Nonlinear free-decay oscillations of a magnetically levitated air bubble in water produced by coalescence

The moderate- to large-amplitude free-decay oscillations of an air bubble in water, formed from the coalescence of two parent bubbles, are studied experimentally and by numerical simulation. Magnetic levitation is used to suspend the air bubble, with a diameter of approximately <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>6</a:mn><a:mspace width="0.16em"/><a:mi>mm</a:mi></a:mrow></a:math>, within the liquid as if under weightless conditions. We find that the method does not significantly distort the shape of the air bubble, facilitating direct comparison with theoretical predictions of oscillations from an assumed spherical equilibrium shape. The coalescence of two equal-sized parent bubbles produces a single bubble undergoing freely decaying nonlinear shape oscillations dominated by the two-lobed shape mode, approximating the problem considered theoretically by Tsamopoulos and Brown, who obtained the corrections to the shapes and frequencies of the linear modes of oscillation to second and third order in amplitude, respectively. The frequency of the two-lobed oscillations decreases with the square of the amplitude, as predicted, but with a coefficient smaller in magnitude than expected. Coalescence of two unequal-sized bubbles, differing in diameter by a factor 1.54, produced a bubble oscillating with comparable contributions from two- and three-lobed shapes. Here the quadratic coefficient is the same (within uncertainty) as that observed in the symmetric case. However, there is also a significant dependence of the frequency on the fourth power of the amplitude in this case, and the quadratic coefficient of the three-lobed mode is approximately three times larger than predicted, which we attribute to coupling between the two- and three-lobed modes. The magnitudes of the various surface harmonics contributing to the shape of the two-lobed mode agree well with Tsamopoulos and Brown's prediction at second order in amplitude and with our calculations at third order. Results from simulation are consistent with those from experiment.