Cavitation dilemma resolved in knuckle-cracking model

Picture this: a dark alley where the suave evil dude has cornered a helpless plot point. Looming behind the evil dude is someone as broad as he is tall. The generic muscle has a shaven head, missing teeth, and an in-grown T-shirt. The plot point cowers as the muscle moves in.

But, before flogging the soon-to-be-dead plot point, the muscle leers and the sound of cracking knuckles echoes off the walls.

Movie clichés not withstanding, popping joints are still something of a mystery. After some experiments and a great deal of thought (and argument, probably involving knuckle cracking), scientists concluded that the popping noise was due to the collapse of cavitation bubbles in the fluid that lubricates joints. That explanation went unchallenged for some 40 years, until more sophisticated experiments showed that, after the popping noise, there were still bubbles in the joint fluid. How could the collapse of bubbles be the source of sound if the bubbles were still there? Now, a pair of scientists has developed a mathematical model of knuckle cracking that shows how both can be true.

Sucking bubbles

Joint popping usually occurs when bones of the joint are suddenly separated by an unusual amount of fluid. As the two bone surfaces move apart, the fluid between the joints suffers a sudden pressure loss. When the pressure drops low enough, the fluid turns to a gas, creating a bubble. This is cavitation.

The high-pressure fluid surrounding the bubble rushes in and crushes the life out of the bubble very quickly. The fast collapse generates pressure waves that leave the fluid as sound waves. The very sharp collapse of the bubbles is what generates the characteristic popping noise. But a key point is that the bubbles end up dead. Yet, experimental evidence says that there are still bubbles in the joint fluid even after the echoes have died away.

The big challenge to understanding this in detail is that it pulls in lots of different physics at lots of different time scales. Modeling the whole mess from end to end is actually very difficult.

Piecewise solutions

To understand cavitation in knuckle joints, a pair of physicists analyzed its processes separately: bubble formation, bubble collapse, and sound-wave generation. The equations for each process were developed and solved using the results from the previous step. So, for instance, the equations that model the collapse of the bubble take the bubble’s properties from the output of equations that model the bubble-formation process. Then the collapse of the bubble is used to generate sound waves in a third, separate model.

The researchers found that bubble formation probably happens exactly as expected: provided the space between the bones in the joint is relatively small, a hefty tug will drop the pressure fast enough to ensure that some of the joint fluid vaporizes and bubbles can form. But these equations only model the pressure drop and not the bubble formation process. So, in fact, the input to the next stage of the model is not a mathematical bubble with calculated internal and external pressures. Instead, the bubble size is guessed and only the pressure is calculated.

In the next step, though, the researchers were in for a surprise. The model bubble collapses rapidly, as expected. But it does not fully collapse—the bubble stabilizes at about half its original size. This, the researchers tell us, means that there is agreement between the cavitation model for joint popping and the experimental evidence. But, again, the story is more nuanced. In the researchers’ model, the bubble cannot vanish, and it cannot collapse to an infinitely small radius because the gas inside would have to reach infinite pressure. Instead, the bubble will oscillate a bit and settle at a radius such that the internal pressure and external pressure are equal. That is exactly what the researchers observe in their mathematical model. So, bubble survival is kind of built in to their model.

The final step is to predict the sound made by the bubble as it collapses in on itself. The results clearly show that the model predicts the popping sound quite well. It should be noted that there is quite a bit of variation in experimental measurements here, so the model predicts the general features that are common across all measurements but not the details of individual measurements.

A pop is a pop is a pop

The agreement between experiment and the model is really quite good, especially considering the simplicity of the model. My big concern is that, by choosing the diameter of the bubble and the geometry of the joint, they had essentially fine-tuned their model to fit. In some ways that is true: the model results do change with joint geometry. And I think I would expect that. I simply can’t imagine that a hip joint would make the same sound as a finger joint when popped.

On the other hand, I had expected that the diameter of the bubble would matter: a bigger bubble would make a different noise compared to a small bubble. But, the researchers tested that, and the sound remains the same. Although they did not investigate the reason, I suspect that it is because, no matter how big the bubble is, it can only collapse to 50 percent of its size, and the rate of collapse appears to be the same in both cases, which might produce the same acoustic spectrum.

The research doesn’t address the ultimate death of these bubbles. There are, I guess, a number of possible pathways that the bubble can take. It could fragment to smaller and smaller bubbles. This would require some sort of instability in the bubble surface, which I don’t think is included in the model. Alternatively, the bubbles vanish because the bubble surface is permeable, and joint-fluid vapor condenses into the surrounding fluid, allowing the bubble to continuously shrink.

Both of these processes are likely to be slow compared to the process of bubble formation and collapse. So we can get persistence of bubbles after the pop without turning them into permanent residents of our joints.

Another interesting conclusion from the model is that there are no stable solutions for joint-popping forces above about 15 Newtons. This is in contrast to experiments, where researchers have found that people often apply about 100 Newtons of force. The researchers suggest that much of that force is taken up by the surrounding tissue so that only a small fraction of the pressure is exerted on the joint.

The model is also quite idealized. It assumes that the pressure waves from one bubble do not influence the generation or collapse of neighboring bubbles. The researchers acknowledge that this is probably not the case because the first bubbles will be formed along the center line of the joint. As those bubbles collapse, more bubbles are formed nearer to the edge of the joint. The pressure waves from the collapsing bubbles will pass through and influence the formation and collapse of the off-center bubbles. These complicated interactions make for a target-rich environment as far as experimental and theoretical physics goes.

In the end, the researchers have done something quite cool here. They have shown how two experimental results that seem contradictory may not be. At the same time, they’ve opened up a whole bunch of new questions: how much force ends up in the joint? Why does bubble size not matter? How do the bubbles influence each other? Where do they go?

The researchers also raise the point that the experimental evidence is rather sparse. The problem is that the cavitation occurs really fast. Typically, scientists use high-speed cameras to observe the dynamics. Unfortunately, the interior of joints are not that amenable to high-speed photography.

, 2018, DOI: 10.1038/s41598-018-22664-4 (About DOIs).

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