has become a popular research issue because of its important and wide applications in various fields, such as microfluidic,⁴¹ biomedicine,^47,48 and noise suppression.⁴⁹ The liquid jetting flow induced by collapsing bubbles is of significance as it can clean wall surfaces,⁵⁰ transport liquid matters, and manipulate micro-objects in liquids.⁴⁰ However, the liquid jets are usually accompanied by high-speed impacts on the solid surface when the bubble collapses in the neighborhood of a solid flat boundary,⁵¹ which may lead to serious cavitation erosion of engineering devices, such as turbo-pumps and marine propellers. Lv et al.⁵² experimentally investigated the interaction between a laser-generated cavitation bubble and a spherical particle. The distance between the cavitation bubble and the particle significantly influences particle motions. If the standoff distance is small, the particle is repelled by the bubble; if the standoff distance is large, the particle is pulled toward the bubble. At a small distance, the microjet impingement is significant, causing the particle to greatly accelerate. Reuter et al.⁵³ recently investigated a single cavitation bubble-induced erosion at ambient pressure close to a metal surface. They postulated that erosion occurs only if self-focusing non-axisymmetric energy conditions prevail. During the collapse, the bubble emits shock waves converging on a single point. The intensifying shock wave leads to a gas phase collapse. Zhang et al.⁵⁴ performed experiments and numerical simulations to investigate the interaction between the jet and the plate structure during underwater explosion. This jetting led to high plate stresses and deflections. Their investigation demonstrated a significant effect of gravity on the jetting-induced stress distribution on the plates. Overall, these cited publications highlight the effect of jetting on the destruction and/or removal of material at micro and macro scales.

Jetting phenomena together with the bubble's bursting behavior is also frequently found near the free surface.⁵⁵ Choi and Chahine⁴⁹ found that noise following the jet from splitting cavities indicates pressure peaks and that this noise is related to the reentrant jets in the sub-bubbles. They experimentally analyzed the splitting of the bubble by generating a bubble between two parallel plates, and they also performed numerical analyses. Their experiments showed an asymmetric collapse along the vertical axis due to the presence of the electrode although their numerical simulations predicted a symmetric collapse. Brujan et al.⁵⁶ investigated a bubble collapsing between parallel plates, with the associated Kelvin impulse being negligibly small. The Kelvin impulse is a factor to determine the bubble's translational characteristics and the direction of its jet. Azam et al.⁵⁷ also investigated the collapse of a bubble in a narrow gap between two parallel plates. They observed two kinds of collapses, namely, one collapse where a bubble splits at its center and collapses at the wall and another collapse where a bubble collapses toward its center. The kind of collapse depends on the gap between plates. Hsiao et al.⁴⁸ presented boundary-element method (BEM) computations of single and tandem bubble dynamics between parallel plates. Quah et al.⁵⁸ presented the dynamics of a spark-induced cavitation bubble in the narrow gap between parallel plates. They investigated the effect of gap width and bubble distance from these plates. In addition, they studied the bubble dynamics between a narrow gap and a free surface. The presence of three boundaries near the bubble influenced the jetting. Phan et al.⁵⁹ numerically simulated the compressible interface flows with and without mass transfers for the case of a bubble collapse between two parallel walls. Recently, Brujan et al.⁶⁰ experimentally and numerically analyzed the collapse of a bubble between two parallel plates. Their bubble's collapse seems to be affected by the presence of the supporting third plate, causing the bubble's surface to execute higher displacements. Tagawa and Peters⁶¹ and Molefe and Peters⁶² experimentally investigated the jet direction for collapsing bubbles inside a square and an equilateral triangular channel. Their analytical model takes into account only the asymmetric flow field at its surface at the beginning of its collapse for the prediction of the jet direction. However, except for the jetting direction at variable angular corners of tingle and square, their prediction did not introduce a relevant information about the collapse phases and asymmetries raised during the bubble's dynamics. Their investigation was also limited to the first collapse and jet direction. In addition, Wang et al.⁶³ experimentally studied the dynamics of a bubble situated inside a rigid circular tube, using spark-generated bubbles and high-speed photography. They observed a disk shape bubble generated after the collapse when the tube diameter exceeds twice the bubble's maximum radius. For a smaller ratio of bubble to tube diameter, they observed counter-propagating jets resulting in an annular liquid jet splitting the bubble. A cloud of microbubbles was generated shortly after the start of the bubble's rebound. Long et al.⁶⁴ investigated the formation, growth, and collapse of cavitation bubbles that ablated at various immersion depths. Principally, they generated a bubble situated only a few millimeters distance (water gap H) from the free surface and the solid surface. For a water layer smaller than the maximum bubble radius (R), the bubble collapses asymmetrically, characterized by a damped implosion. For H $≥$ 2R, the implosive collapse of the bubble may intensify, causing a higher ablation rate and a double-shock peening effect. Han et al.⁶⁵ investigated the dynamics of a bubble between parallel plates using strobe photography and numerical simulations. They postulated that jet speed and jet diameter may be adjusted by the relative size and position of the bubble situated between the plates. They highlighted the role of jetting in micro-injections of biomaterials or living cells and the directional transport of target samples in micro-fluidic systems, which we also discovered in our investigations. However, their investigated jet was not floating. Shen et al.⁶⁶ presented a numerical model of cavitation bubble dynamics in a sealed vessel. They extended their investigation also to a pair of bubbles. For two antiphase bubbles of similar size, the pulsating suppression effect is weak, while for a single bubble or for two matched bubbles, pulsation can be suppressed to a large extent. Recently, Li et al.⁶⁷ studied the bubble dynamics and the associated load transfer to the double-layer cylindrical shells caused by shock waves from underwater explosion. The distribution of the explosion induced loads and the characteristics of interlayer pressure loads were studied using numerical methods. Three stages characterized the loading: the shock wave stage, the collapse jet stage of the interlayer bubble, and the collapse jet stage of the outboard bubble.

Diffuser shaped or non-parallel structures commonly exist in engineering devices. The anatomy of the kidney, veins, and impeller blades of water turbines shown in Fig. 1 are examples of cavitation occurring between two solid oblique walls. In addition, it is well known that the pressure and the associated jet acting on this surface, induced by oscillating bubbles situated in the neighborhood of two solid oblique walls, play an important role also in the cleaning of surfaces. Furthermore, the pressure and the jet impact are closely associated with the erosion of oblique surfaces. Although studies of single and multiple bubbles collapsing between parallel plates have been published, to our knowledge, the dynamics of a bubble collapsing between two oblique plates has not yet been extensively investigated. Recently, Brujan et al.,⁶⁰ Zeng et al.,⁶⁸ and Su et al.,⁶⁹ while examining the bubble dynamics between thin gaps, observed that a splitting bubble generates a jet impacting on a nearby wall. They⁶⁹ postulated that the oscillation period of a bubble collapsing between two parallel plates is larger than the oscillation period of a bubble collapsing in a free field. They also concluded that the oscillation period decreases with the increasing distance between the bubble and the plate. Chen et al.⁷⁰ performed the simplified experiments based on the concept of damaged double layered plates. They generated a bubble near parallel (double layer) plates having a circular hole. Molefe and Peters⁶² briefly investigated the jetting direction of bubble subject enclosed polygonal boundaries, e.g., triangle and square. However, the details of the bubble's dynamics were missing. We studied a bubble in a stationary fluid collapsing between two oblique plates and found that its dynamics differ significantly in the later stages of collapse from the dynamics of a bubble collapsing near a single flat surface or between parallel plates. By varying the position of the bubble between the two oblique plates, we can control the intensity and the direction of its collapse, thereby attenuating the desirable or undesirable cavitation effects. This may be useful in various desired applications. The distance from the diverging edges (DE) of the plate assembly was varied and at a varying distance “b” from the diverging edges (DE) of the plate assembly. The directions shown by the red arrows correspond to the +x, −x, +z, and −z axial directions that describe the topology of a dynamically collapsing cavitating bubble.

Figure 2 shows an overall schematic view of our experimental setup, a three-dimensional view of the glass plate assembly to generate a single laser-induced bubble between two oblique plates, and a top view and a side view of this assembly. Also shown are the computer supported with a laser controller with its laser module having an internal triggering connection with the high-speed camera. The high-speed camera was equipped with extension rings R and an infrared filter F. The cuvette C had an enforced focusing lens L3 in combination with the imaging system with laser line lenses L1, L2, and the beam dumper B. The light source L placed on the opposite side of the cuvette acted as a light source for shadowgraphy. The blade angle for pumps varies between 15° and 45°; for compressors, between 35° and 50°; and for fans, the angle is less than 50°. Impeller efficiencies were estimated for blade angles varying from 20° to 31°.^71–73 For the renal pelvis, we measured the angle of the diffuser, based on the anatomy of the kidney and the CT-scan images. It varied between 16° and 40°.⁷⁴ Therefore, we selected an inclination angle of 20° between the plates (half angle 10°). In addition, we want bubble dynamics to be influenced by the nearby walls (for case of b = 0). Therefore, assuming the $γ≈$ 1.8 at b = 0 mm resulted in the gap “c” at the converging side to 2 mm. The selected values are approximations based on practical applications. However, more parametric studies should be performed in future to study their effects on bubble dynamics between non-parallel plates.

The two oblique glass plates were symmetrically placed at an angle of α = 10° along the longitudinal axis. Situated between edges DE and the converging edge (CE), these plates converged longitudinally to a separation distance of c = 2 mm. The bubble was situated at a distance of a = 13 mm from the vertical supporting plate. The distance from the diverging edges (DE) of the plate assembly was varied. The directions shown by the red arrows correspond to the +x, −x, +z, and −z axial directions that describe the topology of a dynamically collapsing cavitating bubble.

The vertical glass plate supporting the oblique plates allowed the transmission of the backlight through the collapsing region toward the camera sensor. A special fixture was used to assemble the glass plates to avoid spatial and angular errors. The lateral oblique plates were glued to the vertical supporting glass plate using an instant adhesive. Epoxy resin was applied at joints to provide additional strength. All three glass plates had a thickness of 1.0 mm and dimensions of 25 × 15 mm².

