Period-adding route in sparkling bubbles

Chains of bubbles are seen rising along the wall whenever champagne is poured into a glass. The careful observation of a given bubble chain often reveals that the interbubble distance suddenly changes during the degassing process, indicating different bubbling regimes in this elusive phenomenon of effervescence. We report the transitions between these different bubbling regimes that present sequences of multiple periods known as the period-adding route.

Figure 1

(a)–(d) Macrophotographs of the bubble chain issued from a given nucleation site on the wall of a glass poured with champagne. As time proceeds, the bubble nucleation site experiences different bubbling regimes; in (A) bubbles are seen generated from a period-2 bubbling regime, which is characterized by the fact that two successive bubbles rise in pairs. (B) Multiperiodic bubbling. As time passes, a bubbling in period 3 occurs and is shown in (C). Finally, after others various changes in the bubbling regime, the nucleation site ends in a clockwork period-1 bubbling regime shown in (D) $(\text{bar}=1\mathrm{mm})$.

Figure 2

Close-up time-sequence illustrating one period of the oscillation of the gas pocket trapped inside the fiber’s lumen. The nucleation site is in its period-2 bubbling regime. The time interval between two successive frames is $20\mathrm{ms}(\text{bar}=50\mu \mathrm{m})$.

Figure 3

The cellulose fiber blowing bubbles is a tiny “bubble gun” modeled as a system of two coupled oscillators; the course of the bubble embryo oscillating trapped inside the fiber’s lumen is denoted $h(\text{bar}=50\mu \mathrm{m})$ (a). Temporal evolution obtained from the model for a period-2 bubbling regime, in which the bubble radius function has a threshold given by an envelope $\epsilon \cos \left(2\pi \Omega F_b\tau \right)$ (b).

Figure 4

A comparison between attractors obtained from the experiment (left) and from the model (right), respectively; the lines between successive dots show the cycle; in (a) the bubbling regime is in a period 3; in (b) there is a pile-up when the points come into the vicinity of $T=850\mathrm{ms}$, and a very similar behavior is obtained with the model for a period-7 bubbling regime; in (c) the system ends abruptly in a clockwork period-1 bubbling regime. The bubbling dynamics, both in the experiment and in the model, are characterized by a sequence of increasing periodicity always ending in a clockwork period-1 bubbling regime.

Figure 5

Bifurcation diagram of stable periodic points obtained by numerical simulation and by gradually decreasing the frequency ratio $\Omega$.