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 ... 

Thus, substituting (4-203) into (4-201), we obtain the famous Rayleigh-Plesset equation ... 

We shall discuss the solution of the Rayleigh Plesset equation, (4-208), shortly. 

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... 

The relatively mild growth and the violent collapse processes predicted by the asymptotic forms (4-210) and (4-211) are characteristic of the dynamics obtained by more general numerical studies of the Rayleigh-Plesset equation, but additional results for cases of large volume change are not possible by analytic solution. In the remainder of this section, we consider additional results that can be obtained by asymptotic methods. 

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... 

The governing equations for the functions g are obtained from the Rayleigh Plesset equation in the usual manner. We thus substitute (4-244) into the inviscid form of the Rayleigh-Plesset equation (4-230) and collect terms of like powers in s. The first several functions g (t, r) are found to satisfy the dimensionless equations... 

Thus, examining (4-281), (4-283), and (4-285), which is the full problem at 0(1), we see that they are identical to the DEs and boundary conditions that led to the Rayleigh-Plesset equation, except for the neglect of the viscous stress term in (4-285). Thus the solution at 0(1) is... 

Taken together with (4-301), this result for

dynamic equation for the coefficients au(t). For this purpose, we follow the example from the earlier derivation of the Rayleigh-Plesset equation. First, we calculate the pressure p (r, t) by means of equations of motion (4-292), and, second, we apply the normal-stress condition, (4-300), to obtain the dynamic equation for the coefficients au(t). In the interest of brevity all of the details are not displayed here. The result is ... 

Clearly, Gk > 0 for R > 0. Instability corresponding to R > 0 is the analog of Rayleigh Taylor instability. For a flat interface, we could have instability only for R > 0. Here, however, we may still have instability even if R <0, provided R/R is sufficiently large. To determine the condition for G > 0 in terms of controllable parameters, we can substitute for R/R in (4-315) by using the inviscid form of the Rayleigh Plesset equation, (4-204), with y = 0, that is,... 

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]... 

Simplified model based on Rayleigh-Plesset equation and chemical kinetics to predict OH radicals generation rate and the effect of parameters such as reactant concentration, temperature, etc. 

Modeled some trends observed in sonochemistry experiments by coupling chemical reactions with Rayleigh-Plesset equation. 

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. 

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