A supporting rod connected to the three-way translating platform held the glass assembly. The platform allowed moving the assembly in all three directions x, y, and z. A Q-switched Nd: YAG Laser (Nimma-900 Eamtech Optronics) with a wavelength of 1064 nm and a pulse duration of 10 ns generated a bubble in the cuvette filled with distilled water. This cuvette, made of 10 mm thick acrylic glass plates, had internal dimensions of 50 (length) × 50 (width) × 50 mm (height). The cuvette had through holes in the opposite wall to enforce a focusing lens and a flat window allowed the diverging beam to be transmitted. This focusing lens was an aspheric lens with a focal length of 20 mm. It focused the laser beam into the water-generating plasma. An optical system of convex and concave lenses was provided to reduce aberration and to increase the focusing angle of the beam. This facilitated obtaining a single uninterrupted plasma, thereby avoiding the generation of multiple bubbles. Our system with an incident laser energy of 30.5 ± 1 mJ was able to generate a bubble with a maximum diameter of 4.0 ± 0.1 mm. This laser-induced bubble size was considerably larger than the bubbles investigated in the previous studies.

A high-speed camera (Phantom V12.1 Vision Research) equipped with a CMOS sensor having pixel dimensions of 22 μm captured the dynamics of the bubble. The bubble's collapsing events were obtained at a rate of 75.000 frames per second (fps) and a frame size of 156 × 156 pixels for an exposure time of $10 μs$ and an aperture of f/5.6. A Scott glass filter was placed between the high-speed camera and the cuvette to avoid a scattered laser beam entering toward the camera sensor. The collapsing bubble events were captured using a combination of the back- and front-illumination methods. A 100 W high-energy light emitting diode (LED) lamp equipped with an aspherical lens for focusing purpose and a diffuser film provided illumination. The collapsing bubble blocked and scattered the incident light and reflected the front-illuminated focused light toward the camera sensor, thereby producing a shadow of the bubble, which represented an image of collapsing bubble. We purposely rotated the camera between 3° to 5° with respect to the x- and z-axes during tests to visualize the partial lateral (y directional) features of the collapsing bubble. We repeated each test case ten times. We observed that the jetting behavior, the associated symmetries, and the overall bubble dynamics were consistent in all repetitive test cases.

Initially, a glass assembly was immersed into a water-filled cuvette and aligned along the laser's optical axis. We focused the camera on a plane where the plasma was generated, which helped us to obtain the correct bubble diameter. At the plasma location, we laterally positioned a calibration grid to calibrate the resolution of the image in pixels per mm. Initial imaging sequences showed light scattered from the plasma. We assumed that the plasma location represented the bubble's origin, i.e., the center of the bubble. The angled glass plates were symmetrically aligned along the longitudinal x-axis, so that the plasma was located along the centerline. To avoid the presence of the plate affecting the bubble's collapsing dynamics, the bubble was generated at the lateral distance of about 13 mm from a vertical supporting plate, i.e., a in Fig. 2. This distance corresponded to a relative wall distance γ > 6.5, where γ is a ratio between the distance between the bubble center and the plate (a) to the maximum bubble radius (R_max). According to Brujan et al.,⁶⁰ the effect of the vertical plate was significant for a bubble collapsing between parallel plates, especially when the bubble collapses sufficiently close to a solid vertical plate (the corresponding relative wall distance ratio of γ > 3.0). In their investigation, a vertical supporting plate influenced the traveling bubble's collapse and its jet. In the current investigation, we noticed that the laser was reflected from the oblique glass plates when focused from the converging side of the glass plate assembly, thereby resulting in a smaller-sized bubble. Therefore, we focused a laser from the diverging side of the glass plate assembly. We kept the bubble's position always along the centerline between the two oblique plates and varied the longitudinal distance b (see Figs. 2 and 3), i.e., the distance between the diverging side edges of the oblique glass plate and the initial center of the bubble (location of plasma seeding).

Considering the experimental restrictions, such as the framing rate of the high-speed camera, the unidirectional view, and frame size, we performed computations to support our assumptions and explanations wherever necessary. We took into account flow compressibility to consider the shock wave dynamics. Note that compressibility might have significantly affected the quantification of bubble collapse parameters, especially the shock wave propagation at sonic velocities. A detailed numerical analysis was beyond the scope of this article. A detailed description of the numerical method, numerical setup as well as a discretization study can be found in article by Sagar and El Moctar.¹⁷ Here, we are giving a short summary. Computations were performed using the open-source finite volume method based flow solver CavitatingFOAM. This code solved the Navier–Stokes equations for compressible two-phase flows using an Euler–Euler approach, including the barotropic equations of state. An Euler–Euler approach solved the Navier–Stokes equations on a fixed Eulerian grid by considering the gas and liquid phases as a homogeneous mixture of vapor for the gas phase and water for the liquid phase. The time step in the simulation was adapted to an acoustic Courant number of 0.5, corresponding to an average time step of about 1 ns. The simulation time (covering about three bubble collapses) was 0.5 ms. We initiated the numerical computations for the fully grown spherical bubble. For grid sensitivity studies, the CPBR (cells per bubble radius) parameter was already specified for the spatial resolution in Sagar and El Moctar, and here,¹⁷ we used a similar grid resolution to perform our computations.

Our experimental results enabled us to elaborate on the dynamics of a bubble collapsing between symmetrically fixed oblique plates. We systematically varied the position of a bubble in the longitudinal direction. Specifically, we considered the four longitudinal bubble positions b = 0, 5, 10, and 12 mm, measured from the edge of the diverging plates (see Fig. 2) and marked by four red solid dots in Fig. 3(a). Positions b = 0, 5, 10, and 12 mm corresponded to the relative wall distance γ_h $≈$ 1.79, 1.4, 1.1, and 0.86, respectively. As distance b increased from 0 to 12 mm, the maximum size of the bubble decreased from about 2.0 to 1.6 mm. This size reduction could have been related to the restricted passage and the reduced laser beam energy. To monitor the bubble's changing shape during its collapse, we measured bubble contour deformations at the various locations on its surface contour identified by red dots in Fig. 3(b). These positions represent the two end positions of the bubble's surface contour in the longitudinal x direction and the vertical z direction. The images captured here were projections of the shadow in the two-dimensional x–z plane. Therefore, the third dimension was not considered. Nevertheless, a slight inclination of the camera provided images showing three-dimensional (including lateral) features of the bubble's shape. The shadowgraph gray scale image was used to locate an interface (contour of the bubble surface) between white and black pixels using the MATLAB function “edge” as shown in Fig. 3(c). As the images of the bubble's dynamics captured in our experiments reflected the overall vertical symmetry, we measured only the top position $S+z$ of the bubble's surface contour. The longitudinal coordinates $S+x$ and $S−x$ of the bubble's surface contour were obtained as shown in Fig. 3(b). By monitoring these coordinates at each time interval, we estimated the velocities of locations $S+x, S−x$⁠, and $S+z$⁠). We assumed the mean of the bubble's longitudinal end positions as its longitudinal center. Figures 15–18 plot the time series of the bubble's center and the bubble's surface coordinates for the cases of a bubble generated at b = 0, 5, 10, and 12 mm. Alternatively, we used the right, left, top, and bottom terminology with reference to the longitudinal +x, longitudinal −x, vertical +z, and vertical −z directions in the further text while describing the dynamics of the bubble.

Generally, the above-described method was able to reliably predict the bubble's contour up to its second collapse for b = 10 and 12 mm and up to its third collapse in cases for b = 0 and 5 mm. After two or three collapses, the bubble fragmented into multiple, thin, and unrecognizable entities. In such cases, the described image processing ceased to function properly and could no longer be used. Therefore, we limited our measurements up to the second collapse for b = 10 and 12 mm and up to the third collapse for b = 0 and 5 mm. Images of the full grown bubble during the first growth, obtained after the plasma seeding for b = 10 and 12 mm, show the bubble's shadow crossing the imaging frame. Thus, the high-speed camera did not fully capture the left part of the bubble. Under these operational restrictions, the MATLAB function considered the frame edge as a reference defining the coordinates of the point $S−x$⁠, which turned out to be nearly constant values of coordinates $S−x$ over parts of the corresponding time duration in middle subplots of Figs. 17 and 18. Therefore, for the missing part in cases of b = 10 and 12 mm, we estimate $S−x$ coordinates of the missing part based on assuming the longitudinal symmetry of the bubble.

Previous investigations demonstrated that the presence of a solid wall significantly influences the bubble's dynamics when the bubble is generated sufficiently near a solid wall.^9,51 In the case of multiple solid walls, the bubble has an affinity to collapse at the nearest wall.^60,68 This was not always the case. It was shown that although a bubble can be produced near a wall, it collapses on the opposite wall.^75,76

A bubble generated at b = 0 mm was brought to collapse between the diverging edges DE of the glass plates. The vertical distance between the bubble's center and the edge of one of the oblique glass plate was 3.6 mm. This distance corresponds to an approximate relative wall distance γ of 1.8. A previous investigation of Sagar and El Moctar¹⁷ showed that, at γ = 1.8, the bubble's dynamics was significantly influenced by the presence of the wall and that the ability to damage an underlying metal surface exists. However, the sequence of shadow graphic events of bubble dynamics between two oblique plates shown in Fig. 4 demonstrates that the bubble dynamics was hardly affected by the presence of edges of the oblique plates. The bubble did not show an affinity toward the edges of the oblique plates. The sequence depicts various bubble configurations, such as an oval shape, a bubble with a jet pointing in the longitudinal +x direction, an obcordate leaf shape, and an apple-like shape. The sequence covered more than three rebounds and subsequent bubble collapses.

Our investigation demonstrated that the effect of the obliqueness of the plates on bubble dynamics was significant. The bubble's collapse was cumulatively affected by the presence of the glass plate's edges and the converging–diverging liquid volume surrounding the bubble along the longitudinal direction. This asymmetric situation created a pressure gradient in the longitudinal direction acting on the bubble's surface in such a way that its collapse direction was longitudinally extended toward the converging sides of the plates as shown in Fig. 4.

The consecutive sequence of the shadowgraphic events shown in Fig. 4 visualizes the collapsing bubble at b = 0 mm, showing the bubble being generated between the two oblique plates and near edges. On the left side of the bubble has a finite liquid domain, not restricted by oblique plates but bound by cuvette walls. However, along the right side of the bubble liquid domain converged, it was enclosed by a pair of oblique glass plates. At time t = 0.013 ms, scattered light from the seeded plasma is visible along with the gray shadow of the bubble. This shadow is blurred, signifying that the bubble grew faster than the exposure time of the capturing events. Later, the bubble grew spherically until it attained its maximum mean diameter of 3.9 ± 0.1 mm. The bubble's spherical surface partially reflected the image of the pair of oblique glass plates along with their diverging side edges, as seen in the first two rows of the events in Fig. 4. The sharp bubble surface represents the fine interface between the liquid and gas phases. Between approximate times of t = 0.147 and t = 0.266 ms, although it is difficult to distinguish the fully grown bubble, the bubble is nearly spherical, fine surfaced, and stable for a longer period of time. As time progressed, the bubble deformed spherically for a shorter period of time. Later on, its surface deformed at the left-top and the left-bottom sides more than at the right-top and the right-bottom sides, resulting in an asymmetric oval-shaped bubble. This ovality effect became more pronounced as the collapse proceeded. Just before collapsing, the bubble's interface accelerated at a faster rate, visualized by the blurred image captured at times around t = 0.400 and t = 0.413 ms. This first collapse resulted in the formation of the jet directed in the longitudinal +x direction.

