Bubbles Rayleigh-Plesset equation

Now the bubble collapse is discussed using the Rayleigh-Plesset equation. After the bubble expansion, a bubble collapses. During the bubble collapse, important terms in the Rayleigh-Plesset equation are the two terms in the left hand side of (1.13). Then, the bubble wall acceleration is expressed as follows. 

Size distribution plays a major role in the microbubble stability, behavior in vivo, and the microbubble acoustic response. The Rayleigh-Plesset equation which describes the microbubble response to pressure waves suggests that ultrasound scattering is proportional to the sixth power of the microbubble diameter [46]. It is not possible, however, to inject large bubbles (e.g., 0.1 or 1 mm in diameter) in the bloodstream, because they would be immediately lodged in the vasculature as emboli, severely limiting the blood flow. Fortunately, microbubbles with the size of several micrometers are still quite echogenic in the ultrasound... 

The generally accepted explanation for the origin of sonochemistry and sonoluminescence is the hot-spot theory, in which the potential energy given to the bubble as it expands to maximum size is concentrated into a heated gas core as the bubble implodes. The oscillations of a gas bubble driven by an acoustic field are well described by the Rayleigh-Plesset equation.7... 

Various mathematical models have been put forth to describe the rate of bubble growth and the threshold pressure for rectified diffusion.f ° The most widely used model quantifies the extent of rectified diffusion (i.e., the convection effect and bubble wall motion) by separately solving the equation of motion, the equation of state for the gas, and the diffusion equation. To further simplify the derivation, Crum and others made two assumptions 1) the amplitude of the pressure oscillation is small, i.e., the solution is restricted to small sinusoidal oscillations, and 2) the gas in the bubble remains isothermal throughout the oscillations.Given these assumptions, the wall motion of a bubble in an ultrasonic field with an angular frequency of co = 2nf can be described by the Rayleigh-Plesset equation ... 

As a consequence of this nonlinearity, it is impossible to obtain analytic solutions of the Rayleigh-Plesset equation for most problems of interest, in which po j (t) is specified and the bubble radius R(t) is to be calculated. Indeed, most comprehensive studies of (4-208) have been carried out numerically. These show a richness of dynamic behavior that lies beyond the capabilities of analytic approximation. For example, a typical case might have Poo(t) first decrease below p,Xl(()) and then recover its initial value, as illustrated in Fig. 4-10. The bubble radius R(t) first grows up to a maximum (which typically occurs after the minimum... 

If we refer back to the Rayleigh-Plesset equation, (4-208), it is evident that a bubble in equilibrium must have a radius RE that satisfies the condition... 

Now, the dynamics of changes in bubble radius with time, starting from some initial radius that differs slightly from an equilibrium value, is a problem that is ideally suited to solution by means of a regular asymptotic approximation. Of course, the governing equation is still the Rayleigh Plesset equation. Before beginning our analysis, we follow... 

Problem 4-10. An Alternative Derivation of the Rayleigh-Plesset Equation. Find the total kinetic energy Ek of the liquid outside a spherical gas bubble that is undergoing time-dependent changes in volume in an unbounded, incompressible, Newtonian fluid. Show that the net rate of working by the pressure inside the bubble p at the inner side of the bubble boundary is... 

Bubble motions in a sinusoidal sound pressure field can be described, for example, by the Rayleigh-Plesset equation [1, 7]... 

There is always an uncertainty associated with the exact quantification of the collapse pressure generated. Use of the Rayleigh-Plesset equation will dictate the termination condition as bubble-wall velocity exceeding the velocity of sound in the medium, whereas for the case of equations considering the compressibility of liquid, new termination criteria will have to be considered. Thus, for some other conditions with better computational facilities, a different collapse criteria (cavity size lower than 1 or even 0.1% of the initial size) may look feasible. Another collapse criteria based on Vander Wall s equation of state has also been considered [Gastagar, 2004]. The criteria considers that the cavity is assumed to be collapsed when the volume occupied by the cavity is equal to the material volume given by the product of the constant b in the Vander Wall s equation of state and the number of moles. The exact predictions of the collapse pressure pulse is always a matter of debate, nevertheless, a new proportionality constant to avoid this uncertainty can always be developed based on the relative rates of the reaction. 

In an experimental study of explosive vaporization of the bubble on a microheater, by detecting acoustic emissions during explosive vaporization process of the bubble, the bubble volume and the bubble expansion velocity and acceleraticm was reconstructed [17]. The vapor pressure inside the bubble was also calculated using the Rayleigh-Plesset equation. 

The connection between the gas state and the liquid state in the cell is provided by an averaged form of the Rayleigh - Plesset equation [6] which contains also bubble interaction terms... 

By combining the Rayleigh-Plesset equation with a mass and energy balance over the bubble, the temperature and pressure in the bubble can be calculated (16,17). The model also describes the dynamic movement of the bubble wall, which results in a calculated radius of the cavitation bubble as a fimction of time (see Fig. 2). The explosive growth phase and the collapse phase of the bubble can clearly be distinguished. Moreover, in case dynamic effects are more important than the surface tension, the cavitation threshold can be calculated with the dynamic model, while the Blake threshold pressure cannot be used at these conditions. 

Indeed, the P(t) curve resulting from this extended Rayleigh-Plesset equation (7.2) leads to a delayed bubble collapse for the two-bubble case, see Fig. 7.5, in good quantitative agreement with the measurements, despite the fact that the bubbles feel an anisotropic pressure field around their surface, especially in the last stage of their collapse. When the bubbles come closer to each other (middle... 

Equ. 1 being non-linear, in general the solutions are obtained using numerical approaches. However, if the amplitude of the oscillations can be assumed small compared to the bubble radius at rest, Miller [11] has developed an analytical solution for the Rayleigh-Plesset equation ... 

Equation 5.22 is known as Rayleigh-Plesset-Noltingk-Neppiras-Poritsky (RPNNP) [33, 34] equation. This equation determines the temporal evolution of the radius of a bubble subjected to a pressure change at infinity. For the case of a nonviscous liquid, the last term on the right-hand side vanishes. 

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