Supersonic jet dynamics from two-cavitation-bubble interactions: acceleration, tip fragmentation and penetration

This study experimentally and numerically investigates the dynamics of high-speed liquid jets generated by the interaction of two tandem cavitation bubbles, identifying three distinct jet regimes governed by the spatiotemporal impact of the first bubble's collapse pressure wave and establishing phase diagrams to guide biomedical applications like needle-free injection.

The Gist

Imagine you are holding two giant, invisible balloons underwater. Now, imagine you pop the first one, and just a split second later, you pop the second one. What happens when these two "pops" collide?

This paper is a high-speed detective story about what happens when two bubbles, created one after the other, crash into each other underwater. The scientists found that this collision doesn't just make a splash; it creates super-fast liquid bullets that can shoot out at speeds faster than a speeding bullet (over 1,200 meters per second!).

Here is the breakdown of their discovery, explained simply:

The Setup: The "Catapult" Effect

Think of the first bubble (Bubble 1) as a giant, expanding balloon that suddenly deflates. As it collapses, it creates a vacuum and a shockwave, kind of like a giant underwater vacuum cleaner.

The second bubble (Bubble 2) is created just as Bubble 1 is collapsing. Because Bubble 1 is sucking water in to collapse, it pulls Bubble 2 toward it. This stretches Bubble 2 out, making it look like a long, thin egg or a teardrop pointing at Bubble 1.

The Three Types of "Liquid Bullets"

Depending on exactly how far apart the bubbles are and how much time passes between popping them, the tip of Bubble 2 shoots out in three different ways. The scientists call these three "regimes":

1. The Cone Jet (The "Spear")

• When it happens: The bubbles are a bit further apart, and the timing is just right.
• What it looks like: Bubble 2 gets stretched into a sharp point. When Bubble 1 collapses, it hits this point like a hammer, driving it straight into the bubble.
• The Result: A sharp, cone-shaped spear of water shoots out. It's fast, but not the fastest. It's like a well-aimed spear throw.

2. The Umbrella Jet (The "Mushroom")

• When it happens: The bubbles are closer together.
• What it looks like: The tip of Bubble 2 starts to shrink back on itself. Just as it's shrinking, the shockwave from Bubble 1 hits the base of the shrinking tip.
• The Result: Imagine a stream of water hitting a flat surface and spreading out. The water at the back of the jet moves faster than the tip, pushing the tip outward. It looks like a mushroom or an open umbrella.
• Why it's cool: This jet is very stable and travels a long distance, making it great for things like "needle-free injections" (shooting medicine into skin without a needle).

3. The Spraying Jet (The "Supersonic Mist")

• When it happens: The bubbles are very close together, and the timing is perfect.
• What it looks like: The tip of Bubble 2 gets stretched so thin that it snaps, like a rubber band breaking. This is called "neck breakup."
• The Result: This snap creates a massive explosion of pressure right at the break point. It launches a jet of water so fast it goes supersonic (faster than sound).
  • Needle-like: Sometimes it shoots out as a straight, thin needle of water.
  • Mist-like: Sometimes, because it's so unstable, the tip shatters into a cloud of tiny droplets, like a spray bottle.
• The Power: These jets are the champions. They can travel more than 10 times the width of the original bubble. That's like a bullet traveling the length of a football field before stopping.

Why Does This Matter?

You might wonder, "Who cares about bubbles?"

The answer is: Medicine and Engineering.

• Needle-Free Injections: Imagine getting a vaccine or medicine shot without a needle. These bubbles can create a jet of liquid fast enough to punch through your skin and deliver the medicine deep inside, painlessly.
• Micro-Pumping: These jets can move tiny amounts of fluid with incredible precision, useful for delicate lab work or cleaning tiny parts.

The "Liquid Bullet" Model

The scientists realized that once the jet leaves the bubble and enters the water, it acts like a bullet.

• It has a front tip that pushes through the water.
• It leaves a trail of air behind it (like a bubble tunnel).
• They created a simple math formula to predict exactly how far this "liquid bullet" will go based on how fast it starts.

The Big Picture

The scientists created a "map" (a phase diagram) that acts like a recipe book. If you want a specific type of jet (a sharp spear, a stable umbrella, or a supersonic spray), you just look at the map, find the right distance and timing, and you can create it.

In short: By carefully timing two underwater bubbles, we can turn a simple pop into a super-powered, controllable water cannon that could revolutionize how we deliver medicine and clean delicate machinery.

Technical Summary

1. Problem Statement

Cavitation bubbles are known for generating high-speed liquid jets and shock waves during collapse, which are utilized in applications ranging from ultrasonic cleaning to needle-free medical injections. While single-bubble jets near boundaries can reach high speeds (up to ~1000 m/s), they often require stringent boundary conditions and pose risks of shock wave damage to surrounding tissues.

Previous studies on tandem bubble interactions (two bubbles generated with a time delay) have shown they can produce directed, high-speed piercing jets. However, the underlying mechanisms driving supersonic acceleration, tip fragmentation, and long-range penetration in these systems remain poorly understood. Specifically, the transition between different jet morphologies and the factors governing their penetration capabilities in a controlled manner require further elucidation to optimize them for biomedical applications like needle-free injection and micro-pumping.

2. Methodology

The study employs a combined experimental and numerical approach to investigate the dynamics of two tandem cavitation bubbles (Bubble 1 and Bubble 2).