Until its first collapse, the bubble did not change its position significantly, neither vertically nor longitudinally. The end frame of this first collapse at t = 0.413 ms shows that the shape of the bubble and its collapse dynamics was similar to a bubble collapsing near a flat solid surface for γ = 1.8. In both cases, the first collapse occurred in the water and without touching the oblique glass plates. However, the bubble's shape did not reflect the notch formation before its first collapse, as observed by Sagar and El Moctar.¹¹ The speed of the jet generated on the protruded right surface of the bubble was estimated to be about $≈$ 90 m/s, where the maximum jet speed predicted for a bubble collapsing near a flat surface was 120 m/s,⁹ and for a bubble collapsing between parallel walls, it was 35 m/s⁶⁰ and 300 m/s,⁷⁶ respectively. Before its first collapse, the bubble attained an asymmetrical oval or protruded chicken egg shape, also shown in the fourth frame of the third row in Fig. 7 in Brujan et al.⁶⁰ Despite these similar shapes, further jetting shown in our image sequences differed from the jetting observed in the investigation of Brujan et al.⁶⁰ The transformation of the protruded oval shape to the jet generated without the formation of a notch on the bubble's left surface was questionable. To resolve this issue, ultra-high-speed imaging would have been required, which was beyond the scope of our investigation. However, Figs. 5 and 6 show that the numerical investigation supported the associated flow phenomenon during the collapse phase. The numerical results reflected that the notch formation did not occur before the first collapse. Figure 6 shows the pressure field surrounding the collapsing bubble.

The time step as well as the time duration of the simulation was smaller than in the experiment. The time step in the simulation was adapted to an acoustic courant number of 0.5. This corresponded to a time step of about 1 ns. The sampling rate in the experiment was about 75 kHz, which corresponded to a time step of about 13 μs. The simulation time (covering about three bubble collapses) was 0.5 ms, while the time in the experiment was 0.8 ms. The simulation started from a full-grown bubble, and the time shown in Fig. 6 counted from the full-grown bubble condition. In the experiment, the first collapse occurred at time t = 0.413 ms (starting from the plasma seeding at time t = 0 s) and, at times between 0.187 and 0.213 ms, from the full-grown condition of the bubble. The exact time for the full-grown condition of the bubble was difficult to predict, whereas in the numerical simulation, the first collapse occurred at t = 0.187 ms.

The absence of the notch might have been due to the lack of a sufficiently large longitudinal pressure gradient acting on the bubble's surface or its distributed over a larger area of the bubble. The jet's direction pointed toward the converging side of the plates (the right side of the bubble) and along the longitudinal axis between the plates. After its first collapse, the bubble's surface initially remained ruffled until the second rebound. During the initial phases of the second rebound, the jet and the rebounding cavity formed an obscure leaf shaped cavity. At times t = 0.426, 0.440, 0.453, and 0.480 ms, the penetrating jet is visible as a dark line in the interior of the bubble. The bubble rebounded further until t = 0.520 ms. Similar to the first growth (after the plasma seeding), the rebounding cavity remained stable between times t = 0.480 and t = 0.520 ms. The left side face of the bubble remained stationary, while a jet protruded through the right side face of the bubble and translated in the direction of the converging edges, which is also seen in our computations (see Fig. 5). The appearance of a counter jet and the tiny bubble on the left side was pronounced as the second rebound proceeded. As the jet proceeded through the liquid toward the longitudinal +x direction, the protruded right side of the bubble reversed back toward the center of the bubble before the second collapse occurred. At around t = 0.520 ms, the right side of the bubble surface (the side along which the jet protruded out) began to shrink in. Initiation of this shrinking could be because the jet effect has been vanished, thereby reducing the extruding speed of the bubble's left side surface. At t = 0.546 ms, the bubble had an apple-like shape with a wider notch at its left side and a spherized conical surface at its right side. The jet, as it did not physically interact with the nearby oblique glass plates, can be referred to as a floating jet. Here, before the bubble's second collapse, its right side surface reversed back to a spherical (rounded) shape, while on its left surface, a notch was formed causing a bowl-shaped cavity. At the end of the second collapse, it seemed that the jet flow in the longitudinal +x direction pulled on the bubble's left surface. Such an independently floating jet may be useful in microfluidic pumping of fluids through a microchannel or a micronozzle where the actuation is required⁷⁷ without affecting the liquid surrounding the valve.

At t = 0.6 ms, the shape of the bubble during its second collapse was characterized by a crown shape with a base ring on the right side and a top ring on the left side of the crown cavity/bubble as seen in Fig. 4. After the second collapse, the bubble dispersed into a cloud of tiny bubbles, here termed nucleation. The third collapse in our experiments, shown in Figs. 4, 7–9, and 14, was manifested by a suddenly reduced size of the cavity. We visually identified the collapses by the disappearance of cavities or by the cavity/cloud attaining its minimum volume. After the collapse, we visually observed the growth of the cavity again. Therefore, the instance between a decreasing and an increasing bubble/cloud was considered a collapse in our investigations. During the initial phase of the cavity's third rebound, both sub-cavities of the crown cavity rebounded independently, with a base having a ruffled surface accelerating in the longitudinal +x direction and the top ring experiencing radial/toroidal regrowth. Later on, the independence of both the base ring and the top ring was not reflected in the third collapse, i.e., both sub-cavities collapsed concurrently. The third collapse was characterized by a vertical toroid translating in the longitudinal +x direction. Although of short duration, the bubble's fourth rebound and its collapse generated a fragmented toroid and caused intense nucleation in a defined area. This third collapse may be significant for applications where either the microscopic mixing of liquids or the associated immersed nucleation is highly desirable.^78,79 Especially, in micro/mini-scale fluidic systems, the mixing of fluids is often dominated by diffusion due to smaller dimensions and low Reynolds numbers. Mixing of fluid due to diffusion is inefficient and may take longer than expected. Even micro-mixing is affected by the presence of micro molecules, bacteria, and cells. In such cases, intense nucleation and pumping are desirable attributes.

During the bubble's first regrowth, it experienced an almost uniform pressure distribution over its surface. This resulted in a stable bubble size and shape over a relatively longer time duration of about 0.100 ms. Due to the lower pressure inside the bubble, it began to collapse and experienced a radial inward flow over its surface. The distribution of this radial flow affected mainly its shape.¹⁷ For the bubble collapsing near a flat surface, the radial flow at the bottom of a bubble is less than at its top, thereby causing its oval shape. Especially in our current case, the additional flow from the diverging side was expected because of the existing open liquid volume on the bubble's left side and enclosed converging on its right side.

Due to the oblique glass plates, the pressure distribution on the bubble's top-left and top-right surfaces was asymmetric, being higher on its top-left side than on its top-right side. A similar pressure distribution acted on the bubble's bottom-left and bottom-right sides. Thus, overall six principal flow regimes of radial pressures distributions governed the bubble's dynamics. Figure 7 depicts these six flow regimes, comprising two regimes at the bubble's left and right sides, two symmetric inclined regimes at the diverging top-left and bottom-left sides, and two symmetric regimes from the converging top-right and bottom-right sides.

For the bubble collapsing at b = 5 mm, it was positioned longitudinally toward the converging edges of the oblique plates, and at a distance of 5 mm distance from the diverging edge. Figure 8 shows the bubble's collapse sequence comprising distinct four collapses. Similar to the previous case of b = 0 mm, the bubble's first growth reflected the stable bubble shape for a relatively longer time of about 120 μs. Morphological stability made it difficult to determine the exact full-grown condition of the bubble between times t = 0.133 and 0.253 ms. Before the bubble's first collapse, its shape differed significantly compared to the previous case of b = 0 mm. At the beginning of the first collapse process around at t = 0.293 ms, the bubble started to take on an oval shape, where the maxima (vertical oval radii) remained approximately constant, while the minima (longitudinal oval radii) shrank. The bubble's longitudinal shape was asymmetric during the final stages of its first collapse, and its left side was more rounded than its right side. Its oval shape was due to the longitudinally acting asymmetric pressure gradient on its right and left sides generated by the converging flow region, which acted like a diffuser. It seemed that the bubble's top and bottom surfaces had an affinity with respective oblique glass plates. Therefore, the vertical oval radii of the bubble did not change significantly compared to the longitudinal oval radii. The bubble then attained almost a kidney bean-like shape before its first collapse. As expected, the bubble's first collapse was accelerated during the last phase of the collapse, and the interface was blurred at the collapse time of t = 0.426 ms, indicating the highly accelerated bubble's surface against the exposure time of a high-speed camera. It appeared that the first collapse was a toroidal collapse along a longitudinal axis. At this instance of around t = 0.440 ms, a second rebound was initiated, consisting of a toroid growing along the longitudinal axis. Figure 8 shows sequences of the bubble's dynamics, reflecting the change in the inclination of the toroidal cavities during the bubble's first collapse. The red dash-dotted line at t = 0.440 ms marks the axis of such a toroidal cavity. The subsequent second bubble collapsing at t = 0.586 ms was characterized by intense nucleation containing numerous clouds of microscopic bubbles at a concentric location.

Further on, the toroid proceeded to regrow (second rebound). The toroid seems to be inclined about the vertical (z-axis), resulting in an asymmetric toroid having a lateral y-axis of rotation directed normal to the vertical supporting plate. The right side of the rebounding toroid represented the fragmented, thicker, and highly accelerated interface, while the left side of the toroid was thinner and sharper during the initial rebound phases. As time proceeded, both longitudinal sides of the rebounding toroid thickened and became more fragmented. When fully rebounded, the toroid thickened, taking on the shape of a thick diamond ring with two sub-cavities, a shank at its left side and a head at its right side (see Fig. 8, t = 0.493 ms). The rebounded cavity started shrinking at t = 0.520 ms and collapsed radially inward at t = 0.586 ms. The shank of the diamond ring shaped cavity collapsed toroidally, while the head of this diamond ring collapsed as an independent cavity at a concentric region. The second collapse consists of two simultaneous sub-collapses (a shank collapse and a ring head collapse). The collapse instance at t = 0.583 ms clearly reflected a concentric cloud of tiny bubbles surrounded by a thinner toroidal ring. Although initially, the cavity's axis of rotation was directed normally to the supporting plate, i.e., the y-axis, it inclined about the x-axis during its second collapse. The rebounding cavity had then the z-axis as the axis of rotation. The toroid reflected in the collapse instance appears almost in the longitudinal plane of reference having the z-axis of rotation. During its second collapse, the cavity rotated the second time. An inclination dynamics of the diamond ring-shaped cavity is observed in this case of b = 5 mm. The inclination observed here may not be the rotation of cavity itself but the growing and collapsing shapes of the bubble in different directions. However, the vorticial flow induced during the collapse phases might have caused the inclination of the cavity itself. Although this phenomenon seemed to be similar to the toroidal knots illustrated in Kobayashi and Nitta⁸⁰ and Pisanty et al.,⁸¹ the rotational dynamics of such a toroid or toroidal knots were beyond the scope of our investigation.

