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(Color online) Diagram showing the parameters used for the model described here. The two bubbles (with radii r 1 and r 2) are considered axisymmetric about the line joining their centers. R is the radius of the neck of air joining them (which is assumed cylindrical). The inset shows a close-up of the region where the two bubbles touch, and the shaded area shows the cross-section of the liquid annulus under consideration in the model when the neck has reached radius R. The schematic diagrams along the bottom of the panel show how the neck is assumed to change in time, where the solid lines represent the actual bubble walls at any time and the dotted lines show the initial position of the bubble walls.
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(Color online) (a) A plot of the neck width with time. The dotted line shows the results from Thoroddsen et al. and the solid line shows the predictions of the model described in this paper for the same parameters. If the model results are multiplied by 0.65, they exactly overlie the experimental data. (b) The scaling of the acoustic pressure amplitude with a reference pressure, a plot equivalent to figure 13 in Manasseh et al. The circles are the data points from that figure, and the lines show the predictions from the model presented here for θlim of 40° (dotted line), 45° (dashed line), and 50° (solid line).
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(Color online) Results of the model described in this paper, from numerical simulations using the parameters in Manasseh’s paper. The large bubble has a radius of 0.8 mm in all cases, and the results shown here are for small bubble radii of 150 μm (solid line) and 200 μm (dashed line). (a) The function modulating the forcing for the two radii. In each case, the modulating function M(t) passes through 0.5 at t lim. (b) The forcing functions. The thick solid line shows the unmodulated forcing function f(t), and the thin solid and dashed lines show the final forcing functions for the two bubble radii, once that modulating function is applied. (c) and (d) The calculated acoustic pressures with time for these situations. The vertical axis is the same in both plots, and it can be seen that both pressure signals rise initially, and that the amplitude of the pulse associated with the 200 μm radius bubble is much larger than that associated with the 150 μm radius bubble.
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Coalescing bubbles are known to produce a pulse of sound at the moment of coalescence, but the mechanism driving the sound production is uncertain. A candidate mechanism for the acoustic forcing is the rapid increase in the bubble volume, as the neck of air joining the two parent bubbles expands. A simple model is presented here for the volume forcing caused by the coalescencedynamics, and its predictions are tested against the available data. The model predicts the right order of magnitude for the acoustic amplitude, and the predicted amplitudes also scale correctly with the radius of the smaller parent bubble.
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