The Rayleigh-Plesset Equation

As we will see later, the Rayleigh-Plesset description closely matches the actual radial behavior of a bubble as long as the non-radial deformations are small (or of short duration). Since the behavior of a bubble depends on the applied acoustic pressure, Apfel estimated the threshold associated to transient cavitation. A part of this threshold is, of course, common to the Blake threshold (explosive growth of a cavitation nucleus. Fig. 14). 

Transient collapse likely to produce sonoluminescence and sonochemistry may be the end of the road for a bubble grown by rectified diffusion. However, Church 

For example, at 3 MHz, 80% of the bubbles undergoing transient collapse do so because they are above the transient threshold immediately at the start of sonication (initial bubble radii = 1.03 to 1.115 pm). The likelihood of a scenario where a bubble grows by rectified diffusion and then undergoes transient collapse decreases with increasing frequency. Indeed, the rectified-diffusion threshold increases more rapidly with increasing frequency than does the transient-cavitation threshold. 

A simplified description of the phenomenon of bubble collapse, first performed by Besant in 1859, begins with the case of a bubble maintaining perfect sphericity at all times and radially oscillating in an incompressible liquid (the acoustic approximation). Readers interested in advanced developments should refer to specialized papers and books.  

The case of an empty bubble, i.e., a cavity, was considered under the conditions that the cavity remains spherical at all times and the liquid is incompressible, non-viscous, and possesses a negligible surface tension. The solution for the first two questions is as follows. The work done by the hydrostatic pressure po for the contraction of a cavity from (47c/3)Rni (initial) to (4tc/3)R3 (final volume) is found in the form of the kinetic energy of the liquid released to several liquid shells of thickness Ar, mass 47cr2pAr, and speed (dr/dt) (Eq. 18)  

We consider a single bubble that undergoes a time-dependent change of volume in an incompressible, Newtonian fluid that is at rest at infinity. The bubble may either be a vapor bubble (that is, contains the vapor of the liquid) or it may contain a contaminant gas that is insoluble in the liquid (or at least dissolves only very slowly compared with the time scales associated with changes in the bubble volume) or a combination of vapor and contaminant gases. We assume, at the outset, that the bubble remains strictly spherical, and thus that the bubble surface moves only in the radial direction. It follows that the motion induced in the liquid must also be radial, so that 

The assumption of a ID radial flow leads to a great simplification of the fluid mechanics problem. Indeed, the continuity equation, corresponding to (4-193), takes the very simple form 

From a physical viewpoint, this form for ur ensures that the total mass flux is the same across any closed surface that encloses the bubble. 

The time-dependent fiinction Hit ) is determined by the rate of increase or decrease in the bubble volume. The governing equations and boundary conditions that remain to be satisfied are (1) the radial component of the Navier Stokes equation (2) the kinematic condition, in the form of Eq. (2 129), at the bubble surface and (3) the normal-stress balance, (2 135), at the bubble surface with = 0. Generally, for a gas bubble, the zero-shear-stress condition also must be satisfied at the bubble surface, but xrti = 0, for a purely radial velocity field of the form (4-193), and this condition thus provides no usefirl information for the present problem. 

The relationship between Hit) and the bubble radius R(t) is determined from the kinematic boundary condition. In particular, for a bubble containing only an insoluble gas, the kinematic condition takes the form 

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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 ... 

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]... 

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. 

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 ... 

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