The third rebound again generated a horizontal toroid having a vertical axis of rotation together with some thicker cavities at the laterally center location of the toroid. These central thicker cavities merged with the toroid in the later phase of rebound (after t $≈$ 0.640 ms). However, almost half of the toroid in the longitudinal +x direction appears thicker than the rest half of the toroid. Earlier phases of the third collapse revealed that the central cavities merged with the toroid on its right-side, thereby initiating the toroid's third collapse. This third collapse again consisted of two simultaneous horizontal sub-toroidal collapses, also forming a toroidal cloud of tiny bubbles (at t = 0.693 ms). Further on, our observations distinctly identified the fourth rebound causing another toroidal collapse. The second and third (even the fourth) collapses of a toroidal ring and an associated ring head induced a cloud consisting of tiny bubbles at a certain spatial region.

Regarding the importance of microfluidic devices, their requirement includes the controlling of small volumes of fluids, and the liquid–liquid interaction (dispersions), where a high Reynolds number is required. In such cases, a method of producing microscopic bubbles of higher density at the focused location is desirable.⁷⁹ In our current case, the bubble induced an intense nucleation at a average time period of about 250 μs. The intense bubble nucleation was generated at least three times by single bubble collapse: first, after $≈$ 586 μs after the plasma seeding (second collapse); second, after an interval of $≈$ 107 μs (third collapse); and third, after $≈$ 66 μs (fourth collapse). Based on these findings, such a method might be applicable where a shorter duration between microbubble generation or nucleation is required. In addition, the collapsing toroid generated floating nucleation in the water and not nearby the solid surface.

For this case, of the bubble collapsing at b = 5 mm, the pressure distribution during the first regrowth and collapse was still symmetric in the vertical direction. Longitudinally, the pressure was higher from the diverging (left) side of the bubble. Along the converging side of the oblique plates, the pressure gradient was less compared to the diverging side. This negative pressure gradient in the longitudinal direction was caused by the diffuser effect. The effect of pressure asymmetry was noticeable before the end of the first collapse. The asymmetry effect was more pronounced at the end of the first collapse as the increased pressure on the left side created a more rounded shape at the left side of the bubble. The rounded left side topology of the bubble between times t = 0.280 and 0.386 ms remained constant although this surface translated toward the longitudinal +x direction. The oval-shaped right side of the bubble accelerates to the longitudinal −x direction, whereby the right side of the bubble straightened itself from a rounded shape, but only in the last few instances before the first collapse. This straightening of the right surface of the bubble progressed with the curved shape just before the first collapse occurs (after t = 0.4 ms). Overall, the motions of the longitudinal surfaces (left and right) of the bubble are more dynamic than the vertical sides (top and bottom). Even the bubble's top and bottom surfaces did not attach themselves to the respective oblique glass plates although they were quite close to plates (γ_h < 2.0). In investigations of Brujan et al.⁶⁰ and Zeng et al.,⁶⁸ the bubble expansion was limited vertically; hence, the bubble's upper and lower surfaces became flat but did not attach to respective nearby plates. Upon collapse, the bubble split and got attached to plates.

However, the bubble's position did not change until its first collapse occurred. Then, during its second rebound, the bubble started translating in the longitudinal +x direction. The jetting that occurred during the first collapse stretched the bubble's right-sided interfaces. Therefore, the right surface of the bubble is also ruffled as mentioned before. Probably, the jetting also caused the translation of the entire non-spherical entity. Later, during the second and third collapses, the cavity traveled toward the longitudinal +x direction, but it maintained its position during each rebound and the early collapse phase. It seemed that the cavity's translation was significant during the bubble's first collapse compared to the cavity's translation during the second and third collapses. The translation induced during the third collapse was comparatively less compared to the translation that occurred during the first collapse.

Overall, the bubble collapsing at b = 5 mm featured a bean-shaped before its first collapse, a diamond ring shaped during the second rebound, the rotational dynamics of the entire cavity during the second and third collapses (e.g., knotted toroid), and a strong nucleation at the concentric region after the second, third, and fourth collapses. This strong nucleation is useful in microfluidic devices and applications where microscopic mixing is desired. The bubble at the concentric region caused the hydrodynamics interaction among the tiny bubbles, and this interaction stretched the liquid segments causing mixing and homogenization.⁸² On the other hand, the complex fluid dynamic process of the floating toroid and its rotation may be related to the knotted vortices observed by Kleckner et al.⁸³ The results of Kleckner and Irvine⁸⁴ regarding knotted vortices generation are similar to the toroid generated in our experiments, where they mentioned the difficulty in producing such vortices.

For the bubble positioned at b = 10 mm, it was situated longitudinally at a two-third distance of the total test section from the diverging edge. As it was closer to the converging edge, the diffuser effect became stronger. Figure 9 shows the bubble's collapsing and rebounding sequences. The bubble remained spherical during the initial first growth phase. Further on, its growth in the vertical direction was affected by the presence of the oblique glass plates, causing the shape of the bubble's top and bottom side surfaces to spherically tapered toward the longitudinal +x direction after t = 0.080 ms. Later, the tapering effect was mitigated and led to a more spherical bubble shape. Similar to the previous two cases (b = 0 and 5 mm), between times t = 0.160 and 0.240 ms, its shape seemed to stabilize, and it was difficult visually to define its full-grown condition. Until its first collapse, the top and bottom surfaces of the collapsing bubble remained stable and in place. The top as well as the bottom surfaces did not attach to the nearby respective oblique glass plates although they were close enough to these glass plates. Also, the bubble did not split into two parts. In investigations of Brujan et al.⁶⁰ and Zeng et al.,⁶⁸ the splitting of the bubble was observed if the bubble collapses between parallel plates. The upper and lower parts of the bubble were rather close to the oblique plates, as seen in Fig. 8 from t = 0160 to 0.240 ms. Yet, the upper and lower bubble surfaces did not become flat. The shape of the bubble as it reaches its maximum size is different from the shape of the bubble in the article by Zeng et al.⁶⁸ However, as seen in Fig. 4 in Zeng et al.,⁶⁸ the bubble expansion was limited vertically; hence, the bubble's upper and lower surfaces became flat but did not attach themselves to the respective nearby plates. Upon collapse, the bubble split and its parts further attached themselves to the plates. However, the bubble's right and left surfaces moved as a result of the dominant pressure gradient (diffuser effect). Note that in the previous case (b = 5 mm), both these surfaces moved radially inward in the longitudinal direction which was also the case here.

The extent of movement of these side surfaces strongly depended on position b of the cavitation bubble in the diffuser region, i.e., the diffuser effect became more pronounced as the bubble moved toward the converging side of the glass plate assembly. Additionally, the interaction between the bubble's top and bottom sides, and the oblique plates prevented the top and the bottom surfaces from shrinking radially inward toward the center of the cavity. Therefore, during the final stages of the first collapse at t = 0.386 ms, the bubble took on an elongated bean-like shape, with an indentation at its right side. As the pressure on the bubble's right side was lower than on its left side, a notch is developed on its right side. However, this notch formation in the current investigation differed from the notch formation during the bubble's collapse near a flat solid surface. We think that it was related to the bubble being bent, i.e., the bending of the bubble, initiated mainly by the dominant oblique pressure regimes along the oblique plates acting on the bubble's top-left and bottom-left side surfaces (see Fig. 7). These pressures are expected to be higher; therefore, the pronounced bending of the bubble as the bubble position b proceeds toward the converging region. Even so, these pressure at the top-left and the bottom-left surfaces seemed to be higher than the pressures at the middle-left side surface of the bubble because bending was observed but not notching before the first collapse. However, the pressure at the middle of the left side was more pronounced just before the first collapse for this current case b = 10 mm, but not for the case of b = 5 mm. This caused a notch to be formed on the bubble's left side. Thus, for the cases considered until now, the dominance of oblique pressure regimes at the top-left and bottom-left of the bubble caused the bending of the bubble. The stated phenomenon is valid, if the comparison for the bubble bending (curvature of notch) in the case of b = 5 mm and b = 10 mm is considered.

Later on, just before the first collapse at t = 0.4 ms, pressure on the bubble's left side became dominant and caused a deeper notch formation on its left side. This process was comparable to the notch formation in the case of a bubble collapsing near a solid surface, where notching induced the jet formation. Both notch formation as well as bending of the bubble itself caused its middle part to become thinner. It adopted a dumbbell shape with two bulbous protrusions (bulb shaped cavity) at each vertical end. At t = 0.426 ms, three parts of the bubble, a thin and longer vertical cavity in the middle and the two bulbous end cavities, collapsed at the same time. The two collapsing bulbous end cavities got attracted to their respective nearby oblique plates and collapsed in the vicinity of the plates, while the thin vertical-cavity stretched vertically and split into two vertical cavitation poles.

A pair of vertical thin cavities developed during the second rebound. Here, we assumed that, the wider jetting through the thin cavity and the vertical stretching that occurred during the first collapse caused the vertical cavity to split, thereby generating a pair of vertical cavities. Thus, the first collapse consists of four sub-collapses, namely, the collapse of the two bulbous cavities near the oblique plates and the two collapses of the vertical thin cavities. Although these entities collapsed differently/independently, they maintained physical contact not only during the first collapse but also afterward during the second rebound.

