1. Cavitation and Bubble Dynamics Ch.4 Dynamics of Oscillating Bubbles

2. Cavitation and Bubble Dynamics Bubble Natural Frequency Viscous Damping Nonlinear Effects Threshold for Transient Cavitation Rectified Mass Diffusion

3. Bubble Natural Frequency Response of the bubble to oscillations in the pressure at infinity With some mean pressure at infinity and magnitude of oscillation and Where is the complex bubble radius response, is the equilibrium radius The natural frequency of the bubble radius oscillations is defined Where S is the surface tension, k the polytropic constant of the gas

4. Bubble Natural Frequency At small, frequency is a weak function of –No peak frequency exists below a size For larger bubbles the viscous term becomes negligible and peak frequency depends on Pressure amplitude at peak frequency For a 10μm nuclei in water at 300°K, where is about 10 kHz, is about 0.002 bar

5. Viscous Damping Polytropic constant only consistent with linear oscillations Additional damping terms needed; collectively lumped into effective damping term, –3 primary contributors Liquid viscosity Liquid compressibility (acoustic viscosity) Thermal conductivity whereand is the complex polytropic function, is the speed of sound in liquid

6. Viscous Damping Total damping and contribution of separate components of damping as a function of equillibrium radius for water. Plotted as effective damping with a nondimensional unit on X-axis.

7. Nonlinear effects When a liquid is irradiated with sound at a given frequency, the nonlinear response results in harmonic dispersion. Dispersion produces with harmonics of integer multiples of but also subharmonics, with frequencies a fraction of Subharmonics and superharmonics becomes more prominent as the amplitude of excitation is increased. Plot of numerical computations of steady nonlinear oscillations. Top: subresonant frequencies at 83.4kHz Bottom: Superresonant frequencies at 191.5 kHz, P=0.33bar

8. Threshold for Transient Cavitation Need for distinction between cases of steady acoustic cavitation and cases where transient cavitation occur. Define stability of bubble when pressure in farfield is varied. Identify critical radius,, and critical pressure threshold,. Finite time during each cycle wherein growth can occur. –If, liquid inertia is unimportant, bubble will behave quasistatically. –If, inertia will determine the size of the bubble perturbations. –If then the critical pressure can be identified where is defined by

9. Rectified Mass Diffusion Linear contributions of the mass of gas will balance so that the average gas content in the bubble will not be affected. Two nonlinear effects that increase the mass of gas in bubble: 1.Release of gas by the liquid occurs when the surface area is large Net flux of gas into the bubble which is quadratic with amplitude of oscillation 2.Diffusion boundary layer tends to become stretched when bubble is larger Second, quadratic influx of gas into the bubble Bubble growth because of rectified diffusion occurs very slowly compared to other effects. Can be thought of as causing a gradual, quasistatic change in equilibrium radius, Final note: threshold pressure increases with mass of gas in the bubble; it becomes easier to cavitate as the bubble increases in mass.

10. Rectified Mass Diffusion Examples of growth due to rectified diffusion. Four pressure amplitudes shown, lines are theoretical predictions.