Comparison of the numerical and analytical solution of the amplitude of the pressure wave at r = 0 as a function of R ∞. Δp ∞ = 1 atm and f = 10000 Hz.
Maximum bubble radius as a function of the excitation frequency for Δp ∞ = 0.1 and 0.5 atm. Results for the complete model and the model using the RP equation.
Temporal evolution of the bubble radius evolution (up) and radial pressure distribution for a 10 μm air bubble in water excited with a frequency of 3 × 105 Hz in an spherical flask with bubble (middle) and without bubble (bottom). The forcing pressure is set to generate a wave amplitude of 1.5 atm at the flask center when there is no bubble. The current model allows capturing pressure waves emitted during the implosion as well as the pressure disturbances induced by the bubble in the surrounding liquid.
(Color online) Bubble radius and bubble pressure (at r=0) evolutions as functions of dimensionless time for the complete model, the standard RP equation and the RP-type equations with the compressibility corrections suggested by KM, Gilmore and Tomita and Shima. The ratios between the liquid pressure at infinity and the initial bubble pressure, p b,0 are 10 (top) and 100 (center). At the bottom, a zoom of the implosion for a ratio equal to 100 is shown. The model proposed by Tomita and Shima displays the more accurate behavior before the collapse.
(Color online) Peak pressures at the bubble center at the collapse obtained with the five models compared in this work for different ratios of the initial bubble pressure to the pressure at infinity. Models based on the RP equation accurately predict the peak pressures for values of the peak pressures below 1000 atm. For extremely violent implosions, RP models dramatically fail predicting the peak pressures. The modified RP-type models studied in this work significantly improve predictions, especially that proposed by Tomita and Shima, but errors are still of the order of 100%.
Liquid pressure profiles after the bubble collapse every 4 ns. For large pressures, the wave dissipates energy and the amplitude decay is larger than 1/r. As the wave propagates outward from the bubble, its amplitude decreases, dissipation becomes less important and the shock wave amplitude is inversely proportional to the distance.
Parameters of the numerical simulations.
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