The second rebound clearly perceived the two vertical parallel cavities growing, i.e., enlarging cavities and two flattened bulbous cavities on the oblique plate. This vertical regrowing pair of cavities caused the formation of two thin parallel cavities during the first collapse. The bulbous cavities attached to the plates grew independently until t = 0.480 ms, while the pair of parallel vertical cavitation pillars regrew until time t = 0.506 ms. Further on, the right side of the pillars smeared and formed a conical shape, and its top and bottom ends became thinner as the time progressed from t = 0.506 to 0.573 ms. However, the middle part of the thin vertical cavities did not become thinner; instead, they extended toward the longitudinal −x direction. The smearing of the vertical cavities was probably related to the jetting. However, the formation of the conical shaped cavity was due to the pronounced negative pressure gradient at its middle acting in the longitudinal +x direction, thereby thinning the ends of the thin vertical cavity. Jetting-induced flow may have been pulled the interfaces of the cavity in the jetting direction, i.e., in the longitudinal +x direction.

In addition, as discussed above, the diffuser effects also caused smearing of the edges of the cavity in the longitudinal +x direction. Although this is not distinctly visible, the bulbous cavities collapsed on their respective nearby oblique plates. However, the lower part of the flattened cavity was visible, and we assumed that they rebounded and collapsed. The collapse of vertical cavitation pillars occurred at around t = 0.586 ms. During the second collapse, the thinned ends of the pillars disappeared, while the middle conical cavity was still visible as a tiny cloud. The position of the collapsing cavities shifted in the longitudinal +x direction in contrast to its position after the first collapse, which was also noticed in the previous cases during the collapse phases. The bulbous cavities attached to the plates did not see regrowing further, whereas the conical part of the pillars and the pair of cavitation pillars continued their third rebound, separately. The bulbous cavities attached to the glass plates could have been rebounded and collapsed in the vicinity of the plate, where they may have formed a thin toroid or flattened thinner cloud of bubbles, therefore invisible. The bulbous cavities attached to the glass plates almost separated from the vertical thin cavities during the second collapse. Further on, their collapse might have similar dynamics to the collapse dynamics of the single bubble nearby a flat solid surface where the surface attached cavities attain the thin toroidal shape hardly visible from a side view.

Thin vertical string-like cavities regrew during the third rebound until t = 0.653 ms and then instantly collapsed after a duration of $≈$ 27 μs. Meanwhile, the separated part of the thin cavities during the second collapse, i.e., the conical cavity independently rebounded and collapsed further on. It seemed that the conical cavity collapsed at the same time as the thin vertical cavities. Later, thin vertical cavities, consisting of tiny bubbles and segments were visible in all the frames. Nevertheless, it was difficult to predict its collapsing and rebounding phases. Even it was difficult to determine whether the pair of the thin cavity existed or whether they merged. However, the thin vertical-cavity maintained its spatial position as it appeared to be thinning. The dynamics of these tiny cavities was similar to the dynamics of a toroidal cavity collapsing near a flat surface after several collapses.

Note that the initial notch formation at the right side of the bubble continued with dominant notch formation from the left side of the bubble just before its first collapse. Although the notch formed at the right side of the bubble may not really have been a notch, but the bending of the bubble was due to obliquely dominant pressure from the left side of the bubble on its top-left and bottom-left surfaces. The phenomena of bending and notching dynamics were difficult to clarify in the current scope of the investigation. However, a thorough numerical investigation would have been necessary to better understand the underlying physics. Like in the previous cases (b = 0 and 5 mm), the bubble maintained its full-grown condition for a longer time. The stability of the bubble was hardly noticed in the previous investigations of a bubble collapsing near a flat plate or between parallel plates. Therefore, our work might be useful for those who intend to investigate the details during the bubble's first growth and the dynamics of a bubble's surface contour, and the associated shock wave radiation.

After the bubble's first collapse, the cavity maintained its position during the second rebound and collapse. Later, during the second collapse, the thinner cavities traveled toward the converging side (the longitudinal +x direction), whereby the static nature of the vertical cavity during the second rebound and the third collapse formed a thinner nucleation area. Apart from the spherical and toroidal collapses, collapse dynamics of the thin vertical cavity were also of interest from the cavitation dynamics point of view. During the first collapse, two separate bulbous collapses occurred on opposite oblique walls at the same time. This finding might be useful in microfluidic applications, where two similar collapses are desirable. The collapsing cavities on the oblique plates also induced shear stress, a desirable feature in microscopic surface cleaning³³ and drug delivery.⁸⁵ The thinner vertical cavity remained attached to the top and bottom plates, even after its third collapse. Nucleation by thin cavities lasted for a longer duration, a desirable feature when high mixing efficiency is demanded.⁸⁶ Overall, this case of bubble collapse was characterized by the elongated bending and notching of bean-shaped bubble, a pair of parallel vertical cavities (cavitation pillars), and stable thinner vertical cavities.

Figure 10 shows the sequence of cavitation bubble dynamics captured for the bubble positioned at a distance of b = 12 mm from the diverging edges DE. The bubble was close significantly to the converging region. It was at a distance of only 3 mm from the converging edge and 12 mm from the diverging edge. A situation is relevant in identifying the effect of the converging edges on bubble dynamics. Although compared to the previous case of b = 10 mm, the bubble was now generated at an additional distance of only 2 mm further toward the converging edges in the longitudinal +x direction; apart from similar dynamics, we observed some spurious morphological changes.

The bubble was morphologically stable for a relatively longer duration between times around t = 0.187 to 253 ms during its first growth. This was observed for all cases of the bubble collapsing between oblique plates. Until time t = 0.400 ms, the bubble's behavior was similar to the previous case of b = 10 mm; however, major differences were observed at t = 0.413 ms, just before the first collapse occurs. As described above, the notch formation was due to the negative pressure gradient in the longitudinal direction (diffuser effect). While the global bending of the bubble contour was due to the pressure distribution on the bubble's top and bottom surfaces along the oblique glass plates. However, we observed higher bending, i.e., deeper curvature on the right side of the bubble in the current case compared to the previous cases of b = 5 and 10 mm. The diffuser effects resulted in the dumbbell-shaped cavity before the first collapse, while in the current case, the dumbbell ends (bulbous cavities) were larger compared to the previous case of b = 10 mm. In the previous case, the thinner cavity's first collapse at t = 0.426 ms appeared to vertically straighten, while in this case, the cavity has been globally bent in its middle toward the longitudinal −x direction, i.e., diverging edge side. This bending effect was effective for a shorter duration as it lasted only until time t = 0.440 ms.

Compared to the previous case of b = 10 mm, two thinner identical vertical cavities were generated after the first collapse. Independent dynamics of the bulbous cavities led them to attach themselves to the oblique plates. This phenomenon was not clearly visualized in the previous case, where the bulbous cavities attached themselves to the oblique plate. The smaller relative wall distance might have differed the dynamics of the bulbous cavities after they attached themselves to plates, having vertically more pronounced cavities compared to the previous case of b = 10 mm, where the relative wall distance was less. This phenomenon corresponded to the bubbles collapsing near a solid surface. Temporally, the dynamics of the vertical pillar cavities and bulbous cavities attached to the plates were almost the same in both cases. The right side of the pillars looked feathered toward the converging side of oblique plates. The pillar cavities remained stable between times t = 0.480 ms and t = 0.520 ms. Later, the pillar cavities started to shrink and bend toward the converging side just before the second collapse occurred. In the previous case, the pulling of only the middle part of the vertical cavities, where the rest of the vertical cavities remained stable, generated the conical shape directed toward the right side. Even further this conical cavity was separated from the vertical thin cavities during the third rebound and collapse. This might be due to the local diffuser effect at the middle of the thin vertical cavity, whereas in the current case, the pulling of the middle part of the vertical cavity was stronger and vertically wide enough during the second collapse. This was due to the comparatively wider global diffuser effect than in the previous case, resulting in the global bending of the vertical thin cavity toward the longitudinal +x direction. The pulling of the right side of the vertical cavity correlated with the flow induced by jetting with the diffuser effect, where the pressure from the longitudinal −x direction was dominant.

The bulbous cavities attached to their respective nearby oblique plates also started shrinking after t = 520 ms. The second collapse of both bulbous cavities on the nearby oblique glass plates occurred at the same time of t = 0.586 ms as the collapse of the vertical thin cavities. These bulbous cavity collapses generated an asymmetric toroids at t = 0.573 ms, where the right side of each toroid was higher than the left side. The third regrowth and collapse of the thin cavities were similar to the previous case of b = 10 mm, except that the rebounded thin cavities were bent. The shape of the vertical thin cavities was irregular, i.e., not straight as in the previous case b = 10 mm). The bulbous cavities attached to the plates also underwent their third collapse as toroidal collapses. The bottom two rows of the sequences in Fig. 10 show the toroid rebounding and collapsing near the lower oblique plate. The oblique plate also reflected the mirror images of the toroid. The toroidal collapse mainly induced the vorticial flow during their collapse.¹⁷ The shear rate induced by vorticial flow during a bubble's collapse is significant in surface cleaning applications.⁸⁷ Further on, nucleated cavities were generated after the third collapse of vertical thin cavities, but their distribution was less dense than in the previous case of b = 10 mm. On the other hand, the nucleation on the oblique glass plate surface due to the collapse of the bulbous cavity was observed, and it continued even after the third collapse. At around t = 0.759 ms, the fourth regrowth and collapse of the fragmented cavities attached to the plate occurred. In summary, this case was similar to the previous case of b = 10 mm, except for the thin irregularly shaped bent vertical cavities and the dynamics bulbous cavities attached to the glass plate that separated itself from the thin vertical cavities.

Figure 11 shows six collapse instances just before and after the first collapse of the bubble positioned at the four longitudinal distances of b = 0, 5, 10, and 12 mm from the diverging edges DE. These comparative images show that, during the first collapse, the shapes of the bubbles and the collapsing dynamics differed significantly from each other, depending on the initial longitudinal position b. Even the smaller 2 mm difference of longitudinal distance b from 10 to 12 mm influenced the dynamics of the bubble during its first collapse. However, the collapse times were relatively similar for the first collapse of all initial positions b. Although the major difference was the bubble's shape just before its first collapse, the dynamics of the bubble's initial growth and the initial collapse phases were similar (see Figs. 4–10). All cases showed the bubble attaining a morphologically stable shape lasting for a longer duration, which made it difficult to determine its full-grown condition. At about t = 0.40 ms, before its first collapse, the bubble took on an asymmetrical ablated oval shape, a thick bean like shape with a notch on its right side, a thin bean-like shape with notch on both the left and right sides, depending on b. For the bubble at position b = 0 mm, its ablated oval-shape shranks almost uniformly toward its center and radially normal to its surface. For the bubble at position b = 5 mm, it shrank longitudinally and, to a lesser extent, vertically. For the bubble at positions b = 10 and 12 mm, the cavity's vertical shrinkage was negligibly small compared to its shrinkage in the longitudinal direction. This shrinkage depended on the flow-induced pressure distribution acting on the bubble's surface. Specifically, diverging and converging flow volumes were present on the left and right sides of the bubble, respectively. The negative pressure gradient caused by the diffuser effect played a vital role in determining the variable dynamics of the bubble. Primarily, it was the diffuser effect that caused the bubble's collapse dynamics to differ from those of other bubble collapse cases in an open environment or close to a flat surface. Here, we observed that the effect of this asymmetric pressure gradient was more effective than the presence of a nearby wall on a bubble's collapse dynamics, especially for cases of b = 5, 10, and 12 mm. Of course, the interaction of the bubble positioned between the oblique plates became more pronounced in the later phase of the second rebound and collapse. Thus, the diffuser effect is consistently influencing the bubble's dynamics.

