The Blake threshold

The temperature is higher than 5000 K for bubbles of 0.28 3.5 pm in ambient radius. For the linear resonance radius of 11 pm, it is only about 1000 K. In Fig. 1.9b, the rate of production of chemical species is shown. It is above 10s s 1 for bubbles of 0.28 8 pm in ambient radius. The minimum ambient radius coincides with the Blake threshold radius (Rgiake) f°r transient cavitation (active bubbles). It is calculated by the following formula [6]. 

Fig. 1.10 The Blake threshold pressure (1.17) and the linear resonance frequency (1.18) as a function of ambient radius...

The Blake threshold can be represented in an Apfel diagram (Fig. 10),56 where the horizontal axis represents the ratio [applied acoustic pressure]/[hydrostatic pressure], and the vertical axis, the ratio [bubble radius]/[resonance radius]. The resonance radius can be determined by the Minneart equation (see below). In zone A, the bubble cannot grow due to an insufficient quasi-static reduction of the acoustic pressure, and zone B is the zone of explosive growth. 

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). 

After the formation of a cavity, a critical acoustic pressure has to he overcome to initiate the explosive growth of this bubble. The Blake threshold pressure (Pb) describes this critical pressure [eq. 3 (12)] ... 

Equation 3 assumes that the external pressure (Po), the vapor pressure (Pv), the surface tension (ct), and the equilibrium radius of the huhhle (i o) determine the required negative pressure in the liquid to start an explosive growth of a cavity (13). The Blake threshold pressure is based on a static approach and is only valid when the surface tension dominates all dynamic effects, eg, mass transfer and viscosity. 

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. 

Static Pressure. A high static pressure can prevent the formation of cavitation bubbles, as the Blake threshold pressure increases. This implies that less or no cavitations are formed at higher static pressures. To counteract this effect, a higher acoustic pressure is required, which will result in a more violent collapse of a cavitation bubble. 

During pressurization of a liquid, the Blake threshold pressure increases, which implies that higher acoustic pressures are needed to produce cavitations. Obviously, no cavitation occurs when the Blake threshold pressure exceeds the maximum acoustic pressure that can be applied with the currently available equipment. The vapor pressure of the liquid, however, can counteract the static... 

The Blake threshold pressure (P ) describes the critical pressure to initiate the explosive growth of a cavitation bubble [Eq. (4)] [131]. 

The fact that the pressure is negative implies that a negative pressure must be applied to overcome the cohesive forces of a liquid and produce a bubble of radius R. Writing P] = Pjj — Pg allows the estimation of Pg, (known as Blake threshold pressure), which is the negative (or rarefaction) pressure which must be applied in excess of the hydrostatic pressure (Pj ) to create a bubble of radius R. e.g. for large bubbles (i.e. 2a/R, Ph)... 

Figure 7.16 Output images for Blake s algorithm. The threshold was set to three.

Laplacian, the result no longer resembles the input image. Results for Blake s version, which computes the first derivative, applies the threshold operation and then computes the second derivative instead of thresholding the output of the Laplacian, and are shown in Figure 7.16. By thresholding the first derivative, the Laplacian can still be inverted. The result is much more pleasing. 

Horn (1974) / Blake (1985) Described in Section 7.2. Zeros were removed from the input by transforming each channel with data in the range [0, 1] according to y = (255,v + l)/256. The extensions of Blake (1985) are used, i.e. the threshold is applied after computing the first derivative. The threshold is set to 3. The inverse of the Laplacian is computed using an SOR iterative solver. (threshold=3)... 

This question was solved by Blake, who analyzed the growth of a nucleus in equilibrium in the liquid and containing both vapor and gas, with respect to a quasi-static reduction in the liquid pressure. Besides the existence of a threshold for nucleation, a threshold also exists for an "out-of-the-crevice" growth. The conditions associated with this threshold may be much more stringent than those associated with nucleation. Assuming that the effects of buoyancy and dissolution can be ignored and that the volume modifications of the intracavity medium... 

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