Basic mechanisms making millimeter-sized gas bubbles rising in pure water exhibit non-straight paths

What is it about?

Everybody is familiar with the following observation. When one fills a basin from a faucet, air bubbles are entrained downwards into the water and then rise toward the surface. However, when these bubbles are large enough, say with diameters of a few millimeters, they do not rise in a straight vertical line, and rather perform zigzagging or spiraling motions. This note summarizes the current theoretical knowledge concerning the hydrodynamic mechanisms that are hidden behind this intriguing « dance of bubbles » that was already described in several drawings by Leonardo Da Vinci five centuries ago. The paper shows that three distinct mechanisms may be involved in the genesis of the phenomenon: the wake that follows the bubble all along its ascent, the time-dependent deformations of the air-water interface, and the coupling between the bubble (considered as a rigid object) and the surrounding fluid. Over the last 15 years, successive numerical studies based on the linearized version of the equations governing the problem allowed the respective role of each of these three players to be examined separately. The first studies considered the role of the wake alone, with a bubble held fixed in a uniform current and keeping a frozen shape. Then, the next step considered bubbles, still with a prescribed shape, but allowed to move freely in the fluid under the effect of buoyancy. This second step was shown to capture the essential characteristics of the phenomenology observed with real bubbles rising in pure water, namely the switch from the initial vertical path to an oscillatory low-frequency motion when the bubble size exceeds a certain critical size. The most recent step considered the three players altogether and showed that the non-straight motion starts when the bubble diameter exceeds 1.85 millimeters, in excellent agreement with reference experiments. Putting together the conclusions of these successive studies, the paper emphasizes that neither the instability of the wake nor the variations of the interface shape are at the origin of the non-straight path of millimeter-sized bubbles rising in pure water, and indicates why the conclusions may be different in more viscous fluids. In pure water, the origin of the phenomenon stands in the coupling between the unconstrained motion of the bubble and the surrounding fluid; under certain conditions, solid bodies falling under the influence of gravity exhibit non-straight motions governed by a similar phenomenology. In the case of bubbles, time-dependent shape variations only have a secondary role: the critical size beyond which the straight vertical path of a fully deformable bubble becomes unstable is slightly lower than that of a bubble with a similar but frozen shape, but the phenomenology of the instability is the same in both cases.

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Photo by Nancy Hughes on Unsplash

Photo by Nancy Hughes on Unsplash

Why is it important?

Speculations regarding the mechanism governing the path instability of rising bubbles have dotted the history of fluid dynamics, especially over the last 70 years. Recent publications suggest that incorrect views resulting from an incomplete appreciation of the phenomena at play still exist in the community. This note summarizes the main findings of a long series of systematic papers devoted to the origin of non-straight motions of bodies moving freely under the effect of gravity or buoyancy. It provides a rational appraisal of the respective role of each of the available mechanisms in the specific case of small bubbles rising in pure water.

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