For b = 0 mm, the cavity induced a jet directed in the longitudinal +x direction during its first collapse. This extended jetting appeared to be sharper than the jet of the cavitation bubble collapsing near a flat surface (see Fig. 12). In this case, the jetting is not striking to any solid boundary, and the jet continued through the water until it was decelerated or compression of the cavity started. However, in the case of the bubble collapsing nearby a flat surface, the jetting impacted on the wall. Such intense jetting was due to the high pressure gradient acting on the bubble along the longitudinal direction from the diverging to the converging edge. As discussed above, the pressure gradient was partially a diffuser effect. Additionally, for b = 0 mm, the volume of the water on the left half side of the bubble was greater than that on its right half side, generating a higher pressure acting in the −x direction. While the right half of the bubble was surrounded by the oblique glass plates, this half of the bubble collapse was only particularly affected by the oblique glass plates.

The tip of the conical jet was directed toward the longitudinal +x direction. Therefore, we expected this converging jet to push/pump a spatially limited amount of liquid longitudinally through the converging edge of glass plates. For b = 5 mm, after its first collapse, the left side vertical edge of the toroidal became sharp, while its right side smeared toward the right during rebound. These images in Fig. 11 illustrate the complex collapse dynamics of the bubble. As described above (Sec. IV B), the smearing was probably due to the jet induced flow pulling on the bubble's surface causing its to smear.

Only for the bubble positioned at b = 10 and 12 mm, its collapse was characterized by the longitudinally directed bidirectional notches, which penetrated more deeply the left side of the bubble. The curved right side of the bubble characterized its the bending as described in Secs. IV B–IV D. This bending effect was more pronounced as the longitudinal initial position of the bubble b increased, which is evident in the first two columns of sequences shown in Fig. 11. The notch formed on the bubble's left side and the bulbous of the dumbbell-shaped cavity were larger for the case of b = 12 mm than for b = 10 mm. For the bubble positioned at b = 0 and 5 mm, its top and bottom surfaces did not attach themselves to oblique glass plates after its first collapse; that is, its entire collapse occurred in the liquid without the directly interacting with the oblique glass plates. For the bubble positioned at b = 10 and 12 mm, its bulbous top and bottom end cavities were attached to the top and bottom oblique glass plates after its first collapse.

In Fig. 11, although the two-dimensional views of the bubble for b = 10 and 12 mm look like a thin vertical line, we assumed that these images represent a vertical toroid with its top and bottom attached to the top and bottom oblique glass plates. In addition, the surfaces of the vertical cavities were smeared and irregular during the second rebound. This phenomenon occurred also for the case of b = 5 mm. Careful observations revealed the intense jetting flow pulled the bubble's surface in the longitudinal +x direction. However, the widening of the jetting area increased at greater bubble position b. The jet for the case of b = 0 mm was concentric and conical, and for the case of b = 5 mm, it was vertically wider. However, in both cases, the bulbous cavities did not get attached to the oblique glass plates; whereas for b = 10 and 12 mm, the bulbous cavities interacted with their respective oblique close glass plates, and the associated jetting through the cavity was wider. Both effects together resulted in vertical pillar-like cavities. Simply, the bubble was stretched vertically toward the respective oblique glass plates and penetrated by a wider jet during its first collapse generating vertical pillar-like cavities.

For all four bubble positions, the rebounding cavities after its first collapse were smeared toward the longitudinal +x direction, indicating that flow was accelerated in the longitudinal +x direction. This jetting induced accelerated flow and the shrinking of the cavity occurred during the first collapse. For the bubble positioned at b = 0 mm, the regrowing cavity took on an obcordate leaf-like shape. Further on, the first collapse of the bubble and its second rebound resembles a bubble collapsing near a flat surface at a relative wall distance of γ > 1.2, although no notch formed on the bubble's left surface. Especially for the bubble positioned at b = 5 mm, the rotating dynamics of the toroidal ring was observed after the first collapse. At first, the collapsing bubble aligned with the x-axis, rotated itself about the vertical z-axis, and aligned itself with the lateral y-axis of rotation. Further on, the cavity rotates about the vertical z-axis aligning itself to the lateral y-axis of rotation. Only the right side of the rebounded toroidal cavity was smeared, whereby it developed a thick diamond-like ring shape.

For the bubble positioned at b = 0 mm, the generated high-speed jet was not directed normally to the glass plates, but toward the converging opening. Previous investigations dealing with a bubble collapsing nearby a flat boundary γ < 3.0 found that the jetting was directed always toward the nearby the boundary. This postulate the affinity of the bubble to collapse toward the nearby boundary. However, in our current cases, intense unidirectional jetting dynamics along the longitudinal +x direction occurred. The jetting was sharper for b = 0 mm and vertically wider for b = 5, 10, and 12 mm. The jetting occurred completely in the water without the bubble interacting with a nearby boundary, here, the oblique glass plates.

For the case of b = 0 mm, these results might be of practical value not only regarding the impulsive micro-pumping of liquids^77,85 but also for investigating the jetting dynamics of bubbles. The case of b = 5 mm reflected a similar jetting phenomenon, but the jet was wider when penetrated out of the bubble along the longitudinal axis, resulting in the formation of the toroid. For the bubble positioned at b = 10 and 12 mm, Fig. 11 shows cavity thinning and splitting as its top and bottom approach the glass plates during the first collapse. However, with the increasing distance b, the jet induced flow broadened vertically. This broadening may have also been caused by the stretching of the bulbous cavities toward the nearby oblique glass plates. The smaller relative wall distance between the bulbous cavities and the oblique glass plates resulted in the attachment of the bulbous cavities to their respective nearby oblique glass plates. Both bulbous (top and bottom) cavity's rebounds and collapses dynamically mirrored. Their collapse could be similar to the cavitation clouds collapse between pair of non parallel plates application, e.g., impeller cavitation. The individual collapses might induce the structural loading, thereby erosion of both plates like “kill two birds with single stone.” The bubble dynamics were almost perfectly mirrored on both plates and, thus, both oblique glass plates were subject to similar cavitation effects. In addition, bulbous cavities collapsing nearby plates may induce enough shear rates on the oblique glass plates, which is useful in surface cleaning applications. Especially here, a single bubble could be used to clean two surfaces at the same time and at a symmetrically identical place. This phenomenon of the mirrored collapse of bulbous cavities might also clarify the reason behind the healthy tissue ablation by bubbles collapsing in narrower gaps in biomedical applications,²⁶ and how bubble clouds collapsing in a venturi nozzle damage the aluminum foil.^88,89

Figure 13 shows six collapse sequences of the bubble positioned along the longitudinal axis at four distances of b = 0, 5, 10, and 12 mm from the diverging edges DE just before and after its second collapse. Similar to the first collapse, major differences in bubble dynamics were also found in the second collapse. The bubble generated at position b = 0 mm had a horizontal apple shape before its second collapse. Compressed from both sides, i.e., from the left and the right, the apple shape ended up as a thin vertical cavity. In the vertical direction, the shape of the bubble shrank vertically less than in the longitudinal direction. Before its second collapse, the bubble looked like an apple shaped cavity. The dominant longitudinal flow from the left side of the cavity at its center caused a jet to break through the crown-shaped cavity, consisting of a base ring on its right side with facets on its left side. Similar to the collapsing crown, the initial rebounding of the crown cavity consisted of two distinct sub cavities rebound.

The dynamics of the bubble positioned at b = 5 mm differed distinctly from the dynamics of the bubble positioned at b = 0 mm. In the case of b = 5 mm, the diamond ring-shaped cavity had a thick shank with a smeared head at its top (along the longitudinal +x side). The rotational axis of this collapsing cavity was directed normally to the supporting plate, i.e., in the lateral y direction. However, the bubble rotated itself about the longitudinal axis x, where the rotational axis of the cavity transformed to the vertical z-axis after its second collapse. The vortex flow induced during the toroidal collapse might have been caused the change in the inclination of the toroid during its third rebound. During the second collapse, a thick ring-shaped bubble collapsed like a toroid. At the front of this toroidal ring, i.e., on the right side, distinctly smaller collapsing irregular clouds were generated. These irregular clouds may have originated from the collapse of the head cavity of the diamond ring cavity. Two separate entities, an outer toroid, and a cloud, later regrew independently. However, both were thinner than they were before the second collapse. The second collapse for the case of b = 5 mm has characterized mainly the intense nucleation in a certain area (around the collapsing region). The rotation of the toroid could be correlated with the knotted knots discussed above in Sec. IV B of a bubble collapsing at b = 5 mm.

For the bubble positioned at b = 10 and 12 mm, the jet became wider after the first collapse and pierced through the bubble. Further on, the pair of the top and bottom bulbous cavities became attached to the oblique glass plates during the second collapse. Between these bulbous cavities, a pair of thin vertical pillar-like cavities was generated. Apart from this, for the bubble positioned at b = 10 mm, the vertical cavities were thicker than for the bubble positioned at b = 12 mm. However, the right side of both vertical cavities were feathered/smeared in the longitudinal +x direction. For the bubble positioned at b = 12 mm, the bulbous cavities attached to the nearby oblique glass plates were separated from the vertical cavities. Due to the thinner nature of the vertical cavities for the case of the bubble positioned at b = 12 mm, the collapse of these thin cavities occurred earlier than for the case of the bubble positioned at b = 10 mm. The conical nature of the central part of the vertical cavity occurred only for the case of b = 10 mm (see Sec. IV C). For the bubble positioned at b = 12 mm, the vertical cavities were bent toward the longitudinal +x direction. The conical part of the cavity for b = 10 mm separated from the vertical cavities after the second collapse, and the cavities attached to the glass plates also collapse and regrew independently. In both cases of b = 10 and 12 mm, the vertical cavities regrew, forming thin vertical cavities. To our knowledge, such thin, continuous, and linear cavities were rarely observed in experiments of bubbles collapsing near a solid surface. These thin cavities were independent, continuous, and distinct entities, entirely suspended in the fluid. Understanding the independent dynamics of toroidal or cylindrical vapor/gas-filled cavities might be significant for practical applications, mainly where nucleation on a specific area is desirable. The collapse of bulbous cavities at oblique walls might be useful in cleaning a pair of semiconductor surfaces without causing damage. As these bulbous cavities were twin mirrored entities, their identical collapsing dynamics resulted equal shear rates, suggesting that a single bubble induces equal shear rates on similar areas of an obliquely placed pair of plates at the similar time. However, Zeng et al.⁶⁸ also observed such twin entities collapsing between parallel plates.