• Experimental Setup:
  • Generation: Centimeter-scale bubbles were generated using an underwater electric discharge method in a water tank. Two capacitors were discharged with a precise time delay ($\Delta t$) to create Bubble 1 and Bubble 2.
  • Parameters: The study varied the initial spatial offset ($d$) and the initiation time difference ($\Delta t$). These were non-dimensionalized as $\gamma = d / 2R_{max}$ and $\theta = \Delta t / T_{osc}$.
  • Visualization: A high-speed camera (Phantom V2012) operating at 43,000–110,000 fps captured the transient interactions and jet formation.
• Numerical Simulations:
  • Boundary Integral Method (BIM): Used to decouple complex interactions and isolate the effects of pressure waves and surface curvature on jet initiation. It allowed for the removal of Bubble 1 at specific times to test causality.
  • Volume of Fluid (VoF): Utilized an improved OpenFOAM solver (cavBubbleFoam) to simulate the full evolution of the gas-liquid interface, including jet penetration, breakup, and the formation of gas cavities. This method captured topological changes (e.g., neck breakup) that BIM cannot handle.

3. Key Contributions

The paper identifies and characterizes three distinct jet regimes emerging from the interaction of two tandem bubbles, governed by the parameters $\gamma$ and $\theta$:

1. Conical Jets
2. Umbrella-shaped Jets
3. Spraying Jets (including needle-like and mist-like variants)

The study provides a comprehensive phase diagram mapping these regimes and elucidates the physical mechanisms behind supersonic acceleration and tip fragmentation.

4. Key Results and Findings

A. Jet Regimes and Formation Mechanisms

• Conical Jets: Occur when Bubble 1 collapses before the tip of Bubble 2 fully contracts. The pressure wave from Bubble 1 impacts the high-curvature tip of Bubble 2, driving it inward. The jet velocity scales linearly with the impulse of the pressure wave ($\Delta p \Delta t$).
• Umbrella-shaped Jets: Form when Bubble 1 collapses after the tip of Bubble 2 has already begun to contract. The converging flow accelerates the upstream fluid faster than the jet tip. As this faster fluid overtakes the tip, it sweeps the liquid sheet outward, creating a flattened, umbrella-like structure. This can also lead to "double umbrella" structures if a secondary pressure wave hits the jet base.
• Spraying Jets: Result from neck breakup of the elongated tip of Bubble 2.
  • Needle-like: Occurs when the pressure wave from Bubble 1 impacts the neck of Bubble 2 before coalescence, causing a high-pressure stagnation point that accelerates the jet to supersonic speeds (>1200 m/s). The tip is inherently unstable and fragments into droplets, but the continuous jet behind remains stable.
  • Mist-like: Occurs at very small $\gamma$ where the bubbles coalesce. The collapse of dispersed micro-bubbles creates chaotic pressure fluctuations, resulting in a broad, splashing jet with lower velocity.

B. Acceleration and Velocity

• Supersonic Speeds: The study successfully reproduced supersonic jets with velocities exceeding 1200 m/s (dimensionless $U > 120$) in the needle-like spraying regime.
• Acceleration Mechanism: The extreme acceleration is driven by the neck breakup mechanism, where the collapse of Bubble 1 creates a localized high-pressure stagnation point at the neck of Bubble 2.
• Stability: While the jet tips fragment due to capillary instability (Worthington drop-off mechanism), the continuous jet column behind the tip maintains high stability and directionality, supported by the jet base.

C. Penetration Performance

• Liquid-Bullet Model: The authors developed a simplified model treating the jet as a "liquid bullet" subject to hydrodynamic drag. The model predicts that the maximum penetration distance ($S_{max}$) is directly proportional to the external jet velocity ($U_{water}$) upon exiting the bubble.
• Regime Comparison:
  • Spraying Jets (Needle-like): Exhibit the highest penetration capability, traveling >10 times the maximum bubble radius. They are ideal for long-range delivery.
  • Umbrella-shaped Jets: Offer optimal short-range penetration with superior stability and controllability.
  • Conical & Mist-like Jets: Show limited penetration due to lower velocities and rapid energy dissipation.

D. Phase Diagrams

The study established phase diagrams in the $\gamma-\theta$ parameter space:

• Conical jets dominate at larger $\gamma$ and $\theta$.
• Spraying jets (high velocity) are found in regions of moderate $\gamma$ and $\theta$.
• Umbrella-shaped jets occupy a narrow intermediate region.
• These maps provide a practical guide for selecting initial parameters to achieve specific jet morphologies for targeted applications.

5. Significance

• Scientific Advancement: The paper resolves the debate on whether jet formation is driven by surface curvature or pressure waves, concluding that curvature initiates the jet while pressure waves govern its subsequent growth and acceleration. It also clarifies the mechanism of umbrella-shaped jet formation via flow overtaking.
• Technological Application: The findings offer a robust framework for needle-free injection and micro-pumping. By controlling the timing and distance of bubble generation, operators can select between:
  • Needle-like spraying jets for deep, long-range tissue penetration.
  • Umbrella-shaped jets for controlled, shallow, and stable perforation.
• Safety: The ability to generate directed jets that penetrate long distances while minimizing the direct impact of shock waves on surrounding targets addresses a key safety concern in biomedical applications.

In summary, this work provides a fundamental understanding of supersonic jet dynamics in tandem bubble systems, bridging the gap between theoretical fluid dynamics and practical biomedical engineering applications.