The third collapse was clearly perceived in all four cases we experimentally investigated. Figure 14 shows six sequences of the bubble dynamics positioned at the four longitudinal distances of b = 0, 5, 10, and 12 mm from the diverging edges just before and after its third collapse. For the bubble positioned at b = 0 mm, its collapse formed a vertically aligned toroidal shape had longitudinal axis of rotation x. The independently rebounded crown-shaped cavities collapsed together. Clouds of tiny microbubbles were generated during the third collapse which further regrew resulting in more pronounced clouds of tiny microbubbles. For the bubble positioned at b = 5 mm, a thinner circular toroid aligned with the vertical z-axis of rotation represents the bubble's third collapse. For the case of b = 5 mm, the toroid had bulgy nature at its lateral sides, while it was thinner at its left and right ends. The collapse of the toroid at b = 5 mm was similar to the toroidal collapse of a bubble near a solid surface at 1.4 < γ < 2.0. The difference was that, in our current experiments, the toroid was entirely surrounded by the fluid (floating toroid). Neither a solid entity (e.g., an oblique glass plate) nor a neighboring entity (e.g., bubbles) influenced the toroid. Understanding the floating toroid collapses might be useful to determine the energy content of such vapor-filled toroidal cavities. For the bubble positioned at b = 12 mm, the thin complex cylindrical cavities seemed to be linked to each other. Theoretical investigations in Kleckner et al.⁸³ explained the complex collapse dynamics of the associated superfluid knotted vortices.

These linked and knotted vortex loops have been examined for over a century. Understanding the associated physics is of interest in a wide range of physics fields not only limited to fluid dynamics. Knots and links often occur in physical systems, such as shaken strands of rope⁹⁰ and gas-filled vortex structures in fluids.⁸⁴ Kleckner and Irvine⁸⁴ reported that simple principles enable understanding the dynamics of superfluid vortex knots and links. Regarding these phenomena, our experiments might be suitable to visualize the nature of toroidal bubble collapse. The case of the bubble positioned at b = 12 mm reflected also the collapse dynamics of a bubble attached to the wall. The cavities collapsing at the bottom glass plate were thin line toroids, similar to the second and third collapse of the bubble near a flat solid surface. For the bubble positioned at b = 12 mm, vertical thin cavities disappeared after its third collapse, while for the case of b = 10 mm, vertical thin cavities continued to grow and collapse even after the third collapse.

The third rebound and collapse generate gas-filled thin vertical cavities for b = 10 and 12 mm, which might be significant for mathematical modeling and understanding its physics. For the bubble positioned at b = 10 and 12 mm, the thin cavities might be linked to the physics associated with knotted vortices. Theoretical investigations of Kleckner et al.⁸³ explained the complex collapse dynamics of the associated superfluid knotted vortices. Overall, the bubble positioned between two oblique plates perceived at least three distinct collapses. A slight change in the bubble's longitudinal position in the diffuser region highly influenced the dynamics of a cavitation bubble during the observed three collapses. The dynamics of each distinct collapse differed from the dynamics of the other collapses, thus demonstrating the physical significance for various practical applications.

To quantify the bubble's dynamics, we monitored its changing shape during the collapse. This comprises specifically the translation of the bubble's center in the longitudinal x direction and the position of the points on the right, left, and top sides of the bubble's contour, coordinates $S+x, S−x$⁠, and $S+z$⁠, as illustrated in Fig. 3. For the four cases, the bubble generated at longitudinal positions b = 0, 5, 10, and 12 mm, Figs. 15–18 plot, the time series of the bubble's longitudinal position and its changing surface deformation. Each of these figures comprises two graphs. The left graph plots the time history of the longitudinal position of a bubble's center; the right graph, the time history of the right, left, and top positions of the bubble contour. Note that coordinates $S+x, S−x$⁠, and $S+z$ were derived from the so-called edge function of a bubble's surface contour as shown in Fig. 3. We did not plot the coordinate $S−z$⁠, because we assumed the vertical symmetry of the bubble throughout its dynamics. For the case b = 10 and 12 mm, where our high-speed camera images did not cover the left part of the bubble during its first growth and collapse for the interval of 200 μs, we assumed that the bubble's shape was almost symmetrical in the longitudinal direction. The left side surface of the bubble is assumed to be symmetric to the right surface during this interval. The asymmetries in the shape of the bubble were observed during the late collapse phase. Therefore, we estimated the coordinate $S−x$ of the left side contour of the bubble. In these cases, the MATLAB function edge plotted the constant values of $S−x$ considering the frame edge as end position. The symbols FC, SC, and TC corresponds to the first, second, and third collapse, respectively, in Figs. 15–18.

For a bubble generated at b = 0 mm, the left graph of Fig. 15 plots the time series of its center position. In the time interval t = 0 (plasma seeding) to 386 μs, the bubble's center remains at its initial position, i.e., at x $≈$ 0.

Then, it began to translate longitudinally in the +x direction, i.e., toward the converging edges just before the first collapse. The bubble's maximum displacement of about 1.1 mm occurred during its first collapse. It was micro jetting that pulled the bubble cavity in the longitudinal +x direction. Later, the bubble maintained its position until t = 506 μs, which was around the second collapse. The cavity continued advancing in the longitudinal +x direction as it regrew. This translation continued during the third collapse and further rebounds.

The steep translation of about $≈$ 1.1 mm occurred in just $≈$ 54 μs (from t = 386 to 440 μs) during the first collapse, whereas the translation of about 0.8 mm after the bubble's first collapse (until the third collapse) occurred over 213 μs (from t = 480 to 693 μs). The bubble's translation after its first collapse was related to the high-speed jet penetrating out its right side, while the bubble's further translation was associated with the generated vorticial flow surrounding the toroidally shaped cavity during collapses. The effect of this vorticial flow was mitigated at a constant rate, even after the second and third collapses, indicated by the trend of translation of the bubble. Generally, during rebounding, the bubble/cavity remained at its position. However, its collapse induced its translation in the longitudinal direction. Overall, the bubble translated about 2 mm from its origin toward the converging side until its third collapse.

The right graph of Fig. 15 shows that the bubble's surface contour gradually grew until t $≈$ 200 μs, and then, it started to shrink before its first collapse. The time histories of its contour coordinates $S+x, S−x$⁠, and $S+Z$ indicate that the first, second, and third collapses occurred at t $≈$ 413, $≈$ 600, and $≈$ 706 μs, respectively. At the first collapse, three contour plots approach symmetrically at point, which indicated a uniform radial inward motion of the bubble's surface. The point where all three contour positions come closer is offset from the initial position of the bubble, indicating its translation during its first collapse. During its second collapse, the right and left contours approach each other at t = 600 μs, i.e., horizontal surfaces came closer indicating thinning. The top surface of the bubble also shrank during the second collapse indicated by the lowered contour coordinates $S+z$⁠. However, the cavity's left side surface contour $S−x$ shrank more than its right side surface contour $S+x$⁠.

The higher initial velocities up to 20 m/s corresponded to the bubble's initial rate of growth after the plasma seeding at around t = 0.027 μs. During the last phase of its first collapse, the collapse velocities of the bubble's contour was pronounced. The bubble's left surface shrank, attaining a maximum collapse velocity of $≈$ 40 m/s, while its right side of the surface contour advances further, reaching its shrinking velocity of $≈$ 20 m/s. The bubble also shrank vertically; its top side contour attained a velocity of about $≈$ 32 m/s. After its first collapse, the velocity of the right side of $≈$ 90 m/s at t $≈$ 413 μs was due to the jetting in the +x direction. We expected even higher jetting velocities, but our imaging frames were not able to reliably capture the tip of the jet after t $≈$ 413 μs, and this led to constant values of $S+x$⁠. Apart from jetting, the right and top surfaces of the bubble also regrew quickly after the first collapse with velocities $≈$ 19 and $≈$ 30 m/s, respectively. The higher surface velocities also indicated an equally volatile regrowth after the bubble's first collapse. Later, the motion of the bubble surfaces decreased, stabilized at t = 0.506 μs, and then gradually decreased until the third collapse.

The left graph of Fig. 16 plots the time series of the bubble center's longitudinal translation. The bubble translated about 70 μm in the +x direction, i.e., toward the converging edge, linearly until time t $≈$ 70 μs after the plasma seeding. Then bubble remained at this same position of x $≈$ 0.07 mm until t = 320 μs. The bubble translated a distance of 70 μm toward the +x direction during its first collapse phase. Similar to the previous case of b = 0 mm, the bubble advanced rapidly by $≈$ 580 μm in $≈$ 54 μs after its first collapse. During its second rebound, the bubble translated backward, as seen in the left graph of Fig. 16. Determining the longitudinal position of the bubble's center was not precise, because MATLAB we used had to rely on the smeared edges while defining the contour position $S+x$ between t = 480 and 506 μs, thereby predicting the overestimation of the longitudinal +x position of the bubble's center. As shown in the left graph, of Fig. 16, the bubble continued to move longitudinally in the +x direction during the second collapse phase as well as during the entire third cycle without moving backward. The bubble moves a distance of about 70 μm in 40 μs during its second collapse (from t $≈$ 546 μs to t $≈$ 586 μs). During its third rebound and collapse from t $≈$ 586 μs to t $≈$ 693 μs, the bubble moved about 230 μm in around 107 μs. As seen, the bubble moved a significant distance longitudinally during its first collapse after t = 426 μs, where it lost most of its energy, which may have caused it to translate a lesser distance during the subsequent rebounding and collapsing cycles. Jetting in the longitudinal direction during the first collapse may have caused the steep translation of the bubble in the longitudinal direction.

The right graph of Fig. 16 shows the bubble's surface gradually growing and shrinking before its first collapse. As seen in the previous case of b = 0 mm, the increasing and decreasing $S+x, S−x$⁠, and $S+z$ coordinates indicate the occurrence of the first, second, and third rebounds and collapses. In this case, the period in the bubble's first oscillation (growth and collapse) of $≈$ 430 μs was somehow longer than the period of 413 μs for the previous case of b = 0 mm. However, the changing the surface contour coordinates $S+x$ and $S−x$ longitudinally at a late stage of the first collapse indicated that the bubble translated in the longitudinal direction. The cavity had a larger vertical dimension than the longitudinal ones. Interestingly, the bubble's contour along the −x direction (coordinate $S−x$⁠) oscillates about its original position at the first collapse with a decreasingly lower amplitude over the last two oscillations, i.e., during the second and third collapses. However, the bubble's right surface pulsated with a relatively larger motion amplitude, and it simultaneously moved further away, advancing from its original position. Meanwhile, the bubble's top side surface behaved exactly opposite to its right side surface motion, where it kept on dissipating its motion from its original position during the second and third collapse and rebounding cycles. The second oscillation, lasting about 170 μs, was similar to that of the previous case of b = 0 mm.

As seen in graphs of Fig. 16, the overall trend of the translation of the right, left, and top surfaces was similar over time duration from 0 to about 670 μs. Before the end of the bubble's growth at around 200 μs, the three side contours almost had uniform velocities, indicating the nearly spherical expansion of the bubble. After the plasma seeding, the initial expansion velocities were predicted about 20 m/s for three locations on the bubble's contour. After reaching its minimum volume at the collapse time of t = 426 μs, the bubble surfaces began to violently move outward with different velocities, called rebound velocities. The predicted maximum expanding velocity of the bubble's right surface was about 30 m/s, while the velocity of the left surface was only about 7 m/s, and the velocity of the top surface was about 10 m/s. In the second collapse phase, the top surface collapse at a greater rate and lesser rate in the regrowing phase. In this case, the motion of the bubble's contour illustrates also that the expanding and collapsing velocities dissipated with time.

The high-speed camera had to be moved in the longitudinal +x direction to reliably capture the bubble dynamics during its second and third rebound collapse phases. This is because the bubble's translation during its collapses and rebounds was expected in the longitudinal +x direction. Therefore, the bubble generated at b = 10 mm, the captured images, its left portion extended beyond the image frame during the time interval between 80 and 280 μs. Therefore, the MATLAB function we used specified the frame ends for determining the $S−x$ for the mentioned duration. This led to constant values of $S−x$⁠, resulting in flattened part of $S−x$ graph, shown in the right graph of Fig. 17.

The left graph of Fig. 17 plots the time history of he longitudinal position of the bubble's center. It indicates that the bubble's first collapse lasted over a time span of nearly 420 μs. We assumed that the longitudinal position of the bubble's center remained constant before its first collapse. This behavior was similar to the one observed in the previous cases b = 0 and 5 mm. Assuming longitudinal symmetry over the time interval from 80 to 280 μs, we estimated the values of the coordinate $S+x$ while deriving the longitudinal positions of the bubble center. The estimated values of $S+x$ are marked as solid blue dots in the right graph and by hollow blue circles in the left graph of Fig. 17. As seen in the left graph of Fig. 17, after the time of about 370 μs, the bubble's center translated longitudinally in the +x direction.

At the late stage of its first collapse, the right and left surfaces of the bubble came closer together. The closeness differed from the previous cases. The bubble's top and bottom surfaces attached themselves to the upper and lower oblique glass plates, respectively. Therefore, the vertical coordinates of the top surface $S−z$ are constant over time as seen in the right graph of Fig. 17. During the bubble's second collapse, it turned into a vertical pillar shaped cavity that continued to become thinner and thinner. After about 580 μs, vertical cavities broke loose and generated an ensemble of microbubbles, making it impossible to estimate surface contour coordinates.

As seen in the right graph of Fig. 17, during its growth and collapse (after t = 227 μs), the vertical length (height) of the bubble nearly always exceeded its longitudinal length (width). During its first collapse, the bulbous cavities attached themselves to the oblique glass plates. Therefore, the coordinate of the bubble's top surface remained constant afterward. Apparently, the exhibited dynamics of the bubble differed from the previous cases, as its shape was elongated at the beginning of its rebound at about 420 μs. The motion of the bubble's left and right surfaces during its first collapse (at time t = 400 μs) resulted in the velocity of its left side of $≈$ 36 m/s, which exceeded the velocity on its right side $≈$ 14 m/s by about 22 m/s. Between t $≈$ 410 and 560 μs, during the bubble's rebound phase, its right and left side surfaces translated in the longitudinal $+x$ direction. However, after its first collapse until 450 μs, the left surface of the bubble remained relatively motionless, and later on, it starts accelerating during the second collapse phase of the cavity.

When the bubble was generated at b = 12 mm, its dynamics resembled that of the previous case of b = 10 mm. As mentioned above for the case of b = 10 mm, the high-speed camera was moved in longitudinally to capture the bubble dynamics. Therefore, the image frame did not capture the left side of the bubble during the time intervals between 66 and 320 μs. Again, for these image sequences, the MATLAB function assumed the frame end while determining the $S−x$ coordinate. This led to a constant value of $S−x$ and a flattened region of the surface coordinate $S−x$ shown in the right graph of Fig. 18. During this time interval, the bubble was assumed to be longitudinally symmetric. Based on this assumption, we estimated the position of the bubble's left side (⁠$S−x$⁠) and its longitudinal center position x. The estimated values of coordinate $S−x$ are marked by blue solid dots in the right graph and the longitudinal position of the bubble's center by blue hollow circles in the left graph of Fig. 18. Just after the first collapse at around t = 400 μs, the bubble's center position moved consistently over time toward the converging edge of the glass plates as shown in the right graph of Fig. 18.

Until t = 350 μs, we assumed that the longitudinal position of the bubble's center remained constant. A sharp increase to 0.9 mm of the bubble's center in the longitudinal direction occurred over the next 200 μs, i.e., from t = 350 to 550 μs. Although the magnitude and duration of this longitudinal translation was not identical, it had similar trend to that of the last case of b = 10 mm. This continual motion of the bubble's center could have been due to the jetting and vorticial flow around the cavity. At the end of the first collapse (t $≈$ 420 μs), the left and right surfaces were very close to each other, as shown in the right graph of Fig. 18, indicating the thinning of the cavity in the longitudinal direction, as seen also in Fig. 11.

The attachment of the bubble's top and bottom surfaces to the oblique glass plates was confirmed by the time histories of the bubble's top surface contour coordinates $S+z$⁠, as seen by the sudden jump at t $≈$ 420 μs in the right graph of Fig. 18. Similar to the previous cases, the right, left, and top positions on the bubble's contour expanded with higher velocities. After the plasma seeding, the expanding velocity of the bubble's top side coordinate $≈$ 34 m/s exceeded the velocities of the left and right side surface coordinates of $≈$ 22 m/s. Although the bubble's initial growth after the plasma seeding was similar in all investigated cases, it differed in that the velocity of the expanding top surface of the bubble increased as b increased. This was due to the enhanced effect of the upper and lower oblique glass plates. At the end of the first collapse at around t = 413 μs, the bubble's left side surface collapses faster (at about 34 m/s) than its right surface (at about 18 m/s). The higher velocity of the left surface was confirmed in that a dumbbell shape cavity evolved from a thick bean shaped cavity, with a deeper notch from the left side of the bubble shown in Fig. 11 (t = 413 μs). During its second rebound between 410 and 480 μs, the bubble's left surface remains relatively stationary, which later t $≈$ 493 μs began to shrink before the second collapse occurred, whereas the bubble's right side continued to move further in the longitudinal +x direction.

From our images for b = 0 mm, we estimated image-based jetting velocity of about 90 m/s at t $≈$ 413 μs (first collapse). The jetting velocity for the other cases (b = 5, 10, and 12 mm) turned out to be between 20 and 30 m/s although we expected higher jetting velocities. Unfortunately, our imaging frames were not able to reliably capture the interface between water and vapor. Furthermore, the frame rate was too low. Our preliminary numerical simulation for b = 0 mm estimated the maximum velocity of $≈$ 600 m/s at first collapse. We believe that numerical simulations can more reliably predict field parameters and may help in better understanding of flow physics in detail.

Our experiments dealt with the complex dynamics of a cavitation bubble between a pair of oblique glass plates. As it collapsed, the cavitation bubble developed a bean shape, an obcordate leaf shape, a diamond shape with a dual regrowing ring, a rotating diamond ring-shape, a thin vertical cylindrical shape, a thick horizontal toroidal shape cavity, and knotted vortices during its various collapse phases. Regarding the associated formation of jets in the water (floated jetting), practical application might lead to an improved control of microfluidic flow valve and microbubble-based power actuators.⁷⁷ The development of the dosing system, micro-pumping, and underwater micro-propulsion systems may also be beneficial.⁸⁵ 

The collapse dynamics of the bubble was substantially affected by the initial longitudinal position between the oblique plates. The development of an obcordate leaf shaped cavity, which was entirely surrounded by water and did not attach itself to a nearby oblique plate improved our understanding of the jetting dynamics during the cavitation bubble's collapse and can be further investigated. Our experiments revealed the development of several kinds of toroidal bubble shapes and their dynamics. Observed parallel pairs of vertical cavities have shown their multiple collapses and rebounds. Monitoring the generation of such thin and single continuous cavities and the formation of isolated trefoil vortex knots and pairs of linked vortex rings in water was particularly challenging. Kleckner and Irvine⁸⁴ accelerated specific hydrofoil shapes through the water, where the creation of such vortices in the laboratory was an outstanding experimental achievement. Our experimental method using laser-induced bubbles proved its ability to produce macroscopic toroidal and thinner unique gas/vapor-filled cavities. Hopefully, our results might be more useful for the fundamental analysis of the knotted rings and the development of the theoretical background necessary for mathematical modeling.

Ability of our experiments to generate a stable bubble over a relatively long average duration of 100 μs can facilitate the determination of the gaseous content of the bubble itself and the subsequent generation of the toroidal cavities. Our experiments also showed the formation of tiny microscopic bubble clouds for a longer duration. The microscopic bubble generation in our experiments is useful where rapid and repeatable mixing is desired. Apart from this, our method may be helpful to address the quest “toroid or tiny cavity induced damage over the surface.” Future research calls for systematic experimental exploration of, among other things, flow velocities and pressure to explore the associated erosive damage.

Authors acknowledge the support provided by Professor A.-M. Zhang, and Dr. T. Schellin. Authors would like to thank Dr. Z.-Y. Hu for providing support for data extraction; G. Moloudi and E. Katsuno for reviewing the manuscript.

The authors have no conflicts to disclose.

Hemant J. Sagar: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Validation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Ould el Moctar: Supervision (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.